@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Resources, Environment and Sustainability (IRES), Institute for"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Newlands, Nathaniel K."@en ; dcterms:issued "2009-10-01T21:39:39Z"@en, "2002"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The Atlantic bluefin tuna (Thunnus thynnus) is a long-lived, highly migratory species that attains sizes of 2.20 m, and weights of 300 kg or more. Adults undertake cyclic migrations between coastal feeding zones, offshore wintering areas and spawning grounds. During June through October, bluefin tuna are common off the eastern United States and Canada, entering the Gulf of Maine, a semi-enclosed continental shelf area. The population is currently believed to have plummeted to 20% of 1970's levels, yet there is significant uncertainty in their population status and size. This thesis investigates bluefin tuna movement, aggregation and distribution, size and structure of bluefin shoals, and examines how these factors can affect the measurement bias and estimation uncertainty of population abundance. Data analysis methods applied include: interpolation of movement data, Lomb spectral analysis, statistical bootstrap simulation, Kalman filtering, and geostatistics. An automated digital image analysis system (SAIA) is developed for the three-dimensional analysis of fish shoal structure. A theoretical model is also formulated to describe the movement and behaviour of shoaling tuna leading to changes in shoal aggregation, distribution and abundance. The precision in abundance estimation of random, systematic, stratified, and spotter-search aerial survey sampling schemes are simulated under changes in the size, distribution and aggregation of shoals. Correlated and biased random walk models can predict lower and upper limits on displacement and spatial movement range over time. Bluefin tuna move by responding to changes in temperature gradients and to the local abundance of prey, preferring to be situated in the warmest water available, while also showing a weak response to flow and bathymetric gradients. The effect of aggregation on the distribution of shoals considerably reduces precision of population estimates under random transect sampling. Stratified sampling is shown to increase precision to within 5%, with adaptive stratification leading to further increases. Movement and shoal aggregation introduce relatively equal levels of bias and uncertainty in estimating abundance. Results indicate that reliable estimates of abundance can be attained under systematic and stratified survey schemes. However, further reductions in uncertainties associated with the shoal aggregation process are necessary to achieve acceptable precision in abundance estimation."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/13501?expand=metadata"@en ; dcterms:extent "53432559 bytes"@en ; dc:format "application/pdf"@en ; skos:note "S H O A L I N G D Y N A M I C S A N D A B U N D A N C E E S T I M A T I O N : A T L A N T I C B L U E F I N T U N A (THUNNUS THYNNUS) Nathaniel K . Newlands M.Sc , University of Calgary, 1997 B.Sc , University of Guelph, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RESOURCE MANAGEMENT AND ENVIRONMENTAL STUDIES by in THE FACULTY OF GRADUATE STUDIES We accept this thesis as conforming to the reqjLiired standard The University of British Columbia June, 2002 © Nathaniel K . Newlands, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Resource Management and Environmental Studies The University of British Columbia Room 426E, 2206 East Mall Vancouver, B.C., Canada V6T 1Z3 Date: Abstract The Atlantic bluefin tuna (Thunnus thynnus) is a long-lived, highly migratory species that attains sizes of 2.20 m, and weights of 300 kg or more. Adults undertake cyclic migrations between coastal feeding zones, offshore wintering areas and spawning grounds. During June through October, bluefin tuna are common off the eastern United States and Canada, entering the Gulf of Maine, a semi-enclosed continental shelf area. The population is currently believed to have plummeted to 20% of 1970's levels, yet there is significant uncertainty in their population status and size. This thesis investigates bluefin tuna movement, aggregation and distribution, size and structure of bluefin shoals, and examines how these factors can affect the measurement bias and estimation uncertainty of population abundance. Data analysis methods applied include: interpolation of movement data, Lomb spectral analysis, statistical bootstrap simulation, Kalman filtering, and geostatistics. A n automated digital image analysis system (SAIA) is developed for the three-dimensional analysis of fish shoal structure. A theoretical model is also formulated to describe the movement and behaviour of shoaling tuna leading to changes in shoal aggregation, distribution and abundance. The preci-sion in abundance estimation of random, systematic, stratified, and spotter-search aerial survey sampling schemes are simulated under changes in the size, distribution and ag-gregation of shoals. Correlated and biased random walk models can predict lower and upper limits on displacement and spatial movement range over time. Bluefin tuna move by respond-ing to changes in temperature gradients and to the local abundance of prey, preferring to be situated in the warmest water available, while also showing a weak response to flow and bathymetric gradients. The effect of aggregation on the distribution of shoals considerably reduces precision of population estimates under random transect sampling. Stratified sampling is shown to increase precision to within 5%, with adaptive stratifica-tion leading to further increases. Movement and shoal aggregation introduce relatively equal levels of bias and uncertainty in estimating abundance. Results indicate that re-liable estimates of abundance can be attained under systematic and stratified survey schemes. However, further reductions in uncertainties associated with the shoal aggre-gation process are necessary to achieve acceptable precision in abundance estimation. ii Table of Contents Abstract ii List of Tables vii List of Figures xiii Acknowledgments xxx 1 Introduction 1 1.1 Research objectives ' ; 1 1.2 Individual-based spatial models of fish populations 4 1.3 Variability and patterns of fish population abundance 6 1.4 Atlantic bluefin tuna 9 1.5 Study region: Gulf of Maine/Northwestern Atlantic 14 1.6 Fishery-dependent and independent indices of abundance 18 2 Individual Movements 34 2.1 Interpolation of the movement observations 36 2.2 Move-speed and turning angle distributions 44 2.3 Move-speed and turning angle autocorrelations 47 2.4 Spectral identification of movement modes 58 2.5 Space trajectories 69 2.6 Theoretical movement model predictions 69 2.7 Kalman filtering of geoposition data from light-archival data 80 iii 2.8 Significance testing: observations and model predictions 92 2.9 Summary 110 3 Shoal Structure and Behaviour 113 3.1 Supervised automated image analysis scheme (SAIA) 117 3.2 Shoal formations 149 3.3 SAIA calculations of shoal formation structure 189 3.4 Convex hull refinement of ellipsoidal shoal structure 211 3.5 Principal component analysis of structural variables 217 3.6 Shoal dynamics 231 3.7 Summary 243 4 Spatial, Individual-Based Model of Bluefin Tuna 251 4.1 Model and simulation framework 252 4.2 Initial and boundary conditions 263 4.3 Seasonal population 266 4.3.1 Seasonal immigration and emigration of shoals 270 4.3:2 Shoal size frequency distribution 275 4.4 Lagrangian equations 276 4.5 Movement and behaviour dynamics 280 4.5.1 Individual/shoal fitness: foraging rate and predation risk 280 4.6 Multi-layered spatial environment 285 4.7 Model validation tests 309 4.8 Summary and future work 337 5 Abundance Estimation: Measurement and Precision 342 5.1 Survey sampling . . 345 5.2 Analysis of aerial survey data (1994-96) 350 iv 5.3 Survey measurement schemes 411 5.4 Results, summary and future work 423 6 Summary and Conclusions 441 6.1 Regional population abundance 442 6.2 Movement: immigration and emigration 443 6.3 Spatial aggregation and distribution 445 6.4 Shoal size and structure 446 6.5 Movement: foraging, short and long-range searching 453 6.6 Interaction of individuals and shoals 456 Bibliography 461 Appendices 495 A Abbreviations and Notation 495 A. l Abbreviations 495 B Chapter 2: Background, Derivations, Extended Results 497 B. l Move-speed and turning angle distributions 497 B.2 Spectral identification of movement modes 502 B. 3 Space trajectories 512 C Chapter 3: Background, Derivations, Extended Results 522 C. l Shoal size and formation: shoal structure histograms 522 C . l . l Nearest-neighbour distance (NND) 522 C.1.2 Frequency of nearest neighbours 525 C.1.3 Bearing angle between nearest-neighbours (BA) 528 C.l .4 Shoal polarization 531 v C. 2 Convex hull refinement of ellipsoidal shoal structure 534 D Chapter 4: Background, Derivations, Extended Results 540 D. l Seasonal population 540 D.2 Shoal size frequency distribution 545 D.3 Lagrangian equations 548 D.4 Adaptive step-size Runge-Kutta integration 551 D.5 Movement and behaviour dynamics 552 D.5.1 Movement correlations, modes and mode-switching events . . . . 552 D.5.2 Move-speed movement mode (mi ,777,2) variation 552 D.5.3 Move-angle operator 553 D.5.4 Move-speed and turning angle autocorrelation functions 555 D.5.5 Shoal mixing: join/leave/stay decisions, mode alterations 556 D.5.6 Neighbour individuals: attraction and repulsion 564 D.5.7 Movement response to environment and prey 565 D. 6 Multi-Layered spatial environment 568 D.6.1 Observed environmental association of shoals 568 D.6.2 Observed movement response to environmental gradients 577 E Chapter 5: Background, Derivations, Extended Results 588 E. l A review of spatial statistics in survey design 588 F Curriculum Vitae 598 vi List of Tables 1.1 Summary of Objectives, Data Sources and Data Analysis Techniques . 23 1.2 Table 1.1 continued 24 1.3 Table 1.2 continued 25 1.4 Table 1.3 continued 26 1.5 Table 1.4 continued 27 1.6 Summary of Objectives, SIBM Model and Validation/Confidence Tests 28 1.7 Table 1.6 continued 29 1.8 Table 1.7 continued 30 1.9 Table 1.8 continued 31 1.10 Table 1.9 continued 32 1.11 Table 1.10 continued 33 2.12 Summary of geolocation (GPS) and depth records from hydro-acoustic telemetry of B F T (N=l l ) . Start and End times for each record are in format of HH:MM:SS and Elapsed time, ET(s) 37 2.13 Statistics of move-length (k), time duration (TJ ) , vertical inclination (#j), directional ((pi) and turning (c/?j) angles in movement observations of B F T , for rii sampled positions, and shoal size, S 42 2.14 Depth-correlated data summary for hydroacoustic tracking of B F T (N=10). 43 2.15 The number of segments, /V(mi) and N(rri2), for mi and m.2 modes, and the number of mode-switching events, S(m\\^) identified within the individual B F T movement trajectories from the Lomb spectral analysis. 61 2.16 Modal (mi and m^) statistics of move-length (lt), time duration (ri), ver-tical inclination (<9j), directional (fa) and turning (ipi) angles calculated for the observed individual movements of B F T 66 2.17 Estimates of Fork Length (FL)(m), Longitude (LGT)(°W), Latitude (LAT)(°N), elapsed time (ET(d)), mean speed (v(m/s)) and diffusion, D(nm 2/d) for short-term light archival tagging movement observations of B F T (N = 7) (1998-1999), where d denotes days 84 2.18 Long-term light archival tagging movement observations of B F T (N=3, 1999-2000) with time at liberty/elapsed time (ET) ranging from 77-279 days (d). Parameter estimates of advection velocities (u,v) and diffusion (D) obtained from Kalman filtering for BRW, RW models. Estimates of diffusion (D) as would be calculated with a start and end track location without archived geolocation data (i.e., single-point pop-up) (DM) are also shown 85 vii 2.19 Results of fitting move-speed autocorrelations (ACF) observed from hy-droacoustic tracking of BFT to the general form v = (Vj) exp~^TN, for t = nAt, over n successive lags between moves. These results are used to determine values for persistence time, TJV and shoal searching efficiency for each movement path 97 2.20 Searching efficiency and diffusion estimates based on self-intersections of observed R^et o b s over time with Rnet,CR\\v predictions. S - shoal size, T - total foraging time, AT/v - mean foraging time, (T — ATN) - searching time, TTV - persistence time, - characteristic length, D - diffusion, mean dispersal area - (RT-&TN)-> swath width - b^, self-intersection parameter - v, searching distance - LT-ATN, searching efficiency - SN 99 2.21 Testing of observed statistic (A 06 S) for individual movements of BFT to 95% confidence intervals (2cr-intervals) about the expected values, BCRW (see Equation 2.72) for the BCRW, and AT,CRW (see Equa-tion 2.71) for CRW theoretical models 101 3.22 Summary of the spotter observer aerial sampling records of BFT shoals. 120 3.23 Summary of shoal images sorted by background quality: From (Cl)-(C5) in order of decreasing image quality. (*) number of shoal images in class C l were analyzed in the image analysis and results presented. The 1994 images were used in testing of the SAIA image analysis scheme and post-analysis algorithms for which selected results were compiled. Main results were compiled for years 1995-96 122 3.24 Monthly frequencies for analyzed shoal images. The 1994 images were used in testing of the SAIA image analysis scheme and post-analysis algorithms for which selected results were compiled. Main results were compiled for years 1995-96 122 3.25 Definition of variables in post-processing of SAIA digital image analysis used in the measurement and characterization of BFT shoal structure and behaviour. (-) units denote dimensionless measures. 138 3.26 Reduced yf/df for manual (Nm), automated (Nc) and final-corrected, (Ns) of SAIA image analysis school size estimates for each year, 1994-96, shown in Figures (3.44)-(3.46) 145 3.27 Reduced x2/df for manual {Nm), automated (Nc) and final-corrected, (Ns) of SAIA image analysis school size estimates for different shoal formations pooled over years 1994-96. Formations are denoted as: A-cartwheel, B-surface-sheet, C-dome, D-soldier, E-mixed, F-ball, G-oriented shown in Figures (3.51-3.53) 150 viii 3.28 Frequencies of different shoal formations for analyzed shoal images. For-mations are denoted as: A-cartwheel, B-surface-sheet, C-dome, D-soldier, E-mixed, F-ball, G-oriented (H-solitary individuals). The percentage in the number of images for each formation type in each year with respect to the total numbers are provided in brackets 152 3.29 B F T shoal size statistics (mean shoal size, Ns, standard error in the mean (SE), 95% confidence intervals (C.I) and minimum and maximum shoal size for identified structural formations pooled across years 1994-96 (Refer to Figure 3.50) 152 3.30 Monthly mean fork-length(m) for B F T across ages 0-10+ (ICCAT). . . 159 3.31 Reduced x2/df statistics for shoal size-sorted observed histogram fre-quencies of N N D 174 3.32 Same as Table 3.31 of nearest-neighbour distance (NND) for formation type 174 3.33 Reduced x2/df statistics for shoal size-sorted frequency of nearest neigh-bours. Unless otherwise indicated, degrees of freedom (df=36) 178 3.34 Same as Table 3.33 of nearest neighbour frequency for formation type. . 178 3.35 Reduced x2/df statistics for shoal size-sorted observed histogram fre-quencies of nearest neighbour bearing angle (BA) 183 3.36 Same as Table 3.35 of nearest neighbour bearing angle (BA) for forma-tion type 183 3.37 Reduced x2/df statistics for shoal size-sorted observed histogram fre-quencies of shoal polarization. Unless otherwise indicated degrees of freedom (df=20) 188 3.38 Same as Table 3.37 of shoal polarization for formation type 188 3.39 Summary of linear regression of convex hull refinement of ellipsoidal surface area (SAS), and volume (Vs) denoted as SAh and 14, respectively.213 3.40 Surface area and volume estimates for ellipsoidal (SAS, Vs), and convex hull (SAh, 14) approximations to the shape of B F T shoal formations. Estimates of the mean number of edge individuals, Np, and shoal size, Ns (from Table 3.29), for each formation are also provided 214 3.41 Cartwheel formation: P C A correlation matrix and PC1-PC7 eigenvec-tors for shoal variables 221 3.42 Same as Table 3.41 for surface-sheet formation 222 3.43 Same as Table 3.41 for dome formation 223 3.44 Same as Table 3.41 for soldier formation 224 3.45 Same as Table 3.41 for mixed formation 225 3.46 Same as Table 3.41 for ball formation 226 3.47 Same as Table 3.41 for oriented formation 227 ix 3.48 B F T formation frequencies with associated visual shoal size estimates from aerial surveys conducted, 1994-96. These observations provide a larger set of observed frequencies than data set corresponding to quality classed aerial shoal images selected for the SAIA image analysis 233 3.49 Reduced X2/df statistics comparing observed frequency of occurrence at time-of-day (hrs.) between B F T formations. See Table 3.22 for sample sizes of the formations determined by aerial observers 236 3.50 Same as Table 3.49 comparing observed frequency of occurrence at time-of-day (hrs.) between years 1994-96 236 3.51 Classification summary of B F T shoal formations based on approximate means and range in values of internal and external variables. Listed are packing density, ps(BL~3), mean shoal size, N3±5NS, nearest-neighbour distance, NND(BL) , mean number of first nearest neighbours, NNS, modal values for bearing angle between first neighbours, BA(°), max-imum number of edge individuals for maximum observed shoal size, Na, observed range in shoal polarization ( — $ s ) , and a generalized shape description 249 3.52 Summary of P C A analysis of B F T shoal formations. Listed is the per-centage of variance explained by the first two principal components (PC1,PC2) for each formation type, and shoal variables listed in order of decreasing positively correlation associated with P C I (shoal shape) and PC2 (internal structure) 250 4.53 Associations between model variables in forming a reduced model rep-resentation, denoted as model, M . . . . . 254 4.54 Fixed model parameters (N=19 (no grid layers), N=27 (5 grid layers)), aggregated parameter settings, and variables in simulation for process test-results 256 4.55 Definition of SIBM model parameters and variables. (-) units denote dimensionless measures 259 4.56 Aggregation of model variables/parameters into reference categories: en-vironment, population, shoal and individual-scales 260 4.57 Test results of cross-correlation coefficient at zero time-lag (where time-lag interval coincides with mean move-duration) for observed hydroa-coustic movements of B F T (N=10) and each environmental variable. . 306 5.58 Nonlinear least-squares fitting of sigmoidal function for cumulative S P U E versus time (days). r=5 fitting parameters (a,b, c,tQ, SPUEt0), degrees of freedom (d.f.)=(n-r), where k is the number of independent variables (k=l: time) 363 5.59 Monthly sightings-per-unit-effort (SPUE)(individuals/1.8km) for move-ment filters, no depth correction 368 x 5.60 Same as Table 5.59 but with depth correction/calibration 371 5.61 Depth-corrected survey abundance (1994-96) with calibration coefficients based on superimposed behavioural movement modes ( r a i , T O 2 ) and asso-ciated movement depth distribution. V P A Abundance refers to estimates of their abundance (numbers) in the West-Atlantic, for comparison to survey estimates for the Gulf of Maine region 372 5.62 Results of fitting age-specific V P A abundance for the west-Atlantic with transfer parameters and survey calibration coefficients, N=10,000 iter-ations, precision<0.001 using Conjugate-Gradient optimization. Cal-culated x2/df statistics indicate large significant differences using the three available annual estimates (1994-96) in the observed (survey) and predicted (west-Atlantic abundance for ages 7+ and transfer to/from the Gulf of Maine region) time-series. The best-fit, indicated as '*' is obtained for age 7+ of the total western Atlantic abundance, with asso-ciated transfer portions at the end of each year, t 379 5.63 Estimation of aggregation coefficient from fitting of observed shoal size ' frequency distributions to Weibull distribution function (a,b,c,x0,y0), and transformed parameter for power-law/exponential decay function form. Mean and variance of the number of shoals are used to calculate the aggregation coefficient, k (negative binomial spatial distribution of shoals). Parameter standard errors (SE), and associated 95% confidence interval ranges (C.I.) on transformed parameters are provided 392 5.64 Relative abundance estimates, spotter-aerial surveying (1994-96) 405 5.65 Number of shoals and size estimates for BFT in the GOM 405 5.66 Daily observer transects/effort estimates, spotter-surveying (1994-96). . . . 405 5.67 Daily encounter rate of BFT shoals, spotter-surveying (1994-96) 405 5.68 Population density estimates of BFT in the GOM 405 5.69 Summary of diffusion estimates (D) (km2/d) (un-corrected/corrected val-ues) for B F T calculated from various data sources used in analyses: U l -trasonic telemetry/hydroacoustic tracking (UT)(n=10), Short-term light archival (SLA)(n=6), Long-term light archival (LLA)(n=3), Single-point approximation of L L A observations (n=3), and Single-point pop-up tag-ging (SPl)(n=43) 408 5.70 Comparison of calculated B F T diffusion estimates (nm - nautical miles, km - kilometres) with those of other species of tuna available in the literature. B F T - Northern Bluefin (Thunnus thynnus), B E T - Bigeye (Thunnus obesus), Y F T - Yellowfin (Thunnus albacares), A B T - Albacore (Thunnus alalunga), SJT - Skipjack (Katsuwonus pelamis) 410 5.71 Definition of survey model parameters and variables. (-) units denote dimensionless measures 413 xi D.73 Univariate test results for statistical differences in observed movements of B F T (hydroacoustic tracking, N=10) comparing the cumulative distri-butions in relation to sea-surface temperature (SST). Table entries show the value of the test statistic and in brackets the probability (p value) for having the test statistic value (randomized), equal to or greater than the observed value 579 D.74 Same as Table D.73 for water flow 581 D.75 Same as Table D.73 for bathymetry 583 D.76 Same as Table D.73 for chlorophyll-a (phytoplankton concentration). . 585 D.77 Same as Table D.73 for zooplankton (Calanus finmarchicus) abundance. 587 xii List of Figures 1.1 Venn diagram depicting the primary considerations of SIBM models . . 5 1.2 General circulation and bathymetry of the G O M region in the North-western Atlantic Ocean. Modified from [404] 16 1.3 3D perspective of G O M circulation/Georges Bank system [344]. Cir-culation features are superimposed over preferred topographic regions. M C C - Maine Coastal Current, T M F - Tidal Mixing Front, SSF - Shelf-Slope Front. The colour legend indicates vertical depth (m) contours, also indicated near the circulation flow labels 17 2.4 Individual movements of B F T observed during hydro-acoustic tracking experiments within the Gulf of Maine region [216] (from Lutcavage and coauthors [215,216]). Further information on these observations is pro-vided in Table 2.12 38 2.5 Definition of parameters used in the ^o-dimensional interpolation of B F T movements. Each move displacement of variable length, k, has a corresponding directional angle, fa, referenced to a fixed axis, X, and turning angle, ipi, measuring the change between successive move-directions (adapted from [239]) 40 2.6 Definition of parameters used in the three-dimensional interpolation of the B F T movements. 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CD ^ \" f i . - A o QJ SH fi -13 S3 '3 0 -fi CO QJ OH o S3 <0 CO CN^ CN\" CN o CO 43 o a) u S3 CO TJ cfl (3 o CJ (3 43 cd T) > \"QJ TJ O Chapter 2 Individual Movements I present an analysis of individual movements of B F T from hydro-acoustic telemetry and pop-up archival satellite tagging experiments. Sampling of continuous movements of B F T yielded data consisting of a sequence of position estimates. Movement paths were re-constructed in two and three-spatial dimensions by interpolating between successive position estimates. I examined the variation in move-displacement, speed and direction parameters in the movement path reconstructions.1 I examine the form of the move-speed and turning angle distributions, calculating statistics for defined movement parameters in two-dimensions. These distributions show unimodal and bimodal variations according to the scale at which movements are sampled. I also examine the statistical correlation between successive displacements, where values of movement parameters at a given position are dependent on values at previous positions, termed autocorrelation. If attractive and repulsive forces are considered to describe the movement interaction between fish moving in shoals, then autocorrelation in velocity and turning angle indirectly show the variation in the strength of these forces over time. Autocorrelations in movement parameters, such as periodic variation in turning angle, may result from a greater number of neighbours, or on neighbours that have a stronger influence on the movement of the individual being tracked. Cross-correlations between movement parameters are also of interest, and are further confounded by the influence ^ewlands, N . and Lutcavage, M . 2001. Prom individuals to local population densities: North Atlantic Bluefin Tuna (Thunnus thynnus), 421-441, In: Electronic Tagging and Tracking in Marine Fisheries, J.R. Sibert and J.L. Nielsen (eds.), Kluwer Academic Publishers, Dordrecht, 484 PP-34 2.0 Individual Movements 35 of shoaling neighbours. Three-dimensional trajectories are formed by linearly correlating the geopositional data, sampled periodically within the range (60-550)s in the 2D paths, with depth data recorded concurrently at smaller (15s) sampling intervals. This enabled an examination of the full resolution of movements as space trajectories. Alternative movement modes are identified using Lomb periodograms which inversely transform movement trajectories into frequency space [212]. Unlike standard Fourier spectral techniques, the Lomb method does not require periodic sampling at equal inter-vals of experimental time-series data [111]. The Lomb method was used to extract modal statistics for characterizing movement behaviour on the basis of second-order autocorre-lations in the turning angle and move-speed parameters. Move-parameter statistics help to identify characteristic modes of movement and mode-switching events, with the mo-tivational decision-making of individuals making trade-offs between foraging, searching for food, or migrating over larger distances [286]. Changes in turning rate and depth preference are detected between two movement modes (m 1 ,m2) , occurring over (116.7 ± 57.52)s and (109.2 ± 49.05)s respectively. I ex-amine how modal alterations in movement are also important for the interpretation of movements observed at larger spatial extents. Kalman filtering is an efficient, adaptive technique for interpolating between geoposition estimates combining latitude and longi-tude measurement uncertainties. I apply this technique to filter PSAT tagging data that provides information on individual tuna movements at larger spatial scales (e.g., 1000km) than the hydro-acoustic observations (e.g., 10-100)km. I characterize the way in which tuna move, based on movement parameter time-correlations and cross-correlations. I compare the full set of observed movements with theoretical predictions generated us-ing statistical bootstrapping for simple (RW), correlated (CRW), biased (BRW), and biased-correlated (BCRW) random-walk models, and a significance test on the relative contribution of movement persistence and external bias. The results of these analyses on 2.1 Interpolation of the movement observations 36 observed movements of individual tuna provides empirical estimates on movement pa-rameters and correlation dynamics, necessary for formulating a spatial, individual-based model for B F T . 2.1 Interpolation of the movement observations The horizontal and vertical movement and behaviour of giant B F T (110-350) kg tracked by hydro-acoustic telemetry for up to 48 hours in three regions of the Gulf of Maine (GOM) (Stellwagen Bank, Great South Channel, Cape Cod Bay) was recently reported by Lutcavage and coauthors [215,216], shown in Figure 2.4. Giant bluefin rep-resent the primary age/size-class cohort comprising their seasonal assemblage in the study region. Individuals travelling in shoals were tracked using an ultrasonic transmitter and single-directional hydrophone having a detection zone of approximately 1.7 km. Succes-sive geoposition estimates of individuals were approximated as the coordinate position of the tracking vessel determined using GPS (global positioning system). Three consis-tent movement modes were identified: (1) repetitive travel through the thermocline with small move-displacements (<5-40 km/day), (2) travel primarily in the ocean surface layer with large displacements (40-76 km/day), and (3) large, vertical displacements (diving) at dusk and dawn. Further details on the raw data records I compiled for the observed movement paths (N=l l ) are provided in Table 2.12. In this table, two separate move-ment records for the same individual (9604) are denoted as 9604a, 9604b, and individual 9605 was omitted due to an insufficient length in the geolocation record. New archival satellite tags (PSAT's) can sample movements over larger durations than 48 hours. These archival tags are equipped with sensors designed to sample and archive light-intensity [28,164,267,369,420]. A determination of movement path end-points using GPS for pop-up tags is provided by the Argos satellite, which receives and relays the streams of archived data, once a tag, programmed to detached from 2.1 Interpolation of the movement observations i d o o CD c c I.S1 CO co co CO LO LO Tf CO o 00 oo CO LO LO 00 tr-CO 3.450 oi CN co Tf oi CN co Tf Tf' CN 06 rH co Tf 0 co 3.450 LO co 03 ir-Tf cr- co ^ 00 P CO OO H ^ • • CO _ CO CN r i « LO CN LO p2 co CN 00 IT-00 co 00 co Tf LO co CN CN lr-Oi . co co\" 22 '00\" co LO Tf LO Oi rH CN LO CO CN 1^. tr- ^ ^ T—1 ^ ^ CN ^ ^ Tf 0 , „ LO , n LO , „ co , _ co t~- O rH LO CN co Oi rH rH LO LO O LO CN co CN rH LO Oi CO O CN CO LO Tf CO Oi co rH co LO co co co CO Tf CO Oi CO rH LO rH CO CO LO CO O LO CN Tf Tf 00 CO CN Tf CN rH Tf rH Tf CO CN Tf Tf LO LO LO rH O s rH —^' rH -—' H -— rH ~ rH —^' rH —^' CN co rH Oi CN rH IT- Oi CN Oi rH 0O 00 rH CO CN tr- Tf 00 Oi O CO CO Tf CN LO rH rH O CO rH 1-- LO rH CO CN rH 0O rH Oi rH o cj . CD Ice fl o CJ JO \"o CD O co Tf CN Tf Tf rH O Oi 00 oi CO CN CO CN CN Tf co Tf CN Oi Tf CN CN CO o co 06 Tf CO OO Tf CO o o o o o CO LO co o Tf \"* fN 1 - 1 ^ CN rH CN 642) tr- , ^ H O Tf 642) CO CN 43957 LO LO LO O rH CO rH CN 43957 rH 6i CN rH 0 CN Oi o o co 00 O rH CO Tf rH 1 - 1 I~J CN Oi co o o co LO Tf CO tr- CN tr-ee LO tr-LO o co o co —^. o LO co CN ^ Oi LO 3 co LO 4* CN ^ CN O - - 00 , s rH 0 O O CO CO Tf CN co JJ. 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Further information on these observations is provided in Table 2.12. 2.1 Interpolation of the movement observations 39 an individual after a given time duration, rises to the ocean surface. The successive geopositions between GPS starting and end-point position fixes of tagged individuals are then derived with astronomical algorithms and equation fitting procedures to match the observed intensity variations expected on the basis of day-length and time of dusk and dawn (corresponding to a given latitude and longitude respectively) on the earth spheroid for a given time of year (see [421] for relevant astronomical equations). The last few successive position estimates determine the extent to which a tag, once detached from a fish, has drifted on ocean currents before satellite detection. The attenuation of light below the ocean surface can also be considered in such calculations. Additional miniature sensors also record temperature and pressure. Interpolation equations Cartesian (xyz) coordinates, referencing the positions of an individual along its move-ment path, relate to movement parameters, (k, Qi, fa), defined within polar and spherical coordinate systems. Movement parameters are depicted in two-dimensions (X, Y) using polar coordinates, and in three-dimensions (X, Y, Z) in spherical coordinates, as depicted in Figures 2.5 and 2.6. The equations defining the transformation between the cartesian and spherical coordinate systems are, n xn = li sin 9{ cos fa (2.1) i=l n n zn ='Y^k COS 8i (2.3) i=l The corresponding equations for the 2D case are obtained by setting 9i = n/2 in the above equations. For each successive move, i, where i — (1, ...,n) of n total moves; U is 2.1 Interpolation of the movement observations 40 y Figure 2.5: Definition of parameters used in the iwo-dimensional interpolation of BFT move-ments. Each move displacement of variable length, U, has a corresponding directional angle, fa, referenced to the fixed axis, X, and turning angle, (2-5) n . TT2 = - £ l\\ (2.6) n 1 i r ^ 9 <\"> i=i x ' ^ n 1 \" c = cos 09 = — > cos tfi, s = sin sin (pt (2.8) n *7~! n i=i i=i Estimates of these measures for movement observations are provided in Table 2.13. 2.1 Interpolation of the movement observations 42 Table 2.13: Statistics of move-length (k), time duration (r,), vertical inclination directional (fa) and turning (ipi) angles in movement observations of B F T , for sampled positions, and shoal size, S. ID S Ti(s) k(m) Vi(m/s) 1000 16.52 100.41 6.07 88.18 48.37 21.87 9604a 40-200 75.23 89.120 1.18 87.47 54.16 19.10 9604b - 65.67 119.56 1.82 87.40 52.64 17.39 9701 12 89.07 150.34 1.69 88.21 42.36 19.52 9702 6 60.65 85.750 1.41 87.36 44.01 18.75 9703 12 61.57 72.880 1.18 87.50 42.25 20.06 9704 200-300 60.96 115.28 1.89 87.76 45.10 18.36 9705 200 65.77 101.58 1.54 86.49 46.80 38.58 161.64 ± 57.970 1.98 ± 0 . 4 6 0 87.66 ± 0.2500 44.75 ± 2 . 1 5 0 23.27 ± 2 . 7 1 0 2.1 Interpolation of the movement observations 43 7. Figure 2.6: Definition of parameters used in the i/iree-dimensional interpolation of the B F T movements. Each move displacement of variable length, / j , has a corresponding directional angle, fa, referenced to a fixed axis, X , vertical inclination angle, 0, (spherical, azimuthal and polar angles respectively, and turning angle, tpi, measuring the change between successive move-directions) (modified from [293]). Table 2.14: Depth-correlated data summary for hydroacoustic tracking of B F T (N=10). ID Y Y Date(DD/MM) Depth-Correlated Record n Start End Duration(h) 9601 1996 11-12/09 205 11:02:00 (39720) 16:30:00 (145800) 29.47 9602 1999 20-22/99 2787 14:15:43 (51326) 12:49:00 (218940) 48.56 9603 1996 26-27/09 1752 12:51:38 (46298) 20:54:02 (161642) 29.45 9604a 1996 6-8/10 2477 12:12:37 (43957) 10:15:55 (209755) 46.06 9604b 1996 12-13/10 1183 10:25:37 (37537) 07:59:34 (115174) 21.57 9701 1997 15-17/08 1888 12:52:34 (46369) 11:30:00 (214186) 46.62 9702 1997 19-21/08 1820 17:47:48 (64048) 00:26:17 (174376) 30.64 9703 1997 25/08 352 12:51:55 (46315) 18:51:33 (67893) 5.990 9704 1997 16/09 180 14:35:05 (52505) 17:36:49 (63409) 3.030 9705 1997 17-19/09 2567 15:47:23 (56843) 14:39:30 (225572) 46.87 2.2 Move-speed and turning angle distributions 44 2.2 Move-speed and turning angle distributions Frequency distributions in two-dimensions provide an indication of the variability of movement parameters, averaged across the length of an observed path. They are formed by aggregating parameter values for each successive move observed. Mean tendencies in movement observations at a specific spatial or temporal scale are revealed when fre-quency distributions are compared to theoretical probability distributions. Because the motivation of animal's decision making is not usually known, simulation of movement models and significance testing of simulated outcomes often relies on sampling from em-pirical frequency distributions, or the corresponding forms of theoretical distributions for movement parameters. Siniff and Jessen [372] demonstrate how 2D distributions of movement parameters are used in simulating movement model patterns and generating comparisons to telemetry data. Two-dimensional distributions are also useful in identi-fying relationships between animals and permanent or seasonal habitat features. I compare theoretical distributions to the observed frequency distributions for the move-speed and turning angle parameters. This comparison indicates the extent to which individuals move left or right, and the range of their movement speed. The von Mises circular (normal) distribution function occurs when turning angles are concentrated symmetrically about a specific bearing. This distribution has parameters (m, c), where m is the angle of maximum probability (modal direction), c - concentration parameter (alters distribution skewness), \"M - ^wf,{v-m) <2-9> where, ID(c) denotes the Bessel function of order zero (see [15]). This distribution is unimodal for 1, and uniform when c = 1 (i.e., constant f( — 1, 8 > 0,0 < x < oo) and T() is the gamma function [15]. The gamma distribution is a general type of statistical distribution that arises naturally in processes which generate observed events separated in time, termed waiting times. Events are here considered to be associated with changes in the speed of movement. If the occurrence of events is random and rare, then the expected distribution function for an event is expected to follow a Poisson distribution. The Poisson distribution describes a number of discrete events in a sequence, where the number of events over time is exponentially distributed. When a — 1, the gamma distribution reduces to an exponential distribution. The probability of events for a Poisson process is assumed to be small. To avoid having multiple events at same time, the expected number of events that occur is assumed to depend only on the time-interval or duration over which they are counted, but not on time or previous history [330]. Frequency distributions of move-speed and turning angle for the observed movements of individuals were interpolated in two-dimensions. These distribution are provided in Figures 2.7, with additional results for the other tracks provided in Appendix B l . The frequency distributions of turning angle having been defined within the range of [—IT, IT] (Figure 2.5) are shown shifted by a factor of IT, in the positive range of [0, 2TT], measured from the fixed x-axis. 2.2 Move-speed and turning angle distributions 40 These results show that considerable variability exists in the distr ibution of move-speed and turning angle across the observed movement paths, wi th the effects of differ-ences in sampling rate, body weight and shoal size. The parameter distributions show significant alterations in their shape (mean, variance, kurtosis). The general form of these distributions supports theoretical functional forms of the G a m m a and von Mises distributions. Figure 2.7: Left: Observed movement path of individual 9601 in 2D wi th estimated weight, W , and belonging to a shoal of size, a. Right: Frequency distributions of move-turning angle and move-speed, with frequencies scaled between (0,1). 2.3 Move-speed and turning angle autocorrelations 47 2.3 Move-speed and turning angle autocorrelations Autocorrelations of move-displacement, speed and turning angle in the hydroacoustic observations were examined and compared to analytical functions for autocorrelation from a theoretical perspective. Move-speed autocorrelation Niwa has formulated a stochastic dynamical equation for the centroid (shoal-centre) motion of a foraging shoal comprising S individuals (S - shoal size), in 2D space as [271], -_L = K ^1 - V - K(3V2V + r}(t) (2.12) where every individual swims at a steady speed of the shoal, / ? - 1 / 2 , J is the tendency of shoaling individuals to swim parallel to each other (schooling tendency), and « _ 1 is a relaxation movement time-scale for individual fish. The centroid velocity, V, of the shoal is allowed to fluctuate according to «5-correlated noise, fj(t) in Equation 2.12, such that two successive points in time t and t' are correlated as, <»/(«)> = 0, mr?{t')) = 2J8(t-t>)\\ (2.13) where e is a fluctuation measure for this correlation, I is the 2D identity matrix, and 5(t — t') is the Dirac-delta function defined as, , 1 t = t 5(t-t') = { (2.14) 0 t±f In the above equations, (x) denotes the expectation value of x (often denoted as E[x] or x). The first term appearing in Equation 2.12 represents the magnitude of fluctuation in shoal velocity due to changes in the mean and variance of velocity of individual fish. The second term represents the correlation in shoal velocity over time, and the third term is random noise. The main assumptions of this model are that: 2.3 Move-speed and turning angle autocorrelations 48 • individual fish move with speeds that fluctuate about a most probable speed of their shoals. • individual fish in shoals move in directions determined by a fixed coefficient repre-senting the degree of polarity. • changes of individual swimming velocity over time are considered statistically in-dependent - fish interact randomly such that the rate of change in velocity and tendency to swim parallel to each other is the same. Under these assumptions, the collective movement of individuals in shoals is main-tained by correlation in their velocities. This theoretical model shows that for individual fish, the correlation in movement parameters over time, such as velocity, has a large effect on the collective motion of their shoals. The model demonstrates that correlation between velocity and other movement parameters, such as turning angle, are important. However, the model does not consider: (1) a relative ability of individual fish to respond to their neighbours to maintain distinct formations while moving collectively, and (2) non-random movement responses to changes in their environment that would require re-alistic assumptions on how movement and behaviour influence each other. Nonetheless, Niwa's model provides a basis upon which new models can be formulated. Models which consider more realistic assumptions may be compared to the predictions of this model. Niwa derives an analytic solution for the model in Equation 2.12, and derives a steady-state solution for mean-square shoal velocity, (V2)st, The persistence time, r§ (s), defined as the tendency of a shoal to continue moving in the same direction, is related to the steady-state shoal velocity as, (2.15) (V2) S t (2.16) 2e/S 2.3 Move-speed and turning angle autocorrelations 49 and its characteristic length is, st (2.17) Using an analytic solution for V(t), and the relations for rj(t) in Equation 2.13, the two-time correlation coefficient of a shoal's move-speed in steady-state is, Substituting Equation 2.16 for the steady-state solution in Equation 2.18 yields the fol-lowing expression for move-speed correlation in shoal velocity, While Equation 2.19 is the form of the velocity autocorrelation expected for steady-state shoal movement, a similar form for the velocity autocorrelation is expected for individuals travelling within a shoal under steady-state assumptions. The magnitude of the autocorrelation coefficient indicates the extent that individual fish accelerate and decelerate over time. A value of one indicates that the individuals move at a constant acceleration, either maintaining a steady-state velocity or increasing/decreasing their move-speed monotonically [290]. In Figures 2.8 and 2.9, I show calculated move-displacement and move-speed auto-correlation functions (ACF) for the individual movements, as the correlation coefficient versus number of moves. Plots of the sample autocorrelation, A C F (correlogram) for each of ith time-lags (moves) were calculated to a time-lag of y/n + 10 [79]. The re-sults show that the speed of movement between successive moves is positively correlated across a range of 15-50 moves, except in one instance (9602), where it oscillates between positive and negative. Associating the fluctuations of move-correlation shown in these plots alongside shoal size in Table 2.16 identifies how move-speed correlations vary with shoal size and move-speed fluctuations induced from shoaling neighbours. Shoal size was t > t! (2,18) t > t' (2.19) 2.3 Move-speed and turning angle autocorrelations 50 estimated by observers at tracking start-times [46]. As shoal size, S, increases at a fixed fluctuation level, e, the exponential decay in the autocorrelation function of velocity is predicted to decrease, thereby maintaining positive correlations out to longer time-lags. These results do not provide support for any clear trend between shoal size and velocity autocorrelation, although both shoal size and the behaviour of fluctuations resulting from cross-correlation in move-velocity between neighbours influences the individual ACF's. The predicted exponential decay in the velocity-correlation is verified, yet additional variability between move-speed correlation across shoal size is evident, possibly a re-sult of differences in shoal shape and structure. Individuals travelling in different shoal formations must rely on lateral lines for monitoring swimming speed. Evident from the velocity-ACF fluctuations, the role of lateral lines in perceiving and mediating the attrac-tive and repulsive forces between neighbours may have a large influence on move-speed correlations [290]. The speed autocorrelation results are fitted to the functional form of Equation 2.19. The fitted results provide empirical estimates for the persistence time (rs) and char-acteristic length (Is) for shoaling individuals. Calculated values for these movement characteristics are provided in Table 2.20. I calculated the mean time and dispersal area from the movement observations, and a coefficient indicating the degree of random movement (diffusion coefficient) in the tracking observations. Later in this Chapter, I discuss how the observations can be compared to theoretical predictions that assume fish move randomly. At times in the sequence of moves, the observations are shown to match random movement predictions, termed self-intersections. The observed number of self-intersections indicates the degree that individuals are able to orient while searching for food [290]. 2.3 Move-speed and turning angle autocorrelations 51 Figure 2.8: Sample autocorrelation functions (ACF) (correlation coefficient versus lag in the number of moves) for move- displacement in the observed movements of B F T individ-uals 2.3 Move-speed and turning angle autocorrelations 52 Figure 2.9: Sample autocorrelation functions (ACF) (correlograms) (correlation coeffi-cient versus lag in the number of moves) for move-speed in the observed movements of B F T individuals 2.3 Move-speed and turning angle autocorrelations 53 Turning angle autocorrelation A theoretical equation for turning angle autocorrelation representing the alteration in the turning probability density, p(r, 0 is defined by, Hj(hj - h) cos - r) | I Y,j(hi ~ h) sin - r) 2a 2 V • cos2 - r) V • sin 2 wfo - r) (2.24) having an independent offset parameter, r defined by the relation, sin 2utj tan(2cvr) = = ± (2.25) v 1 XV cos 2 ^ v 1 The power estimate, Pn{\\S), defined above, has several useful properties which the usual discrete Fourier transform does not have. First, the inclusion of the r parameter makes the periodogram invariant to a shift of the origin of time. Second, this form makes periodogram analysis equivalent to least-squares fitting of sine curves to the data. The sampling offset, r ensures that Pn(u) is independent of shifts in the actual observation times, U by any constant, thereby resolving the harmonic or oscillatory content within an observed time-series. The Lomb method is superior to Fourier transform methods (FFT) for the sampling of uneven data, as observations are weighted on a per-point basis rather than per-time interval, resolving the problem of a fixed sampling rate and aliased information above the Nyquist critical frequency. Statistically, P„(a>) is defined so that, if the time series of (hj — h) is purely noise, then the power in Pn(uj) follows an exponential probability distribution. This exponential distribution provides a convenient estimate of the probability that a given peak is a true signal, or whether it is the result of randomly distributed noise. Using the Lomb method, I re-sampled the movement data at different rates. This procedure re-samples the data, scanning over frequencies above and below / c , generat-ing power spectra for move-turning angle, P(f)v, and move-speed, P(f)v, as spectral power versus / = U>/2TV(radians) (Figures 2.12- 2.14). The Nyquist critical frequency corresponds to equal re-sampling of the movement data at / = fc = 0.5. The resulting spectra show the relative contribution of changes in movement parameters across a range 2.4 Spectral identification of movement modes 60 of M independent sampling frequencies for each movement path. Inspection of these spectra reveal the level of background noise, single and multiple peak frequency signals. By examining how the maximum peak intensity for turning angle and move-speed varies along the observed movement path, I measure how the maximum peak intensities cor-relate between parameters as second-order autocorrelations. The autocorrelation results for the observed movement path, 9601 are shown in Figure 2.15. Results from this anal-ysis for the other observed movements are contained in Appendix B2. The paths were then segmented according to two separate modes based on correlation and de-correlation of the rate of change in turning angle and move-speed, denoted mi and mi respectively. Mode mi is identified with cross-correlation in turning rate and move-speed, and mi, with an absence of such correlation. The number of occurrences of each mode and mode-switching events for each movement path are provided in Table 2.15. B F T made deep daily dives near dusk and dawn. These dives are identified as a third characteristic of their movement, denoted m^, as a mode-switching event that separates the mi and mi movement modes. I aggregated the segments of each mode type for all the observed movement paths calculating modal statistics. Based on computed standard error estimates, the vertical inclination and move-direction angles do not reveal significant differences between move-ment modes (Table 2.16). Mean turning angles for observed movement according to mode mi are generally smaller than mode mi. The results indicate consistent ranges for the movement parameters. However, significant differences in rate of turning, and depth preference between movement modes were found. Examples showing these difference's are provided in Figures 2.16 and 2.17. Figure 2.16 compares the observed data with theoretical probability distributions of turning angle, p(r, if) and time rate of change in turning angle, (3(T). The probability distribution (q(r)) of move-lengths equal to or 2.4 Spectral identification of movement modes 61 Table 2.15: The number of segments, N(mi) and iV(m 2 ) , for mi and m 2 modes, and the number of mode-switching events, S^m^) identified within the individual B F T move-ment trajectories from the Lomb spectral analysis. I D J V ( m i ) N(m2) 5 ( m l i 2 ) 9601 1 2 2 9 6 0 2 3 2 4 9 6 0 3 1 1 1 9 6 0 4 a 2 3 4 9 6 0 4 b 0 1 0 9701 1 2 2 9 7 0 2 2 2 3 9 7 0 3 1 2 2 9 7 0 4 1 1 1 9 7 0 5 2 1 2 greater than the mean move duration (time-interval between successive moves), r , de-cays exponentially. The frequency distribution of mode-switching events over time of day shows higher frequencies coinciding with dusk and dawn as shown in Figure 2.17. More movement data is required to determine the significance of mode-switching events occurring near dusk and dawn. 2.4 Spectral identification of movement modes 62 50 40 * 30 s PH 20 10 0 -9601 0.0 0.2 50 40 > 30 s PH 20 -| 10 0 9602 0.0 0.2 0.4 0.6 0.8 1.0 40 30 20 10 0 -9603 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2.12: Lomb normalized periodograms (Power spectra) of P(f) (spectral power) versus frequency (Hz=l/s), / = o>/27r for all moves n* = ( l , . . ,n) . The spectra show relative peak intensities in the signals of rate of change in turning angle, P(f)v>, and move-speed, P(f)v relative to background noise fluctuations. Peak intensities change across an individual's movement path according to the autocorrelation of these parame-ters between successive moves. Results are shown for individuals 9601-9603. 2.4 Spectral identification of movement modes 9604a 50 40 HI > 30 S &. 20 -J io A 9604a 0 _ ^ 4 r j J t l M V U 0.0 0.2 0.4 0.6 0.8 1.0 9604b 0.0 0.2 0.4 0.6 0.8 1.0 80 60 H > S 4 0 H 9701 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2.13: Same as Figure 2.12 for observed movement paths 9604a-9701. 2.4 Spectral identification of movement modes Figure 2.14: Same as Figure 2.12 for observed movement paths 9702-9705. 2.4 Spectral identification of movement modes 65 9601 Figure 2.15: Autocorrelation results for individual 9601. (Top to Bottom): (a) Amplitude variation of the maximum signal, P(f)max with greater than 99% significance above Gaussian white noise versus reference move position, n*, where n* = (l,...,n) and n is the total number of successive moves. This plot shows variation in the second-order correlation of turning angle and move-speed (turning rate and acceleration/deceleration) across the observed movement trajectory, (b) and (c) Lomb normalized periodograms, P(f) (spectral power) versus frequency (Hz=l/s), / = U>/2TT profiled for the second n* = (l,..,n = 300) total moves showing separable signal peaks (movement modes) above a noise background. 2.4 Spectral identification of movement modes 66 Table 2.16: Modal (mi and m 2 ) statistics of move-length (/,), time duration (TJ), verti-cal inclination (#;), directional (0,) and turning (ipi) angles calculated for the observed individual movements of B F T . Mode m i ID h(mi)(m) Ti(mi)(s) Vi(m/s) fi{°) 9601 815.4 600.0 1.35 89.59 13.46 9602 103.6 60.16 1.73 86.32 13.49 9603 71.53 16.52 5.42 87.36 13.14 9604a 91.62 62.77 1.45 87.30 16.44 9604b - - - - -9701 95.15 61.69 1.53 88.52 17.11 9702 79.60 60.69 1.31 86.98 19.82 9703 74.52 60.68 1.23 87.07 17.12 9704 103.8 60.70 1.71 87.30 20.14 9705 92.61 66.64 1.42 86.88 21.63 169.8 ± 8 0 . 8 0 116.7 ± 5 7 . 5 2 1.91 ± 0.443 87.48 ± 0.3278 16.93 ± 1.049 Mode m,2 ID Zi(m 2)(m) Ti(m2)(s) Vi(m/s) 0i(°) 9601 672.2 548.0 1.31 89.53 29.82 9602 93.50 60.15 1.56 87.16 19.57 9603 125.6 16.54 8.60 88.87 18.15 9604a 86.11 73.65 1.31 87.39 20.55 9604b 119.6 65.68 1.83 87.40 17.39 9701 137.6 80.12 1.59 88.24 18.89 9702 87.87 60.62 1.45 87.76 18.20 9703 53.38 62.08 0.86 86.98 33.72 9704 122.5 61.13 2.01 88.05 15.16 9705 125.0 63.83 2.00 84.60 18.59 162.3 ± 5 7 . 2 2 109.2 ± 49.05 2.25 ± 0 . 7 1 4 87.50 ±0 .4168 21.00 ± 1.871 2.4 Spectral identification of movement modes 07 Figure 2.16: Observed turning statistics derived for the B F T movements. Here, p(r, ip) is the probability distribution of turning for a move of duration r (frequency with which moves of duration r are observed). The frequency distribution for p(T,(ri) (rad), turning angle, i + Pi) (2.26) i = l n i = l where Pi is defined as, 0i = (2.30) where U is the move length. Substituting Equations 2.9 and 2.20 for the probability distributions of turning angle and turning angle correlation between successive moves over time, into the above equation yields, 7T TT TT OO (5xi) = (v) J p((pi-i)d(pi-i - - - J p{ip\\)dipi • Jpi^dfa x cos(0i + Pi) J Tp(T,fa + p{)dr —TT —TT —TT 0 (2.31) The initial distribution of directional angle, 0, is distinguished in the above expression as p((j)\\)d(bi. The above equation relates the displacement of a move i to the initial ori-entation angle i, the expected value of move-speed, (v), and the successive integrations 2.6 Theoretical movement model predictions 74 of the probability of turning over all angles (—7r, 7r) for previous moves, p( + V^i) (2-32) i = l i = l j'=l which is the net-squared displacement of an individual after n successive moves. Here-after, I denote net-squared displacement as R 2 e t — (x2) for i = (1, ...,n) moves, where 'net' refers to the total number of moves observed. R2et therefore denotes the square of the total displacement between the start and end of a movement trajectory, whereas the square of the displacement across a smaller number of moves is denoted as R^. A general expression for the variance of movement along the X axis coordinate (sim-ilarly for all coordinates in a given reference system) is, o2xx(n) == [(x2)n - (x)l] (2.33) and similarly for covariance (variance between coordinate axes), v2xy(n) = [(xy)n-(x)n(y)n] • (2.34) which requires values of the expectations in Equations 2.31 and 2.32. The link between individual movement parameters and those that characterize the consequence of move-ments at larger spatial scales requires extrapolating the expressions for the expected value, net-square displacement, and the variances for each coordinate axis in time to 2.6 Theoretical movement model predictions 75 their asymptotic limits. Asymptotic expressions for population parameters, expressed in terms of movement parameters at the individual level are, for population advection, = (vx,vy), vx = lim (x)t, vy = lim (y)t (2.35) t—»oo t—»oo for population mobility, = (fJ.x,iJ,y), px=\\im-!-o-2xx(t), fiy = lim ±-o2yy(t) (2.36) t-*oo Zt t—>oo Zl and for population diffusion, D, D = lim ^ = lim j^f (2.37) t—>oo Zl t—>oo z l Net-displacement, .R 2 ^, is defined by Equation 2.32. Patlak has investigated the two main alterations leading to alternative random walk models, and by solving partial differential equations has shown how movement persistence and external bias assumptions alter a diffusion process with varying degrees of uncertainty [292,293]. The effect of movement persistence and external bias at the diffusion level is discussed by Turchin [396,397]. For correlated random walk models, Marsh and Jones have derived an expression for R 2 e t from Equation 2.32 [239] in two-dimensions. Similarly, Skellam has derived a formula for R 2 e t for a correlated random walk that was generalized by Nossal and Weiss to include the external bias, and further simplified by Karieva and Shigesada [189,272,373]. The correlated and biased correlated random walk models are distinguished according to the width of the turning angle distribution, with the correlated model having a narrow width, and the biased correlated model distinguished by broadening in width. Correlated-random walk (CRW model) This model is derived with move-length, / j , and move-turning angle, cpj sampled randomly from theoretical probability density functions (pdf). The turning angle distri-bution is assumed symmetric for the CRW model (i.e., the mean sine of turning angle, 2.6 Theoretical movement model predictions 76 s — (simp) = 0). A lack of such symmetry implies a directional bias. A movement path is formed by successive independent sampling, forming sequences k and cpi. As defined in Equation 2.4, the sequence of move-direction angles, fa is not independent. The second term (cross-coordinate term) in the general equation for (a;2) in 2D, as Rlet = (x2) = E<(<^ )2>+2 E E w (2-38) i = i t = i j = i becomes, E xjy*= E lJlk e x p(^ - = E lJlk e x p v ( E - E ^m I f (2-39) j^ fc I \\m=l m=l / J and, E Xjj/fc = ^ ZjZfc < exp I i ]P (2.40) j^k j>k \\ \\ m=k / \\ m=k / J with the complex form of the harmonics using Euler's formula, substituting Xj = lj exp(i(j)j) and yj — /^ exp(—z0fc). The expected value, i ? 2 e t = (x2) is then, ( E = (o2 E (c+is)3~k + (c - i s Y~ k ( 2 - 4 1 ) where c and s are the mean cosine and sine of turning angle respectively, in Equation 2.8. For the assumed, symmetric turning angle distribution where c = (cos if), = (o2E2cJ\"fc = 2™> = <*> \\ 2 { l - c - i s ) \\ n - l - c - i s ) + 2(l-c + i s ) { n - 1-c + is JJ (2.45) and evaluating this term for the B C R W model [189] yields, 2 \\ ( c - c 2 - s 2 ) n - c \\ W 1 ( l - c ) 2 + s 2 J )(n+l)/2[((! _ C ) 2 ((1 - C ) 2 + S 2 ) 2 2 j 2s 2 + (c2 + s 2)(\" + 1)/ 2[((l - c) 2 - s2) + (cos(n + l)a) - 2s(l - c) sin(w + v 2 j (c - c 2 - s2)n -c 2s2 + (c2 + s 2 ) ^ 1 ) / 2 W 1 ( l - c ) 2 + s 2 ( ( l - c ) 2 + s 2 ) 2 7 where, 7 is defined as, 7 = {(((l - c ) 2 - s 2) cos(n + l)a - 2s(l - c) sin(n + l)a)} , a = tan _ 1 (s/c) (2.46) With the derived cross-term expression for the B C R W model, R„et is, R2 - IR2) nil2) I 2 m 2 f ( C \" °2 ~ S 2 ) U - C I 2 ^ + + ^ ) ( n + 1 ) / 2 , . | r 2 4 7 l n^e* - (Rn) + 2 j ( i _ C )2 + ,2 + ( ( 1 _ C ) 2 + ,2)2 T j ( 2 \" 4 7 ) Therefore, i ? 2 e t for the correlated theoretical movement models (CRW and BCRW), in Equations 2.43 and 2.47, both predict that fluctuations in move-turning angle, cp,, alter the ratio c / ( l — c), that increases for small turning angles (persistence), increasing the net-squared displacement. Assuming a population of non-interacting individuals, population diffusion, in its linear form, is also expected to increase. 2.6 Theoretical movement model predictions 78 Simple-random walk (RW model) The simple-random walk model (RW) samples the move-length and move-direction distributions independently forming sequences, {k}, and {&} respectively. Here, the move-direction angle, is assumed to be independent of turning angle. The sequences of move-length and move-direction angle follow state-space equations in 2D, describing the process by which an individual moves, oii = + rji, i = l,...,n (2.48) where time, t = (0, iAT,nAT) across the ith move, is the true position and rji is a serially un-correlated random vector with expectation of its mean, (r]i) = 0, and covariance, Qi represented in 2D as a 2 x 2 matrix, ° ) (249) V < < J \\ 0 2 D ) A n equation describing the measurement process can similarly be formed as a first-order Markov statistical sampling of the movement RW process of an individual moving according to process, yi = ai + €i, i = l,...,n (2.50) where yi is a vector of observed position for an individual, a; is its actual position deter-mined from the assumptions of a RW process, and is a serially un-correlated random vector with expectation of its mean, (e) = 0, and covariance, Hi represented in 2D as a 2 x 2 matrix in the diffusion limit, H i = f oL < \\ ( oL o \\ ( 2 5 i ) \\ < °yy ) V 0 °lv J Although a theoretical RW process for individual movement is fully described by Equation 2.48, the measurement of such a process given by Equation 2.50 is used in 2.6 Theoretical movement model predictions 79 the next section, as a necessary part of the Kalman filtering of systematic measurement uncertainties in latitude and longitude geoposition estimates for the analysis of light-archival observations. For the RW model, evaluation of the cross-term in Equation 2.41 yields, (^x3yk) = {l)2{n(n-l)a'2} (2.52) where, a' = (cos (f>) as the expected value in the cosine of move-direction. Rlet = (Rl) = n(l2) + n(n - 1) 0. Biased-random walk (BRW model) For a BRW model, bias terms are introduced in the move-sequence, and Equa-tion 2.48 is re-written as, ai = ati-i + Vi + i = l,...,n (2.54) where Vi is a vector of bias in movement at the diffusion limit whereby, advection in each coordinate direction is used, V ; = (vx,vy)i = (u,v)i, (2.55) Bias in the movement of a RW, giving rise to a BRW process, introduces a corresponding term in the measurement equation (equation 2.50), which is re-written as, yi = ai + di + €i, i = l,...,n (2.56) with di as the vector, (2.57) 2.7 Kalman filtering of geoposition data from light-archival data 80 where dx and dy are zero when no systematic measurement bias exists. The use of the Kalman method to filter measurement errors requires numerical fitting of these model types to the observations, and the use of an objective minimization (log-likelihood function) involving the asymptotic movement parameters of advection and diffusion. Bootstrap simulation to compare the predictions of the correlated random walk models (CRW and BCRW) was next conducted for the PSAT tagging observations of B F T movements. 2.7 Kalman filtering of geoposition data from light-archival data Kalman filtering is a Bayesian statistical inference and forecasting method used in many fields with diverse theoretical and practical application. [127,250]. Theoretically, the Kalman filter (KF) method provides estimation of linear and quadratic Gaussian problems - the problem of estimating the 'state' of a dynamic system perturbed by Gaus-sian white noise. The spectral analysis of movement parameters identified correlation peaks in the power spectra above Gaussian white noise. Using a separate equation to describe the measurement process linearly correlated to changes of a natural dynamic system, the K F method provides an estimator that is statistically optimal with respect to any quadratic function of measurement error [127]. It is commonly used in statisti-cal regression and time-series analysis. Kalman filtering enables the control of processes occurring within natural, dynamic systems. The method involves coupling a descrip-tion of an underlying process (process model) to the estimation of uncertainties in the measurement of process parameters (measurement model). For many applications, it is not possible or desirable to measure every dynamic variable, and the K F method provides a means for inferring missing or excluded information from indirect and noisy measurements. For this reason, this method is applied in predicting the state of systems and forecasting behaviour in the future (e.g., earthquake zones, river flow during floods, 2.7 Kalman filtering of geoposition data from light-archival data 81 planet trajectories, and trends in financial markets. Sibert and Fournier have recently applied the K F method in the analysis of individual movements obtained from recent light-archival tagging experiments on tunas [366, 369]. This new application of the K F method increases estimation accuracy in geoposition estimates from tracking and tagging devices, and enables a consideration and comparison of the assumptions and predictions of theoretical movement models. The set of K F recursive equations [366] that were used to filtering the PSAT data are, state estimate extrapolation : = ojj- i + Vi (2.58) uJi^yi- CKi|i_i - di (2.59) error covariance extrapolation : P i K _ ! = + Qi (2.60) Fi = Pm-! + Hi (2.61) state estimate update : cxi = Oi\\i-i + Pm-iF-'ui (2.62) error covariance update : Pi = - P^F-'Pm-! (2.63) Pi is the covariance matrix of geoposition estimation error, and a^-i is the successive estimate of the actual position at move i, given that it is actually located at position ojj. Reference [127] provides a detailed description of the theory of the method and the derivation of the above recursive equations. I implemented this method to examine a selected set of the light-archival tagging observations, using code provided by Sibert and coauthors that utilizes ADModel Builder numerical routines (Otter Research Inc.). The authors also examine full set of observations in terms of the numerical performance of this method using numerical simulation testing. The limited number of observations 2.7 Kalman filtering of geoposition data from light-archival data 82 used in my analysis are taken from a larger set of observations collected by M . Lutcavage (unpublished data). Short-term light-archival tagging observations used in testing these archival tags are listed in Table 2.17. A summary of the selected set of observations obtained from long-term light archival tagging experiments is provided in Table 2.18. The short-term observations were not analyzed having been conducted as tests on the deployment and geoposition estimation algorithms for these archival tags using light-intensity data. Application of the K F method yields estimates on the geoposition uncertainty, and the advection/diffusion movement parameters (cr2,, a 2 , u, v, D). Estimates are obtained through numerical optimization of a goal function that assumes normal or Gaussian distributed errors and parameter distributions [161]. Synonymous with minimizing error terms in a cost or likelihood function, optimization involves minimizing the negative, natural logarithm of the likelihood function, n 1 n - In L = n In 2TT + - ^ In \\Fi \\ + - ^ u; Oi t— o co m CN 00 O O) C i CI Oi O Oi O) O O O O rt rt rH o> TP TP m CN O CN i—l TC m o> co co CN CO CN CN CN rt CN rt T f T f T f T f T f rt Oi T f T f O lO CN CN CN CO Oi Oi O Ol lO CO CO CO CO CO Oi 00 Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi o o o © T f o m O i CN CN CN CN o o o CO Oi T f 00 CO CO CO CO t— o pi CN CN H CN in T f in co t- in CO CO O Oi Oi rt CN CN O T f T f 00 T f T f O CO CO CO 2.7 Kalman filtering of geoposition data from light-archival data -3 o CN co N b N H (0.4334, 2.821) (0.2500,2.364) CM O l O l k ° O LO T-I CM oo ^ CO ^ CO J-. o O LO r-l CM O O - o 10.1942 11.120 5.41 9.804 -42.144 -50.00 N S 3, Q 95.7696 227.67 638.74 826.79 819.22 12224.82 1626.70 1217.54 2145.27 (p/rau) A -1.47411 -1.4272 2.429 4.3505 -3.118 -2.629 (p/rau) n 1.61221 1.5992 9.905 9.5967 6.721 6.734 -InL -1025.08 -969.27 -281.51 -488.49 -174.21 -178.14 a. e 6 (BRW) 4 (RW) (DM) 6 (BRW) 4 (RW) (DM) 6 (BRW) 4 (RW) (DM) ET(d) O l i v CN co o CM I v t-Pop-Up 30/06/00 15/04/00 24/12/99 Release 24/09/99 25/09/99 08/10/99 ID.(n) 4235 (171) 4364 (68) 4745 (51) 2.7 Kalman filtering of geoposition data from light-archival data 86 I .onuitude (decrees W.) Figure 2.19: GIS display of observed movement path of individual (4235) wi th in the Atlant ic Ocean based on geopositions derived from archived light-intensity. 2.7 Kalman filtering of geoposition data from light-archival data 8 7 7 0 6 0 5 0 4 0 3 0 2 0 • observed predicted (BRW) predicted (RW) predicted (DM) • initial-release location (GPS) • final - pop-up location (GPS) 4235 -74 -72 -70 -68 -66 -64 -62 -60 -58 -56 Longitude ( °W) Figure 2.20: Ka lman filtered of observed movement path of individual B F T , 4235. The optimized paths representing filtered geoposition estimates based on the theoretical pre-dictions of the R W and B R W models are shown in relation to the observed geoposition estimates. Latitude is °N, and longitude in °W in reference to the equator and prime meridian, respectively. 2.7 Kalman filtering of geoposition data from light-archival data 89 6 0 30 I L _ _ L _ _ _ | _ 1 1 1 1 -80 -70 -60 -50 -40 -30 -20 -10 Longitude (°W) Figure 2.22: Same as Figure 2.20 for observed movement path 4364. Figure 2.23: Same as Figure 2.19 for observed movement path 4745. Kalman filtering of geoposition data from light-archival data 4 2 28 1 ' —>— —' 1 -80 -75 -70 -65 -60 Longitude (°W) Figure 2.24: Same as Figure 2.22 for observed movement path 4745. 2.8 Significance testing: observations and model predictions 92 2.8 Significance testing: observations and model predictions Statistical significance testing of the theoretical movement models based on observed variations of (x2) requires estimating its variance in the form, o2(R)n = R2n- (Rn)2 (2.65) 2 where the corresponding equations for each theoretical model determining Rn are used. To determine the variance above, theoretical equations for Rn are also necessary. Mc-Colloch and Cain have developed a numerical scheme for calculating the variance of Rn using harmonic expansions leading to a complex series of calculations. Alternatively, Rn can be determined using statistical bootstrapping techniques. This later approach was used to predict confidence intervals for the theoretical models. Statistical bootstrapping Bootstrap simulation can have parametric or non-parametric assumptions using ob-served data to generate a large number of possible simulated outcomes. Both approaches lead to relative improvements in the accuracy of statistical tests [97]. The parametric bootstrap procedure involves randomly sampling from the theoretical parameter distri-butions and generates large numbers of simulated movement paths. This procedure is classed as a Monte-Carlo randomization approach, where parameter values are sampled from respective distributions with replacement. The non-parametric approach does not rely on theoretical assumptions of how move-parameters are distributed, but instead per-mits random sampling from observed frequency distributions of the movement parameters to generate simulated outcomes. Non-parametric statistical bootstrapping, in estimat-ing statistical measures such as variance and standard error, therefore assumes that the observed data consists of independently- and identically-distributed observations, de-noted as y—(xx,X2, ...,xn). Using a Monte-Carlo algorithm, n independent samples are 2.8 Significance testing: observations and model predictions 93 drawn with replacement from an empirical distribution, F, yielding a bootstrap sample, y*=(x\\,x*2, . . . ,£*). The bootstrap procedure involves three steps: (1) a random number generator used to independently draw a large total number of bootstrap samples, B ~3> n, denoted as yl, y\\,...,y*B, (2) for each bootstrap sample, yl a statistic of interest is eval-uated, as 9*(b) = #(y*(b)) for b = (1, ...,B) samples, (3) the mean and variance of the statistic is then calculated from the bootstrap samples as, o B = - i - (J2Hb)-H-)2) >(•) = j\\flHb) (2-66) \\b=l / 6=1 As B —> oo, OB will approach the true variance, b = o(F). 100(1 — 2a)% confidence intervals (CI) for a statistic, 9*(b), denoted \\9(L),%/)]> a r e obtained by ordering the bootstrap samples required to calculate CPs estimated from the distribution percentiles of a statistic. Efron and Tibshirani have examined how many bootstrap replications or samples must be drawn to generate bootstrap estimates of statistical measures to achieve de-sired values of statistical power [97]. The following approximation for the coefficient of variation (CV) (i.e., ratio of standard deviation and mean), is a measure of statistical precision of the bootstrap method, { \\ 1/2 C V { b f + ^B+l I (2.67) where 8 is the kurtosis of the bootstrap distribution of 9* given the observed data y, and E(S) is the expected value averaged over y. C V values considered to provide sufficient statistical precision, lie in the range (0.10-0.30). Assuming E(8) = 0, Efron and Tib-shirani have calculated that to achieve CV(b) > 0.10 there is little improvement past B = 100 bootstrap samples, and B = 25 typically provides reasonable results. How-ever, to accurately estimate the confidence intervals of a statistic, a far higher number of bootstrap samples, B, are required. Calculations available from the literature indicate 2.8 Significance testing: observations and model predictions 94 B = 1000 is an approximate minimum to compute reasonably accurate confidence inter-vals. Confidence interval estimation is a more sensitive measure of statistical accuracy than standard errors, and therefore also requires more computational effort. I used the bootstrap procedure with 10,000 simulations to calculate the variance of Rn, calculating 5% and 95% percentiles of the Rn distributions as 95% confidence intervals for the theoretical random walk models. I compared these results to each movement observations for the hydroacoustic data, as shown in Figures (2.28- 2.29). At the spatial scale of these observations (daily movements), the tuna are observed to move with Rn ranging between a lower limit, corresponding to the predicted CRW model, and an upper limit, corresponding to the BCRW model. The linear and quadratic increase of Rnet for the CRW and BCRW models respectively is verified in these results, in agreement with Equations (2.43-2.47). Considerable external bias in their movements is shown with the observed Rnet deviating away from the predictions of the BCRW model for individual paths; 9601, 9602, 9603, 9701, 9702, 9703, and 9705. Across a variable number of successive moves (move lags), the results for paths 9601 and 9703 show movements dominated by external bias (BCRW), whereas for 9603 and 9604a, movement persistence (CRW) dominates. In the case of 9604a, the observed movement fluctuates between CRW and BCRW model predictions. The fluctuation in external movement bias may be explained by the attraction of tuna to local abundances of prey, and was further investigated. Efficiency of shoals searching for food I investigated the searching efficiency of shoals as related to the number of self-intersections between the observed variation in net-squared displacement over time (num-ber of successive moves) and theoretical random walk predictions. Niwa has examined 2.8 Significance testing: observations and model predictions 95 how a foraging shoal is, in theory, expected to move statistically under the external at-traction of preferable regions or prey feeding areas (patches or arenas), whereby they move according to Equation 2.12 [270]. In the absence of external attraction to foraging patches, a shoal is considered to move according to correlated random-walk predictions. To examine the resulting RNET characteristics when a shoal forages, I followed arguments similarly used by Niwa [270]. Let a shoal of size N forage for a total time, T=l day. The time individuals in the shoal spend feeding on prey is A^T. Therefore, the total time spent feeding between forage patches is, T — A^T. The travelling distance, LT-ANT, calculated using the shoal's steady-state speed, VC is then VC(T — A^T). The travelling distance is much larger than the shoal's characteristic length, lN, defined in Equation 2.17. The expected value of the net-square displacement for the time duration when a shoal travels denoted as, (P2\"_AArT) is its mean dispersal area. Mean dispersal area is dependent upon the total foraging time (T) and relies on the assumption that the number of local prey patches per unit area in their vicinity, c, satisfies c {RT-ANT)- T n e P a t n o f a shoal that is searching between prey foraging patches is then expected to intersect itself at a rate proportional to L\\_ANTI (RT-ANT) The rate of self-intersection is expected to vary inversely proportional to shoal size. With reference to the defining equations for persistence time (r^) and characteristic length (/yv) for a shoal in Equations 2.16 and 2.17, the net-squared displacement ((R2)) predicted by a CRW model [271] is, Defining the searching efficiency of a shoal, SN, as the encounter rate of a shoal with patches of its prey, Niwa derives the following equation, (2.68) SN — cV LT-ANT — v 'T-ANT (T - ANT) - l (2.69) 2.8 Significance testing: observations and model predictions 96 T-A NT f ( I l T Number of moves, 11 (or time, t) Figure 2.25: Schematic showing how mean dispersal area, RT-ANTI t i m e spent searching, (T — ANT), and foraging time, A^T is calculated from fluctuations in RNET. These estimates are used to explain the observed variation in their external movement bias on the basis of the attraction of their shoals to areas where they forage on their prey. where u is the self-intersection parameter, c is the number density of prey patches, and 6jv is the width of the surface area (swath width) of a' shoal perpendicular to its move-direction. The overlapping area of a shoal's search path is assumed to be proportional to the number of self-intersections. In my calculations, shoals were taken to have mean centroid speeds of 1.5 m/s, a shoal swath width of 6JV=1.5 m, assumed to scale linearly to shoal size, and shoal size estimates were assigned 5% standard error. The number density of prey patches, c was fixed at 100, with a persistence time, TJV of (17.65 ± 2.845)s deter-mined from the results of fitting the velocity-ACF's to a negative exponential function v = (y2t) exp - */ T N . The fitting plots are shown previously in Figures 2.9, with fitted parameter estimates and uncertainties provided in Table 2.19. I calculated the number 2.8 Significance testing: observations and model predictions 97 Table 2.19: Results of fitting move-speed autocorrelations (ACF) observed from hydroa-coustic tracking of B F T to the general form v = (V2t) e x p _ t / T N , for t = nAt, over n successive lags between moves. These results are used to determine values for persistence time, T/v and shoal searching efficiency for each movement path. ID N ±SN (shoal size) Vs\\±8(Vs\\){m/s) b±5b(s~l) TN ± 6(TN)(S) Adjusted R2 9601 500 ± 25 0.827 ± 0.066 0.133 ± 0 . 0 1 4 7.542 ± 7.544 0.885 9602 175 ± 8 . 7 5 0.440 ± 0 . 1 2 8 0.231 ± 0 . 0 8 4 4.330 ± 8.365 0.314 9603 1000 ± 50 0.784 ±0 .031 0.061 ± 0.003 16.44 ± 6.302 0.931 9604a 100 ± 5 0.672 ±0 .031 0.049 ± 0.003 20.35 ± 0 . 5 1 7 0.884 9604b 100 ± 5 0.505 ± 0.069 0.045 ± 0 . 0 1 0 22.38 ± 0.656 0.341 9701 12 ± 0 . 6 1.100 ± 0 . 0 2 9 0.036 ± 0.002 28.03 ± 0.044 0.943 9702 6 ± 0 . 3 0.623 ± 0.037 0.042 ± 0.004 23.65 ± 0.028 0.785 9703 12 ± 0 . 6 1.020 ±0 .051 0.083 ± 0.006 11.99 ± 0 . 1 0 7 0.923 9704 250 ± 12.5 1.114 ± 0 . 0 5 5 0.096 ± 0.007 10.37 ± 2 . 5 6 5 0.933 9705 200 ± 10 0.738 ± 0.028 0.032 ± 0.002 31.39 ± 0 . 6 6 5 0.876 of self-intersections of the observed R n e t path with the theoretical C R W prediction, as shown in Figures (2.28- 2.29). In movement paths 9604a and 9604b the self-intersecting fluctuation of their movements with respect to the C R W prediction is most evident. The movements of 9602, 9603, 9604a show more rapid self-intersections over smaller number of moves. For each individual path, I derived estimates for the searching efficiency of their shoals using Equation 2.69. These estimates are listed in Table 2.20. Figure 2.25 pro-vides a schematic illustrating how I related the movement fluctuations observed between the C R W and B C R W model predictions of R n e t to differences in the relative search-ing efficiency of their shoals. Calculated efficiencies show that shoal size increases the efficiency of searching when individuals shoal, where shoal size and percent searching efficiency scale in ratio of approximately 1:0.0357%. This ratio estimates that the addi-tion of twenty-eight individuals to a shoal, increases the searching efficiency of shoals by approximately 1%. 2.8 Significance testing: observations and model predictions 98 Significance testing In addition to statistical bootstrapping, I applied a significance test, developed by Marsh and Jones, to compare the theoretical predictions of the C R W and B C R W models to the movement observations from hydroacoustic tracking [239]. This test statistically compares the relative contribution of persistence and external bias on the directionality of movement. The statistic for this significance test, hereafter termed the A statistic is, A„fc= = — n (2.70) where fa and CP CQ O co fl O • rH CP CP CQ SH CP CP CQ fl O T3 CP CQ CO - O CO CP -8 a • r H +^ CO CP fl . 2 'co eg fe; r t H H — T3 co < ~ - 0 5 T f c O T f c O C O O C O O i C O C O m co t> T f C O O C N O O C O r H O r H H C N C N c N C N H H W c o m o i c N T t m N c N h w \"oionioiooHN ' \" ' 6 d P P cd HH co in C N T f CO CO Oi rt CN T f CN T f in co 00 rt CO t ~ _ CN T f T f Oi rt S T f 00 CN O £5 t> CO CO rt T f . . O O O rH rt 1 - 1 d o d d d co m co m T f o t CN T f O O Is-i n co co oi co oo • • ' O l O j i T f Oi 00 Oi CD rt CO O CN T f CN T f CN T f co m CO 2 ~ o o in o CN CN cd X> CN CO T f T f o o o o o CO CO CO CO CO C N co T f m o o o o t- o l> t~ O l O l O i O l O i O i O l O i O i O l -H -H -H 1^ 8 m co m C N T f P T f O T f m CN T f CD oo § B P O i 00 t -co ,-c O o O i oo rr o o o °0 O i CN T f Oi CO O i Q H H CO H N O I N N C O M O I H l O T f O l T f T f T f O i C O N l O C O c N N C O ° C O C N H H C C N H I O I O O C M r t t - T f O r t O C N O o in o P d C N o d m co 1 0 m t^- CN rt O O O O O P g P in d oo H oo iv o rt O i rt CO CO O O O O O o 0 0 0 © o d d d d o O c - j p o C O o c O C N O O T f ^ Q O O C N H N C N C N i n C N H ( N H oo co in co P ^ o i CN CO CO CO N CO CO rt rH CO N N CD CO s d H ° ! so CO O O CO CM CN rH 00 CN ce - o rtCNCOTfTfrHCNCOTfin O O O O O O O O O O i C D C O C O C O C D N N b - N N \\Oi O i O i O i O i O i O i O i O i O i -H -H -H H -H H 2.8 Significance testing: observations and model predictions 100 the possible range of observed values for the A 0 & S statistic within [-1,1] are depicted in Figure 2.27. Changes in the distributions about the origin illustrate how internal orientation (persistence) (left pdf) and external orientation (right pdf) in movement contribute to bias in the mean and uncertainty in the variance of the observed value of this statistic, A 0 ( , S . Confidence intervals of the A T test-statistics for the C R W and B C R W models (E(Ab), E(Aa)) are obtained by calculating standard deviations (cra,ab) from simulated values for the test-statistic, A T , generated under the assumptions of each theoretical model. These CI's with 95% confidence are termed '2s-intervals' for the A T test-statistic, for r observations and a — 1.96 « 2 (95% confidence). Results of applying the A significance test across m successive moves, where m = (10, 20, n) and n is the total number of moves observed, are provided in Table 2.21. The 2s-interval ranges for the test-statistics, E(Ab) — AT,CRW and E(Aa) = AT,BCRW that contain values of the observed statistic, A0bs after m successive moves, are identified as (, )*. These results show that for seven of the ten movements observed, external bias is detected after only ten successive moves. In four of these paths (i.e., paths 9602,9603,9701,9704), the external bias was significant over twenty moves. For paths 9601 and 9702, external bias was present across the duration of the movements. (2.73) 2.8 Significance testing: observations and model predictions 101 Table 2.21: Testing of observed statistic (A 0 fe s) for individual movements of B F T to 95% confidence intervals (2<7-intervals) about the expected values, AT,BCRW (see Equa-tion 2.72) for the B C R W , and L\\T,CRW (see Equation 2.71) for C R W theoretical models2. ID Observed statistics (i,A 0(, s) 2200 Li Frequency (10 individuals) Figure 3.40: Depth frequency distributions for B F T individuals across shoal size range categories, 1995, zm = 10. 3.1 Supervised automated image analysis scheme (SAIA) 1996 Frequency ( l (r individuals) Frequency (10 individuals) 10 15 20 25 Frequency (103 individuals) x: 5 -J 30-39 1 10 - 1 15 -J 20 -25 -30 - i i — i 1 1 0 1 Frequency (IO 3 individuals) S p . Q 5 10 15 - T 20 -L 25 H 30 50-99 o 12 16 20 Frequency (10 individuals) 0 5 ? 10 •S 15 cx s Q 20 25 30 100-199 \" 1 1 [ 1 1 0 2 4 6 8 10 12 Frequency (10 individuals) 8 10 *5 Frequency (10J individuals) Figure 3.41: Same as Figure 3.40 for year 1996. 3.1 Supervised automated image analysis scheme (SAIA) 135 Object processing: manual and automated tuna identification and detection Selected output of the pre-processing and object identification steps of the SAIA scheme are shown in Figure 3.42. A simple pattern-matching algorithm is employed to filter objects, using both visual and automated object identification, based upon their spatial position. The centroid positions of all objects identified are associated. This procedure establishes a local neighbourhood criteria having a radial distance threshold of 0.5 BL, denoted as Rcrit, where Rcrit is fixed appreciably less than the expected distance between nearest-neighbours. This condition allows for the case where individual fish may be positioned on top of each other in the vertical, depth dimension. This threshold is used to test the correspondence or match the spatial locations of tuna identified in the visual and automated methods. Object positions from the two identification procedures that do not lie within this fixed radial threshold range are discarded. More complex pattern-matching algorithms in the analysis of static image data or spatial-point processes use Bayesian methods for associating and classifying objects based on shape and other measurable attributes [193]. 3.1 Supervised automated image analysis scheme (SAIA) 136 (C) 17* <> * fit mi . , .y VfE * * • „ \" • « % * rig I mt V » \\-'t n Ml Ufl) - PUB (I) PS BT UMte-(Fi f fi it 11 1U 11! Xi (3-84) y° = N-s^yi ( 3 - 8 5 ) i i N s i (3.87) The mean polarization angle of a shoal (direction of mean shoal velocity) is defined in the image xy plane as, # s - tan\" 1 (3.88) 3.1 Supervised automated image analysis scheme (SAIA) 140 In Cartesian (x, y, z) orthogonal coordinates, the general equation describing the quadratic surface of an ellipsoid is, 4+if+4= 1 (3-89) a2, ¥ cz according to polar radius, c along the z-axis, and equatorial radii a and b along the x and y axes respectively. Corresponding parametric equations for each coordinate axis for 9 G [0,7r), and 0 G [0, 2n] angles as, xs = a sin 9 cos

a). When all three axes dimensions are equal an ellipsoid is a sphere. Eccentricities or ellipticities (ei,e2) relating a ellipsoids's axial dimensions are, e 2 _ a ~ c a? 4 = b2 c b2 (3.90) (3.91) k~— (3.92) e i where k in the above equations is the ellipsoidal modulus. The apparent length (Ls), width (Ws), and depth (Ds) of an ellipsoidal shoal is determined by calculating the maximum distance between the measured positions of identified objects in a shoal along the x, y, and z axes. In this procedure, apparent shoal length is initially determined as the larger shoal axial dimension in the horizontal (xy) image plane. Apparent shoal length and width are then transformed by rotation about the shoal's center (centroid) in the xy plane by the shoal polarization angle (^ s ) , so that 3.1 Supervised automated image analysis scheme (SAIA) 141 the axis of shoal width, W's and shoal length, L'a is measured parallel and perpendicular to the direction of shoal orientation, respectively. The rotation transformation equations describing the conversion of apparent shoal dimensions (LS,WS), to actual (L'S,W'S) measures are, Conversion of apparent length and width with respect to shoal orientation is depicted in Figure 3.43. The converted length (L's), width (Wa), and depth (D'a) of an ellipsoidal shoal is related to its axial dimensions as: L's = a/2, W's = b/2 and D's — c/2, respectively. Cross-sectional surface areas in the horizontal xy plane (CSAxy = nba) and vertical yz planes (CSAyz = irbc) are calculated. Ellipsoidal volume is calculated as Vs = (4/3)7tabc with ellipsoidal surface area (SAS) calculated for the case of oblate (c < a) and prolate (c > a) spheroidal shoal shape using the following formula [36], The arc-length of an ellipse in the xy plane representing a horizontal slice of a 3D ellipsoid, is calculated as, L's = 2 v / ( a 2 - 6 2 ) c o s 2 Vs + b2 (3.93) (3.94) o / x/(asint9') 2 + (6cos69')W o 2TT b J y/b2 + (a 2 - b2) sin 2 B'dQ' (3.95) o b 7 y/l - k2 sin 2 6'dQ' o = b-E{9',k) 3.1 Supervised automated image analysis scheme (SAIA) 142 (A) 2D plane - ellipsoidal (B) shoal dimensions transformed by shoal geometry rotation ui direction of mean shoal polarity Figure 3.43: Conversion of apparent length (Ls) and wid th (Wa) under transformation rotation by shoal polarization angle, tys aligned wi th respect to shoal velocity, v to shoal length (L's) and wid th (W's). 3.1 Supervised automated image analysis scheme (SAIA) 143 where E(8', k) is an incomplete elliptic integral of the 2nd kind with modulus k defined in Equation 3.90 and k = (\\ — o?/b2). E(6', k) can be approximated by a rapidly converging Gauss-Kummer series to the incomplete elliptic integral, n =o y n J where h = ((a — b)/(a + b))2. Truncation of this series provides an approximation used in calculating a shoal's perimeter length in the image, xy-plane as, with relative error of ~ 3 - 2 _ 1 7 / i 5 for small values of h. Additional shoal variables relating the internal organization of individuals with their external shape, assuming ellipsoidal shoal structure, are: packing density (calculated as the number of individuals, Ns per unit volume), area-to-volume ratio (AVRS), shoal elongation (a/c), and compactness CS — (ATTSAS/P2). Shoal compactness is a shape statistic denned as the ratio of area to the area of a circle with the same perimeter. Shoal compactness is invariant under changes in shoal polarization angle/mean orientation. Results of these calculations are provided in Section 3.3. Shoal size estimation The resulting visual Nm, automated Nc and corrected Ns estimates of shoal size (in-dividual counts) are shown in Figures (3.44-3.46) for (1994-96). Reduced x2/^/-statistics were used to compare two observed frequency distributions and to compare observed and expected frequencies. x2/4f estimates were calculated to compare shoal size estimates between survey years, by adapting available statistical subroutines [330]. In the case of (3.96) P ~ ny/2(a2 + b2) 3(a + b) - y/(a + 3b)(3a-+b) (3.97) 3.1 Supervised automated image analysis scheme (SAIA) 144 comparing two distributions of observed frequencies( [330] and references therein), the X2-statistic is, i where Ri and Si are the number of shoal size estimates for the first and second data sets for statistical comparison. Each term in this summation approximates the square of a normally-distributed quantity having unit variance - whereby the variance of the difference of two normal distributed quantities is the sum of their individual variances, not their average. Given the unequal number of total annual shoal count estimates, a formula for the X2-statistic analogous to Equation 3.98 that may be used to discriminate between the annual shoal size estimates is, 2 _ (VS/RRj - y/R/SSi)2 where the respective number of data points in each observed series are, R = ^ R i (3.100) i S = Y^Si (3.101) i The reduced x 2 statistic, denoted as x2/df is obtained by dividing a x2 value by the number of degrees of freedom, df. Since the mean of the x 2 distribution is equal to df, one expects in a reasonable experiment to obtain x 2 ~ n. However, caution is necessary because the width and skewness of this distribution, so calculation and comparison of the reduced x2/df measure (scaled by the degrees of freedom) was used in the analysis. Under the null hypothesis HQ that the observed frequency distributions are the same, if x 26 S > xlrit then H0 is rejected and significant differences are present, while if xlbs < xlrit the H0 is accepted. The critical value of the reduced statistic, X2rit/df, at the 95% confidence level for df = 40, is 1.394. Reduced x 2 statistics, denoted as, (x2/df), are provided 3.1 Supervised automated image analysis scheme (SAIA) 145 Table 3.26: Reduced x2/df for manual (Nm), automated (JVC) and final-corrected, (Ns) of SAIA image analysis school size estimates for each year, 1994-96, shown in Fig-ures (3.44)-(3.46). Count Type Year x7d/(df) 1994 1995 1996 Nm vs. Nc Nm vs. Ns Nc vs. Ns 0.343(42) 0.321(44) 0.296(40) 0.253(40) 0.282(45) 0.254(42) 0.395(41) 0.297(46) 0.316(40) in Table 3.26. Values of the statistic indicate the significance in the relative precision between successive corrections in shoal size estimation from the image analysis. Resulting corrected estimates of shoal size Ns are shown not to be significantly different (p < 0.05) for all survey years. The critical value in this test is 1.394, and since xlbs/df < 1-394, the test indicates that no significant difference between the size estimates of the visual and automated count procedures (p < 0.05) (95% confidence level). This result is also relatively uniform across survey years. A more accurate test of visual and automated shoal size estimation could involve comparison of estimates across a large set of images associated with known shoal sizes. Kolmogorov-Smirnov statistics could be used, in this case, to test the precision of the visual and automated counting methods. Nonetheless, the agreement between the vi-sual (manual) and automated counts from the statistical comparison of shoal size his-togram frequencies verifies that accurate shoal size estimates were obtained. The pattern-matching algorithm used to reduce bias between the manual and automated methods leads to x2/df values between values of the test statistic. This indicates that use of the pattern-matching technique helped to reduce bias in shoal size estimation without contributing to significant over-estimation or under-estimation of shoal size. 3.1 Supervised automated image analysis scheme (SAIA) 146 Figure 3.44: SAIA shoal size estimation: Comparison between estimates from final, corrected (Ns), manual (Nm) and automated (Nc) object identification for 1994. 3.1 Supervised automated image analysis scheme (SAIA) Figure 3.45: Same as Figure 3.44 for year 1995. 3.1 Supervised automated image analysis scheme (SAIA) o U -i 1 1 1 r 0 100 200 300 400 500 600 Automated Count ( N c ) o n 1 1 r 0 100 200 300 400 500 600 Manual Count ( N m ) 8 \"8 3 n 1 1 1 r 0 100 200 300 400 500 600 Manual Count ( N m ) Figure 3.46: Same as Figure 3.44 for year 1996. 3.2 Shoal formations 149 3.2 Shoal formations The analyzed set of shoal images (N=463) was qualitatively sorted according to seven distinct shoal formations shown in Figure 3.47. As suggested by Partridge and coauthors in an analysis of soldier shoal formations for this tuna species, the high degree of organization and rigid shoal structure enable their shoals to be classified or categorized into distinct formations. I classified the shoal observations into the following formation types: (1) cartwheel, (2) surface-Sheet, (3) dome or 'packed dome', (4) soldier, (5) mixed, (6) ball, and (7) oriented, on the basis of their overall shape [288]. The frequencies of the different formations identified across survey years are listed in Table 3.28. Shoal size statistics were calculated by pooling the images associating across the three year period (1994-96) for each formation type. Calculated three-dimensional positions of individuals are provided in Figures (3.48-3.49), demonstrating the ability of image intensity calibration to profile each shoal formation in the vertical, depth di-mension. Histograms of the frequency of shoals of size, Ns resulting from classifying the observations according to formation types are shown in Figure 3.50. Distribution statis-tics of shoal size are summarized in Table 3.29 for each of the formations. While the range of shoal size between observed minimum and maximum values for the formations over-lap, the formations are distinguished according to increasing shoal size. The cartwheel, soldier, and oriented formations comprised a smaller mean number of individuals than the mixed, dome, ball and surface-sheet formations. Distinct from other formations, the cartwheel, soldier, and oriented were estimated to have smaller variance in the number of individuals. Comparison of the x2/df and shoal size statistics pooled across years are shown in Figures (3.51-3.53) and listed in Table 3.29. The x2/ci/-statistics comparing the estima-tion of shoal size for each formation type indicate an increased precision of the automated estimation procedure in comparison to visual shoal size estimation for the surface-sheet, 3.2 Shoal formations 150 Table 3.27: Reduced x2/df for manual (Nm), automated (Nc) and final-corrected, (Ns) of SAIA image analysis school size estimates for different shoal formations pooled over years 1994-96. Formations are denoted as: A-cartwheel, B-surface-sheet, C-dome, D-soldier, E-mixed, F-ball, G-oriented shown in Figures (3.51-3.53). Count Type Formation Type x2/df(di) A B C D E F G Nm vs. Nc Nm vs. Ns Nc vs. Ns 0.036(6) 0.048(6) 0.048(6) 0.273(24) 0.203(24) 0.185(22) 0.230(33) 0.185(37) 0.300(33) 0.119(7) 0.120(7) 0.004(7) 0.264(30) 0.266(34) 0.181(29) 0.250(27) 0.258(29) 0.186(25) 0.212(16) 0.163(18) 0.166(17) soldier, mixed, ball formations. A l l of these formations with the exception of the soldier formation, comprise a substantially large number of individuals - the automated method is able to reduce the considerable bias in shoal size estimation expected for these forma-tions. For the cartwheel formation, no difference between the statistics is detected for the small number of degrees of freedom (df=6). The automated method did not perform better than the visual one for the dome and oriented formations. However, the deviation between the x2/df statistics for the oriented formation is relatively small. Deviation in the test statistic for the dome and oriented formations might best be explained by large observed variability in the shoal size for these formations, or possibly that the set of image observations contained significant background variability that contributed to mea-surement bias in the automated method. For the dome formation, the deviation in the values of the test statistic; visual, 7 V m : 0.185, compared to automated, Nc: 0.300, and the final corrected estimates, Ns, was significant at (33-37) degrees of freedom. However, the deviation between the visual and automated counts alone is within the range of deviation measured for the other formations. This may also indicate that the pattern-matching algorithm favoured counts from the visual identification over the automated method for the dome and oriented formations. 3.2 Shoal formations 151 Figure 3.47: Structural formations identified in digital image database from aerial survey observations of B F T shoals, 1994-96: (A) Cartwheel, (B) Surface-Sheet, (C) Dome/Packed Dome, (D) Soldier, (E) Mixed , (F) B a l l , (G) Oriented. 3.2 Shoal formations 152 Table 3.28: Frequencies of different shoal formations for analyzed shoal images. Forma-tions are denoted as: A-cartwheel, B-surface-sheet, C-dome, D-soldier, E-mixed, F-ball, G-oriented (H-solitary individuals). The percentage in the number of images for each formation type in each year with respect to the total numbers are provided in brackets. Year Formation Type Total A B C D E F G H 1994 7(64) 11 (31) 24 (29) 18 (22) 17 (23) 26 (51) 57 (49) 0(0) 160 1995 1(9) 20 (55) 46 (53) 40 (49) 45 (60) 16 (31) 34 (29) 1 (25) 203 1996 3 (27) 5(14) 17 (18) 24 (29) 13 (17) 9(18) 26 (22) 3(75) 100 Total 11 36 87 82 75 51 117 4 463 Table 3.29: B F T shoal size statistics (mean shoal size, Ns, standard error in the mean (SE), 95% confidence intervals (C.I) and minimum and maximum shoal size for identified structural formations pooled across years 1994-96 (Refer to Figure 3.50). Formation Type Shoal Size Statistics Sample(N) Shoal size (Na) Error (SE) 95% C.I. Minimum Maximum (A) Cartwheel 11 37.36 6.650 14.81 10 84 (B) Surface-Sheet 36 130.6 17.03 34.57 15 421 (C) Dome 87 77.59 7.500 14.90 5 339 (D) Soldier 82 11.59 0.8300 1.660 1 32 (E) Mixed 75 50.88 4.790 9.520 4 210 (F) Ball 51 84.25 11.32 22.75 18 502 (G) Oriented 117 31.70 2.140 4.230 4 122 Figure 3.48: Selected SAIA analysis output revealing 3D structure of B F T formations based on relative intensity, Iz of objects in the 2D images. 3.2 Shoal formations Figure 3.49: Same as Figure 3.48 for formation types (D) to (G). 3.2 Shoal formations 155 50 100 150 200 Shoal Size (Ns) 0 100 200 300 400 500 600 Shoal Size (Ns) 0 100 200 300 400 500 600 Shoal Size (Ns) 0 100 200 300 400 500 600 Shoal Size (N s) 50 100 150 Shoal Size (N s) 200 0 100 200 300 400 500 600 Shoal Size (N s) 50 100 150 200 Shoal Size (Ns) Figure 3.50: Histograms of shoal size for B F T formations: frequency versus corrected estimates of shoal size, Ns. 3.2 Shoal formations 156 100 300 „ 200 100 0 140 120 100 80 60 40 20 0 (E) Mixed / y 7 / y y y 0 / ^ O o o 100 200 N _ 300 (G) Oriented 7 y / y o ^ * 8 8° o n 1 r i 1 r 0 20 40 60 80 100 120 140 N „ 400 50 -| (D) Soldier y 40 - y 30 -° v 20 -10 - # ^ ° 8°o o 0 - o ^ o 1 1 1 i Figure 3.51: SAIA shoal size estimation: Comparison of estimates from automated (Nc) and manual (Nm) object identification between different B F T structural formations, 1994 data. 3.2 Shoal formations 157 100 80 60 40 20 0 (A) Cartwheel y y y y 400 300 Z\" 200 100 0 (B) Surface-sheet y y y 0 20 40 60 80 100 N . 100 200 300 400 N, 300 -| 200 100 0 (C) Packed-dome y y y * 0 100 200 N . 300 250 200 150 100 50 0 (E) Mixed y y y y 4 y y ^y A 0 50 100 150 200 250 100 200 300 400 N 140 H 120 100 80 60 40 20 0 (G) Oriented 7 y y y y A y y X 0 20 40 60 80 100 120 140 N„ Figure 3.52: SAIA shoal size estimation: Comparison of estimates from final, corrected (Na) and automated (Nc) object identification between different B F T structural forma-tions. 3.2 Shoal formations 158 100 80 60 40 20 0 (A) Cartwheel 400 300 200 100 0 (B) Surface-sheet o y y 0 20 40 60 N _ 80 100 100 200 300 400 N „ 300 200 100 0 (C) Packed-dome o • S O /• \"<> o o 50 100 150 200 250 N „ 100 200 300 400 N 120 100 80 60 40 20 0 \" (G) Oriented 7 / s / / / o 5 i i i i 0 20 40 60 80 1 1 1 100 120 140 m Figure 3.53: SAIA shoal size estimation: Comparison of estimates from final, corrected (Ns) and manual (Nm) object identification between different B F T structural formations. 3.2 Shoal formations 159 Table 3.30: Monthly mean fork-length(m) for B F T across ages 0-10+ (ICCAT). Age Month 01 02 03 04 05 06 07 08 09 10 11 12 0 0.27 0.29 0.31 0.34 0.36 0.38 0.40 0.43 0.45 0.47 0.49 0.51 1 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.73 0.75 0.77 2 0.79 0.81 0.83 0.85 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 3 1.02 1.03 1.05 1.07 1.09 1.11 1.12 1.14 1.16 1.18 1.19 1.21 4 1.23 1.25 1.26 1.28 1.30 1.31 1.33 1.35 1.36 1.38 1.39 1.41 5 1.43 1.44 1.46 1.47 1.49 1.50 1.52 1.53 1.55 1.56 1.58 1.59 6 1.61 1.62 1.64 1.65 1.67 1.68 1.69 1.71 1.72 1.74 1.75 1.76 7 1.78 1.79 1.80 1.82 1.83 1.84 1.86 1.87 1.88 1.89 1.91 1.92 8 1.93 1.94 1.96 1.97 1.98 1.99 2.00 2.02 2.03 2.04 2.05 2.06 9 2.07 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.18 2.19 2.20 10+ 2.07 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.18 2.19 2.20 Shoal size and formation: individual length and age distributions Distributions of total fish length(m) as a function of shoal size are calculated in the image analysis by converting calibrated measurements of individual body-length (units of BL) according to the mean total length of individuals across age and month. Compar-ative mean estimates of body fork-length (m) for B F T across ages 0-10+ and month are shown in Table 3.30 [11]. Figure 3.54 shows the result of converting the calibration mea-surements of body-length, forming histograms in the frequency of individuals across the age range (0-10) years and total length (m). For shoals comprised of ages 3-10+, variation in their fork length(m) across the months of July-October is within the expected range of (1.86-2.18)m, indicating that the assumption of 1 BL=1.5m slightly underestimates their total body-length(m). The relationship of the total length(m) of individuals according to the number of individuals in their shoals for (1994-96), identifies a consistent trend of larger shoals containing individuals of smaller size and more uniform length distributions. Frequency histograms of total length within interval ranges of increasing shoal size are 3.2 Shoal formations 160 shown for (1995-96) in Figures (3.55-3.56). As shoal size increases, the mean and vari-ance of their length distribution decreases. A n interesting feature of the changes in the length-frequency distributions in both the 1995 and 1996 survey years is their symmetry about mean total body-length for shoals comprised of < 10 or > 50 individuals, with high asymmetry evident in shoals within the intermediate range of 40-49 individuals. Frequency histograms of total length sorted by formation type (Figures 3.57- 3.58) for the 1995 and 1996 survey years show similar shifts about the mean individual length across formations. Estimates of the total body-lengths of individuals were converted to approximate fork-length estimates by age and month. This conversion was used to gen-erate shoal age structure distributions as a function of shoal size and formation type. Shoal age structure distributions shown in Figures (3.59-3.58) indicate that small shoals comprise larger individuals within the older age categories 7-10+. This trend is not so evident in the histogram distributions across formation type. 3.2 Shoal formations 161 -i 1 1 1 1 r 0 20 40 60 80 100 120 140 160 180 200 Shoal Size(Ng) 20 40 60 80 100 120 140 160 Shoal Size (Ng) 4.0 i 3.5 -3.0 -1 25 -s 20 -1 1.5 -H 1.0 -0.5 -0.0 -1996 (N=32) 0 20 40 60 80 100 120 140 160 180 Shoal Size (Ns) Figure 3.54: B F T total length(m) of member individuals across shoal size for years 1994-96 obtained from measurements in the image analysis. 3.2 Shoal formations Total Length (m) 1 2 3 Total Length (m) (E) 40-49 1 2 3 Total Length (m) 70 60 50 H 40 30 20 10 0 40 30 20 10 0 (B) 10-19 Total Length (m) (D) 30-39 l t h _ 0 1 2 3 4 Total Length (m) Total Length (m) Figure 3.55: S A I A analysis histograms of total length(m) of individuals for interval of shoal size, shown in the upper right-hand corner of each histogram, 1995. 3.2 Shoal formations 1 2 3 Total Length (m) 1 2 3 Total Length (m) 1 2 3 Total Length (m) 20 15 10 -5 0 Total Length (m) (D) 30-39 n D_ 1 2 Total Length (m) 1 2 Total Length (m) io H (F) 100-199 rr 1 0 1 2 3 Total Length (m) Figure 3.56: Same as Figure 3.55 for year 1996. 3.2 Shoal formations 164 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Total Length (m) Total Length (m) 80 60 40 20 (E) Mixed 1 — V f I I— 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Total Length (m) 80 -60 -40 20 (B) Surface-sheet Itb-i v 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Total Length (m) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Total Length (m) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Total Length (m) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Total Length (m) Figure 3.57: SAIA analysis histograms of total length(m) of individuals corresponding to shoal formations, 1995. 3.2 Shoal formations 10 8 6 H 4 2 (A) Cartwheel In m 1 2 Total Length (m) 1 2 3 Total Length (m) 1 2 Total Length (m) 10 8 6 4 H 2 0 60 40 20 (C) Packed-dome 1 2 Total Length (m) 1 2 3 Total Length (m) (G) Oriented JH 1 2 Total Length (m) Figure 3.58: Same as Figure 3.57 for year 1996. 3.2 Shoal formations 166 35 30 25 20 15 10 5 0 (A) < 10 - i — ' i ' i i 4 6 S 10 Age (yr) 70 60 50 40 30 20 10 (D) 30-39 -J— — r - — T -4 6 8 10 Age (yr) Age (yr) Figure 3.59: Age structure of B F T shoal sizes across ages 0-10+ years obtained by con-version of individual lengths using monthly means of length-at-age (m) (see Table 3.30), 1995. 3.2 Shoal formations 25 20 15 10 5 0 ( A ) < 10 _nr^ 4 6 8 10 Age (yr) Age (yr) 60 50 -40 -30 -20 -10 0 (G) 100-199 Age (yr) Figure 3.60: Same as Figure 3.59 for year 1996. 3.2 Shoal formations 168 10 8 6 4 2 0 (A) Cartwheel 10 12 Age (yr) 120 SO 40 (B) Surface-sheet 6 8 Age (yr) 10 12 Age (yr) Figure 3.61: Age structure of B F T formations across ages 0-10+ years obtained by con-version of individual lengths using monthly means of length-at-age (m) (see Table 3.30), 1995. 3.2 Shoal formations 150 120 90 60 30 0 Age (yr) (F) Ball 4 6 8 Age (yr) Hi 20 15 H 10 5 0 xo 60 40 20 H 0 (C) Packed-dome - 1 — 10 Age (yr) (E) Mixed - P - I 1 1\"-\"\" •' • r 1 i 2 4 6 8 10 Age (yr) Figure 3.62: Same as Figure 3.61 for year 1996. 3.2 Shoal formations 170 Shoal size and formation: shoal structure histograms In this section, I present the image analysis results of histograms of nearest-neighbour distance (NND), frequency of nearest-neighbours, bearing angle between nearest-neighbours (BA), and shoal polarization. I discuss statistical analysis results for each of these vari-ables across shoal size and formation type. Selected results for shoals observed in the year 1995 are provided in this section, with additional results for 1996 contained in Appendix C l . Nearest-neighbour distance (NND) For each survey year, histograms of nearest-neighbour distance (3D distance between first nearest-neighbours) for intervals of shoal size and formation type were generated in SAIA's post-analysis stage. These results are provided in Figures 3.63 and 3.64 over shoal size intervals and formation types. Mean values and 95% confidence intervals associated with these frequency distributions were calculated and are detailed in Sec-tion 3.3. Reduced x2/df statistics results are calculated for the purpose of comparing the formation-and shoal-size sorted histograms (Tables 3.31 and 3.32). No significant difference was found between NND and shoal size in a previous analysis by Partridge and coauthors of 141 B F T shoal images with estimated sizes in the range of 2-79 individuals [288]. In the study, the images were sorted into 26 parabola, 23 echelon, 16 straight-line and 8 pairs (all containing < 15 individuals), and 68 unclassified shoals having more than 20 individuals. These formation classification types (parabola, echelon, straight-line and pairs) are sub-classes of the soldier formation type characterized in the image analysis presented in this thesis. Inspection of the variation in N N D for the shoal size categories of < 10, 10-19, and 20-29 individuals, for the years 1995-96 (Figures 3.63 and C.201 in Appendix C ) , indicate a trend towards decreasing N N D for increasing shoal size, in contrast to previously published results [288]. Reduced x2/df statistics 3.2 Shoal formations 171 comparing the < 10 category with 10-19, and 20-29 are 2.57 and 3.99, respectively, for 1995, and 2.28 and 3.33 for 1996. These results indicate a significant difference X2/df > (xLo .05/ d/)crit=(1.36-1.42) for (35-40) degrees of freedom at the 95% (1 -a) 100% confidence level. Larger differences in N N D are found as shoal size increases in 1996, but are not reproduced in the results for 1995, where a decrease in N N D with shoal size alternates back to an increasing trend for shoals of size > 200. Note that these results do not represent sorting of NND's according to formation type. When this is performed, large differences in the variation of NND with increasing shoal size, between the formations are found. These differences are most significant for the dome, ball and surface-sheet formations (1995-96). In Section 3.3, NND variation in shoal structure is examined in more detail for each formation type over shoal size. 3.2 Shoal formations 172 0 1 2 3 4 5 0 1 2 3 4 5 NND (BL) NND (BL) Figure 3.63: SAIA analysis histograms of nearest-neighbour distance (NND) for individ-uals across interval range of shoal size, shown in the upper right-hand corner of each histogram, 1995. 3.2 Shoal formations 173 (D) Soldier 2 3 NND (BL) 200 160 120 H 80 40 0 0 (G) Oriented 2 3 NND (BL) Figure 3.64: SAIA analysis histograms of nearest-neighbour distance (NND) for B F T formations, 1995. 3.2 Shoal formations 174 Table 3.31: Reduced x2/df statistics for shoal size-sorted observed histogram frequencies of NND. Year School Size Interval (x7df)(df) Interval 10-19 20-29 30-39 40-49 50-99 100-199 >200 1995 <10 10-19 20-29 30-39 40-49 50-99 100-199 2.57(40) 3.99(35) 4.72(34) 3.75(50) 5.15(37) 7.43(35) 7.17(32) 4.32(37) 3.46(36) 2.81(53) 8.19(38) 13.96(38) 4.50(35) 3.16(30) 2.57(48) 12.59(35) 23.79(33) 3.39(30) 2.06(45) 12.88(34) 24.43(32) 3.13(28) 9.72(52) 17.76(49) 2.72(43) 36.56(36) 4.21(34) 6.71(32) 1996 <10 10-19 20-29 30-39 40-49 50-99 100-199 2.28(32) 3.33(34) 3.07(31) 2.96(30) 7.84(33) 3.44(34) 6.76(36) 2.55(32) 2.35(30) 2.15(28) 18.13(32) 4.21(33) 13.93(34) 2.82(29) 2.33(27) 22.82(31) 5.56(32) 18.54(33) 2.44(27) 8.15(29) 2.43(30) 6.24(32) 13.64(27) 4.72(30) 10.16(32) 16.43(31) 59.48(32) 37.33(33) Table 3.32: Same as Table 3.31 of nearest-neighbour distance (NND) for formation type. Year Formation Type (x 2/df) (df=200) Formation Type (B) (C) (D) (E) (F) (G) 1995 (A) Cartwheel (B) Surface Sheet (C) Dome (D) Soldier (E) Mixed (F) Ball (G) Oriented 0.245 0.107 0.076 0.100 0.120 5.71 5.12 1.90 2.91 2.82 4.42 2.21 0.695 0.240 5.71 0.765 0.827 1.92 0.398 1.59 5.71 1996 (A) Cartwheel (B) Surface Sheet (C) Dome (D) Soldier (E) Mixed (F) Ball (G) Oriented 0.543 1.19 0.458 1.11 0.847 4.02 1.38 2.07 0.600 1.65 0.781 2.22 0.830 2.23 4.02 0.745 1.43 1.05 0.735 0.356 0.408 3.2 Shoal formations 175 Frequency of nearest neighbours Results from the image analysis for the frequency of nearest-neighbours (1995-1996) are shown in Figures 3.65 and C.203 over shoal size, and in Figures 3.66 and C.204 by formation type. These distributions are obtained without any restriction on the distance between individual fish. The restriction of a critical distance between nearest-neighbours (NNDcrit) determines the mean and variance in number of first nearest-neighbours (NNS), partitioning these frequency distributions. Individuals in shoals of size < 10 have 2-5 nearest neighbours. As shoal size increases, the NNS for individuals increases forming a unimodal distribution for shoals of size 10-19 individuals, bimodal for shoal sizes in the range of 20-99, and a reversion to a unimodal form in shoal sizes > 100. Frequency distributions for each formation type indicate a preference of 4-6 nearest-neighbours for the soldier and oriented formations, 2-4 for cartwheel, 8-14 for ball, 2-10 for mixed. The preferred NNS of individuals in the dome and surface-sheet formations shows significant variability. Calculation of reduced x2/df statistics (Table 3.33), for both the years 1995 and 1996, indicate that for shoals of size < 49, variation in NNS is far less significant than for larger shoal sizes (p < 0.05). Similarly, statistics indicate consistently large NNS variation between formation types. In Section 3.3, NNS variation is examined in more detail for each formation type over shoal size. 3.2 Shoal formations 176 1200 IOOO H 800 600 400 200 H 0 (A) < 10 t k 0 5 10 15 20 Frequency of Nearest-Neighbours 25 350 300 250 H 200 150 100 50 0 (C) 20-29 n o 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 100 80 60 H 40 20 0 (D) 30-39 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours Figure 3.65: SAIA analysis histograms of frequency of nearest-neighbours for individuals across interval range of shoal size, shown in the upper right-hand corner of each his-togram, 1995. These distributions are obtained without any restriction on the distance between individual fish. The restriction of a critical distance between nearest-neighbours (NNDcrit) determines the mean and variance in number of nearest-neighbours (NNS), partitioning these frequency distributions. 3.2 Shoal formations 177 (A) Cartwheel n - r i 0 5 10 15 20 25 Frequency of Nearesl-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearesl-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearesl-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours Figure 3.66: Same as Figure 3.65 frequency of nearest-neighbours for formation types 1995. ' 3.2 Shoal formations 178 Table 3.33: Reduced x2/df statistics for shoal size-sorted frequency of nearest neighbours. Unless otherwise indicated, degrees of freedom (df=36). School Size Interval (x7 d f ) (df) Year Interval 10-19 20-29 30-39 40-49 50-99 100-199 >200 <10 2.35 3.04 2.08 2.63 4.07 5.39 2.62 10-19 4.55 2.11 2.19 8.62 12.56 1.33 20-29 3.24 2.20 13.30 20.57 1.77 1995 30-39 2.95 12.37 18.72 1.65 40-49 50-99 100-199 12.40 18.74 33.99 2.13 3.19 2.23 <10 1.30 2.36 1.78(33) 4.18(35) 1.60 3.58 1.21 10-19 2.83 2.35 1.80 15.13 3.78 12.77 20-29 2.65 1.19 18.98 4.52 15.86 1996 30-39 1.27 5.93 1.87 4.88 40-49 50-99 100-199 10.20 2.63 8.40 8.24 49.40 33.12 Table 3.34: Same as Table 3.33 of nearest neighbour frequency for formation type. Year Formation Type (x2/df) (df=20) Formation Type (B) (C) (D) (E) (F) (G) 1995 (A) Cartwheel (B) Surface Sheet (C) Dome (D) Soldier (E) Mixed (F) Ball (G) Oriented 4.04 5.26 8.40 8.51 11.14 74.00 26.27 31.55 6.95 26.63 9.62 87.40 25.42 35.47 15.47 58.13 45.37 38.50 22.17 75.93 80.64 1996 (A) Cartwheel (B) Surface Sheet (C) Dome (D) Soldier (E) Mixed (F) Ball (G) Oriented 3.39 4.44 10.07 13.34 5.28 37.30 6.09 10.62 15.26 15.11 5.85 43.18 12.30 16.51 20.56 22.47 24.80 74.10 19.62 20.43 7.52 3.2 Shoal formations 179 Bearing angle between nearest-neighbours (BA) Results from the image analysis for the horizontal bearing angle (BA) between nearest-neighbours are shown in Figures 3.67 and C.205 over shoal size, and formation type, in Figures 3.68 and C.206. Preliminary examination of the variation between these frequency distributions provide an indication of whether the distributions in nearest-neighbour distance (NND) presented in the proceeding section (Section 3.2), reflect an angular bias of preferred separation distance. In general, bearing angle over shoal size, between individuals (xy plane) is relatively uniform, with weak modes present in the range of (10 — 40)° and (120 — 160)°. For shoals of size < 19 individuals, nearest-neighbours show a preference not to be located behind or in-front of each other (where BA=(0,180)°), and as shoal size increases, individuals take up these positions with larger frequency. The frequency distributions for bearing angle preference are uniform across formation types for both the 1995 and 1996 data. Sorting by formation type provided a better indication as to the extent of variation of B A in their shoals structure. Calculated statistics in Table 3.35 indicate B A for the cartwheel, soldier, mixed, ball and oriented formations is uniform, but large variation is evident for the dome and surface-sheet formations. To further examine the B A variation, plots of nearest-neighbour distance (NND) versus the bearing angle (BA) were generated, shown in Figure 3.69. These plots provide a better indication of how the bearing angle varies in B F T shoals. Even though the preferred range of separation distance NND varies between their formations, these plots show, that across all types, bearing angles of approximately 30°, 90° and 120° are preferred. Cartwheel, ball and surface sheet formations show a clear trend of increasing bearing angle with NND. Similarly, oriented, dome, and mixed formations show this increase, however a larger range of B A is apparent. The result for the soldier formation shows the weakest dependence of bearing angle with NND. While B A varies, 3.2 Shoal formations 180 NND distances are kept fairly rigid at mean separation distances much larger than for any other formation. In Section 3.3, variation in bearing angle (BA), is examined in more detail for each formation type over shoal size. These results for bearing angle between first nearest-neighbours confirms earlier published results by Partridge [288]. 3.2 Shoal formations 181 16 14 12 io H 8 6 4 2 0 (A)< 10 30 60 90 120 150 ISO BAO 50 40 30 20 H 10 (C) 20-29 0 30 60 90 120 150 180 BA(°) 50 40 30 4, 20 10 0 (E) 40-49 0 30 60 90 120 150 180 BAO BAO 30 60 90 120 150 180 BAO 60 90 120 150 180 BAO 120 150 180 BAO 30 60 90 120 150 180 BAO Figure 3.67: SAIA analysis histograms of bearing angle (BA) between nearest-neighbours for individuals across interval range of shoal size, shown in the upper right-hand corner of each histogram, 1995. 3.2 Shoal formations 182 4 H (A) Cartwheel 0 30 60 90 120 150 180 BA(°) 30 60 90 120 150 180 BA(°) 0 30 60 90 120 150 180 B A O 100 30 60 90 120 150 180 B A ( ° ) 0 30 60 90 120 150 180 B A O 0 30 60 90 120 150 180 B A O 0 30 60 90 120 150 180 B A O Figure 3.68: SAIA analysis histograms of bearing angle (BA) between nearest-neighbours for the B F T formations, 1995. 3.2 Shoal formations 183 Table 3.35: Reduced x2/df statistics for shoal size-sorted observed histogram frequencies of nearest neighbour bearing angle (BA). Year School Size Interval (x'7df) (df) Interval 10-19 20-29 30-39 40-49 50-99 100-199 >200 1995 <10 10-19 20-29 30-39 40-49 50-99 100-199 2.28(42) 2.07(46) 4.72(45) 2.44(46) 2.58(51) 7.99(48) 4.71(44) 4.29(49) 2.31(49) 1.94(48) 7.23(52) 18.92(50) 2.81(49) 2.84(48) 2.04(51) 10.23(52) 29.04(49) 3.69(50) 2.88(51) 12.24(51) 32.86(50) 2.58(50) 10.71(53) 31.06(50) 3.08(51) 47.37(53) 4.83(53) 10.57(50) 1996 <10 10-19 20-29 30-39 40-49 50-99 100-199 1.82(31) 1.62(42) 7.04(21) 4.44(27) 1.79(49) 1.29(47) 1.24(52) 2.60(44) 2.48(32) 1.89(34) 13.66(50) 2.66(47) 10.28(52) 2.18(44) 1.30(42) 14.79(50) 2.28(50) 10.34(53) 3.68(31) 3.59(51) 1.38(47) 1.96(52) 8.84(49) 1.77(47) 6.36(52) 9.44(52) 46.66(52) 24.15(53) Table 3.36: Same as Table 3.35 of nearest neighbour bearing angle (BA) for formation type. Year Formation Type (x 2/df) (df=36) Formation Type (B) (C) (D) (E) (F) (G) 1995 (A) Cartwheel 1.46 1.30 3.30 1.49 1.56 2.09 (B) Surface Sheet 12.07 8.34 3.91 1.63 2.96 (C) Dome 8.81 3.92 1.51 3.39 (D) Soldier 2.27 0.84 2.11 (E) Mixed 1.76 3.80 (F) Ball 2.86 (G) Oriented 1996 (A) Cartwheel 2.52 2.43 7.53 1.36 1.50 1.68 (B) Surface Sheet 8.15 10.99 2.23 2.78 2.37 (C) Dome 11.18 1.79 2.39 2.94 (D) Soldier 1.35 2.04 2.42 (E) Mixed 2.36 2.41 (F) Ball 2.41 (G) Oriented 3.2 Shoal formations 184 Figure 3.69: Nearest-neighbour distance, NND (BL), versus bearing angle between near-est-neighbours, B A (°) for B F T shoal formations. 3.2 Shoal formations 185 Shoal polarization The orientation angle of individual tuna (tpj) with respect to the mean direction or polarization angle (<&s) of their shoals is plotted as a function of shoal size in Fig-ures 3.70 and C.207, and formation type, in Figures 3.71 and C.208. Turning angle is the time-dependent or dynamic equivalent of orientation angle in this analysis of static shoal structure. Variation in orientation angle averaged over all shoal structures in the analysis provides an indication of how their turning angle varies in time about the mean orientation of their shoals. For both survey years, shoals of size < 10 individuals are highly polarized. Ori-entation bias in the individual orientation angle about mean shoal polarization angle is represented by broadening about the central peak. As a function of shoal size and formation type, peak broadening was relatively symmetrical, indicating that in a time-averaged sense, individual B F T match their orientation angles collectively to alter the mean direction of their shoals. The results show that small groups of individuals within a shoal are more cohesive. As shoal size increases, continued peak broadening indicates that maintaining orientation angle is more difficult. X2/df statistics provided in Ta-bles 3.37 and 3.38 show, consistently, significant differences in shoal polarization across shoal size (p < 0.05). Across formation type, however, variation was far less. Significant differences were detected between the soldier and both the surface-sheet and dome for-mations (p < 0.05). The surface-sheet formation had significantly different mean shoal polarization than the other formations, having also the largest mean shoal size. 3.2 Shoal formations 186 1600 1400 H 1200 1000 800 600 400 200 0 -100 •50 0 50 100 CiV200 1995 <10 10-19 20-29 30-39 40-49 50-99 100-199 147.21(19) 156.82 123.52 107.73 86.20 80.15 99.58(15) 56.10(19) 67.72(19) 67.38(19) 46.67(19) 56.75(19) 81.45(18) 59.68(19) 64.44(19) 40.25(19) 54.71(19) 77.08(19) 53.85(19) 35.29(19) 45.92(19) 76.48(19) 30.93(19) 40.71(19) 75.86(19) 46.27(19) 80.56(19) 75.24(19) 1996 <10 10-19 20-29 30-39 40-49 50-99 100-199 80.79(19) 75.83 66.82(18) 52.20 53.58 44.42 38.93(19) 35.48(19) 54.15(19) 45.74(19) 18.25(19) 33.19(19) 31.04(19) 51.17(19) 43.22(19) 23.04(19) 31.35(19) 30.59(19) 36.79(17) 8.20(19) 25.47(17) 26.50(18) 9.41(19) 24.99(17) 23.44(18) 28.52(19) 28.00(19) 24.15(18) Table 3.38: Same as Table 3.37 of shoal polarization for formation type. Year Formation Type (x'Vdf) (df=36) Formation Type (B) (C) (D) (E) (F) (G) 1995 (A) Cartwheel (B) Surface Sheet (C) Dome (D) Soldier (E) Mixed (F) Ball (G) Oriented 1.12 1.10 1.93 1.03 0.94 1.10 9.12 5.60 3.63 1.81 4.00 6.44 3.07 1.41 3.94 2.53 1.79 5.26 2.24 2.49 3.96 1996 (A) Cartwheel (B) Surface Sheet (C) Dome (D) Soldier (E) Mixed (F) Ball (G) Oriented 2.59 0.64 6.40 1.09 2.26 2.51 6.80 9.96 6.51 6.23 6.03 5.42 2.70 4.07 1.80 1.48 1.65 1.38 3.29 2.11 2.28 3.3 SAIA calculations of shoal formation structure 189 3.3 SAIA calculations of shoal formation structure Cartwheel formation Shoal structure results, in Figure 3.72, are summarized for the cartwheel formation as a function of shoal size, Na for nearest-neighbour distance NND, number of nearest neighbours, NNS, shoal polarization, <3>s, bearing angle between neighbours, B A , ellip-soidal shoal perimeter length, Ps, and number of edge individuals of the shoal convex hull. Due to the small number of observed shoals in this formation (N=4), scattergrams for other shoal measurement variables are not presented. The results show that NND increases with shoal size within a range of approximately (0.5-1.5) B L . The number of nearest neighbours ranges from 1-4 individuals, with their bearing angles centered about 90° and 120°. No clear trend of shoal perimeter is evident, however, the number of edge individuals lying on the perimeter of this formation increases exponentially with shoal size to a limiting value of approximately 25 individuals for shoals of size 100. Surface-sheet formation Scattergrams, labelled (1)-(12), between selected shoal structure variables listed in Table 3.25 are shown in Figure 3.73. As shoal size increases, ellipsoidal shoal length (1) and width (2) both increase, with shoal width increasing more rapidly than length. For ellipsoidal shoal volume, packing densities, ps, for this formation is estimated to be (0.0508 ±0.0278) BL\"3. The maximum packing density estimated from the observations of this formation is 0.2698 BL-3. A log-log plot of shoal elongation and compactness in (6) shows that this formation is kept compact by increases in shoal elongation (or shoal length in the horizontal plane), as the number of individuals increases. Comparison of ellipsoidal volume, Vs, and convex hull volume, Vh, with shoal depth 3.3 SAIA calculations of shoal formation structure 190 CO tf o •a a m 100 (A) Cartwheel Formation 20 40 Shoal Size (N s) 40 60 Shoal Size (N.) 20 40 60 Shoal Size (N.) 80 100 20 40 60 80 Shoal Size (N,) Z 30 A 20 H | z 10 100 V / / / / f 20 40 60 Shoal Size (N ) 100 20 40 60 Shoal Size (Ns) 80 100 Figure 3.72: Shoal size variation of B F T shoal structure, 1995-96: cartwheel formation. 3.3 SAIA calculations of shoal formation structure 191 < length, indicates that this formation is approximately an oblate spheroid (c < a). Shoal structure results are summarized for this formation as a function of shoal size, Ns, for nearest-neighbour distance, NND, number of nearest neighbours, NNS, shoal po-larization, s > •a . 100 200 300 400 500 Shoal Size (Ns) 100 200 300 400 500 Shoal Size (Ns) 40 30 20 10 (8) • • • • 100 200 300 400 500 Shoal Size (Ns) a (ft b a. fi 100 200 300 400 500 Shoal Size (N s) (6) Log • > < 100 200 300 Shoal Size (N,) 400 1 2 3 Total Length (m) a a •J2 100 200 300 400 500 Shoal Size (N ) Log(Vh) ( I0 3 BL 3 ) Figure 3.73: Shoal variable measurements obtained from SAIA image analysis: face-sheet formation. 3.3 SAIA calculations of shoal formation structure 193 (B) Surface-Sheet Formation o H 1 1 1 1 — 1 o -f 1 1 1 r -0 100 ' 200 300 400 0 100 200 300 400 Shoal Size (Nj) Shoal Size (Ng) Figure 3.74: Shoal size variation of B F T shoal structure, 1995-96: surface-sheet forma-tion. 3.3 SAIA calculations of shoal formation structure 194 packing density estimated from the observations of this formation was 1.0162 BL 3. Comparison of ellipsoidal volume, Vs, and convex hull volume, Vh, where shoal depth < length, indicates that this formation is also approximately an oblate spheroid (c < a), as the surface-sheet formation, but shows greater shape variability. Shoal structure results, in Figure 3.76, are summarized for this formation as a func-tion of shoal size, Na, for nearest-neighbour distance NND, number of nearest neigh-bours, NNS, shoal polarization, $ s , bearing angle between neighbours, B A , ellipsoidal shoal perimeter length, Ps, and number of edge individuals. Np, of a shoal's convex hull. The results show that shoal size increases lead to reduction in the variation of NND, with rigid, mean separation distances between nearest-neighbours of 1.0BL. A n increase in the mean number of nearest-neighbours compared with the surface-sheet formation is evident, with NNS ranging between 1-6 individuals for shoal sizes < 50. For shoal sizes 50 — 100, the NNS values are limited to approximately one individual. Associated with NNS approaching one individual for large shoal size, inspection of the bearing angles variation indicates that neighbours prefer to position themselves at bearing angles, B A ~ 90° and B A ~ 120°. This translates into neighbours preferring to be positioned beside each other and aligned in a similar fashion to the surface sheet formation. Unlike the surface-sheet formation, even though shoal width increases with shoal size, the length of its perimeter is constrained consistent with a more rapid decrease in packing density for this formation as shoal size increases. The number of edge individuals, Np lying on the shoal perimeter increases exponen-tially with shoal size, reaching a limiting value of approximately 30 individuals at a shoal size of 200. Consistent with the results for the cartwheel and surface-sheet formations, the number of edge individuals for a shoal size of 100 is approximately 25. Shoal polariza-tion ranges between (—18°, 18°), having a greater variation range than the surface-sheet, but less than for the cartwheel formation. 3.3 SAIA calculations of shoal formation structure 195 (C) Dome Formation 70 60 50 40 30 20 10 0 • (1) • • 'tf • '. • • p • 100 200 300 400 500 Shoal Size (N s) 0 100 200 300 400 Shoal Size (N s) > E > 20 15 5^ 0 (7) 1 • U W P ^ ' l 1^1 0.5 1.0 1.5 2.0 2.5 Tolal Length (m) 1.0 1.5 2.0 Total Length (m) 0 100 200 300 400 500 Shoal Size (Ns) 0 100 200 300 400 Shoal Size (N,) > E 0 100 200 300 400 500 Shoal Size (Ns) 1.0 • (BL\" 0.8 • i CL 0.6 • • • • c & 0.4 • • tcking ] 0.2 • / • .1 o. 0.0 • (11) 2.5 100 200 300 400 Shoal Size (N.) B Q n r— 0 100 200 300 400 500 Shoal Size (N s) (6) 't Sm • OS > < 0 100 200 300 400 Shoal Size (N s) > Log(V c)(10 3BL 3) Figure 3.75: Shoal variable measurements obtained from SAIA image analysis: Dome formation. 3.3 SAIA calculations of shoal formation structure 196 Figure 3.76: Shoal size variation of B F T shoal structure, 1995-96: dome formation. 3.3 SAIA calculations of shoal formation structure 197 Soldier formation Scattergrams, (1)-(12) for this formation are shown in Figure 3.77. As shoal size increases, ellipsoidal shoal length (1) and width (2) both increase proportionately. Unlike the surface-sheet, and dome formations, shoal width scales with length in a ratio of approximately 1:1. A log-log plot of shoal elongation and compactness in (6) indicates that this formation can be further classes into two sub-classes evident from two clusters in the scattergram. These sub-classes are characterized, as: (i) constrained elongation and varying or un-constrained compactness (as with the surface-sheet formation), and (ii) unconstrained elongation and constrained compactness (as with the dome formation). This result may indicate that as shoal size increases for this formation, that shoal structure directly trans-lates into a similar structure as with the dome or surface-sheet formation depending upon preferred nearest-neighbour bearing angle. Mean packing density, ps is (0.126 ± 0.0773) BL~3. As shoal size increases, packing density is relatively constant. The maximum packing density estimated from the obser-vations of this formation was 1.871 BL~3. Ellipsoidal approximations for shoal surface area as a function of shoal size show that surface area is relatively constant, varying within (1-1000) BL2 over the shoal size range of 1-35 individuals. Comparison of ellipsoidal volume, Vs, and convex hull volume, Vh, in (12) show substantial bias in the ellipsoidal approximation to shoal shape. Shoal structure results, in Figure 3.78, are summarized for this formation, as a func-tion of shoal size, Ns, for nearest-neighbour distance, NND, number of nearest neigh-bours, NNS, shoal polarization, . < U 3 • 2 • (4) 0 5 10 15 20 25 30 35 16 -r -1 CQ 14 -o 12 -10 -> 8 -i 6 -> 4 -O 2 • 0 -0.5 Shoal Size (Ns) (7) 1.0 1.5 2.0 Total Length (m) 2.5 0 10 20 30 40 50 Shoal Size (Ns) 5 4 05 \"2 3 N 2-1 < \" 1 0 > E o > (5) 0 5 10 15 20 25 30 35 Shoal Size (N s) 12 10 8 6 4 2 H 0 0 (8) 5 10 15 20 25 30 35 Shoal Size (N.) Q K. & Ik © of > < 10 20 30 40 50 Shoal Size (N s) (6) -l o 1 Log(Cs) 3 H (9) T 1 1 1 0 5 10 15 20 25 30 35 Shoal Size (N s) 0.3 0.2 op 0.1 (10) 1 * U*I-»H Total Length (m) 0 5 10 15 20 25 30 35 Shoal Size (N ) > \"oo o Log(Vh) (103 BL 3 ) Figure 3.77: Shoal variable measurements obtained from SAIA image analysis: Soldier formation. 3.3 SAIA calculations of shoal formation structure 200 (D) Soldier Formation Shoal Size (Ns) Shoal Size (Ns) 200 150 100 50 H «• • •Vi .r • • I I* -10 —I 1— 20 30 Shoal Size (Ns) 40 | Z 10 20 Shoal Size (N ) Figure 3.78: Shoal size variation of B F T shoal structure, 1995-96: soldier formation. 3.3 SAIA calculations of shoal formation structure 201 of this formation was 1.2016 BL~3. Comparison of ellipsoidal volume, Vs, and convex hull volume, Vh, in (12) show that this formation is best approximated as a convex polygon. Shoal structure results, in Figure 3.80, are summarized as a function of shoal size, Na, for nearest-neighbour distance, NND, number of nearest neighbours, NNS, shoal po-larization, <3>s, bearing angle between neighbours, B A , ellipsoidal shoal perimeter length, Ps, and number of edge individuals, Np of a shoal's convex polygon. Consistent with the surface-sheet, dome and soldier formations, N N D decreases as shoal size increases to a mean separation distance between nearest-neighbours of (1.0-1.5) BL. The number of nearest-neighbours lies within the same approximate range as the dome and soldier formations and larger than the surface-sheet or cartwheel formations: NNS ranges between 1-6 individuals. A strong preference of nearest-neighbour bearing angles of 90° and 120° is evident, with most neighbours positioned directly beside each other. Similar to the soldier formation, shoal perimeter ranges to a maximum value of approximately 80 B L . The number of edge individuals, Np lying on the shoal perimeter also increases exponentially with shoal size, consistent with the surface-sheet, dome and soldier formations, with 25 edge individuals at a shoal size of 100, approaching a limiting value of 30 individuals at a shoal size of 250. Shoal polarization ranges with most shoals between (—20°, 20°), as with the dome formation. Ball formation Scattergrams, (1)-(12) for the B F T ball formation are shown in Figure 3.81. Increas-ing shoal size is associated with proportionate increases in both shoal length (1) and width (2), scaling in a ratio length:width at approximately 1:1. The log-log plot of shoal elongation and compactness(6), unlike the soldier and mixed formations showing alternation of constrained and unconstrained compactness 3.3 SAIA calculations of shoal formation structure 40 30 - J 20 J 10-1 (1) — i — i — i — i — 0 50 100 150 200 250 Shoal Size (N„) (£) Mixed Formation - i r -0 50 100 150 200 250 Shoal Size (NJ 12 - r 10 -3 B 8 -o 6 -JZ 4 -U J 2 -0 -(3) 0 50 100 150 200 250 Shoal Size (N.) 1 2 Total Length (m) 0 50 100 150 200 250 Shoal Size (N,) > 1 3 O > 2.5 2.0 1.5 1.0 0.5 0.0 (5) T 1 0 50 100 150 200 250 Shoal Size (N s) 0 50 100 150 200 250 Shoal Size (N.) B CO 2 > < o-i - l 2.0 1.5 LO + 0.5 0.0 (6) i i MTII| 1 \"i'Mirri|— 2 -1 0 Log(Cs) • • (9) • • • 0 50 100 150 200 250 Shoal Size (N.) 1.5 1.0 0.5 4 0.0 $ m i HHMlWd (10) 1 2 3 Total Length (m) c •J2 50 100 150 200 250 Shoal Size (N s) 5 j m J CD 4 ; cn o 3 i > 2 i Log 1 1 0 • (12) • > • I I r-TTTTTTIp— I 2 Log(V n)(10 3BL 3) Figure 3 . 7 9 : Same as Figure 3 . 7 7 for mixed formation. 3.3 SAIA calculations of shoal formation structure (E) Mixed Formation 50 100 150 200 250 0 20 40 60 80 100 120 Shoal Size (Ns) Shoal Size (Ns) 0 50 100 150 200 250 0 50 100 150 200 250 Shoal Size (Ns) Shoal Size (Ns) Figure 3.80: Same as Figure 3.78 for mixed formation. 3.3 SAIA calculations of shoal formation structure 204 and elongation, variation in compactness is favoured. This characterizes the ball forma-tion, whereby its structure increases radially outward about the shoal centroid (circular). Mean packing density is (0.108±0.0871) BL~3. The maximum packing density estimated from the observations of this formation was 0.9158 BL\"3. A wide variation between ellipsoidal volume, Vs, and convex hull volume, Vn shown in (12) is evident. This variation is larger than for the surface-sheet, dome, soldier, or mixed formations, indicating that this structure is best associated as a sphere (ellipsoid having axes c = a). Shoal structure results are shown in Figure 3.82 as a function of shoal size, Ns, for nearest-neighbour distance, NND, number of nearest neighbours, NNS, shoal polariza-tion, & s , bearing angle between neighbours, B A , ellipsoidal shoal perimeter length, Ps, and number of edge individuals, Np, of a shoal's convex polygon. Consistent with the cartwheel formation, and differing from the surface-sheet, dome and soldier formations, NND increases with shoal size in the range 0-200 individuals. A single shoal of size 350, indicates that N N D does then decrease for large, ball forma-tions. Mean separation distance between nearest-neighbours is reduced from (1.0-1.5) B L , within the range (0.5-1.0) B L , and the number of nearest-neighbours is between 1-3 individuals. The preference for 1-3 nearest-neighbours is similar to the cartwheel and surface-sheet formations. As with the mixed formation, a strong preference of nearest-neighbour bearing angles of 90° and 120° is evident. Shoal perimeter ranges to a maximum value of approximately 80 B L with large variation as shoal size increases. The number of edge individuals on the shoal perimeter is consistent with other formations, increasing exponentially with shoal size with roughly 25 edge individuals at a shoal size of 100, approaching a limiting value of 30 individuals. Shoal polarization ranges between (—30°, 30°). 3.3 SAIA calculations of shoal formation structure (F) Ball Formation 100 200 300 400 500 Shoal Size (N s) 0 100 200 300 400 500 600 Shoal Size (Ns) Total Length (m) 0.5 0.4 0.3 0.2 H 0.1 0.0 (10) 1 2 Total Length (m) © < 2 u CL, 50 40 30 20 10 0 * (2) • s 100 200 300 400 500 Shoal Size (N.) 0 100 200 300 400 500 600 Shoal Size (N s) 100 200 300 400 500 600 Shoal Size (N_) 0 100 200 300 400 500 600 Shoal Size (N ) B Q 0 100 200 300 400 500 Shoal Size (N s) 0 100 200 300 400 Shoal Size (N s) > 3 Log(Vh) (10 3 BL 3 ) Figure 3.81: Same as Figure 3.77 for ball formation. 3.3 SAIA calculations of shoal formation structure 206 (F) Ball Formation x 10 100 200 300 Shoal Size (NJ 400 40 Shoal Size (N s) 40 60 80 Shoal Size (NJ 40 60 80 Shoal Size (NJ 100 80 60 40 20 1 i 2 O % 20 40 60 80 Shoal Size (N J 100 20 40 60 80 Shoal Size (NJ 100 Figure 3.82: Same as Figure 3.78 for ball formation. 3.3 SAIA calculations of shoal formation structure 207 Oriented formation Scattergrams, (1)-(12) for the oriented formation are shown in Figure 3.83. Increasing shoal size is associated with proportionate increases in both shoal length (1) and width (2). Similar to the mixed and ball formations, and unlike the soldier and mixed for-mations that show alternation between constrained and unconstrained compactness, the log-log plot of shoal elongation versus compactness in (6), shows that variation in shoal compactness is favoured. Mean packing density is (0.194 ±0.0783) BL~3. The maximum packing density estimated from the observations of this formation was 1.1949 BL~3. As in the case of the ball formation, a wide variation between ellipsoidal volume, Vs, and convex hull volume, 14 shown in (12) is evident. This variation is larger than for the surface-sheet, dome, soldier, and mixed formations and within the same range as the ball formations. This formation shows alteration between ellipsoidal structure and structure best approximated by convex hull fitting. This indicates that the shape of this formation possesses large edge or shape effects. A n approximately equal frequency of shoals observed, in this formation, are well approximated either as an ellipsoid or convex polygon. Shoal structure results are shown in Figure 3.84 as a function of shoal size, Ns, for nearest-neighbour distance, NND, number of nearest neighbours, NNS, shoal polariza-tion, $ s , bearing angle between neighbours, B A , ellipsoidal shoal perimeter length, Ps, and number of edge individuals, Np, of a shoal's convex hull. Consistent with the surface-sheet, dome and soldier, and differing from the cartwheel and ball formations, N N D decreases with shoal size in the range 0-120 individuals. Mean separation distance between nearest-neighbours is within the range (1.0-2.0) B L , and the number of nearest-neighbours is between 1-6 individuals. Nearest-neighbour bearing angles are centered about 70° and 110°, and shoal perimeter ranges to a maximum value of approximately 80 B L with large variation as shoal size increases. 3.3 SAIA calculations of shoal formation structure 208 The number of edge individuals on the shoal perimeter is less than in other forma-tions, increasing exponentially with shoal size, approaching a limiting value of 20 indi-viduals. As in the ball formation, shoal polarization ranges between (—30°, 30°). Most shoals with this formation are strongly polarized, being less polarized than the soldier formation. 3.3 SAIA calculations of shoal formation structure 209 (G) Oriented Formation B u S i 3 0 25 50 75 100 125 150 Shoal Size (NJ B 25 50 75 100 125 150 Shoal Size (NJ B o -1 T -0 25 50 75 100 125 150 Shoal Size (NJ < u 0 20 40 60 80 100 120 140 Shoal Size (NJ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Total Length (m) 0 20 40 60 80 100 120 140 Shoal Size (NJ 4 H 21 • (8) • * • \\ • • • 50 100 Shoal Size (NJ 150 w > < 2.5 2.0 1.5 1.0 0.5 0.0 (6) 111ITTI|—t-rriwq—i i MITII| i i1 - 1 0 1 2 3 I,og(CJ (») n 1 1 1 1 1 0 20 40 60 80 100 120 140 Shoal Size (NJ © OO c 3 1.4 • (10) m 1.4 -1.2 • w m 1.2 • 1.0 -M h«H a 1.0 - • 0.8 • >, 0.8 • 0.6 • Hr»H c 0.6 • t 0.4 • t—i oo 0.4 • / * 0.2 • c O 0.2 • *t 0.0 • • ••• .'-J»g~ • 5 a. 0.0 • 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Total Length (m) 0 20 40 60 80 100 120 140 Shoal Size (NJ 5 | 4 i BL' o 3 i sT 2 ; Log 1 i 0 -\\ (12) • • • nq—i\"vrrmi|- T i i iini|—i~r 0 1 2 3 4 Log(Vn) (103 BL 3 ) Figure 3.83: Same as Figure 3.77 for oriented formation. 3.3 SAIA calculations of shoal formation structure (G) Oriented Formation 100 120 140 5? 10 .9 Z Shoal Size (N s) ri O H •a s 0 10 20 30 40 50 60 Shoal Size (N s) 10 20 30 40 50 Shoal Size (N s) 200 150 100 50 •• • # • •a 1 I Z 10 20 30 40 50 60 70 Shoal Size (N.) 10 20 30 40 50 Shoal Size 0NS) Figure 3.84: Same as Figure 3.78 for oriented formation. 3.4 Convex hull refinement of ellipsoidal shoal structure 211 3.4 Convex hull refinement of ellipsoidal shoal structure As highlighted by Pitcher and Partridge [319], if one assumes an ellipsoid provides a good approximation for the shape of a shoal, further problems remain in calculating the occupied volume per fish (packing density). The results detailed in Section 3.3 for B F T , confirm that the proportion of individuals on the periphery of shoals is greater for shoals having a small number of individuals, as previously reported in laboratory studies on saithe, herring and cod [319]. This means that the estimated volume per fish increases (packing density decreases) as shoal size increases, until the proportion of edge individuals falls below a given value. This edge effect biases estimates of shoal surface area, volume and packing density as the total occupied volume of a shoal should include space within which fish will not take up position. The true boundary of an ellipsoidal shoal can therefore be considered to expand by an additional factor to include edge individuals and additional space. This additional factor, when assumed to be half of the mean nearest-neighbour distance outside of individuals farthest away from the centre of a shoal, can reduce measurement bias introduced by shoal edge effects. The different shoal formations are, however, distinguished by their shape determined by consistent positions of individuals on the edge of their shoals. A convex hull approxi-mation to shoal shape and structure, unlike the ellipsoid approximation, considers shape distortion by variation in the position of individual fish forming a shoal's boundary or perimeter (edge-effects). In addition, a ellipsoidal assumption may under-estimate or over-estimate shoal surface area and volume. To examine both the characteristics of B F T shoals in terms of the number of individuals on the shoal perimeter (edge-individuals) and the uncertainties in surface area and volume under shape distortion, a procedure was implemented to provide estimates of surface area and volume by determining the convex hull's of each shoal analyzed in the SAIA image analysis. 3.4 Convex hull refinement of ellipsoidal shoal structure 212 In computational geometry, for a polygon, P, a convex hull is defined as the smallest convex polygon containing P. To construct a convex hull for the shape of a fish shoal, a subset of individuals is identified forming the hull's vertices through the permutation of all possible polygons formed using individuals in a shoal as vertices. Hull vertices are identified using a procedure whereby a distinction is made between individuals that are internal or external as different polygons are formed. A n individual E e P is termed an extreme point of P, if no triangle containing E can be constructed that used the other points in the set of polygon vertices [358]. A n algorithm was implemented to find the convex hull for each shoal structure analyzed using the SAIA scheme [21]. The algorithm used, termed a QuickHull algorithm for convex hulls, is similar to randomized, incremental algorithm approaches for convex hull and spatial Delaunay triangulation. Aspects of this algorithm improve the reliability and efficiency for finding the convex hull of a set of points in multi-dimensional space. Automation of the QuickHull algorithm was performed to iteratively find the convex hulls for each analyzed shoal structure. Resulting convex hull geometries were found and the surface area (SAh), volume (V / J , and number of edge individuals (Np), was calculated for each shoal. Np is the number of vertices forming the convex hull for a shoal structure. The surface area and volume values were sorted by shoal formation type, and used to compare with values estimated by assuming shoal structure is ellipsoidal. The comparison showed the degree in which the assumption of ellipsoidal geometry holds for each B F T formation. The number of edge individuals as a function of shoal size was used to characterize B F T shoals based on their shape. Figure 3.85 illustrates the calculation of a shoal convex hull in three-dimensions with vertices as the individual members on its edge. The edge individuals of the resulting convex hull for the shoal structure are shown in red in both the 2D and 3D perspectives. Comparison of the ellipsoidal surface area (SAa) and volume (Vs) with the refined values of surface area (SAh) and volume 3.4 Convex hull refinement of ellipsoidal shoal structure 213 Table 3.39: Summary of linear regression of convex hull refinement of ellipsoidal surface area (SAS), and volume (V^) denoted as SAh and Vh, respectively. SAS versus Vs SAh versus Vh Formation a (slope) CV(%) a (slope) CV(%) (A) Cartwheel - - - 0.384 ± 0.0360 0.946 9.27 (B) Surface-Sheet 0.206 ± 0.0220 0.503 10.8 0.376 ± 0.0200 0.785 5.20 (C) Dome 0.234 ± 0.00900 0.886 3.89 0.348 ± 0.0120 0.881 3.55 (D) Soldier 0.242 ± 0.0280 0.476 11.7 0.925 ± 0.0560 0.678 6.09 (E) Mixed 0.138 ±0.0110 0.682 7.70 0.431 ± 0.0180 0.830 4.14 (F) Ball 0.203 ± 0.00900 0.945 4.27 0.369 ± 0.0250 0.820 6.73 (G) Oriented 1.75 ±0.0120 0.998 0.666 0.611 ±0.0300 0.789 4.83 3.4 Convex hull refinement of ellipsoidal shoal structure 214 CO 2 « S -o ^ o3 < a ~ •fl o> X N 2 -a it o H H ' — - CO -e co ^ --rt g x -a CP -rn EH O CJ SH _ , .S CD OJO CO r H CP HO a A a a CD co T J 1 • i-H O «9 'o CD CD T J o3 ^ a a -r-= + ^ 00 M CD H CO fl O a 1 fl o3 > O r H o CO r—H o3 CD o3 a - 2 r H . O 03 r - H T f C N O l CM r ^ 1257. 3266. 1022 in C O r H CM r-~ C O C O C O O l T f in r H 1033 in in T f o oq c q r H ^ C O CM oi 0 0 Iv T f d C O o 0 0 in r H T f in c o CM CM r H C O r H CM CQ 0 0 -H -H -H -H -fl -H -H -a CO o C O .68.3 C O oi d O l C O c q d iq od d .68.3 T f CM t~ 0 0 r H r H 15 r H C O C O c o m C O C O 'o CO C N r H C O o 0 0 C O cn c d CO d 0 0 d d id r H W cn I O 0 0 in r H C O C O 0 0 C i r H IV 0 0 r H t- O l in r H C O C N T f r H CM r H CM cq -H -H -H 41 \"fl -H -H 41 O l 0 0 O l C N 0 0 c q O l C O 0 0 i v C O C N od od d r H t~ CM C O o T f T f C O O l oo in C O 0 0 r H r H C N CO T f T f CM T f C O T f CP CP i—H r f l CO -o r f l 0 ) U C T3 cfl c o o 1^ m oi 3.4 Convex hull refinement of ellipsoidal shoal structure 215 Figure 3.85: Left: S A I A analysis output structure of a shoal in the horizontal (xy) plane, and corresponding vertical projection of individual positions in the vertical, z-plane (depth), Right: Calculat ion of shoal convex hull in three-dimensions wi th vertices as individual members on its edge or shoal perimeter. Edge individuals comprising the fitted convex hull to the shoal structure appear outlined as larger circles, in both the 2D and 3D perspectives. 3.4 Convex hull refinement of ellipsoidal shoal structure 216 (B) Surface-Sheet Formation 4 Figure 3.86: Comparison of ellipsoidal surface area (SAS) and volume (Vs) and refine-ment of surface area (SAh) and volume (14) with fitted convex hull for the surface-sheet formation. 3.5 Principal component analysis of structural variables 217 (Vh) from the convex hull procedure are presented for the surface-sheet formation type (Figure 3.86). Results for the other formation types in the analysis are contained in Appendix C l . Linear regression results are calculated to examine the differences in surface area and volume between formations in Table 3.39. Surface area and volume estimates for the ellipsoidal (SAS, Vs), and convex hull (SAh, Vh) approximations to the shape are listed for each formation type in Table 3.40. Estimates of the mean number of edge individuals, Np, and shoal size, iV s (from Table 3.29), for each formation are also provided. Coefficients of variation (precision) (CV%) between the fits of surface area and volume for the ellipsoid and convex hull approximations were compared. Gains (in order of increased precision) with the use of the convex hull method was achieved for: (1) soldier, (2) surface-sheet, (3) oriented, (4) mixed, (5) ball, and (6) dome. For the cartwheel formation a comparison could not be made, however its precision in the convex hull method is lower than for all other formations. For all formations, use of the convex hull method lead to gains in shape precision. The cartwheel, soldier, surface-sheet and oriented formations were best approximated with convex hull geometry having significant variation in the xy plane. The ball formation was best approximated as a sphere, while the mixed and dome formations were best approximated as oblate ellipsoids (c < a). 3.5 Principal component analysis of structural variables Shoal structure variables (number of individuals, length, width, depth, volume, NND and BA) were selected as input variables for a principal component analysis (PCA) to compare their variances. The P C A results helped to characterize structural variation between each formation. Shoal length, width and depth and volume are shape-related, and NND, and B A are internal shoal structure variables. Shoal size (number of individ-uals) was included to associate its variance with the leading shape and internal structure variables. 3.5 Principal component analysis of structural variables 218 Principal component analysis (PCA) is a multi-variate technique used to generate optimal partitions of the total variance of all input variables, using a least-squares ap-proach. The central idea behind the P C A method is to reduce the dimensionality of a data set in which there are a large number of interrelated variables, while retaining as much as possible of the variation of the whole data set. The technique involves trans-forming each variable linearly according to the correlation between the input variables (cross-correlations) forming an un-correlated set of principal components (PC). These components are then ordered by decreasing variance, whereby each component extracts a maximal share of the total variance. Empirical orthogonal function analysis (EOFs) is a closely related application of the P C A method for examining correlation and vari-ance in time series data (autocorrelations). A detailed discussion of the E O F method is provided in [98]. Let Zi, i = ( 1 , N ) denote the input matrix containing the set of N shoal variables, z4. Let n denote the number of samples for each input variable. The P C A linear equation transforms the input variables into the set of principal components yp, p = (1,...,N), termed the P C A matrix with dimensions (nXN), denoted as Y . The P C A transformation equation is, 2 Y = ' Z Q \" ' (3.102) Q is an NXN orthogonal matrix of transformation coefficients termed eigenvectors, where each column of Q is denoted qp. Each column of Y is a principal component vector y p , and the variance for the values for each of these vectors is termed the eigenvalue, where for all columns of Y , (Ai,.. . ,A p) eigenvalues are calculated. Let V be the NXN correlation matrix of Z. Relating each eigenvector (i.e., from matrix Q) with the corresponding eigenvalues (i.e., from matrix Y ) , we have 2 A correlation matrix was chosen to standardize the shoal structure variables having different units. 3.5 Principal component analysis of structural variables 219 V q p = A p q p (3.103) The correlations between the input variables, Zi, and the P C A components, yp, is computed in the P C A or so-called eigen-analysis as, over i elements q i P , corresponding to each eigenvector, q p . From Equation 3.104, we see that each eigenvector q p reduces the correlation matrix (V) into a single scalar, Xp. The P C A analysis was implemented using the ViSta, Visual Statistics Software Sys-tem developed by Forrest W. Young, (www.visualstats.org). For each of the seven B F T shoal formations, separate P C A analyses were performed. The P C A correlation matrix, V (NXN), of correlation coefficients between each of the N=7 shoal input variables, and the seven eigenvectors qip denoted as (PC1-PC7) are provided in Tables (3.41-3.47) for each of the formations. The first principal component (PCI) is associated with shoal shape, and the second (PC2), with shoal internal structure. The input variables were, in general, positively correlated with each of these principal components. However, the components did not describe shoal shape and internal structure independently, but instead showed that their shoal formations vary according to the relative influence of both internal and external variables. The proportion of the total variance explained by these two components assigned the variability observed in the three-dimensional structure of each formation. At a desired level of statistical confidence the contribution of these components should explain > 95% of the total variance observed. The relative percentage of total variance explained by the P C I , PC2 components in the analyses conducted for B F T shoal formations indicated that the addition of other shoal variables can improve the P C A results. corr(zi,yp) = rip = yj Xpqip (3.104) 3.5 Principal component analysis of structural variables 220 PCA results: cartwheel formation The first principal component, P C I , has an eigenvalue of 3.99, which corresponds to 57% of the total variance, while the eigenvalue of PC2 is 2.21, accounting for 31.2% of the total variance. The first two components account for 88.6% of the total variance. Listed in order of decreasing correlation strength (magnitude): P C I is positively correlated with shoal volume, width, depth and length, and nega-tively correlated with bearing angle, shoal size, and nearest-neighbour distance. PC2 is positively correlated with depth, and negatively correlated with length, width, shoal size, depth, volume, and bearing angle. For this formation, the strongest positive correlations, greater than expected by random variation, are between: length and width (0.958), shoal size and NND (0.820), and shoal size and bearing angle (0.986). PCA results: surface-sheet formation The first principal component, P C I , has an eigenvalue of 2.77, which corresponds to 39.5% of the total variance, while the eigenvalue of PC2 is 1.66, accounting for 23.7% of the total variance. The first two components account for 63.3% of the total variance. Listed in order of decreasing correlation strength (magnitude): P C I is positively correlated with shoal volume, length, width and shoal size, depth, bearing angle, and negatively correlated with nearest-neighbour distance. PC2 is positively correlated with nearest-neighbour distance, bearing angle, width, length, and negatively correlated with depth, shoal size and volume. For this formation, the strongest positive correlation, greater than expected by ran-dom variation, is between: length and width (0.834). 3.5 Principal component analysis of structural variables 221 Table 3.41: Cartwheel formation: P C A correlation matrix and PC1-PC7 eigenvectors for shoal variables. Shoal Variable Variable Shoal Size Length Width Depth Volume NND BA Shoal Size 1.000 -0.180 -0.418 -0.643 -0.660 0.820 0.986 Length 1.000 0.958 -0.406 0.733 -0.021 -0.317 Width 1.000 -0.288 0.774 -0.308 -0.526 Depth 1.000 0.324 -0.288 -0.608 Volume 1.000 -0.227 -0.777 NND 1.000 0.741 BA 1.000 PCA Eigenvectors PCI PC2 PC3 PC4 PC5 PC6 PC7 Shoal Size -0.4590 -0.2613 -0.0973 0.6587 -0.1268 0.2382 0.4526 Length 0.2772 -0.5584 -0.0708 0.2770 -0.3006 -0.6273 -0.2137 Width 0.3655 -0.4535 0.1228 -0.0203 -0.1957 0.7294 -0.2738 Depth 0.2036 0.5347 -0.5044 0.4447 -0.2571 0.1181 -0.3747 Volume 0.4364 -0.1813 -0.4575 0.1109 0.6986 0.0326 0.2569 NND -0.3375 -0.2534 -0.7116 -0.5041 -0.2423 0.0512 -0.0028 BA -0.4828 -0.1767 0.0160 0.1575 0.4924 0.0089 -0.6842 3.5 Principal component analysis of structural variables Table 3.42: Same as Table 3.41 for surface-sheet formation. Variable Variable Shoal Size Length Width Depth Volume NND BA Shoal Size 1.000 0.291 0.008 0.225 0.633 -0.364 0.128 Length 1.000 0.834 0.159 0.684 -0.048 0.183 Width 1.000 -0.025 0.424 -0.077 0.119 Depth 1.000 0.630 -0.273 -0.073 Volume 1.000 -0.219 0.132 NND 1.000 0.413 BA 1.000 PCA Eigenvectors PCI PC2 PC3 PC4 PC5 PC6 PC7 Shoal Size 0.3729 -0.2496 0.3673 -0.6066 0.3016 -0.1874 -0.4120 Length 0.5010 0.3295 -0.2359 0.0025 0.1432 0.7277 -0.1863 Width 0.3823 0.4037 -0.4861 0.0134 -0.1759 -0.6198 -0.1985 Depth 0.3147 -0.3669 0.2534 0.6992 -0.1914 -0.0402 -0.4183 Volume 0.5618 -0.0726 0.1878 0.1385 0.2212 -0.1463 0.7445 NND -0.2047 0.5430 0.3391 0.3234 0.6276 -0.1596 -0.1561 BA 0.0854 0.4810 0.5990 -0.1385 -0.6150 0.0518 0.0500 3.5 Principal component analysis of structural variables 223 Table 3.43: Same as Table 3.41 for dome formation. Variable Variable Shoal Size Length Width Depth Volume NND BA Shoal Size 1.000 -0.156 0.015 -0.211 -0.180 0.036 0.346 Length 1.000 0.866 0.480 0.875 -0.321 -0.034 Width 1.000 0.255 0.773 -0.425 0.018 Depth 1.000 0.501 -0.163 0.050 Volume 1.000 -0.231 -0.095 NND 1.000 0.207 BA 1.000 PCA Eigenvectors PCI PC2 PC3 PC4 PC5 PC6 PC7 Shoal Size -0.1268 -0.6588 0.3817 0.0257 0.6310 -0.0283 -0.0678 Length 0.5356 -0.0939 -0.0414 -0.1926 -0.0290 0.2574 -0.7736 Width 0.4942 -0.2059 0.2316 -0.2539 -0.1614 0.4785 0.5833 Depth 0.3312 0.0234 -0.5144 0.6318 0.4018 0.2051 0.1498 Volume 0.5169 -0.0400 -0.1311 -0.2487 0.1345 -0.7784 0.1676 NND -0.2676 -0.1308 -0.6496 -0.6209 0.2448 0.1954 0.0757 BA -0.0660 -0.7039 -0.3084 0.2264 -0.5792 -0.1330 -0.0227 PCA results: dome formation The first principal component, P C I , has an eigenvalue of 3.14, which corresponds to 44.8% of the total variance, while the eigenvalue of PC2 is 1.36, accounting for 19.3% of the total variance. The first two components account for 64.2% of the total variance. Listed in order of decreasing correlation strength (magnitude): P C I is positively correlated with shoal length, volume, width and depth, and nega-tively correlated with NND, shoal size and bearing angle. PC2 is positively correlated with depth, and negatively correlated with bearing angle, shoal size, width, NND, length and volume. For this formation, the strongest positive correlations, greater than expected by random variation, are between: length and width (0.866), and length and volume (0.875). 3.5 Principal component analysis of structural variables 224 Table 3.44: Same as Table 3.41 for soldier formation. Variable Variable Shoal Size Length Width Depth Volume NND BA Shoal Size 1.000 0.441 0.451 0.424 0.197 0.074 -0.077 Length 1.000 0.953 0.497 0.751 -0.092 -0.079 Width 1.000 0.526 0.751 -0.077 -0.084 Depth 1.000 0.404 0.032 -0.050 Volume 1.000 -0.069 -0.143 NND 1.000 0.256 BA 1.000 PCA Eigenvectors PCI PC2 PC3 PC4 PC5 PC6 PC7 Shoal Size 0.3175 -0.2176 -0.6630 0.3076 -0.4687 -0.3130 0.0002 Length 0.5198 0.0045 0.1946 -0.0003 -0.1957 0.4050 0.6997 Width 0.5242 -0.0076 0.1710 -0.0094 -0.1580 0.4019 -0.7137 Depth 0.3772 -0.1702 -0.2927 0.1649 0.8455 0.0170 0.0289 Volume 0.4490 0.0973 0.3775 -0.3023 0.0239 -0.7447 -0.0015 NND -0.0473 -0.7128 -0.1237 -0.6817 -0.0345 0.0913 0.0113 BA -0.0904 -0.6372 0.4999 0.5675 -0.0230 -0.1155 -0.0072 P C A results: soldier formation The first principal component, P C I , has an eigenvalue of 3.26, which corresponds to 46.6% of the total variance, while the eigenvalue of PC2 is 1.27, accounting for 18.1% of the total variance. The first two components account for 64.7% of the total variance. Listed in order of decreasing correlation strength (magnitude): P C I is positively correlated with shoal width, length, volume, depth, shoal size, and negatively correlated with bearing angle and NND. PC2 is positively correlated with volume and length depth, and negatively correlated with NND, bearing angle, shoal size, depth and width. For this formation, the strongest positive correlation, greater than expected by ran-dom variation, is between: length and width (0.953). 3.5 Principal component analysis of structural variables 225 Table 3.45: Same as Table 3.41 for mixed formation. Variable Variable Shoal Size Length Width Depth Volume NND BA Shoal Size 1.000 0.238 0.366 0.188 0.038 -0.081 -0.030 Length 1.000 0.871 0.510 0.714 0.031 -0.034 Width 1.000 0.508 0.718 -0.040 -0.185 Depth 1.000 0.789 -0.170 0.012 Volume 1.000 -0.070 0.053 NND 1.000 0.048 BA 1.000 PCA Eigenvectors PCI PC2 PC3 PC4 PC5 PC6 PC7 Shoal Size 0.1920 -0.5191 -0.0541 0.7565 -0.2556 -0.0918 0.2112 Length 0.4984 0.0441 -0.2086 0.0173 0.4443 0.6223 0.3481 Width 0.5106 -0.1545 -0.1930 0.0134 0.3419 -0.3314 -0.6718 Depth 0.4434 0.1487 0.2768 -0.0849 -0.6734 0.3841 -0.3107 Volume 0.5019 0.2801 0.0948 -0.1841 -0.1191 -0.5886 0.5159 NND -0.0616 0.3488 -0.8706 0.0892 -0.3274 -0.0175 -0.0331 BA -0.0410 0.6946 0.2696 0.6149 0.2134 -0.0179 -0.1386 PCA results: mixed formation The first principal component, P C I , has an eigenvalue of 3.15, which corresponds to 45.0% of the total variance, while the eigenvalue of PC2 is 1.11, accounting for 15.8% of the total variance. The first two components account for 60.9% of the total variance. Listed in order of decreasing correlation strength (magnitude): P C I is positively correlated with shoal width, length, volume, depth and shoal size, and negatively correlated with bearing angle and NND. PC2 is positively correlated with bearing angle, NND, volume, depth and length, and negatively correlated with shoal size and width. For this formation, the strongest positive correlations, greater than expected by random variation, are between: length and width (0.871), and depth and volume (0.789). 3.5 Principal component analysis of structural variables 226 Table 3.46: Same as Table 3.41 for ball formation. Variable Variable Shoal Size Length Width Depth Volume NND BA Shoal Size 1.000 0.498 0.375 0.463 0.445 -0.045 0.054 Length 1.000 0.936 0.655 0.850 0.315 0.090 Width 1.000 0.614 0.782 0.303 0.162 Depth 1.000 0.717 -0.014 0.272 Volume 1.000 0.254 0.326 NND 1.000 -0.017 BA 1.000 PCA Eigenvectors PCI PC2 PC3 PC4 PC5 PC6 PC7 Shoal Size 0.3047 0.2939 0.4677 0.7643 -0.0763 -0.0576 0.0937 Length 0.4900 -0.1650 0.1239 -0.1421 -0.2743 -0.0459 -0.7873 Width 0.4686 -0.1736 0.0083 -0.2682 -0.4191 -0.4475 0.5499 Depth 0.4169 0.2930 0.0272 -0.2374 0.7790 -0.2762 -0.0101 Volume 0.4798 0.0060 -0.1170 -0.0873 0.0161 0.8308 0.2406 NND 0.1460 -0.7762 -0.2756 0.4321 0.3251 -0.0853 0.0238 BA 0.1486 0.4097 -0.8219 0.2705 -0.1745 -0.1431 -0.1024 PCA results: ball formation The first principal component, P C I , has an eigenvalue of 3.71, which corresponds to 53.1% of the total variance, while the eigenvalue of PC2-is 1.16, accounting for 16.6% of the total variance. The first two components account for 69.7% of the total variance. Listed in order of decreasing correlation strength (magnitude): P C I is positively correlated with all variables: length, volume, width, depth, shoal size, bearing angle and NND. PC2 is positively correlated with bearing angle, shoal size, depth and volume, and negatively correlated with NND, width and length. For this formation, the strongest positive correlations, greater than expected by random variation, are between: length and width (0.936), and depth and volume (0.850). 3.5 Principal component analysis of structural variables 227 Table 3.47: Same as Table 3.41 for oriented formation. Variable Variable Shoal Size Length Width Depth Volume NND BA Shoal Size 1.000 -0.042 -0.022 0.050 0.034 -0.311 0.043 Length 1.000 0.430 0.248 0.178 0.213 0.025 Width 1.000 0.513 0.834 0.192 0.090 Depth 1.000 0.464 -0.021 0.066 Volume 1.000 0.107 0.168 NND 1.000 -0.371 BA 1.000 PCA Eigenvectors PCI PC2 PC3 PC4 PC5 PC6 PC7 Shoal Size -0.0193 -0.4545 0.7541 -0.1778 -0.3381 0.2780 0.0344 Length 0.3429 0.1764 -0.0062 -0.8788 0.0266 -0.1738 -0.2192 Width 0.5969 0.0059 0.0103 0.0959 -0.2019 -0.2511 0.7284 Depth 0.4449 -0.1396 0.1518 0.0970 0.7707 0.3941 -0.0289 Volume 0.5469 -0.1107 -0.0110 0.3773 -0.3154 -0.1756 -0.6448 NND 0.1444 0.6704 0.0383 0.0494 -0.3104 0.6553 0.0090 BA 0.0886 -0.5302 -0.6376 -0.1807 -0.2331 0.4626 0.0593 PCA results: oriented formation The first principal component, P C I , has an eigenvalue of 2.45, which corresponds to 35.0% of the total variance, while the eigenvalue of PC2 is 1.55, accounting for 22.1% of the total variance. The first two components account for 57.2% of the total variance. Listed in order of decreasing correlation strength (magnitude): P C I is positively correlated with shoal width, volume, depth, length, NND and bearing angle, and negatively correlated with shoal size. PC2 is positively correlated with NND, length and width, and negatively correlated with shoal size, bearing angle, depth and volume. For this formation, the strongest positive correlations, greater than expected by random variation, are between: width and volume (0.834). 3.5 Principal component analysis of structural variables 228 PCA correlation and variance of shoal structure The P C A correlation and variance results of each the selected variables used to describe B F T shoal formations are illustrated in Figure 3.87 and Figure 3.88. In Sec-tion 3.7, these results were further interpreted to provide a summary of the structural characteristics distinguishing each of their formations. 3.5 Principal component analysis of structural variables 229 Ns L's W's D's Vs NND BA \\ 1-Ns L's W's D's Vs NND BA -I 1 1 1 1 1 \\ Ns L's W's D's Vs NND BA \\ 1 1 1 1 1 1-Ns L's W's D's Vs NND BA - \\ 1 1 1 1 H Ns L's W's D's Vs NND BA \\ 1 1 1 1 1 h Ns L's W's D's Vs NND BA H 1 1 1 1 1 h Ns L's W's D's Vs NND BA Figure 3.87: Box-Wisker plots of correlation coefficient of P C A components versus shoal structural variables for the B F T formations, showing median, quartiles, 5% and 95% percentiles, where Ns is shoal size, L's is length, D's is depth, Vs is volume, NND is nearest-neighbour distance, and BA is bearing angle. 3.5 Principal component analysis of structural variables 230 PI P2 P3 P4 P5 P6 P7 Principal Component PI P2 P3 P4 P5 P6 P7 Principal Component PI P2 P3 P4 P5 P6 P7 Principal Component PI P2 P3 P4 P5 P6 P7 Principal Component PI P2 P3 P4 P5 P6 P7 Principal Component PI P2 P3 P4 P5 P6 P7 Principal Component PI P2 P3 P4 P5 P6 P7 Principal Component Figure 3.88: P C A Scree plots showing the proportion of variance explained by each principal component across B F T formations. These plots show the leading components (PC1,PC2,PC3) are sufficient to explain up to 70% of the observed variance in shoal structure for each of the formations. 3.6 Shoal dynamics 231 3.6 Shoal dynamics Shoal size dynamics of shoal formations From the calculated mean shoal size estimates (Table 3.29) obtained by complete sorting of observed shoal structures into seven formation types ((A) cartwheel, (B), surface-sheet, (C) dome, (D) soldier, (E) mixed, (F) ball, (G) oriented), an heuristic linkage between formations (how one formation may turn into another) was depicted (Figure 3.89). This linkage relates to shoals increasing and decreasing in size. Different linkages between the formations may occur as a result of mixing interactions whereby individuals exchange between shoals, leading to the formation of new shoals that have different sizes. This process would translate the linear process into a non-linear dynamic one that is more realistic. Diurnal frequency of occurrence of shoal formations A primary factor affecting motivation of fish in shoals is time of day. Previous studies have found changes in how fish respond over the diel cycle [113,175,287,318]. Changes in foraging/feeding motivation may be the major influence of time-of-day cycles in shoaling behaviour. Time-of-day differences in the observed frequencies of occurrence of the shoal formations was investigated. Formations frequencies associated with visual shoal size estimates by Lutcavage and coauthors [220] from the aerial surveys (1994-96) were examined. Table 3.48 lists the observed frequencies for each of the formations in the observations. Differences between the occurrence of formation types were revealed. The frequencies were not highly dependent on the time of day that survey observers sighted B F T shoals because the time component of survey effort is relatively uniform across day-length. However, the spatial locations of shoal formations did vary in the survey sightings suggesting that the comparison of formation frequencies may be biased. 3.6 Shoal dynamics 232 Polar plots of the observed variation in formation frequency versus time of day (hours) are shown in Figure 3.90 for the cartwheel, dome, soldier and ball formations. Other formations were not classed. Corresponding x2/df statistics were calculated for time-of-day variation in observed frequencies in each year of (1994-96) and are listed in Table 3.49. In addition, statistics were calculated to compare frequencies between the years of the surveys. These results are provided in Table 3.50. Accompanying each x2/df are the test p-values as the significance probability. Significant differences are indicated by small p-values. For the 1994 data, significant differences were found between the fre-quencies of occurrence of soldier and cartwheel shoals (p < 0.001). A significant difference was also found between dome and ball formations. For year 1995, differences between soldier and ball formations are found at the 95% significance level. In 1996, significantly different frequencies (p < 0.001) were found between the dome and cartwheel, and soldier and cartwheel formations, respectively. These results provide evidence for daily varia-tions in the occurrence of different shoal formations. At varying levels of significance, shoal formation alterations were found to occur with dome and cartwheel formation fre-quencies varying with respect to the soldier and ball formation. Similarly, the soldier formation varied mostly with the cartwheel and ball formations. The most consistent time-of-day frequencies were found for the soldier and ball formations. This result pro-vides an indication that the leading variation in the time-of-day formation occurrence exists between shoals having mean sizes <80 individuals (soldier), and a mean size of >80 individuals (ball formation). Across survey years, variation was significant for the dome formation (p < 0.001) between the 1995 and 1996 years. No identifiable patterns in formation occurrence across years was evident, even though some annual variation did exist. Nonetheless, the results showed that frequencies of occurrence for B F T shoals formations over time of day and across years was similar for two types: the soldier and ball formations. 3.6 Shoal dynamics 233 Table 3.48: B F T formation frequencies with associated visual shoal size estimates from aerial surveys conducted, 1994-96. These observations provide a larger set of observed frequencies than data set corresponding to quality classed aerial shoal images selected for the SAIA image analysis. Formation Type Year Shoal Size (A) Cartwheel (C) Dome (D) Soldier (F) Ball Classed 1994 yes 26 40 41 76 183 (N=1730) no 40 94 63 113 310 1995 yes 15 118 67 110 310 (N=2927) no 18 120 92 183 413 1996 yes 17 155 72 27 271 (N=1429) no 17 155 72 56 300 3.6 Shoal dynamics 234 (F) Figure 3.89: Top to Bottom: Depiction of the dynamic linkage of the B F T shoaling for-mations on the basis of increasing mean shoal size between formations: (A) Cartwheel, (B) Surface-Sheet, (C) Dome/Packed-dome, (D) Soldier, (E) Mixed, (F) Ball, (G) Ori-ented, see Table 3.29. Arrows, representing a linear linkage between each formation, are shown oriented in the direction of increasing mean shoal size. 3.6 Shoal dynamics 235 (I)) Soldier (F) Ba l l Figure 3.90: Frequency of occurrence of B F T formations for shoals observed in aerial surveys across years 1994-96. Circular numbers are time of day (hrs). 3.6 Shoal dynamics 236 Table 3.49: Reduced x2/df statistics comparing observed frequency of occurrence at time-of-day (hrs.) between B F T formations. See Table 3.22 for sample sizes of the formations determined by aerial observers. Year Formation Type Formation Type (x2/df) (p value) (D) Soldier (A) Cartwheel (F) Ball 1994 (C) Dome (D) Soldier (A) Cartwheel 0.640(0.904) 0.619(0.920) 1.83(0.009) 0.269(<0.001) 0.849(0.700) 0.921(0.570) 1995 (C) Dome (D) Soldier (A) Cartwheel 1.15(0.284) 0.626(0.907) 0.833(0.783) 0.820(0.709) 1.54(0.046) 0.618(0.920) 1996 (C) Dome (D) Soldier (A) Cartwheel 0.970(0.501) 0.227(<0.001) 0.696(0.942) 0.235(<0.001) 1.37(0.113) 1.00(0.458) Table 3.50: Same as Table 3.49 comparing observed frequency of occurrence at time-of-day (hrs.) between years 1994-96. Formation (x'7df) (.P v a l u e -94 versus 95 94 versus 96 95 versus 96 (C) Dome (D) Soldier (A) Cartwheel (F) Ball 2.79(<0.001) 0.736(0.813) 0.865(0.648) 0.551(0.959) 3.97(<0.001) 0.763(0.781) 0.553(0.958) 1.02(0.436) 0.544(0.962) 1.05(0.400) 0.498(0.978) 1.40(0.096) 3.6 Shoal dynamics 237 Virtual reality visualization of shoal dynamics A visualization extension was developed for the SAIA analysis scheme that is able to generate both static and dynamic visualizations based on the automated geometric reconstructions of shoal structure. The extension was developed and coded in the virtual-reality modelling language (VRML) , and can be used to compare and visualize results from the three-dimensional geometric reconstruction of shoals in the automated image analysis of static shoal structure with theoretical predictions or observed data on dynamic changes in shoal structure and movement [7]. V R M L is an object-oriented language. The V R M L scheme developed creates each individual tuna as a main object within the code. The code for the three-dimensional tuna object was designed and modelled by FlashFire Designs, Inc. (Staunton, VA, U.S.A.) and references a further nine objects each body part of a tuna onto a set of 1244 separate polygons. The positions and orientation angles of each individual tuna are determined from data output of the SAIA image analysis routines, and the code re-constructs the structure of a shoal. The reason for developing the V R M L scientific visualization extension in the SAIA system was to provide a framework to enable visualization of the three-dimensional shoal structures output from the data analysis. Furthermore, this extension enables visualiza-tion of shoal structure dynamics. Using this extension, changes in shoal structure can be either simulated by a shoaling dynamics model or in reference to observational data where available. The code developed allows one to examine how changes in shoal struc-ture over time emerge as a result of individual movement and interactions. A n overview of this framework is provided in Figure 3.91. Three-dimensional shoal formations of B F T using the V R M L model extension are shown in Figures (3.92-3.95). 3.6 Shoal dynamics 238 Movement observations: dynamic individuals SAIA shoal analysis: static shoals: Processing of parameter arrays: variable time ^(tijtj+tj,... ,t,i) or static formations at fixed t=tj SLBM-BFT model: dynamic shoals/coupled individuals Interpreter module: encoding of movement parameter arrays into VRML© VRML© environment module: spatial and temporal interpolation VRML© objects: BFT individual Static 3D visualization: observed shoal structure, formations Dynamic 3D visualization: predicted shoal structure, formations, space-time trajectories Figure 3.91: Framework overview of the virtual-reality ( V R M L ) extension to the SAIA image analysis scheme. 3.6 Shoal dynamics Figure 3.92: V R M L visualization of B F T shoal structure: Cartwheel formation 3.6 Shoal dynamics Figure 3.93: Same as Figure 3.92 for soldier formation. 3.6 Shoal dynamics 241 Figure 3.94: Same as Figure 3.92 for mixed formation. 3.6 Shoal dynamics 242 Figure 3.95: Same as Figure 3.92 for oriented formation. 3.7 Chapter 3 Summary 243 3.7 Summary A new supervised automated digital image analysis system (SAIA) was developed for the three-dimensional analysis of fish shoal structure and behaviour, shoals. The scheme integrates a user interface for supervising automated measurements and con-ducting theoretical extrapolative calculations of shoal shape and structure. The image analysis method can be used to calibrate, measure, sort, compare and archive information on shoal structure. The method was tested and applied to 463 digitally-processed aerial survey photographic observations of B F T shoals for a three-year period. The aerial sur-veys were conducted by fish-spotter observers and the New England Aquarium (NEAQ). The digital image database consists of shoals detected in July-October in the Gulf of Maine. Testing of the method was performed using observations of the first survey year (N=160, 1994), with a complete analysis performed for observations from subsequent years (1995-96). The components of the image analysis scheme are: (1) data initialization (i.e., data archiving and database management, spatial and intensity image calibration), (2) pre-processing steps (i.e., user environment settings, image area-of-interest delineation and image histogram output), (3) individual object identification (i.e., intensity filtering and thresholding, manual image measurement, (4) post-processing (i.e., automated object sig-nature detection/rejection, object measurement calculations and object statistics, shoal structural statistical discrimination and classification), (5) three-dimensional visualiza-tion of output shoal structure for real-time simulation, and (6) archiving of results. The SAIA scheme, applied to the survey observations of bluefin tuna shoals, enabled an efficient and detailed calculation of the structure of their shoals. Future improvements could include the hardware, software and image database algorithm extensions making the method a fully automated digital image capture system, and enable the rapid re-trieval, efficient archiving, and analysis of large sets of observational data. 3.7 Chapter 3 Summary 244 The automated analysis of shoal structure, together with future direct observations of fish shoaling dynamics, now provide a method to calibrate shoal size estimates, shape and structure observations in fisheries surveys using aerial, acoustic or radar technologies. The shoal structure results obtained from the SAIA image analysis method can also be used to compare large sets of observations of shoal structure to theoretical models of emergent behaviour in fish shoals resulting from individual fish movements and interactions. Reduced x2-statistics were calculated to compare the relative precision of visual and automated method for identifying individuals in shoals, and shoal structure pattern-corrected estimates of shoal size. Calculated visual, automated and corrected estimates of shoal size Ns were not significantly different (p < 0.05) across survey years. The shoal observations were further sorted into seven characteristic formation types. (A) cartwheel, (B) surface-sheet, (C) dome, (D) soldier, (E) mixed, (F) ball and (G) oriented. The x 2/d/-statistics for the shoal size estimates of each formation type indicated an increased precision of the automated procedure in comparison to visual estimation for four of the seven formations; surface-sheet, soldier, mixed, and ball formations. With the exception of the soldier formation, all of these formations were found to comprise a substantially large number of individuals, for which the automated image analysis method was able to substantially reduce bias in visual shoal size estimation. Separation distance between first nearest-neighbours (NND) ranged from ~ (0.1 — 2.0) B L . The smallest NND's in this range occur in the cartwheel and ball formations, with the largest occurring in the surface-sheet and oriented formations. This range can be compared to a mean three-dimensional NND of 0.9 B L as a reasonable assumption for a typical fish shoal [319]. As a function of shoal size, N N D in their shoals increased to a maximum and then decreased. Shoaling B F T individuals were found to have between one to six first nearest-neighbours (NNDcrit — 1.5BL). The surface-sheet and ball formations had between 3.7 Chapter 3 Summary 245 one to three neighbours. Approximate modal values of preferred bearing angles between neighbouring individuals were 30°, 90° and 120°, indicating that individuals prefer to be positioned horizontally directly beside each other, in a range of (30 — 60)° from the horizontal position. Bluefin tuna show highly polarized shoal structure. Differences in the mean polar-ization of their shoal formations was evident by shoal orientation angles varying in a range of (5 — 30)°. The mean packing density of bluefin tuna shoals across all forma-tion types was (0.80 ±0.072) BL~3. Systematic measurement uncertainty introduced on packing density associated with the fixed shoal depth assumed in the intensity calibration of the images was estimated as 5p = ±0.0267 BLrz. For both random and systematic measurement uncertainty, shoal packing density was (0.80 ± 0.098) BL~3 in a range of (0.71-0.90) BL'3. Pitcher and Partridge have conducted an extensive analysis of data on cruising shoals of saithe, herring and cod [319]. Packing density estimated for saithe (Pollachius virens) and herring (Clupea harengus) laboratory shoals were 0.71 BL~3 and 1.43 BL~3 respec-tively [319]. They indicated that on the basis of the results of their experiments and evidence from other work, packing density, approximated as 1.0 BL~3, is supported for fish when they shoal, although volume may increase (packing density decrease) in more loosely organized shoals. Estimates of packing density from laboratory experiments can be compared t o those from observations of shoals in the wild using acoustics instruments. Misund [254], for observations of saithe and herring shoals in the wild, reports that pack-ing densities are generally one order of magnitude lower than the prediction of 1.0 BL~3 of Pitcher and Partridge [319]. The estimated packing density for B F T shoals was (0.80±0.098) BL'3 and lies within the estimated range of previous laboratory experiments for other species. Comparing packing density estimates obtained from the SAIA image analysis with the approximation 3.7 Chapter 3 Summary 246 predicted by Pitcher and Partridge of ps = 1.0 BL~3 suggests that the estimates for B F T under-estimate true packing density. Some other systematic bias contributing to measurement uncertainty, in addition to error introduced from extrapolating shoal depth in the analysis, may not have been considered. However, packing density estimates for shoals in the wild obtained by Misund were less than those estimated in the laboratory. Under this consideration, packing density estimated for B F T shoals in the wild obtained in this analysis is also less than available laboratory estimates (e.g., saithe, herring, cod). Given the differences previously seen in estimates of packing density between laboratory and wild measurements, the estimated packing density of B F T shoals is close to the approximation of ps = 1.0 BL'3. The most realistic theoretical aggregative packing model predicts a volume per fish of 0.60 BL3, or packing density of 0.84 BL~3. Theoretical optimum packing of spheres yields a volume per fish of 0.64 BL3 (i.e., packing density of 0.78 BL'3) [319]. Mean packing density for B F T was estimated to be greater than these theoretical estimates. The maximum packing density for B F T estimated from the observations is (0.94±0.561) BL'3 at the 95% confidence level, corresponding to a range of (0.3740 — 1.4956) BL'3. As summarized by Misund [254], if individuals pack denser at greater speeds or higher levels of arousal, then this could cause variation in packing density between regions within a shoal [291, 319]. Separate from the scaling assumptions of ellipsoidal shoal structure and volume, relatively small changes in nearest-neighbour distances (NND's) may create large changes in packing density. Such variation may be especially apparent in free-swimming shoals due to variation in shoal speed and density as shoals change their orientation, encounter food patches, or respond to predators. The results of the image analysis also show that, even without a consideration of individual velocities and turning angle variations, B F T maintain rigid shoal shape and structure. For example, for all formation types, the number of edge individuals increased 3.7 Chapter 3 Summary 247 exponentially to a maximum of 25-30 fish, independent of shoal shape and size. This result suggests that individual B F T tuna on the edge of the shoal were well-determined and that their first neighbours are positioned with respect to the edge individuals at preferred distances, number of nearest-neighbours, and bearing angles. Moreover, this result suggests that the internal arrangement of individual B F T in their shoals may be determined by the number of edge individuals. The well-defined formations together with each formation type having a specific range of shoal size, suggests that when the number of edge individuals is restricted, the overall shape and internal structure of their shoals is also constrained. Preferred ranges in the numbers of first nearest-neighbours, separation distance, and relative bearing angles in B F T may be related to a wide-range of factors, including: (1) sensory lateral line perception, (2) movement efficiency in relation to reductions in hydrodynamic drag and increased propulsion, (3) the need to maintain an angular range of vision for both independent and inter-fish coordinated movement, (4) hydrostatic stability due to positive and negative buoyancy (swim-bladders), and (5) the need to maintain an efficient, regulated supply of water (oxygen) passing through their gills [1,2,57,290,307,409,410,414,416,418,419]. Principal component analysis of a set of seven variables relating to shape and internal structure explained approximately 70% of the total variance observed in their shoals. A summary of these results is provided in Table 3.52. Scatter plots of the first two prin-cipal components (PCI: shoal shape, PC2: internal structure) are useful for developing a formation signature test in the automated analysis of shoal images from aerial sur-veys. By combining a signature test for shoal formation type within the SAIA scheme, threshold settings for NNDcrit, and the best pattern-matching threshold, Rcrit for com-paring individual positions between visual and automated shoal size estimation specific to each formation type could be determined. Extending the SAIA analysis to include an 3.7 Chapter 3 Summary 248 automated feedback of P C A analysis results would enable the method to recognize shoal formations and categorize static digital images of shoals according to structural variance. This addition would make it possible to automate analysis of time-lapsed observational data of fish shoaling dynamics. In this way, the automated analysis method would be then able to track real-time changes in individual movement velocities and turning an-gles, correlations between individual movement parameters, and shoal shape. The results obtained could provide a basis for comparing real-time observations of shoaling dynamics with theoretical models of individual and shoaling behaviour. Significant differences in the time-of-day frequency of occurrence between their soldier and ball formations (p < 0.001) were found. These formations had the smallest (soldier, Ns = 11.59 ± 1.66) and largest (ball, Ns = 84.25 ± 11.32) observed mean shoal sizes, where mean shoal size was < 100 individuals. For shoal sizes > 100 individuals, their shoals form surface-sheets, with a mean size of Ns = 130.6 ± 34.57. This suggests that shoal size is a driving factor influencing alterations in their shoal formation or vice versa. Differences in shoal structure associated with each of their formations was dictated by both internal and external shoal structural variables. B F T formations were characterized by having different levels of variation in shoal length, width, depth, nearest-neighbour distance (NND) and bearing angle between neighbours (BA). The SAIA scheme, applied to the survey observations of B F T shoals, enabled an efficient and detailed calculation of shoal structure. 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CN' ^ r H r H O l CN ^ -H -H -H -H -H -H -H r H 0 0 L O r H CO C O C O n . © l^ - CO CO I - H o i o i oo m o I D IO OO CM N O rt 1 r H L O oo co o T * o o CN 0 0 CN o o -H oo o L O o (M N O O 0 0 CN CO T - H Oi O co co o-CO N H (D h N O O O O O O oo co co T * CN O r H T - H CN o o o co 0 0 t -o o -H \"H r t 1 CD CD r f l CO I CD CD M H SH fl CD CD r f l CD a o Q SH CD O T J 32 8 r f l .2 cd OQ < OQ O Q fa fa O o O o L O co o o ,*—*\\ o o o L O o t>-r H r f CN CM T-H •§ -u cd cd cd cd cd cd L O o o 0 0 o L O o CN CM CO CO 1 — 1 co CN fl o td a • t-h s SH OH OH < < CO OH cd r f l zn co o O r4 o £2 ° ° ° CN ~ CN 1 1^ CN CN r H t - H C O T—H f—-\\ T-H T-H T-H - CN - ~ -o r- i o O O O C7i - O) o ^ ® N CJ \"2 'o cn V V T J T J •i-h .t-h O O SH SH 0> CP fl O OJO >> \"o OH X CP > fl O CJ -fl -fl OH OH cn cn 1-2 ^ - f l 1 - H U O O v ' -H <» OH OH T J CJ X r f l CD fl H J 0 cd cc3 cd CP bO SH a> -fl a cn O CJ . cd cd OH bC a * o CJ CP fl cd b O fl O o co OH CD 3 S3 CP o- -CO OH a o CJ o cd OH a o CJ o CNL ^ ^ O 00 o o o o T—I T - H C N C N C O C O O 0 0 • O O O , CN CO CO L O L O ~ CD CD r f l (* O zn +3 CD CD r f l CO I CD o HH SH fl SH CD CD a 3 Q cn T J CD T J CD CD cd OQ < OQ U Q fa fa O 3.7 Chapter 3 Summary CD . +J <+H CO O ;fl bp CD °J \" S & CD CD T 1 cu •> o a _ ^ » § -is •-< T) C) CD Co ^ .22 a f ; - , >-Q ft ™ w O fl OH • 2 O _ , III o 3 . J f l . f l O O co _fl _ W CD 13 p o r f l PQ r: CN *3 u o 2 .-a o S .3 n fl <•> PH O O & 8 « S ° o S O S CN a . ^ \"+3 ^ » r W S H O EH cm a CD a3 13 o J3 co CD fl 13 'ft w CD CJ fl a3 • i-H ^ as CO CO IT -? !> Q ^ ^ 52; CQ g ^ ^ ^ «) CO «S WS CO 93 v Q ^ Q Q Q Q ^ © C O C N N O ) N C N 0 0 C O \" t f O ci t>-co to ( O ( O ( O © m CD CD \"co CD | CD £ 2 . c3 ^ O co S H CD . 2 i——I O CO CD X o3 PQ CD fl* _CD o < C Q O Q H f c O Chapter 4 Spatial, Individual-Based Model of Bluefin Tuna Mathematical models can us help to understand the mechanisms that regulate changes in the spatial distribution and abundance of fish populations. Theoretical models can provide an integrated description of fish populations by considering how individual move-ment, behaviour, shoaling, predator-prey interactions and changes in the marine envi-ronment are coupled in space and time. I present a spatially-explicit, individual based model (SIBM) of Atlantic Bluefin Tuna, resident seasonally in the Gulf of Maine(GOM)/Northwestern Atlantic Ocean. This SIBM model integrates the movement and behaviour of individuals and shoals in response to their prey and changes in the marine environment, and is called the FIESTA (Fish Estimation, Schooling and Tuna Abundance) model. A Lagrangian representation of the model is formulated, whereby individuals and shoals move in space by adjust-ing their orientation, turning angle, velocity, strength and sensitivity in their response to spatial concentration of prey and gradients in oceanographic variables. The model distin-guishes how movement and behaviour are coupled, and considers two modes of movement associated with foraging (intensive search) and travel (extensive search). The dynamics of shoals is governed by individual movement mode alterations and conspecific attraction and repulsion. The exchange of individuals between shoals is regulated by individual and shoal fitness functions that depend on perceived feeding rate and predation risk. Fitness controls whether individuals decide to join, leave, or stay in shoals, and how shoals inter-act. Information obtained from the analysis of hydroacoustic tracking, satellite tagging 251 4.1 Model and simulation framework 252 and aerial survey observations supports the structure of the model, and is used to provide estimates of model parameters. Results from the individual and shoal structure analyses are used to formulate processes movement and behaviour of individuals, shoal structure, formation and mixing, and movement with respect to scalar concentrations and gradi-ents of environmental variables within an environment grid. This grid can be referenced in two and three-dimensions, and includes phytoplankton and zooplankton spatial con-centration (i.e., prey-correlates), sea-surface temperature (SST), temperature-at-depth, bathymetry, and ocean circulation. Seasonal immigration and emigration of shoals is also considered. The results of model validation tests are presented. The model provides an analytical framework for examining and quantifying the extent to which the size and structure, inter-action, spatial aggregation and distribution of shoals may affect measurement precision, bias and uncertainty. Additional work aimed at improving the model, and comparing its predictions with observations at the individual, shoal and population scales is needed. Continued development and testing of the model will allow a comparison of the precision of different aerial survey sampling schemes in measuring spatial and temporal changes in abundance under simulated changes in the population. 4.1 Model and simulation framework Model structure, variables and fixed parameters A n overview of the SIBM model and simulation framework is shown in Figures 4.96-4.97. Fixed parameters and variables (Table 4.55) are specified in the model initialization stage. The model has a total of fourty-three parameters not counting the environmental grid. The addition of a grid layer adds three parameters. Hence, with a grid of five reference layers and no fixed parameters, the full model has fifty-eight parameters. In 4.1 Model and simulation framework 253 Table 4.56, the full set of variables are listed, organized according to the categories: environment, population, shoal and individual-scales. Figure 4.97 presents the model and simulation structure after the initialization stages where tracking of movement, behaviour and interactions are simulated at the individual or shoal-levels. The simulation unit can either be a shoal or an individual, and the model can be simulated in either two- or three-dimensions. With the set of associations and fixed parameters, the model (with an environmental grid) has the following parameters, M = M[r ' , A', e', Ra, Rr, Zs, Ys, T, ft] (4.105) Parameter Description T' movement correlation time-lag A' movement correlation strength e' sensitivity in movement response to environment Ra social attraction range Rr social repulsion range Zs shoaling advantage for foraging Ys shoaling advantage for dilution of predation risk r shoal fusion/merging Q shoal fission/splitting To reiterate, the full model listed in Table 4.55 contains fourty-three referenced variables (no grid layers) or fifty-eight variables (with five grid layers). By associating variables, leading variables (r',A',e') are formed, reducing the totals to twenty-five (no grid layers) and thirty-six (five grid layers). The following associations can provide a reduced model representation: Empirical estimates are used to fix certain parameters leaving nineteen variables (no grid layers), and twenty-seven (five grid layers), further reducing the total to six (no grid layers) and nine (five grid layers). Table 4.54. lists values assigned to model variables forming fixed parameters. The 4.1 Model and simulation framework 254 Table 4.53: Associations between model variables in forming a reduced model represen-tation, denoted as model, M . Time-lags consistency in movement correlation: T' = TO = = TV = T M Consistent strength in movement response and correlation: A' = (A p ,Vp) = Xe = \\ = Xv = Xg(ml) = Ag(m2) Movement mode harmonics: ua(ml) = ug(ml) W b ( m l ) = ui,(m2) wa(m2) = ujg(m2) c j 0 ( m l ) ^ ua(m2) Movement environmental response: e' = ( e P , V P ) population of B F T in the study region was considered to be composed of fish >7 years old. A seasonal population of B F T in the study region was considered to have a uniform age distribution, with p^=80% of the fish at each age >7 in the Western Atlantic region transferring to the G O M region. The critical range at which shoals can interact was fixed at 7S'D c rj t=200m, with neighbouring shoaling fish interacting within a range of 1.5 body-lengths or 3m (where 1 BL=2.0m). In order to simplify the model, the non-linear scaling parameter, /?p, in the movement response of fish and shoals to the model environment was set to unity. This simplification allowed sensitivity and strength in movement response to be varied in the model, without confounding effects being introduced by the non-linear scaling parameter, (3P. The maximum depth of movement, pp was fixed at 10m below the ocean surface. Simulations were only performed in two-dimensions so that this parameter is only used when three-dimensional simulations are run with the reduced model representation. A value of rmax = 6m was considered to be a reasonable estimate for the maximum visual range of individual tuna. A variable modifying the decaying exponential form of correlation in turning angle, av, was not considered, and was set to zero. A n explanation of the fixed parameters assigned for variables relating to foraging 4.1 Model and simulation framework 255 and predation is provided later in this chapter. In reference to the list of aggregated fixed parameters in Table 4.54, ranges were fixed to constrain these variables in model simulations. The number of moves where movement parameters and modes are considered to be correlated was fixed within r'=4 moves. Similarly, the strength and correlations and sensitivity in movement varies within fixed ranges 0.02-0.06 and 0.25-2.25, respectively. These ranges were assigned from a comparison of observed and model predictions of movement response to environmental variables. The amplitude and oscillation frequencies for the behavioural variation of turning angle and move-speed (i.e., p\\2, u^2, pi'2, ul'2) were set to empirical estimates obtained from a spectral analysis of hydroacoustic tracking observations (refer to Figures 2.13 and 2.14 in Chapter 2). The range of attraction and repulsion for neighbouring shoaling fish (Ra,Rr) were assigned fixed ranges with maximum values of 6.67m and 1.67m respectively. Fixed assignment of these parameters were made under the condition that Rr < Ra- These ranges were determined from test simulations of the model. Shoal fusion and fission rates were assigned the fixed maximum range corresponding to 0-100% of B F T to join or leave their shoals. 4.1 Model and simulation framework 256 Table 4.54: Fixed model parameters (N=19 (no grid layers), N=27 (5 grid layers)), aggregated parameter settings, and variables in simulation for process test-results. Fixed Parameters Value P'a 0.80 (80%) ISDcrit 200m a,b,sc,d 1.0,-1.5,150,1.0 NNDcrit 1.5 BL=3.0 m, for BL=2.0m pp 1.0 Pp 10.0 rmax 6.0m 0.00 (2.0,0.003,10.5,0.5) foo,Qf,Tf,M (0.003,10.5,0.5,0.5) Aggregated Parameters Range (Simulation) r ' [0-4] A' [0.02-0.06] e' [0.25-2.25] pa(ml),pa(m2) [0.7-0.8] pb(ml),pb(m2) [0.2-0.3] • uja(rnl),ua(m2) 2TT • [0.10 - 0.50] ub(ml),ujb(m2) 2TT • [0.10 - 0.50] Variable Range (Simulation) max(A^) [10 min] or [600 s] max(i?a) [6.67]m max(i?,.) [1.67]m Zs [4.0,8.0] Ys [1.0-2.5] r [0.0-1.0] n [0.0-1.0] 4.1 Model and simulation framework 257 Model Iiiitialization: Fixed Parameters/Initial Values of Variables Dynamic Re-Dimensioning of Arrays Simulation Initialization: Annual VPA Population Time-Series, Natural/Fishing Mortality Vectors Population Age/Length/Weight Structure West Atlantic Spatial Mixing Annual Transfer Rates Derived G O M Seasonal Model Population Abundance Initial Shoal-Size Frequency Distribution Seasonal Immigration and Emigration Schedule Environmental Grid Initialization: 2D/3D Spatial, Environmental Grid (Concentration and Gradients) Annual, Monthly SST and Zooplankton Abundance Variability Lagrangian Movement Trajectory Tracking: Spatial Grid and Shoal/Individual Move-Duration Resolution Numerical Accuracy and Sensitivity Thresholds Shoal-Level Tracking hidividual-Level Tracking Figure 4.96: SIBM model and simulation framework: initialization stage. 4.1 Model and simulation framework 258 Shoal-Level Tracking Iiuliviilual-Level Tracking Initial Shoal Movement Modes (rrii.mj) Daily Schedule of Mode-Switching Events (mj) Initial Shoal Positions on Grid Boundary 1 Initial Individual Movement Modes (m,, m>) Daily Schedule of Mode-Switching Events (mi) Initial Individual Positions/Shoal Structure Seasonal Immigration 2D/3D Lagrangian Trajectory Equations: (Adaptive Step-Size, Runge-Kutta) Intrinsic Movement: (Move-speed, turning angle correlation and harmonics) External Bias: (Movement response to gradients and prey concentration/correlates) Shoal Intersection Detection Shoal Mixing (Shoal Fusion/Fission and Individual Exchange) Nearest-Neighbour Detection Individual Attraction/Repulsion: Individual Join, Leave, Stay Decisions (Foraging Rate and Predation Risk Trade-off) Shoal Movement Mode Alteration i_ Individual Movement Mode Alteration (Self-Similar Mapping of Nearest-Neighbours) Seasonal Emigration I Shoal/Individual Trajectories and Spatial Distribution Population Abundance Time-Series Figure 4.97: SIBM model and simulation framework, continued. 4.1 Model and simulation framework 259 leg P CP ci J CP CP a o 3 Q. o ' • H .^3 i —t 03 TJ TJ fl 13 3 T> \"> T) fl bO ^ O -3 nfl G O bO 2 QJ >H cp cti O O o3 TJ .A -A TJ • cp cj o c3 TJ fl 3 - O 03 fl O | 3 o a fl o 'SO co fl SH oi cj xi 3 13 3 a g s o tg O '5b S -Td £ 1 5 c V *H , S I -\"3 03 •3 T5 > « c - 03 o T> v CP co « cp cu „ > £ 03 B 13 » 5 c§ S *7i fl S V o o3 a> S s O -2 § O O 03 -a .8 6 03 I , o 2 3 a s g S O § bC bO C y fl CP 2 <2 3 cp 03 S x - A cp •3 .s .a -3 SJ. 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, bO tH CP fl CP cp\" CP . a. d. cp\" « '3 o. a C 5 o3 a o 13 T3 cj o3 \" 3 '? TJ .5 > O TJ s ° e ^ 3 CP\" S bO cp fl \" 03 (H 13 ^ 3 .SP co \"cp •> ? I* a .a '•S l l 3 - c>. ti +o -w -w 1^4 r e 1 ^ C N ^ ? Cfc ^ . 5 O T PH QJ' T o a, PH 3 9-b ^ 1 a. 4 4.1 Model and simulation framework a .2 CO cp < g . . co ^ T3 LO CO A H • > fl I§.I fl fl CS rfl CS bO CO cS l l H 0 £ 1 ^ CO T EH D 3*. T fl o> s fl o S-l CD > fl o 3 OH bO a \"a. A o u B tH O §1 • f l I a) a; fl o c cfl o o w cu fl _ fl fl D, fl S .H § 1 2 £ < 9 - 3 ; £ - 3 . a. 4.1 Model and simulation framework 261 Model assumptions The key assumptions of the SIBM model are: • Model Representation: movement is assumed to follow a Lagrangian representation, with reference to variable time and space. External bias is attributed to changes in local environmental gradients. In the current formulation of the model, it is assumed global attraction does not contribute to external bias. Move-speed is assumed to be proportional to forces in steady-state motion, with inertial motion neglected. • Shoal Immigration/Emigration: Immigration/emigration is assumed to be directed from the southern boundary of the study region, with shoals moving northward. This assumption is based on tagging observations which show B F T shoals immi-grating northward into the G O M region, and is consistent with reported northward migration of B F T from the Gulf of Mexico, as their identified spawning ground in the West Atlantic [10,26,158,240-242,249,269,274]. Shoals are assumed to emi-grate out of the region by orienting their movements directly southward. Emigra-tion is assumed to be independent of shoal size, age-structure or space. Following the onset of emigration at time, tm, shoals/individuals movement is re-directed, but still maintained according to the model's movement and behaviour charac-teristics. This assumption is based on recent satellite tagging observations ob-tained by Lutcavage and coauthors (1999; 2000) recording shoals moving south out of the region, and then eventually turning eastward towards the central At-lantic [13,159,214,215,217]. • Shoal Mixing (shoal-level): The fusion of shoals is assumed to be more likely for larger shoals in proportion to shoal size, s, with fission increasing non-linearly with shoal size (as per Equation D.193 for Case 2 detailed in Appendix D2). 4.1 Model and simulation framework 262 • Shoal Mixing (individual-level): For two interacting shoals (j,f), individual i in shoal j, and individuals i! in shoal f are assumed to either join, leave or stay (JLS) in their shoals as a function of their fitness or net reproductive output. • Visual Range: B F T visual range is assumed to increase linearly over time at a con-stant rate of v, during dusk and dawn, and a maximum visual range is maintained otherwise, as during the day increases/decreases in light intensity are substantially less than during dusk and dawn. • Attraction/Repulsion: Re-arrangements in the relative positions of individuals about the centroid of a shoal is assumed to occur by mutual attraction/repulsion and the spatial extent or range of repulsion is assumed to be less than the range of attraction (Rr < Ra). • Alteration of Movement Modes: Shoaling is considered a self-organizing process, whereby individuals in a shoal move and interact over time by deciding to move according to a choice of two behaviours: forage (mj) or search (m2). • Movement Stochasticity: Stochastic variation in turning angle and move-speed are assumed to be distributed according to the forms of the Von-Mises circular normal and Gamma distribution, respectively. Additional stochasticity is assumed to exist, being independent of time and space, and distributed as Gaussian white-noise. • Movement and Behaviour: Persistence in movement is assumed to be coupled dif-ferently to behaviour according to two modes: (mi) foraging (intensive-search), and (m2), travel (extensive-search). Movement during foraging is assumed to have a more rapid turning rate than during travelling. Support for this assumption is based on results of an analysis of observed movements that show that these two modes are distinguished by an alteration of their rate of turning. Move-speed is also assumed to vary between the modes, supported by empirical data. 4.2 Initial and boundary conditions 263 • Fixed Model Parameters: Model variables are assumed fixed according to assump-tions detailed in Section 4.7. Fixed values are determined from results of separate analyses of B F T movement and shoal structure. Model implementation The model is programmed in the Visual B a s i c ^ language. Restrictions on the size of the array dimensions require that a reduced number of shoals and individuals are tracked. Depending on the initial size of a simulated population, and whether shoals or individuals are tracked in the model, computational arrays are re-dimensioned to increase computational speed and memory-handling efficiency. 4.2 Initial and boundary conditions Initialization of the model involves specification of the G O M seasonal population abundance from age-specific V P A estimates and model transfer rates. Simulations were performed with V P A stock assessment data, and estimates of G O M population abun-dances, natural and fishing mortality, and population age-structure. The age-composition of a model population can be varied to include a range of age-classes within 1 — 10 + , however in the simulations performed the population was consid-ered to comprise individuals of 7 — 10 + years, in agreement with age-specific abundance estimates obtained from aerial survey data (see Chapter 5) . Individual fish are assigned ages by random selection from the total input numbers of individuals in each population age-class. The assignment of ages to individuals is constrained so that the population age-structure is maintained. On the basis of individual age-assignments, corresponding length/weights are determined. Length/weight for each age-class can be set to have monthly and/or annual variation with estimated values assigned from B F T stock assess-ment length/weight matrices. Assessment estimates of age-specific natural and fishing 4.2 Initial and boundary conditions 264 mortality matrices for the Western Atlantic population are also used in the model. Initially, the sizes of shoals are determined by sampling of a parameterized size-frequency distribution (normalized) that is then re-scaled to the size-frequency distri-bution corresponding to the size of an input population. Seasonal immigration and emigration schedules are also specified. Additional initialization stages include the input of biological and physical layers of the model's environmental spatial grid. The layers of the environmental grid reference chlorophyll-a, denoted ChlA, as a measure of phy-toplankton abundance, and zooplankton spatial abundance, both that are assumed to correlate to the net-abundance of B F T prey-species. The model construction permits addition or removal of layers in the environmental grid used in a simulation. To better characterize the distribution of prey species, the phytoplankton and zooplankton layers could be removed, and replaced with empirical or theoretical spatial abundance estimates of different known prey species. The spatial abundance of B F T is more likely to be more correlated to the abundance of their prey than the abundance of phytoplankton and/or zooplankton. A n input estimate of mean move-duration (e.g. 10 minutes in the simulations per-formed) sets the minimum temporal resolution of the tracking integration in the model. If shoal-level tracking is desired, then shoals (i.e., the primary model unit) are tracked without reference to individual variability. If individual-level tracking is instead desired, then individual and shoal-level processes are all simulated, whereby individuals are the primary unit in the model. For the primary units of the model, initial movement modes are randomly assigned, with initial shoal positions being randomly determined from a normal distribution, bi-variate in the x and y coordinate directions, within a designated/input area along the boundary of the environmental grid. 4.2 Initial and boundary conditions 265 After initialization of the model, the seasonal immigration of individuals/shoals pro-ceeds, and their movement trajectories are tracked over time using an adaptive step-size numerical integration procedure. This procedure uses the input minimum threshold of step-size (i.e., move-duration), and a desired numerical accuracy. The numerical accu-racy was set to 0.1% for all model simulations. The model's state-space equations are iterated over successive moves referenced to parameterized move-speed and turning an-gle correlation functions, and movement mode harmonics (mi,7/12). These considerations describe the movement and behaviour of B F T , with external movement bias determined by movement response to gradients over layers in the environmental grid. Dynamic time-stepping of alterations in movement and behaviour At each time-step, individual attraction and repulsion, shoal intersection, mixing, and exchange of shoal members can occur. Shoals increase and decrease in size according to shoal fusion and fission processes. The intrinsic movement of both individuals and shoals is assumed to follow two main modes, (mi ,771-2). Mode-switching occurs as a discrete event coinciding with dusk and dawn in the diurnal cycle, and is denoted m 3 . Mode-switching represents a shift in the motivation of shoals or individuals to forage, travel or search over larger distances at certain times of day. This diurnal mode-switching provides an underlying fluctuation in how shoals or individuals move. When shoals or individuals intersect they can exchange members. In mode mi , shoals are assumed to increase in size as they aggregate to feed, whereas in mode m 2 , shoals tend to maintain or decrease in size. At the level of individuals, modes alternate in time according to perceived changes in movement and behaviour of nth order nearest-neighbours. Individuals in shoals can decide to join, leave or stay (i.e., JLS decisions) at each time step. When shoals intersect, their movement mode alternates between (mi,777,2) according to whether shoal size increases or decreases. As shoals interact they can also 4.3 Seasonal population 266 maintain their movement mode. Shoal interaction constrains the underlying process of movement mode-switching within the diurnal cycle. Therefore, JLS decision-making not only constrains shoal sizes and their interaction, but also how individuals and shoals move within the ocean environment. 4.3 Seasonal population The Virtual Population Analysis (VPA) method used in B F T population assessment is summarized in Appendix D l . Modifications and additional considerations relating to fish movement provided by fishery-independent mark-recapture tagging data and fishery survey abundance time-series indices have been introduced within the V P A methodology for improving the estimation of fishing mortality in estimating population abundance over time (i.e., within the tuned-VPA method) [55,178,274,337]. In the V P A formulation, Porch and coauthors have recently considered movement and tagging data for estimating B F T regional abundances [325,327,328]. Their Atlantic-wide distribution is comprised of Atlantic populations separated at 45°W. Highly advective movement is assumed to occur annually between each region. The abundance for each population within each region (west/east Atlantic), is denoted as Na+itt+i,i where i = (1,2) for age-class, a and year, t, and represented as, AUM+U = N a < t < 1 e - ^ + F ^ \\ l - 6lr2) + N a ^ e - ^ + F ^ 9 2 - 1 (4.106) where, 9lt~,J ( i , j=l ,2) , are the fraction of total individuals transferring between the two (east/west) populations at the end of each year, t. The values, Ma^\\ and Ma^ are age-specific natural mortality rates, and F a i t ] i and -Fa,t,2 a re fishing mortality rates for each region. Natural mortality is assumed to vary between age-classes and is constant in time. However, fishing mortality is assumed to be dependent on both age and time. Tuned-VPA annual estimates of abundance in the western Atlantic were used [10,11]. 4.3 Seasonal population 267 Age-specific fishing mortality was assumed to vary over time on an annual and/or monthly basis. Numbers-at-age (0—10+) for the western Atlantic population from V P A assessment estimation for a constant natural mortality of M=0.14/year and variable age-specific fishing mortality (Fajt) were used to simulate their seasonal residency in model. The V P A estimated time-series of abundance for each age-class in the Western Atlantic population of B F T shows a wave of cohorts moving through time (Figure 4.99). The west-Atlantic population was sub-divided into the west-Atlantic sub-region ex-cluding the Gulf of Maine (WR), and the Gulf of Maine (GOM). This required a consid-eration of annual population transfer between these two sub-regions at the end of each simulation year {Y^OM-^WR a n ( j pvv#->GOM^ a n ( ^ ^ n e a s s u m p t i o n of an expected annual equilibrium fraction of the west-Atlantic population (NttwR), seasonally occurring in the G O M study region, denoted as NTTGOM- For year t, In Equation 4.107, a = 10 + denotes a terminal-age class as a plus group representing a pooled age-class consisting of all age-classes ^10 [339]. The choice of the terminal age-class to include all fish of age ten years or older follows a similar assumption used in population assessments [178]. Assuming a seasonal fraction for each age-class (a) in the Gulf of Maine, with respect to the total west-Atlantic population (p'a), then, 10+ (4.107) a=l 10+ 10+ (4.108) 10+ 10+ (4.109) The annual abundance for the G O M and W R sub-regions across years is, Nt+i,GOM = (i — r t iGOM^WR )Nt,GOM + 17 WR^GOM Nt,WR (4.110) Nt,WR = (i - r; WR^GOM )Nt{t — U-i) (4.114) PM-e V«(t-ti)) t i - ! < t < t i expressed in terms of the parameters rju, (to = 1, ...,U>T) that varies the levels of injection or population immigration over each immigration wave, u. The immigration of shoals into the region requires the additional consideration of their shoal size-frequency distribution. The initial distribution was assumed fixed over the time-interval during immigration (t0, ...,U), where U is the starting time of the simulation for full population size. During immigration, the sizes of immigration shoals are fully determined by random sampling from the initial parameterized form of this distribution. Emigration Emigration was scheduled to occur after a pre-specified time, tm, when fish respond by turning directly south. Alternative ways of depicting emigration are possible, and may involve ordered-sequencing of emigration movements with respect to the distance between shoals. Emigration was assumed to be independent of shoal size, age-structure or space. Following the onset of emigration at time tm, fish movement is re-directed, but still maintained according to the model's movement and behaviour characteristics. A sufficient time-interval between tm and the end of a simulation each year tn allows all shoals/individuals to leave the region. Use of a smaller time-interval, At— (tn — tm) allows a greater proportion of shoals to remain resident with the region. Figure 4.100 depicts the profile of the scheduled immigration and emigration of shoals for u = 3, and emigration time, tm. Depictions of scheduled immigration and emigration across the grid boundary are shown in Figures (4.101-4.102). Immigration in the model is assumed to be directed from the southern boundary of the study region, with shoals moving northward. This 4.3.1 Seasonal immigration and emigration of shoals 272 assumption is based on tagging observations which show shoals immigrating northward into the region, and is consistent with reports that they northward migration from their spawning ground in the West Atlantic (Gulf of Mexico) [10, 26,158, 240-242,249, 269, 274]. Shoals are assumed to emigrate out of the region by orienting their movements directly southward. This assumption is based on recent satellite tagging observations, recording shoals moving south out of the region, and then turning eastward heading towards the central Atlantic [13,159,214,215, 217]. 4.3.1 Seasonal immigration and emigration of shoals 273 t - to tfw=l) t(w=2) ^=tl-t(w=3) Time, t II II z . N(res ident) the integral can be further simplified. For example, in Equation 4.117, if u is constant over the time-interval, then, :{tk+1) = (tk+1,tk)x(tk) + f $(tk+1,o)C{o)u(o)do- (4.117) x(tk+1) = ${tk+1,tk)x(tk) + T{tk)u(tk) (4.118) (4.119) tk+i T(tk)= J (tk+uo-)C{o)do (4.120) tk Note that T(tk) here bears no relation to the annual transfer rates in Equation 4.110, although similar notation is used in the above generalized equation. In the model, dif-ferential equations generate trajectories in the movement and behaviour of individuals and shoals. These equations contain certain terms that are made constant, with others allowed to vary over a time-interval, taken as a mean move-duration. The specifica-tion of the time-interval used in propagating the D E equations, sets the lower limit of spatial and temporal resolution in the model, and reduces the terms involved in the numerical integration. The physical and biological layers in the spatial environmental 4.4 Lagrangian equations 278 grid are sufficiently smooth with respect to the mean move durations/time-integration steps. Consequently, the spatial/temporal resolution of a movement trajectory recognizes smoother gradients in the simulated ocean environment. Numerical methods For numerical integration of the model, without the assumption of a fixed mean-move duration, the adaptive step-size Runge-Kutta integration involves all terms in the state-space equations. Without the assumption of a fixed mean move-duration as the minimum integration step for the Runge-Kutta method, the range of move-duration is controlled to achieve a predetermined numerical accuracy with minimal computational effort [330]. In this way, full integration would move shoals/individuals over small move steps through regions where environmental gradients change rapidly, and larger move-steps over smoother terrain. However, full integration requires monitoring of the sen-sitivity of all model variables introduced in the time-integration for model validation. Further details of the adaptive Runge-Kutta method used are provided in Appendix D4. The general discrete state-space equations describing the movement and interaction of the ith individual in the jth shoal, in the ^-coordinate direction is, Xijfa+i) = xij{tk)+ J (Ax(a) • (Cx{a)ux(a)) + Fx(a) + Ex{a))da + nt (4.121) tk where A = (Ax,Ay,Az) is a vector of the angular move-direction (6), and turning angle dependence (ip) of movement in each coordinate direction. Movement in the y— and z—coordinate directions follows the same generalized equation above (Equation 4.121). The vector C = (Cx, Cy,Cz) is move-speed autocorrelation coupling, F = (Fx, Fy,Fz) is the attraction/repulsion (AR) coupling between nearest-neighbour individuals in a shoal. E — (Ex, Ey, Ez) is the external bias in movement response to environmental 4.4 Lagrangian equations 279 concentration of tuna prey-correlates (i.e., zooplankton, phytoplankton), and gradients in physical, oceanographic variables (i.e., SST, bathymetry, flow). The term, r]t, is a random white-noise movement dispersion contribution. Together, the Et and r\\t terms represent movement taxis that can be either directional and non-directional, and a kinesis response, respectively. The vector u was previously used to denote only a vector of total inputs or input coupling vector to a dynamic system (Equation D.204). In the above set of equations, u = (ux, uy,uz) denotes a vector of initial move-speeds in each coordinate direction. Ini-tial values for other model variables are specified and together these form the system's input coupling vector. Equations 4.121 are modified to describe movement of shoals, with movement and behaviour approximating the mean fluctuation of individuals. At the shoal-level, the attraction/repulsion term (F) was omitted. Also, for both the indi-vidual and shoal-level simulations, the depth component of the attraction/repulsion (Fz) was omitted. Therefore, no vertical component of attraction/repulsion occurred, and individuals were constrained to interact in the horizontal plane. In Chapter 2, a stochastic equation formulated by Niwa to describe the evolu-tion of shoal velocity involves various forces that affect their acceleration. The attrac-tion/repulsion and environmental movement response in the SIBM model are instead formed as time-averaged quantities proportional to movement velocity [275]. Change in velocity are represented using harmonic functions constructed on the basis of the move-ment analyses (Chapter 2). These functions oscillate move-speed and turning angle to match the actual variation observed for the m i and m respectively, visual range Tij{tk) is, Tmax ~ \"(td ~ tk) 0 < tk < td ry(*fc) = { rmax t d < t k < td, (4-124) fmax — ^{tk — td') td' < tk < tdny Equation 4.124 assumes visual range increases linearly over time at a constant rate of v during dusk and dawn and a maximum range is otherwise maintained [164, 267]. For individuals, feeding rate is a function of visual range (r^) and weight (Wj,-), having parameters, / o o , (the asymptotic value of feeding rate), Qf (per-capita food in-take), and rf (the visual range of B F T prey). For shoaling prey, rf scales with the size (structure and numbers of individuals) of their shoals. Similarly, individual predation risk in Equation 4.126 is related to an asymptotic value based on two scaling parameters, (pi, pi)-, per-capita food-intake, Q M , of BFT ' s predator, and their visual range, rM . These functional forms are termed Holling Type III responses [68,170]. These functional forms have been used in modelling the movements of juvenile sockeye salmon [66]. Because B F T are a predatory species and have few predators as adults, foraging might be expected to dominate considerations of predation risk. However, both consid-erations are represented in the model to relate the trade-off between foraging and preda-tion risk to their movement and shoaling dynamics. Feeding rate and predation risk of individuals and shoals were represented, respectively, with the following functional forms; / / ( r - m ) = h+rJ?(W\\ *\\ ( 4 J 2 5 ) (1 + Qj/irij + rf) ) and, Arij,Wi)=, \" i / t t + ttW) (4.126) Following Mangel and Clark [234], the question of the optimal trade-off between food intake and risk of predation can be characterized as a short-term decision taking place 4.5.1 Individual/shoal fitness: foraging rate and predation risk 282 within the time spent feeding (feeding interval, At). Total food intake and mortality risk over a time-period At is then, Iij(At) = J /'(r&WJdt (4.127) A t Myi&t) = 1 - exp' ^ - ^ n'inj, Wi) j dt (4.128) For shoals, feeding rate and predation risk are described using simple functions that introduce a dependency on shoal size (SJ + 1) (number of individuals), where shoal size, S j > 1 [143]. Scaling of individual feeding success by shoaling is controlled by a parameter Zs representing the degree that a shoal of a given size is able to capture prey. For low Zs, a solitary individual has a high probability of successfully capturing prey. In this case, additional individuals forming a shoal have a small effect on increasing shoal for-aging success. For high Zs, solitary individuals have a low probability of capturing prey, whereby increases in shoal size have a substantial effect on shoal foraging success [143]. The parameter, Zs, is termed individual foraging failure since it scales associated with a decreasing probability for an individual in capturing its prey. Feeding success while shoaling follows the function, ft-'M-z^&T) <4'129) Per-capita predation risk for a shoal is scaled by Ys. As Ys increases, the addition of an individual to a shoal j of size Sj leads to a significant decrease in shoal per capita risk. When Y = 1, perfect dilution of predation risk is maintained across increasing shoal size. For Y > 1, shoal predation risk is reduced by shoal confusion and dilution effects [262,398]. The parameter Ys is termed dilution of predation risk. Per-capita 4.5.1 Individual/shoal fitness: foraging rate and predation risk 283 predation risk for a shoal is expressed as, ^ W ^ Y a ) = ^ f ^ (4.130) Figure 4.103 profiles the feeding rate and predation risk functions used to characterize individual and shoal fitness with parameter values chosen to scale the dependent variable in each function within the range [0,1]. The trade-off of feeding rate and predation risk is considered an age-specific, long-term strategic objective to maximize reproductive success [65]. The relative benefits for choosing a foraging or predation risk strategy can be represented in terms of net reproductive output (w). Net reproductive output is an individual's residual reproductive value at any age, and is formed as a ratio of foraging net energy returns to the probability of mortality per unit time [143]. Net reproductive output, w, can be expressed for both individuals and shoals, denoted and for individual i in shoal j , and shoal j respectively. 4.5.1 Individual/shoal fitness: foraging rate and predation risk 284 Figure 4.103: Individual and shoal-level profiles of fitness functions referenced in the SIBM model (Qf = 10.5,/«, = 0.003, rf = 0.50, = 10.5, ^ i = 2.0, fi2 = 0.003, 4.6 Multi-layered spatial environment 285 4.6 Multi-layered spatial environment Environmental grid layers Tuna move by varying their turning angle and move-speed, but they also respond to concentrations and gradients in environmental variables (both abiotic and biotic) termed external movement bias. Changes in these variables were spatially referenced to an oceanographic grid for the G O M region (Figure 4.104). This finite-element (FEM) grid is also used in particle tracking, shallow water hydrodynamics, continental shelf circulation and turbulent boundary layer oceanographic modelling by the Numerical Methods Labo-ratory, Thayer School of Engineering, Dartmouth College, Hanover, N H , USA [223, 224]. In the SIBM model, each of these variables correspond to a data layer of the environ-mental grid. The sea-surface temperature (SST), residual and M2 tidal flow, and bathymetry are referenced to horizontal and vertical nodes in the oceanographic grid, and were ob-tained from output of a finite-element formulation of the non-conservative form of the vertically-integrated advection/diffusion/reaction (ADR) equation (called the A C A D I A model). The M2-tide is one of many principal tide-producing force constituents, being semi-diurnal, principal lunar, with a period of 12.42 solar hours [324]. The surface flow circulation results of these oceanographic models are in general agreement with surface drift observations, and profile changes in water circulation at depth under the combined influences of these variables with deep-basin topography and baroclinicity [224]. Reported tests of the agreement between the oceanographic model predictions and observational data have been primarily focused on regional flow/circulation. These data layers are shown in Figures (4.105-4.112) for annual mean-averaged, bi-monthly periods. 4.6 Multi-layered spatial environment 286 Figure 4.104: Three-dimensional mesh/grid for referencing environmental data layers in the SIBM model from a comprehensive coastal circulation model for the G O M re-gion (Numerical Methods Laboratory, Thayer School of Engineering, Dartmouth Col-lege, Hanover, N H , USA (see [223,224], grid GS2). The grid comprises horizontal fi-nite-element nodes, with vertical reference layers below each horizontal node. Observed environmental association of shoals Oceanographic variables I compared the SST predictions of oceanographic models for the study region by ex-amining the correspondence between the mean and variance values generated by matching observations of movement and shoal sightings onto the temperature grid layer from the oceanographic model. Hydroacoustic tracking and aerial survey observational data was used in this procedure. Figures (D.215- D.218) show the frequency distributions of SST 4.6 Multi-layered spatial environment 287 for shoals observed in each survey month in the G O M region that were obtained. These range of these distributions are in the range (15 — 20)°C, in agreement with the distri-butions for the shoals from AVHHR-satellite observations provided in Figure 4.110. In addition to SST, selected results of monthly frequency distributions of the association of shoals to environmental variables were also obtained. Monthly frequency distributions for each environmental variable are given for the 1994 survey year. Results for the 1995-96 survey years, available for statistical analysis, are given in Appendix D6. Changes in the range of each environmental variable where shoals are observed is expected to vary each month according to seasonal trends. Shoals were observed in re-gions of reduced current flow, with no clear consistent association to a preferred range of chlorophyll-a concentration, zooplankton abundance, or bathymetric depth. The results suggest that the movement of shoals may better correlate to gradients in these vari-ables. However, the location of shoals does appear to be strongly correlated to monthly SST's within each survey year. Shoals appear to prefer the warmest water available, approximately > 15°C, supporting previously reported findings on ambient temperature as an important factor affecting their thermoregulation, hydrodynamic and metabolic efficiency [8,40,44,120,216,229,268]. The results obtained can be compared to the results of a separate analysis conducted by Schick and coauthors (unpublished data) that also examined the association as sta-\" tistical correlation between B F T presence and environmental variables. A set of 390-796 shoals from the aerial survey data was selected. This study examined the spatial cor-relation between shoal presence and the variables sea surface temperature, distance to a sea surface temperature front, frontal density, and depth. For each day that schools were sighted, the authors performed twenty-four Mantel tests, yielding over 2,600 results. The mean SST associated with shoals was (18.1 ± 2.80)°C, in agreement with my model results of a SST range for the presence of shoals as (15 — 20)°C. 4.6 Multi-layered spatial environment 288 The analysis conducted Schick and coauthors (unpublished data) also provides new results on the relationship of shoals to sea surface temperature fronts in the study region. Shoals were located at a mean distance of 19.7km from SST fronts. The mean bathymetric depth of the shoals was estimated as (139 ± 70.2)m. The results did not identify a consistent relationship of shoals to temperature fronts. However, while the presence of shoals was found to exhibit weak spatial correlation to environmental features, Schick and coauthors suggest that prey abundance may correlate more strongly to the presence of shoals. Such information may provide an ability for models to predict where shoal aggregations are likely to form. My results indicate that the movement of shoals may better correlate to environmental gradients. A major gradient not analyzed or used in the model is prey abundance. Under the assumption that the abundance of phytoplankton, zooplankton and prey are correlated, my results indicate that B F T respond to local gradients in zooplankton and phytoplankton abundance. In future studies, the correlation of B F T presence and movement response to environmental variables could also consider the degree of cross-correlation between environmental variables. Prey abundance Observed data on the spatial distribution of B F T prey species within the G O M is in-cluded within a Groundfish Atlas compiled by the East Coast of North America Strategic Assessment Project (ECNASAP) . The distributions are derived from trawl survey data (1970-94) by the Department of Fisheries and Oceans (DFO), Canada, and National Marine Fisheries Service (NMFS), USA. [49]. This survey data requires standardization for possible effects between different survey gears. This data was not available for the current investigations, so mean data of chlorophyll-a (phytoplankton) concentration and zooplankton abundance, were used as a proxy for the abundance of prey species. 4.6 Multi-layered spatial environment 289 Phy top lank ton abundance and chlorophyll-a Phytoplankton are microscopic plants that live in the ocean. Collectively, phyto-plankton grow abundantly in oceans around the world and are the foundation of the marine food chain. Small fish, and some species of whales, eat them as food. Since phytoplankton depend upon certain conditions for growth, they are a good indicator of change in their environment. For these reasons, and because they also exert a global-scale influence on climate, phytoplankton are of primary interest. Phytoplankton require sunlight, water, and nutrients for growth. Because sunlight is most abundant at and near the sea surface, phytoplankton remain at or near the surface. Also like terrestrial plants, phytoplankton contain the pigment chlorophyll, which gives them their greenish color. Chlorophyll is used by plants for photosynthesis, in which sunlight is used as an energy source to fuse water molecules and carbon dioxide into carbohydrates-plant food. When surface waters are cold, water at deeper depths upwells, delivering nutrients toward the ocean surface for phytoplankton. However, when surface waters are warm, regulated by larger climatic changes, they do not allow the colder, deeper currents to upwell and effectively block the flow of life-sustaining nutrients. As phytoplankton are depleted, the local abundance of fish and mammals that depend upon them for food, also decline. Chlorophyll-a is the principal photosynthetic pigment and is common to all phy-toplankton. Chlorophyll-a concentration is used to estimate phytoplankton abundance or biomass. It has several advantages as a measure of phytoplankton biomass, including: (1) the measurement is relatively simple and direct, (2) it integrates cell types and ages, (3) it accounts to some extent for cell viability, and (4) it can be quantitatively coupled to important optical characteristics of water. However, the concentration of chlorophyll-a is an approximate measure of phytoplankton biomass, as the cellular content of this pig-ment depends also on local composition of different phytoplankton species and ambient environmental conditions. 4.6 Multi-layered spatial environment 290 Mean data of chlorophyll-a concentration was assumed to correlate with the abun-dance of B F T prey species, and was complied from available historical observational, survey data (1977-87) for the study region. These observations are based on 5966 samples collected during 106 surveys over the 10-year period formed from sampling conducted by the Marine Resources Monitoring, Assessment and Prediction Program, ( M A R M A P ) [96,251,280]. The M A R M A P sampling stations are ~ (25 - 35) km apart. This data is available from the U.S. G L O B E C (GLOBal ocean ECosystems dynamics) program, for scholarly use by the academic and scientific community according to their data management policy. This data does not provide the best temporal or spatial match to the B F T observational database, but is the best data available. Zooplankton abundance The spatial and seasonal variation in zooplankton abundance and age structure is reported to be strongly coupled to major hydrographic regimes and circulation patterns in the G O M [280]. In the Northeast continental shelf, which includes the G O M , twelve copepod species comprise 85% of the total zooplankton abundance. The species Calanus finmarchicus, Pseudocalanus spp. and Centropages typicus account for 75% of the to-tal abundance and are important prey for cod and herring larvae [362-364]. Coupling of an individual-based population dynamic model of Calanus finmarchicus to a G O M circulation model for the Georges Bank region was performed using the oceanographic grid of the SIBM model for B F T [223,253]. Results for the SIBM model of Calanus finmarchicus predicted that higher abundances form on and near Georges Bank and the north entrance of the Great South Channel in the study region [253]. 4.6 Multi-layered spatial environment 291 Figure 4.105: Bimonthly mean sea-surface temperature SST (°C) for the Gulf of Maine, from G O M oceanographic model [224]. 4.6 Multi-layered spatial environment 292 July 1994 o c o 3 cr u 0.1 0.2 0.3 Flow magnitude (m/s) 0.5 0.0 0.5 1.0 1.5 2.0 2.5 Mean concentration (u.g/1) -50 -100 -150 -200 -250 -300 Depth (m) o c w 3 cr B u. 80 60 40 20 0 (b) SST 10 15 20 25 30 Mean sea-surface temperature ( C) 3 cr - i — ' 1 1 1—' r 0.00 0.01 0.02 0.03 0.04 0.05 0.06 2 2 Mean concentration (N x 10 Im ) Figure 4.106: Frequency distributions of environmental variables (Current flow veloc-ity, SST, chlorophyll-a concentration, zooplankton abundance, and bathymetry for B F T shoals observed in spotter-surveying (July, 1994). 4.6 Multi-layered spatial environment August 1994 0.1 0.2 0.3 0.4 Flow magnitude (m/s) 0.5 0.5 1.0 1.5 2.0 Mean concentration (|xg/l) 2.5 o c 3 cr 35 30 25 20 15 10 5 0 (c) Bathymetry •J n -50 -100 Ik u c o 3 cr o 10 15 20 25 30 Mean sea-surface temperature ( C) 0 200 400 600 800 1000 1200 2 2 Mean concentration (N x 10 An ) 200 -250 -300 Depth (m) Figure 4.107: Same as Figure 4.106 for August, 1994. 4.6 Multi-layered spatial environment September 1994 0.1 0.2 0.3 0.4 Flow magnitude (m/s) 0.0 0.5 1.0 1.5 2.0 Mean concentration (ug/1) 0.5 o c u 3 30 25 20 15 10 (b) SST 1 , , A 1 0 5 10 15 20 25 30 Mean sea-surface temperature ( ° C ) 0 200 400 600 800 1000 1200 2 2 Mean concentration (N x 10 Im ) Depth (m) Figure 4.108: Same as Figure 4.106 for September, 1994. 4.6 Multi-layered spatial environment October 1994 0.1 0.2 0.3 0.4 Flow magnitude (m/s) cr e 4 3 2 l H 0 (c) Chlorophyll-A 0.5 1 — r r 0.0 0.5 1.0 1.5 2.0 2.5 Mean concentration (ug/1) -50 -100 -150 -200 -250 -300 Depth (m) 10 15 20 25 30 Mean sea-surface temperature ( C) 0 200 400 600 800 1000 1200 2 2 Mean concentration (N x 10 /m ) Figure 4.109: Same as Figure 4.106 for October, 1994. 4.6 Multi-layered spatial environment 296 JS (50 o o JS o o >. o H u 3 cr u 0 5 10 15 20 25 30 0 5 10 15 20 25 30 60 40 20 0 Aug = 20.56, a = 0.4054 i i r 0 5 10 15 20 25 30 60 A 40 20 1995 = 19.00, o = 0.2698 40 30 -20 -10 0 Jul 60 40 20 0 i 1 1 1 r 0 5 10 15 20 25 30 = 21.13, o = 0.1956 i i i — i — i — 0 5 10 15 20 25 30 - Aug = 21.13,0 = 0.1956 i L 0 5 10 15 20 25 30 20 15 10 5 A 0 Sep = 14.60, o = 0.9807 1 1 0 5 10 15 20 25 30 60 40 A 20 0 Sep = 18.98, o = 0.0400 — i 1 — * H I r 0 5 10 15 20 25 30 Sea-surface temperature (°C) Figure 4.110: Observed frequency distribution of shoals and AVHRR-derived sea-surface temperature (°C) for survey years 1994-95, in July-September. Distributions for October are not fitted due to insufficient sample sizes. Normal distributions are fitted to the observations for estimating the mean and variance in SST associated with the annual and monthly survey shoal sightings. 4.6 Multi-layered spatial environment 297 Jan - F R e s i d u a l '»-w-»--»- _ ///<•'-M r/>, % Y / / ' / \" - M I \\ \\ \\ \\ dtd*t t H I H H . l l l l f f s/M%t|lt« \\> tfi\\^\\$ * ' *\\*t ****** t\\\\\\\\\\ \\ W v**~o* \\W, yjjjf -\\-.\\UW.\\>mil}\\ lttl>t> l\\ i \"~\"\"inii//t//ftf//irrr 0 . 2 8 0 . 2 4 0 . 2 0 0 . T 6 0 . 1 2 0 . 0 8 0 . 0 4 May -R e s i d u a l s S I K - i . i i • i i i \" - . fi . , , 1 I x . , | > . | I < 1 | < \" \" . i n \\ ) ^ •» ^»* \\ t - * I J * . M \\ t ^ U w ( ^ •t t . */r . W W f 1/ / . — A I / A - \" ( W 7 ...,,14111 , v \\ / ^/^V/ I M H H -\\\\*.NVVv>£w* > • • > < • • <' /) Y'l 11 it 11 > / f \\ \\ / / 1 \\ i i n ( ; t l / I ' it II Itr' » / M ! - • > « / > \\ > i n / f f ' / n i l l n < //, it\\-~\"f//r I M M ' I • 0 . 3 2 • 0 2 8 • 0 . 2 4 • 0 2 0 • 0 1 6 • 0 . 1 2 • 0 . 0 8 • 0 . 0 4 Figure 4.111: Bimonthly, vertically-averaged residual and M2-tide flow velocity vectors and magnitude (see legend) from oceanographic model predictions, referenced as a data layer in the model for January-June. 4.6 Multi-layered spatial environment 298 S e p Residual. i H M / l ) >....< n | /w 411 ' * 11 * V \\ W — ^ - N V w * ^ *11 . - >^u\\W f 11 /1/ / (i tf*~-\\ 'l\\\\ll\"Ul!/t' fcr_Wtf \\ w///11\"-'lUfffM/jntttii ,,W///»^U 111 /////////'•Mil - -///^/i«, . i irrl j jn.^ 'tit l l l l I Ii • ^ : : : : : : : ! M ( m ' \" 0.32 0.24 ' I T I T V •0.32 •0.24 Figure 4.112: Same as Figure 4.111 for July-December. 4.6 Multi-layered spatial environment 299 Figure 4.113: Observed distribution of chlorophyll-a in the upper 75 m of the water column from shipboard surveys, 1977-88. In addition to zooplankton spatial abundance, chlorophyll-a is considered to correlate with mean abundance in major B F T prey species. Distributions are formed from sampling observations with the transects used in the Ma-rine Resources Monitoring, Assessment and Prediction Program, ( M A R M A P ) (See [280]). 4.6 Multi-layered spatial environment 300 4.6 Multi-layered spatial environment 301 Environmental variables as cues for movement The environmental grid was used to simulate how shoals and individuals move and respond to their environment. The integration of environmental information referenced to an oceanographic grid was also used to match observed movements and the spatial dis-tributions of shoals to each of environmental variable/grid layer. The model represents a framework for integration of predictive modelling and observational data. This approach is particularly useful to independently validate geo-position estimates derived from light-intensity variation records of PSAT tags. In such situations, concurrent archived data from sensors measuring such variables as temperature, pressure and water flow can be matched on the environmental grid, and a comparison of position estimates from the light-intensity record with each of these archived data streams can be made at an appro-priate resolution. Observed movement response to environmental gradients Using the model's environmental grid, hydroacoustic tracking observations were matched to each grid layer. The aim of this procedure was to identify preferred ranges of each environmental variable. Gradients in each variables across the observed movements are extracted from the grid and related to the rate of change in their vertical movement. These results reveal how bluefin move in response to environmental gradients, and are associated with the form of Equation D.244 formulated in the model. Gradient response curves from this relation are also provided. 4.6 Multi-layered spatial environment 302 Significance testing Cumulative frequency plots representing the percentage of time in the observed movements in relation to each environmental variables are shown in Appendix D6 (Fig-ures (D.223-D.227). Statistical analysis was performed using a univariate Kolmogorov-Smirnov (KS) test to compare the cumulative frequency variation between the obser-vations [330]. The KS test statistic is defined as the absolute difference between two cumulative distribution functions. In comparing, one data set, 5/Vi to a second set, S^2, the KS statistic is, D = m a a ; _ 0 0 < x < 0 0 | S 'Ar 1 (2 ; ) - SN2(x)\\ (4.131) The KS statistic is invariant under reparameterization of the independent variable, x, where x:(SST, flow, bathymetry, phytoplankton, zooplankton). The function for the significance denoted, QKS, is given by [330], oo QKSW = 2 ^ ( - i r 1 e \" 2 ^ 2 (4.132) 3 = 1 as a monotonic function with limiting values, QKS{0) = 1, QKS(OO) = 0 (4.133) The significance level for an observed value of the test statistic, D, under the null hy-pothesis that the distributions are the same, is given by, p(D > observed) = QKS Therefore, small values of the probability, p, indicate that two compared cumulative dis-tributions are significantly different. In the above equation, Ne is the effective number of data points. For the case of a comparison between two cumulative distribution functions, Ne is, NiNo 'Ne + 0.12 + 0.11/V We ]) (4.134) 4.6 Multi-layered spatial environment 303 where Ni and JV2 are the number of data points in the first and second distributions, respectively. Statistical results of the KS test for the observed movements are provided in Ta-bles (D.73-D.77) in Appendix D6. Table entries show the value of the KS test statistic, denoted D, and in brackets the probability (p value) for having the test statistic value, equal to or greater than the observed value. Corresponding to each series of empiri-cal cumulative distribution functions, linearly-interpolated move-speed is plotted against the gradient for each independent variable. This comparison was made by intersection, and grid-extraction of the observed movements on the environmental grid data layer for each variable at a temporal resolution equal to the move sampling duration (AT). The relationship between vertical movement and each environmental variable was compared to the model's functional relationship formulated for movement response (see Appendix D5). Sea-surface temperature Results of the KS test between the observed individual tuna movements (N=10) and SST were significantly different. However, the range of preference in SST is approximately (10 — 15)°C, and includes all movement observations with percentage of time between (50-100)%. The range of SST varies in time, corresponding to the conditions during the hydroacoustic tracking experiments (1996-97). Large variance between the movement depth and SST gradient was evident, but stronger correlation between these variables was more pronounced as the gradient increased. Water flow Results of the KS test for current indicate significant differences among all empirical cumulative distributions. The range of preference in flow magnitude is approximately 4.6 Multi-layered spatial environment 304 (0.01—0.70)m/s, and includes all movement observations with percentage of time between (0-100)%. The cumulative distributions increase fairly rapidly, with an observed shift for two observations (9604a and 9604b). The relationship between vertical movement, and the flow gradient, shows a weak response to the gradient indicated by a clustering at low range of gradient (0.0-0.l)m/s. The results indicate that flow gradient and vertical movement is non-linearly related. A range of gradient response curves corresponding to Equation D.241 are plotted across the observed scatter-points, intersecting a few observed movement responses across larger flow gradients. Bathymetry Results of the KS test indicate that the observed movements are strongly correlated to changes in bathymetry. The movements occur across a range of (0-250)m depth. The relationship between vertical movement and bathymetry gradient, however, did show variation in terms of the strength of correlation to their movement response. The results indicate that bathymetry gradient and vertical movement is non-linearly related. A range of gradient response curves corresponding to Equation D.241 are plotted across the observed scatter-points. Series of the observed movement responses, as individual points in the scatterplot, are intersected by the response curves. These results suggest that B F T respond to changes in bathymetry with partial support for the movement response function in the model considered to be a function of relative distance to the maximum change of a gradient and B F T response sensitivity. 4.6 Multi-layered spatial environment 305 Phytoplankton and zooplankton The KS test results indicated that the observed movements of B F T responded differ-ently to phytoplankton and zooplankton concentration. The range of movement associ-ation with phytoplankton was more constrained (0.7-1.4)pg/l than for zooplankton con-centration, even though the preferred range did vary considerably between the movement cumulative distributions. The scatterplots of vertical movement response to gradients in these variables indicated a strong sensitivity in their movement response, greater than with the SST, water flow or bathymetry variables. Time-delayed movement response to environmental variables Time-lags in the movement response to the variables of SST, flow, bathymetry, phy-toplankton and zooplankton were also examined. Two selected results corresponding to the observed movements (9601,9703) are provided. In Figures (4.115-4.116), the coeffi-cient of cross-correlation for each variable was plotted across the number of successive moves, or move time-lag/move duration, Ar(s ) . Significant variation at the 5% and 95% confidence levels are indicated for the cross-correlation coefficient with horizontal lines. A summary of the test results comparing the cross-correlation coefficient at zero-time lag for the observed movements for each environmental variable is provided in Table 4.57. For 9601, a significant correlation between movement and SST occurred between a lag range of (-3,-1-3) moves. Response to flow showed a time-lag of four moves. Sig-nificant differences were not detected in the cross-correlation of vertical movement re-sponse with phytoplankton, zooplankton abundance and bathymetry. For individual 9703, movement response showed a quite different relationship to phytoplankton, zoo-plankton and bathymetry. However, a consistent response was found for individual 9601 to SST and water flow. Significant movement response to SST and flow was detected 4.6 Multi-layered spatial environment 306 Table 4.57: Test results of cross-correlation coefficient at zero time-lag (where time-lag interval coincides with mean move-duration) for observed hydroacoustic movements of B F T (N=10) and each environmental variable. Variable 9601 9602 9603 9604a 9604b SST 0.3543 0.9189 0.8001 0.7928 0.6134 Flow Magnitude -0.1783 -0.1087 0.7615 0.8397 0.8709 Phytoplankton -0.1024 0.1609 -0.1464 0.1913 0.0013 Zooplankton 0.0210 -0.2079 0.0739 -0.0973 -0.0872 Bathymetry 0.0005 0.1222 -0.2896 -0.2298 -0.1658 ± 9 5 % C.I. 0.1356 0.0371 0.0468 0.0394 0.057 Variable 9701 9702 9703 9704 9705 SST 0.7988 0.8954 0.7457 0.8476 0.7482 Flow Magnitude 0.3813 0.1057 0.6636 0.7249 -0.1975 Phytoplankton -0.0381 -0.1240 -0.2796 -0.0728 0.0324 Zooplankton 0.1257 -0.2423 0.0806 0.0677 -0.1471 Bathymetry 0.0296 -0.0393 0.2083 0.1348 0.0872 ± 9 5 % C.I. 0.0451 0.0459 0.1043 0.1057 0.0387 within a small number of successive moves, whereas, response to the prey-correlate vari-ables and bathymetry was associated with a longer time-lag or number of moves. Across successive moves, the correlation between their movement and chlorophyll-a, zooplankton and bathymetry was found to switch between negative and positive. The alternation be-tween negative and positive correlation occurred within approximately twenty successive moves. Due to the limited number of movement observations analyzed, more movement data for B F T is required to substantiate the results of this correlation analysis. The results support the use of the derived response function in the SIBM model. A prefer-ence for a restricted range in SST within the warmest water available and changes in their movement at depth was linearly correlated to SST within five successive moves. Response to flow was non-linear, and less sensitive to flow gradient, occurring across five moves. The strongest responses were detected to gradients of the prey-correlate variables and bathymetry. 4.6 Multi-layered spatial environment 307 9601 0.4 0.2 0.0 -0.2 -0.4 r w - f T - r r - r r i IT SST —i 1 1 1— -20 -15 -10 -5 0 n i 1 5 10 15 20 0.4 0.2 A -0.2 -0.4 Vertical Flow 0 0 \"IT^-MJT-I M jZTL LJ \">— -20 -15 -10 -5 0 10 15 20 c u '3 £ o U c o 1 i o U 0.4 0.2 H 0.0 -0.2 A -0.4 Chlorophyll-A Concentration TTJTTTTJTJJT^ — i 1 1 1 20 -15 -10 -5 0 ~i 1 1 5 10 15 20 0.4 Zooplankton Concentration 0.2 0.0 -0.2 -0.4 — i 1 — i — i 1 1 1 — 20 -15 -10 -5 0 5 10 15 20 0.4 0.2 0.0 -0.2 -0.4 Bathymetry jJJJ I I I I L I J J J J - U - M - ^ — i J J j j j j j J J - L i i J j - ^ 1 1 1 1 1 1 1 -20 -15 -10 -5 0 5 10 15 20 Move time-lag, Aj(s) Figure 4.115: Cross-correlation coefficient for individual movement trajectory 9601, across range in time-lag (number of moves) between successive moves. 4.6 Multi-layered spatial environment 9703 c CJ 1.0 0.5 0.0 -0.5 -1.0 0.8 0.4 0.0 •0.4 -0.8 0.4 c o i o U SST T f T T - f l H - T i a - T T - x l T - T i a ^ T - F K -20 -10 ( ) 10 20 i-i Vertical Flow -1 -20 -10 i i — i 3 10 20 CJ 9 0.2 Chlorophyll-A Concentration 0.0 , -0.2 --0.4 --40 -30 -20 -10 10 20 30 40 111111111L Zooplankton Abundance -40 -30 -20 -10 0 10 20 30 40 \"ur mniil it im Bathymetry !Trlrri>flr,n _ m _ r r n n -40 -30 -20 -10 0 10 20 30 40 Move time-lag, A, (s) Figure 4.116: Same as Figure 4.115 for observed movement trajectory, 9703. 4.7 Model validation tests 309 4.7 Model validation tests Validation tests were conducted for the reduced model, M . These tests were per-formed by running the model for shoals (i.e. at the shoal-level). The variables, Zs and Ys for shoaling advantage due to foraging and dilution of predation risk relate to the level of individual fish. Model validation tests examining shoaling advantage due to foraging and dilution of predation risk were not considered. Validation tests were selected to examine the model where empirical estimates were available from the analysis of indi-vidual movement and shoal structure data, presented in Chapters 2 and 3. These tests were conducted by simulating the model over relatively short time-periods, but produced sufficient results to validate key processes in the model. Since the model characterizes spatial distribution on the basis of individual decision-making in shoals, the processes of individual movement, shoal interaction and movement mode-switching were consid-ered key aspects that required a detailed examination. These processes were considered crucial because the coupled simulation of the model's processes over short time-periods enabled a more straight-forward interpretation of the test results. Moreover, the time-scales of the simulation tests covered both a sufficient temporal scale and enabled the most important processes of the model to be examined with a high degree of resolution with respect to the time-scales over which they occur. Environmental grid spatial resolution Testing of the spatial resolution of the environmental grid in three-dimensions was conducted with respect to observed individual tuna movements from hydroacoustic track-ing. Results of this procedure are shown for track 9602 in Figure 4.117. Sub-figure (A) shows the spatial trajectory of 9602 in three-dimensions, with the corresponding refer-enced grid points. The spatial resolution of the grid was sufficient in referencing changes in both the horizontal and vertical movement in the observations. Sub-figures (C)-(F) 4.7 Model validation tests 310 show profiles of each of the data layers in the model's environmental grid for the tra-jectory of individual, 9602. The profile for the flow data layer was uniform across the number of moves observed. For 9602, movement response in the depth dimension was insensitive to changes in bathymetry, zooplankton and phytoplankton abundance, while this individual responded stronger to the SST gradient, as shown in sub-figures (C) and (F). Immigration process The immigration process of the model at the shoal-level was tested (Figure 4.118). The movement of twenty shoals was simulated across a time-period of ten days. The dis-tance between shoals is relatively small compared to the distance in their movement over the simulated time-period. Immigration into the region started from the southwestern edge of the environmental grid and shoals moved in a northerly direction, indicated by the arrow in Figure 4.118. The positions of the shoals after five, eight, nine and ten days are shown. These positions were calculated by matching the final positions of movement trajectories onto the environmental grid after each simulated day. Grid-matching also re-quired geoposition estimation. The SIBM model integrates a geolocation algorithm based on a Mercator projection of the earth's spherical geometry. These results are generated without scheduled immigration whereby a proportion of the total number of shoals would be injected into the region across simulation run-days. These results involve testing of the spatial component of the immigration process. During the immigration process, shoals were set to move without responding to environmental gradients or changes in prey abundance. This ensured that the shoals immigrated a sufficient distance into the region, so that movements could be referenced to the model's environmental grid. This also reduced the possibility of shoals encounter-ing the grid boundary. At set time intervals, boundary conditions were checked as the 4.7 Model validation tests 311 movements were simulated. Both the coastline boundary and the grid boundary were cross-referenced by the model, whereby shoals within a critical distance to the boundary were reflected away. These boundary conditions reflect shoals and individuals symmet-rically away from the boundary perimeter by reflection of their move-directions. While the immigration process results appear to simply show shoals moving linearly, the shoals had also to respond to the boundary conditions that caused them to move around the island coastline. Emigration process The model's emigration process was tested by simulating the movement of two shoals. Results are provided under the assumption of random (Von-Mises distributed) turning angle, and harmonic coupling variation in turning angle with correlation in turning angle and move-speed according to the model equations in Figures 4.120 and 4.121. No move-ment response to the changes in the environmental variables was assumed. The random distributions for turning angle and move-speed assumed in the model are shown in Fig-ure 4.119. In this figure, parameterized gamma-distributed move-speed and Von-Mises circular normal turning angle distribution are shown. The distributions are kept fixed for each movement mode (mi,m 2) with gamma-distributed move-speed having parameters a = (1.50, 2.25) and 8 = (0.5, 0.5) for m 1 , m 2 ) respectively. Turning angle distribution has direction (0°, +10°) and concentration parameter, c = (1.0,1.0) for movement modes, mi and mi respectively. Randomly sampled deviates are obtained from these distributions to simulate movement trajectories (see Equations 2.9 and 2.11 in Chapter 2.). In the case of random movement, the simulated trajectories for two shoals, (labelled a and b) moved in the directions indicated as (la,lb) before the onset of emigration in the model. After the emigration time, tm, the two shoals are re-directed directly south (the fixed direction for emigration out of the region), moving in directions indicated by (2a,2b). 4.7 Model validation tests 312 As the shoals emigrate, they maintain the characteristics in their movement. For this case, turning angle is sampled randomly from a Von-Mises distribution, whereby emigration out of the region is expected to occur over a longer time-period, to the time where all shoals have left the region. The results confirm this expectation. For the non-random case, harmonic variation in turning angle with correlation leads to more rapid emigration out of the region. Before the onset of emigration, the trajectories in two-dimensions for two shoals are shown moving in directions indicated as (la,lb) in Figure 4.121. After the pre-scheduled emigration time, tm, in the model, the movement of the shoals was re-directed southward, and they maintained their movement characteristics. Turning rate process Process testing of variation in turning angle and speed was performed. These tests involved simulating an individual movement trajectory with non-random, harmonic varia-tion in turning angle and move-speed, and random turning angle and move-speed sampled from the Von-Mises and gamma-distributions as provided in Figure 4.119. In addition, movement resulting from pure-random variation in turning angle was examined. In this case, pure-random refers to random sampling from a uniform distribution of turning an-gle within the range [-180,180]°. The mode-switching event m^, occurring at dusk and dawn is seen in the simulated trajectories. In these simulations, movement during the mode-switching event was artificially set to occur in two-dimensions so that these events are seen as exaggerated displacements, whereas, in realistic simulations, movement would take place at depth, and displacement during the events would instead take place in the depth, 2-axis movement dimension, as rapid vertical dives. 4.7 Model validation tests 313 Non-stochastic turning angle and move-speed Figure 4.122 shows the results for variation in turning angle under different assump-tions, with move-speed as non-random/non-stochastic. Sub-figure (A-l) shows a simu-lated trajectory according to the model's harmonic function for variation in turning angle corresponding to the movement modes (mi ,m2). Sampling from the Von-Mises distribu-tion with a restricted range in turning angle in the range [-10,10]° shows that variation in turning angle in the simulated trajectory reduces, as shown in sub-figure (A-2). In sub-figure (A-3), pure-random variation in turning angle leads to movement occurring within a restricted spatial range with intervals where movement is more directed, and variation in turning angle was reduced. With correlation in turning angle across ten successive moves in sub-figures (B-l)-(B-3), corresponding to each case of sub-figures (A-l)-(A-3), variation in turning angle appeared further reduced. For the pure-random case, correlation lead to a more visible separation between periods of movement more directed and restricted in spatial range. Non-stochastic turning angle, stochastic move-speed Under the assumption of random/stochastic variation in move-speed sampled from the gamma distribution, results corresponding to the turning angle assumptions used in the process tests shown in Figure 4.122 are provided in Figure 4.123. With stochastic variation in move-speed, the results show that correlation in turning angle between ten successive moves maintains variation in turning angle. For the pure random case, the variation in move-speed reveals larger move-displacements. During intervals where spatial range is small due to rapid variation in turning angle between successive moves, variation in move-speed lead to larger displacements than the case where move-speed was non-random. 4.7 Model validation tests 314 Turning angle and speed correlation processes Process testing was next performed on the model's correlation functions for turning angle and move-speed. These results are provided in Figures 4.124 and 4.125. The simulated trajectories were compared to those generated with correlation set to occur across the duration of fifty successive moves. Move-speed was fixed as non-random, varying according to the model's harmonic coupling function. A comparison of the results generated under correlation in turning angle and correlation in move-speed identified that turning angle correlation leads to more directed movement than move-speed correlation. This suggests that movement trajectories with correlation in move-speed have larger total displacements than those with correlation in turning angle. Movement mode variation Example harmonic variation in turning angle (C v ) , and move-speed (C„), in the model corresponding to each movement mode, mi and m 2 is shown in Figure 4.126. Turning angle frequency between each of the movement modes varied with no change in the frequency of variation in move-speed between the modes. Mode mi is associated with a higher rate of variation/turning angle frequency. Both the variation in turning angle frequency (i.e., turning rate) and its correlation between successive moves was evident. Correlation in each variable was set to extend across ten successive moves. These results correspond to the harmonic coupling variation of Cv, C9 shown in Figure 4.126. Simulated model trajectories following each movement mode m i , m 2 are shown in Figure 4.128. The first-half of the movement trajectory follows m 2 , with an alteration in turning rate between m 2 and mi taking place, as indicated. Simulation of a similar trajectory where turning angle is pure-random and correlation is still present between ten successive moves, a clear division between the characteristics of each movement mode was evident. These cases show how simulated and observed trajectories of B F T at the non-random and 4.7 Model validation tests 315 random limits can still reveal movement mode characteristics distinguishable by different turning rate and move-speed variation. Figure 4.129 shows the percentage of total number of simulated shoals (N=200) in each movement mode, m i , mi versus time of day (hrs) for elapsed run times within a day, for one day and after two days. (A) shows the initial alteration between the modes. During mode-switching events, such as m%, switching stopped, and all shoals in the model under took deep vertical dives near times of dusk and dawn. As time elapsed, shown in (B) and (C), the shoals adopted one of the two movement modes as a result of their spatial interaction. The results confirm that during times of shoal mixing and exchange of individual fish, they altered their movement mode as a function of the shoal size. After two days, shoals were distributed with a reduced ability to interact and the population separates into collections of shoals in each movement mode. In cases where the interactions of shoals remain high as time elapses, variation in the percentage of shoals before and after the switching events at dusk and dawn occurred. The simulated results leading to mode separation (i.e., shoals in either of the two movement modes) were relatively uniform, due to a relatively small number of interactions. As the number of interactions between shoals increases, it is expected that mode separation may become more non-uniform (i.e., a significant difference in the proportion of shoals in either mode). Movement mode switching events and shoal interaction processes Process testing of mode-switching and shoal interaction are provided in Figure 4.130. Results are shown for N=200 shoals simulated in the model. Sub-figure (A) profiles the shoal mixing term dependent the shoal sizes of two interaction shoals for a single encounter. No environmental movement response was assumed. Sub-figure (B-l) shows an example profile of a shoal with movement alterations occurring as a result of its interaction with other shoals as profiled in (B-2), moving according to either m\\ or m^. 4.7 Model validation tests 316 Shoal interactions that lead to an increase in shoal size for a given shoal are set to correspond to m i , and smaller shoals, to m 2 . As time elapsed time, larger shoals moved in the mi mode, and smaller shoal sizes moved according to the m 2 mode. Sub-figure (C-l) is the resulting mode-switching profile showing the alteration in shoal movement modes by shoal interactions as in (B-2) and the underlying process for switching of modes that is independent of shoal interaction in the model. Sub-figure (C-2) is the shoal interaction profile resulting from the superposition of shoal size independent and dependent mode-switching. Individual attraction and repulsion processes Testing of the attraction and repulsion process of the model was performed. Simu-lation results are shown in Figure 4.131 for a shoal of five individuals. Case (A) (Left Figures) show non-random move-speed and turning angle, with correlated in these vari-ables. Case (B) shows pure random turning angle and move-speed, with no correlation in move-speed and turning angle. Simulation runs were generated by varying the scaling between the attraction and repulsion radii, Rr, i?a(m)=(0.83,1.30),(1.67,3.33),(1.67,6.67), where Rr < Ra. The last two fixed radii settings correspond to the SIBM model's con-straint that Rr < NNDcru < Ra where A / 'A /\"D c r i i=3.0m. The attractive and repulsive forces were parameterized with (ajg(l),ug(2))—100, (A g ( l ) , A9(2))=1.67. Shoal fusion and fission processes Shoal fusion and fission was tested. Shoal size frequency distributions generated for different rates of shoal fusion and fission. A total of N=200 shoals were simulated. The results are shown in Figure 4.132. This test compared the shoal size frequency distributions after a simulation time of two days. During the first day, no interactions were allowed to occur. The initial distribution of shoal size was obtained by sampling a 4.7 Model validation tests 317 shoal size frequency distribution truncated at the maximum allowable shoal size of 200 individuals. Sampling from a distribution with a maximum shoal size of 1500 produced an initial distribution that was more constrained and uniform (refer to Table 4.54 for the initial shoal size frequency distribution parameters). Here, the rate of shoal fusion and fission is defined in terms of shoal interaction, not the time duration during which two shoals interact. For the case of shoal fission, set at 10% and fusion at 0%, the resulting distribution of shoal sizes after two days shows a reduction in the number of shoals of larger sizes, with a peak frequencies in the range (20-30) and (100-120) individuals. For the case of shoal fusion set at 50%, with fission at 0%, the resulting distribution shows a significant number of shoals having larger shoal sizes. The two peaks at smaller shoal sizes result from the formation of larger shoals, causing corresponding reductions in shoal sizes during interactions. For shoal fusion and fission set at 50%, the resulting distribution after nine days is shown. Both a broadening in the initial range occurs from shoal sizes of (0-200), to approximately (0-400), with peak frequencies scaling from shoal sizes of (40-400). Resulting spatial distributions of the shoals for varying fusion and fission rates are also provided in Figures 4.133. The radius of the circles in this Figure scale with shoal size. Spatial aggregation Figure 4.134 demonstrates the use of the model to predict regions where B F T shoals aggregate. A total of 200 shoals were simulated. This simulation was for the reduced model, having fixed parameter values listed in Table 4.54. Shoals moved by responding to all environmental layers in the simulation grid for a time period of ten days. The aggregations shown in the predictions correspond to five, eight and ten days elapsed. The predictions identify locations of shoal aggregation that are similar to those in aerial surveys. Model predictions suggest that the annually observed shoal aggregations may 4.7 Model validation tests 318 be explained as having resulted from the movement of shoals moving responding locally to environmental gradients, with northward migration along the G O M coastline being the dominant underlying trend. 4.7 Model validation tests 319 Figure 4.117: Testing of the environmental grid spatial resolution in three-dimensions. Horizontal and vertical nodes of the grid are referenced and values from each grid layer are profiled, corresponding to the individual movement trajectory, 9602. 4.7 Model validation tests 320 -71 -70 -69 -68 -67 -66 Longitude (°W) Figure 4.118: Immigration of shoals into the G O M region from the south-western edge of the environmental grid. The movement direction is indicated. Simulation of 20 shoals and the corresponding locations at 5,6,9,10 run days are shown. During the immigration process shoals are set to move without responding to environmental gradients or prey con-centration/abundance level, and they immigrate in aggregations. At a pre-scheduled end of the immigration process, the movement direction angle and response to environment gradients leads to partitioning of shoal aggregation. 4.7 Model validation tests 321 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 - l o u u l u u move-speed, v turning-angle, cp Figure 4.119: Gamma-distributed move-speed and Von-Mises circular normal turn-ing angle. The distributions are kept fixed for each movement mode (m i ,m2) with gamma-distributed move-speed having parameters a = (1.50,2.25) and (3 = (0.5,0.5) for rai,ra2 respectively. Turning angle distribution has direction (0°,+10°) and concen-tration parameter, c = (1.0,1.0) for movement modes m i and m 2 respectively. Randomly sampled deviates are obtained from these distributions to simulate movement trajecto-ries. 4.7 Model validation tests 322 Figure 4.120: Top: Emigration process test for two shoals containing ten individual fish moving with random turning angle (sampled from a Von-Mises distribution at each move). Before the onset of emigration shoals move in the directions indicated as (la,2a). After emigration time, tm, the shoals are re-directed south (lb,2b) maintaining their movement characteristics. The accompanying figures provide a higher resolution of the simulated movements for each shoal (lb,2b). The random variation here differs from pure random due to a contraction in the deviation range of turning angle and move-speed associated with their assumed distributions. 4.7 Model validation tests 323 140 i 1 1 1 1 1 1 1 1 90 100 110 120 130 1 40 150 160 170 3 X(10 m) Figure 4.121: Top: Emigrat ion process test for two shoals moving wi th non-random movement parameters. The harmonic form for alteration in move-speed and turning angle, and correlation in each of these movement parameters is maintained. Non-random and correlated variation in their movement, in contrast to random assumptions, leads to highly advective movement trajectories. Before the onset of emigration shoals move in the directions indicated as (la,2a). After emigration time, tm, the shoals are re-directed south (lb,2b) maintaining their movement characteristics. Bottom: The simulated movement trajectories of the two shoals shown at higher spatial resolution. 4.7 Model validation tests 324 c >> JS o X (ui £.01) A (ui 01) A C o I t o o (ui £ 0 I ) A (ui 01) A X X m > o C J CP ' c ? CO - -fl 5 3 a. ^ 3 3 CO CQ bO fl TJ C O Q. 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| 1 1 [ 1 | 1 1 \\ mode m.| mode rri2 1TI3 switch 0 2 4 6 8 10 12 14 16 18 20 22 24 Time of Day (hrs) Figure 4.129: Percentage of total number of simulated shoals (N=200) in each movement mode (mi,7712) versus time of day (hrs) for elapsed run times within a day, for one day and after two days. (A) shows the initial alteration between modes. During mode-switching events, such as 7713, the switching stops and all shoals in the model move according to the characteristics of 7713, set as a deep vertical dive near times of dusk and dawn. As time elapses, in (B) and (C), the shoals adopt one of the two movement modes as a result of their spatial interaction, whereby during times of shoal mixing and exchange of individual fish, shoals alter their movement mode as a function of the shoal size that results. After two days, shoals are sufficiently distributed in the spatial dimension whereby interactions are reduced and the population of shoals is separated into movement modes. In cases, where interactions of shoals remain high as time elapses, variation in the percentage of shoals before and after the switching events at dusk and dawn occurs. The simulated results show uniformity in mode separation due to a sufficiently small number of interactions. 4.7 Model validation tests 332 500 1000 1500 Number of moves 2000 & 2 H c 2 i 2 0 500 1000 1500 2000 Number of moves (lag 20 moves) 0 500 1000 1500 2000 Number of moves (lag 20 moves) 250 500 1000 1500 2000 Number of moves (lag 20 moves) CD £ 13 o 0 500 1000 1500 2000 Number of moves (lag 20 moves) F i gure 4.130; Process testing of mode-switching and shoal interaction. Results are shown for N=200 shoals simulated in the model. (A) shoal mixing over time dependent on the sizes of two interacting shoals. No environmental movement response was assumed. (B-l) shows an example profile of a shoal with movement alterations occurring as a result of its interaction with other shoals as profiled in (B-2), moving according to either mi or m^. Shoal interactions that lead to an increase in shoal size for a given shoal are set to correspond to mi, and smaller shoals, to m,2. This setting, over elapsed time, leads to larger shoals moving in the mi mode, and smaller shoal sizes moving in the m.2 mode. (C-l) is the mode-switching profile that results when alteration in shoal movement modes occurs, not only by shoal interactions (B-2), but with the underlying switching of modes independent of shoal interaction. (C-2) is the shoal interaction profile resulting from this superposition. 4.7 Model validation tests 333 s o non-random, with correlation -15 -10 -5 0 10 IS 20 X(lfrm) pure random, no correlation 15 20 25 X(10 Jm) a f> o s o 0 10 20 30 40 50 60 70 X(10 Jm) 5 10 15 20 25 30 35 XOO^m) IO-3 10-2 10-x(irrm) X(10 Jm) Figure 4.131: Attraction/repulsion process. Simulation results are shown for a shoal of five individuals (each line), with move steps scaled sufficiently large to reveal attraction/repulsion differences along the trajectories. Case (A) (Left Figures) show movements with non-random move-speed and turning angle, with these variables correlated between successive moves, Case (B) (Right Figures) show movements with pure random turning angle and move-speed, with no correlation in speed and turning angle between successive moves. Simulation runs are shown for varying scaling between the attraction and repulsion radii, i^(m),flo(m)=(0.83,1.30),(1.67,3.33),(1.67,6.67), where i? r < Ra. The last two fixed radii settings correspond to Rr < NNDcrit < Ra where NNDcrit=3-0m. The attractive and repulsive forces were parameterized with (w g(l),w g(2))=100 and ( A g ( l ) , A f l(2))=1.67. 4.7 Model validation tests 334 20 15 CJ a CD 3 10 a* e 5 Shoal Fission Rate = 10% 100 80 200 300 400 500 Shoal Size 100 200 300 400 500 Shoal Size Shoal Fusion Rate = 5 0 % t=l day 100 200 300 400 500 Shoal Size 100 200 300 400 500 Shoal Size Shoal Fusion/Fission Rate = (50%, 50%) 60 4 40 4 200 300 Shoal Size 400 500 20 4 200 300 Shoal Size 400 500 Figure 4.132: Shoal size frequency distributions for varying shoal fusion and fission rates (i.e., the number of individual fish mixing per interaction as a percentage of shoal size), N=200 shoals. 4.7 Model validation tests 335 23 22 21 20 19 18 t = 1 day 0 & o o $ T ^ y ^ O o Shoal Fusion Rate = 50% 46 i 1 r — i 1 1 r 44 42 40 38 36 t = 2 days \"T 1 1 r -25 26 27 28 29 30 31 32 X ( k m ) 24 23 22 21 20 19 18 t = 1 day o > A T V Q o 0 o 53 54 55 56 57 58 59 60 61 X ( k m ) Shoal Fusion Rate = 50%, Fission Rate = 50% 192 -\\ 188 J l 184 180 176 t = 9 days o . .oP o O 1° 0 o 5 0 25 26 27 28 29 30 31 32 X ( k m ) 244 248 252 256 X ( k m ) 260 Figure 4.133: Spatial distribution of shoals (N=200) for varying shoal fusion and fission rates. The radius of the circles scales linearly with shoal size, used to depict the relative difference between the sizes of each shoal. 4.7 Model validation tests 336 (A) FIESTA SIBM-model predictions 45 -71 -70 -69 -68 -67 -66 Longitude (°W) 45 4 4 | 4 2 41 (B) 1994 survey (observed) (479 shoals) • 1 • a i 1 ' \" -O 0 ^ ° MM - > • • • I x.. . '* JM . . . > . < K (C) 1995 survey (observed) 4 5 4 4 -71 -70 -69 -68 -67 -66 Longitude (°W) (D) 1996 survey (observed) -71 -70 -69 -68 -67 -66 Longitude (°W) 4 2 41 (527 shoals) do o °(-i 8 ° * ^1* . si -71 -70 -69 -68 -67 -66 Longitude (°W) Figure 4.134: Model predictions of the spatial distribution of shoals (N=200) using fixed parameter values in Table 4.54. Shoals move by responding to all environmental layers in the simulation grid for a time period of ten days. The aggregations shown in the predictions correspond to t=(5,8,10) days elapsed in the model, as shoals move northward, following the coastline. The predictions identify locations of shoal aggregation that approximately correspond to those seen in aerial surveys (1994-96) across years. Model predictions explain the location of aggregations seen in the survey distributions on shoals moving by responding to environmental gradients, with predominately northward advection along the coastline. 4.8 Summary and future work 337 4.8 Summary and future work Summary A spatial, individual based (SIBM) model was formulated for the seasonal assemblage of bluefin tuna in the Gulf of Maine/Northwestern Atlantic. The model was constructed on the basis of new results from detailed analyses of B F T movement and shoal structure. The model considers movement and interaction, shoal size and structure. The model provides a description of how these considerations are coupled to the formation of spatial aggregations of shoals. The model was developed to enable the prediction of emergent patterns in B F T shoal distribution to be characterized on the basis of individual decision-making, seeking to maintain survival and improve evolutionary fitness. Specific rates of shoal fusion and fission in the model can be further examined and may distinguish optimal trade-offs between feeding rate and predation risk as a function of shoal size. As leading changes in the size of shoals occurs primarily when they are in relative proximity to each other, the model may also delineate separate foraging and travel zones according to movement mode adjustments made by shoaling individuals. Continued refinement and improvements to the model will enable spatial simulations to be conducted that can compare the precision of different survey measurement schemes in estimating B F T population abundance. The key results from the research work in this chapter are: 1. Simulation test cases reveal that simulated and observed trajectories at the non-random and random limits both show movement mode characteristics based on turning rate and move-speed variation. 2. Model predictions show that the assumptions of random variation in turning angle and move-speed does not prevent shoals from emigrating from the study region, but that as process variance in turning angle and move-speed increases, the rate of 4.8 Summary and future work 338 emigration and immigration decreases. 3. Under the assumptions of a correlated random walk model unbiased by movement in response to environmental gradients, immigration and emigration is associated with a high degree of correlation in turning angle and move-speed. 4. The directionality of the simulated movements of fish is maintained under harmonic turning angle and move-speed varying in the range of observed movements. Under the assumptions of the model, the alteration of directionality in movement (external bias) is directly associated with fish responding to gradients in their environment. The model predicts that directionality and external bias are coupled, but separable mechanisms. These two mechanisms, when coupled produce large variation in movement predictions, similar to patterns seen in observations. 5. As the degree of randomness in the variation in turning angle increases, shoals become concentrated and residency increases. 6. As the degree of correlation in turning angle increases, under random variation, the number of zones within which their movements become concentrated increases, with more directed movement taking place as they move between spatial aggregation regions (zones). 7. Random variation in move-speed, in addition to random variation in turning angle, tends to increases the extent of regions becomes concentrated. 8. The effects of correlation in turning angle and move-speed are most evident when these parameters are assumed to vary randomly 9. Correlation of turning angle and move-speed, when both parameters vary randomly, can cause fish to either move directly away or move directly towards a given region. 4.8 Summary and future work 339 10. New predictions are generated on the basis of two movement modes: foraging ( m i ) and travel ( m 2 ) . Process testing shows that movements can be simulated by apply-ing differences in two movement parameters: the range and frequency of variation in turning angle and move-speed, and correlation between these parameters. Process testing shows that the movements of fish can still become concentrated. 11. New predictions estimating the degree of shoal interaction and mixing necessary to generate changes in the frequency distribution of shoals and to match observations are generated by the model. Process tests demonstrate the ability of the model to describe and simulate individual and shoal movement, behaviour, and interaction population processes. Together, these processes produce shoal size frequency distri-butions associated with high fusion, and fission. Both fusion and fission must take place to produce distributions that match observational data. When shoal fusion dominates, larger shoals form. However, for a fixed population size, the number of smaller shoals is also predicted to increase. This same effect is evident when shoal fission dominates over fusion. Process tests predict that for the number of shoals at a given size, the degree that shoals exchange fish must be regulated, ei-ther or a combination of the rate that shoals intersect as they move, by individual decision-making, or by shoal size and structure. 12. Model predictions identify locations of shoal aggregation that correspond closely to those seen in aerial survey observations. Across the three survey years, the predic-tions and observations show similar aggregation regions forming. Model predictions explain the location of aggregations detected by surveys as shoals respond to envi-ronmental gradients, with predominately northward advection along the coastline. The model predicts that over a period of two-three days fish move between aggre-gation separated by distances similar to those seen in survey observations. 4.8 Summary and future work 340 Future work Further testing involving sensitivity and cross-correlation between model parameters (either the reduced or full models) is required. The total number of simulated individuals and/or shoals, and the duration of simulation time used in generating the test results was restricted due to computational speed/time restrictions. For the reduced model, simulation tests for N=200 shoals over a month take about four-six days to run. Annual simulations require a considerable amount of time, especially in the case where movement is referenced to the data-layers of the environmental grid, and where attraction and repulsion interactions are set to occur for simulations run at the individual-level. Future research goals using this model are summarized as follows: 1. Further development and testing of the model requires a supercomputer or network of clustered computers enabling parallel processing for efficient computation [213] 2. Simulation runs to generate daily, monthly and annual spatial distributions of shoals predicted by the reduced model from both the individual and shoal-levels 3. Simulation runs to generate daily, monthly and annual distributions of shoals pre-dicted by the full model from both the individual and shoal-levels 4. For the reduced and full models, simulation runs should be performed, under the following scenarios: • no movement response (no referenced grid layers) • individual foraging and predation risk trade-off and nearest-neighbour interactions • shoal/individual movement in response to each data layer • shoal/individual movement in response to all grid layers • shoal/individual movement simulations to examine the effects of climate change on distribution 4.8 Summary and future work 341 5. Fishing spatial dynamics could also be considered in the model. By modelling and simulating the removal of individual bluefin tuna from their shoals, the effects of reductions on the sizes of shoals and their structure and movement could be explored. This would address how the fishing process may impact a shoaling fish population and would test the null hypothesis that fishing does not alter movement and shoaling behaviour. This test may show new effects - an effect whereby shoals compensate for the sequential removal of fish by moving to aggregation regions following environmental gradients is plausible. This effect may occur as a result of reductions in the size and number of fish interacting in shoal formations, as formu-lated in the model, leads to changes in their movement. This may occur because the removal of fish from shoals impacts the shoal's ability to provide individual tuna with foraging and predation risk benefits. Chapter 5 Abundance Estimation: Measurement and Precision A central focus of fisheries stock assessment is to obtain estimates of abundance that reflect true trends and responses of populations in time and space, under varying lev-els of human exploitation and management regimes [163]. Without accurate population (i.e., stock) assessments and their proper use in management, exploited fish populations can collapse, creating severe economic, social and ecological problems [279]. Fisheries management currently recognizes that the consequences of fishing and the underlying population dynamics of fish are both inherently uncertain [201,348]. Stock assessment methods must therefore consider factors that contribute to uncertainty, and how these factors interact alongside the dynamics of fishing. Fish movement and behaviour, spa-tial aggregation, predator-prey trophic interactions, environmental, oceanographic and climate change are all considered important biological and physical factors that produce variability in population abundance and uncertainty in its measurement. To improve our understanding of how these factors behave, interact and lead to changes in abundance, requires modelling and spatial simulation of the underlying population dynamics, with a comparison of the precision of alternative schemes for spatial sampling and measurement. Fishery-independent data from tracking, tagging, and aerial surveying, combined with the use of theoretical model and simulation predictions, is a methodology that seeks to increase our understanding of fish population dynamics and to improve the estimation, prediction and forecasting of abundance. Alternative aerial survey schemes are devised for spatial sampling of the seasonal 342 5.0 Abundance Measurement and Precision: Aerial Surveying of Bluefin Tuna 343 population of B F T in the study region. The results of these simulations provide bootstrap statistical confidence percentiles, and estimates of survey bias and uncertainty. The sampling schemes examined are: (1) random line-transect, (2) systematic line-transect, (3) systematic stratified transect, (4) adaptive stratified, and (5) spotter-pilot search. The precision of five different spatial sampling designs is examined through simulation-based inference. Uncertainty in abundance estimates are obtained with survey measurement parameters: the area of the survey region, number of daily observers, number and width-length dimensions of replicate transects within each survey day, and allocation of survey effort based on spatial stratification of the survey area. A n analysis of aerial survey data obtained by fish spotter observers over a three-year period (1994-96) is also presented. Spotters are pilots in small aircraft employed by commercial fishermen to locate, identify, and estimate the size of shoals. They aid in directing fishing vessels towards regions where shoals are visible near the ocean surface, and where shoals aggregate on a daily basis. Aerial surveys of B F T conducted by fish spotter-pilots provide a large quantity of information on the size and structure, inter-action, spatial aggregation and distribution of shoals. The results of separate analyses of hydroacoustic tracking and pop-up satellite tagging data of B F T movements provide detailed information on the foraging and searching behaviour of individuals, and seasonal variability in the immigration and emigration of their shoals for the G O M region. This investigation demonstrates how tracking and tagging information can be used in the design of efficient and more accurate surveys. Tracking data is used to calibrate sur-face shoal sightings in the spotter aerial surveys according to the expected proportion of time individuals spend at depth. Surface sightings and shoal size estimates are calibrated using different movement speed filters to reduce bias introduced by potential repeated detection within each survey day. This bias is termed multiple-counting bias of shoals 5.0 Abundance Measurement and Precision: Aerial Surveying of Bluefin Tuna 344 and contributes to over-estimation of abundance. A n index of abundance as sightings-per-unit-effort (SPUE) is derived, and its variability is examined at daily, monthly and annual scales of variation. Serial correlation in the daily shoal sightings is also exam-ined to identify the extent that the repeated detection of shoals between survey days contributes to over-estimation of abundance. Calibrated survey abundance estimates are fitted to fishery-dependent, age-specific abundance estimates from V P A stock assess-ments, using estimates of spatial transfer rates and spatial horizontal and vertical survey calibration coefficients. Seasonal abundance, the number of shoals, the distribution of shoal sizes, survey effort coverage, observer transect lengths, observer-shoal encounter rate and population density estimates are calculated. Statistical fitting of shoal size frequency distributions is performed to derive esti-mates of a coefficient representing the degree of aggregation in the annually observed spatial distribution, termed the aggregation coefficient. Geostatistical techniques are applied to identify the range and spatial extent of the aggregation of shoals. Spatial autocorrelation in the observed shoal distributions describes the aggregation of shoals. Corrected estimates of population diffusion and advection, and its variability over time, are obtained by integrating data on individual fish movements across a range of spatial and temporal scales. This information aids in developing spatially-stratified survey de-signs, where survey effort is allocated in proportion to the expected population density (i.e. based on the number of shoals and variance of shoal size). Spatially stratified survey designs can also consider how the expected abundance of fish is associated with changes in environmental variables. The survey measurement schemes are simulated and results of the effect of spatial aggregation and shoal size distribution are presented. Future work could involve use of a spatially-explicit, individual-based model (SIBM model) developed for B F T and the concurrent simulation of survey measurement schemes to further investigate survey abundance measurement biases and uncertainties. Using the 5.1 Survey sampling 345 model, spatial population simulations over time generate the distributions of shoals under the effects of various factors contributing to variability in their abundance. Coupling each of the five survey measurement schemes presented here with the spatial simulation pre-dictions of their underlying SIBM model's population dynamics provides an analytical framework for continuing to examine the effects and interaction between shoal move-ment, mixing, aggregation, association to preferred ranges in environmental variables, and population impacts of climate variability. 5.1 Survey sampling Survey sampling A recent examination of alternative models applicable to marine fisheries manage-ment provides key recommendations for improving the quality, sampling, and integration of different types of data in fisheries stock assessment methods [222,226,279]. The sci-entific review committee compared actual and simulated data with a variety of different characteristics to evaluate the relative performance of current stock assessment method-ologies through testing of their sensitivity and robustness to different assumptions [279]. The ability of available assessment methods to identify trends in population variability over time due to fish growth, maturity, natural mortality, age composition and recruit-ment, fisheries catchability, gear selectivity and mortality was examined. The simulations devised demonstrated that current assessments are appreciably sensitive to the type and quality of data, and the underlying assumptions regarding changes in populations in space and time. A further review of fish stock assessment uncertainty and abundance forecasting, identifies that fisheries science must aid in developing an approach to provide conditioned choices for guiding management decisions, and realistic and adequate testing of scientific assumptions to ensure that the results of past management decisions actually 5.1 Survey sampling 346 translate into desired outcomes [294]. Perhaps the most important and challenging focus of current fisheries science, assessment and management is to develop an effective, syner-gistic means to devise, implement and apply assessment methodologies that integrate a wide range of factors that contribute to uncertainty in population estimation, forecasting and prediction. A n important management action to assist in mitigating the use of inaccurate data, the over-estimation of abundance, and over-fishing of populations, identifies that assess-ment models should express population and measurement related uncertainties explicitly. A n example of this approach is a spatial, environmental population dynamic model (SE-P O D Y M ) that has been tested and applied in skipjack tuna population assessment. This model was developed by the Oceanic Fisheries Programme (OPC) of the Secretariat of the Pacific Community (SPC) for exploring the underlying mechanisms by which environmental variability affects the pelagic ecosystem and tuna populations [23,207]. Spatially-explicit population models (SIBM's) provide a framework for integrating the-oretical assumptions, and a wide range of experimental data collected at various spatial and temporal scales. These frameworks support the integration of direct-observational data from tracking, tagging and surveying experiments. S IBM models aim to develop a detailed understanding of the variability of fish populations in time and space, and to explore how processes operating at various scales are coupled together and interact to produce a variety of emergent outcomes at the population scale. These models aim to view populations as realistically as possible. To attain such realism, ideally, these models are constructed on the basis of the results of detailed analyses of both historical and new experimental data, and rely on a close association to theoretical models and their predictions. These models can be used to identify gaps in knowledge and specific areas where both continued theoretical investigation and/or experimentation is required. Such approaches provide a basis upon which measurement uncertainty and bias may be made 5.1 Survey sampling 347 explicit, so that they can be interpreted, estimated and systematically reduced. Survey design Optimal survey methods may be characterized as those that deliver rapid informa-tion, with acceptable statistical precision, at low cost, while minimizing potential survey biases. A review of spatial statistics in survey design is provided in Appendix E l . De-pending on the particular survey method employed, fish behaviour and movement dynam-ics will introduce alternative levels of bias in survey data. The only way that survey biases can be estimated is by comparing independent estimates of density or population abun-dance from different experimental techniques [137]. Commonly, population estimates ob-tained through catch-at-age analysis are compared to those obtained from a direct survey or multiple surveys. Unlike bias, variance in survey factors can be estimated directly from data collected during a survey. Standard statistics can be then be applied to estimate vari-ances, and bootstrap simulation provides a means to extrapolate the observations to infer theoretical distributions of mean probabilities that would be obtained at larger sample sizes, or alternatively, if an infinite number of surveys were conducted. Statistical meth-ods applicable to spatial data obtained from shipboard, aerial, remote-sensing surveying provide different methods for dealing with bias and uncertainty in population abundance, related to: shoal size estimation (number of individuals), observer-shoal encounter rates, multiple-counting of shoals, observer visibility, object sightability, and estimation of error in spatial interpolation [74,100,102,106,154,173,176,181, 245,246,323,343]. Traditionally, the assessment of fish populations and estimation of population abun-dance have relied on fishery-dependent data such as catch-per-unit-effort (CPUE). Rose and Kulka demonstrate how, in the case of northern cod (Gadus morhud), elevated C P U E data was interpreted incorrectly, leading to over-estimation of stock size, inflated fishery 5.1 Survey sampling 348 quotas, and unsustainable fishing mortality levels [347]. This study shows how the de-cline of northern cod off Newfoundland and Labrador occurred with C P U E correlated to abundance at local spatial scales of mobile fishing activities, but uncorrelated to regional or population/stock abundance. In both clupeid and tuna populations, local densities are likely determined by behavioural factors, and it is acknowledged that C P U E is unlikely to be correlated with abundance. The relationship between fishery-dependent abundance indices (CPUE) and true abundance for highly migratory, shoaling populations is rarely, if ever, known, and there are serious challenges in characterizing temporal and spatial dynamics within CPUE-based assessment frameworks [64,67,114,116,163, 318,334,359]. Surveying with fishing vessels can generate stimuli that affects the presence and velocity of nearby fish by generating low frequency sounds with their peak energy within the fre-quency range of teleosts [114]. Interactions between vessels used to conduct surveys and fish behaviour may introduce extensive bias to abundance or biomass predictions made using survey data [112,114,117,147, 375]. Hara has compared aerial line-transect and acoustic surveying abundance estimates for sardine, concluding that direct enumeration of their shoals, due to improved detection abilities in the aerial surveys, greatly improved the population estimates. The clear disadvantage of ship surveys was their speed limitations, while aircraft could provide coverage of larger areas in a fraction of the time, and could survey abundance of fish shoals with minimal disturbance affects. Fish are often not caught for various reasons from vessels due to use of proper net equipment, inability to efficiently locate small concentrations of shoals, or long travel times required to reach surfacing shoals necessary to conduct efficient sampling. Previous aerial surveying of tuna schools by commercial spotter-pilots has been shown to improve the detection of shoals from 2-20 times that of surveys conducted on fishing/research vessels [377,378]. Gaertner and coauthors, in an logistic regression analysis of the influence of fishers' behaviour on the purse-seine 5.1 Survey sampling 349 catchability of tuna shoals, indicate that as a result of fishers using thresholds to decide the spatial locations for net-setting, they modify their search behaviour on an annual basis. This effect introduces serious biases in the analysis of abundance derived from the resulting catch rate data [117]. A n examination of biomass estimates obtained from conventional acoustic surveys are shown to contain measurement biases related to the spatial distribution of fish shoals, and the presence of considerable vertical and horizontal avoidance of shoals from the survey vessel [375]. Fishery-independent survey data may provide more accurate estimates of population or stock trends because this data contains more, and less biased, information on the present status of a stock than data derived from fishery catches. Fishery-independent surveys are currently identified as offering the best opportunity for controlling sampling conditions over time, and the best choice for achieving a reliable index of population abundance, if they are designed well with respect to sampling location, timing and other considerations of statistically valid survey design. Recent research investigations have been directed towards the development, testing and application of different aerial and acoustic survey spatial sampling strategies in providing fishery-independent estimates of population abundance [20,46,48, 51, 75, 77,122,147, 211, 244, 322,332,379]. Fishery-independent sampling methods aim to integrate a variety of observational data and aid in providing a further understanding of how fishing impacts populations. The use of aerial surveying in directly observing and assessing pelagic fish populations is an experimental method that also provides information on movements and shoaling behaviour that can be used in the formulation and testing of spatial population dynamic models [17,75,147-150,220,281,381]. Other methods of fishery-independent sampling include: tracking and tagging of individuals from passive (mark-recapture), and active (ultrasonic, satellite) methods, the detection of shoals and aggregations using acoustic transducer and streak-tube light-detection and ranging (LIDAR), and synthetic aperture 5.2 Analysis of aerial survey data (1994-96) 350 radar (SAR) technologies from aerial and satellite platforms [63,86,128,166,168,176, 238,247,248,276,277,354,385]. These sampling techniques extend the use of fishery-dependent (e.g. C P U E index) measures of abundance using information on how popula-tions change, independent of fishing dynamics (i.e., where and when fish are caught). 5.2 Analysis of aerial survey data (1994-96) In 1993, the New England Aquarium (NEAQ) initiated an aerial survey program in collaboration with member fish spotters of the United States East Coast Tuna As-sociation (ECTA). Spotter pilots are employed in the B F T fishery within the study region to monitor tuna shoals and aggregations and aid in directing fishing activities to target shoals [35,283,377]. The survey program continued its investigations on move-ment, shoal structure, and associations with other marine life, with hydroacoustic track-ing technology, single-point and archival (PSAT) tagging experiments. These tracking and tagging studies provide detailed data and information of the vertical and horizontal movement, shoaling behaviour, environmental relationships, and post-season migrations of tuna within the study region [73,214-221]. Depending upon the type of aircraft used by an observer (e.g., Cessna, Cub, Citabria), survey pilots view exclusively from one side of the aircraft and engage in rapid horizontal turning, covering a full 360° in viewing angle to sight shoals in their vicinity, or have central seating, permitting the sighting of shoals from both sides of the aircraft. Even though search flight paths vary substantially between observers and across survey days, spotters are reported to maintain a relatively consistent trend according to various rules -one of which is to search so that the sun is behind them for reducing glare. While spotting for tuna shoals that might be visible from the ocean surface, pilots select various flight altitudes to optimize viewing conditions, flying between (213-305)m. Typical sighting distances on either side of the aircraft range from 0.5-1.0 nautical miles or ~(1.0-2.0)km 5.2 Analysis of aerial survey data (1994-96) 351 (pers. comm.). Acoustic surveys by the Southwest Fisheries Center (NMFS) use tran-sect widths of 250 m, and typical values between (0.4-1.0)km are reported [102]. These ranges vary based on weather and sea-state conditions. The aerial survey program for B F T provided the spotters with digital and photographic equipment to enable them to observe and record shoals of primarily adult fish [218-220]. Pilot aircraft geoposition was recorded every 15s using an on-board data-acquisition system installed on laptop computers, except during GPS transmission failures [219]. The aerial survey employed aerial photography using hand-held 35mm cameras (Nikon N8008s) with auto-focus zoom lenses (70-210)mm, circular polarizing/haze filters for glare reduction, and date and time information. Colour slide film (Ektachrome 400 ASA, Kodak), with selected depth pen-etration and contrast characteristics was used. Data collected in surveys repeated each year during (1994-96) consist of aerial images of surface schools detected along spotter transect lines. The data includes the following information: pilot identification, detection times and global position system (GPS) lo-cations, quantitative environmental and surface anomalies, and local surface biological associations. These records contain a small number of replicate search paths, by each observer each day, but in general, replication in these surveys is associated with each sur-vey day. Records of flight altitude and tilt angles are not available for associating with the shoals that were photographed. For these reasons, comparisons between observers could not be conducted to quantify observer sighting bias. If indices of abundance are not corrected for differences between spotter observers, then temporal changes in abundance may be confounded by changes in observers (i.e., observer bias). Changes in the number of observers was determined in a previous analysis of spotter aerial search data to be the main contribution of uncertainty in abundance [211]. The aerial survey data of B F T obtained by fish spotter observers for the three-year period (1994-96) were analyzed. Survey data for the years (1993-95) was previously 5.2 Analysis of aerial survey data (1994-96) 352 analyzed by Lutcavage and coauthors [218-220]. To verify results for the seasonal and annual abundance of B F T in the survey region, the survey data of the 1994-95 survey years was re-examined. New results were obtained for the 1996 survey year. Visual shoal size estimates were used in this analysis. However, a comparison shoal size estimation between visual identification and automated detection of individuals in shoals sighted in the aerial surveys has also been performed, involving the development of a supervised digital image analysis system (SAIA). Selected information on movement characteristics and behaviour, shoal structure and size were integrated in this analysis. Simulation of different survey schemes that include the spotter-search scheme are compared. The results of this analysis provide important recommendations for aerial surveying of B F T that can improve the estimation of regional population abundance. 1994 survey In 1994, surveying of tuna shoals was conducted between July 7th-October 7th. The total number of spotter observers was 13, covering a total of 82,666 nautical miles (nm) of flight trackline search effort. A total of 479 shoals representing a cumulative count of 47,810 individual fish were sighted and photographed by pilot observers on 39 of a 92 possible survey days. The number of observers each day varied between 1-11 and was relatively inconsistent across survey days. The total number of search paths (i.e., total number of observers each survey day across all survey days) was 236. The total area covered by these replicate tracks was approximately 16,000km2. Filtering of the shoal sightings assuming shoal movement speeds of 1.5 knots (kt) (0.77 m/s), and 4.0 kt (2.06 m/s), applied to minimize multiple-counting bias, eliminated (65.7±7.4)% and (47.5±7.2)% of shoals respectively, reducing the total number of shoals from 479, to 231 and 163, respectively. This corresponds to (76.6 ± 5.6)% and (59.3 ± 6.8)% of individuals, reducing the total number of individuals from 47,810 to 28,065 and 5.2 Analysis of aerial survey data (1994-96) 353 20,208 for the 0.8 and 2.1 speed filters, respectively. Tuna spotters and fishers report maximum travel speeds of (l-5)m/s, and more typical values of (l-2)m/s for large fish in the region, that are also supported by published estimates [410]. A detailed analysis of tuna movements observed in hydroacoustic tracking experiments (N=10, 1995-97) provide mean estimates of movement speeds in the range of (1.91 ± 0.440)m/s associated with area-restricted (foraging), and (2.25 ± 0.710)m/s directed search (travel) movements. 1995 survey In 1995, surveying was conducted between July 3-October 13. A total of 12 pilot observers participated in the survey. The number of observers each day varied between 1-7, covering a total search effort of 107,306 km, during a total of 60 survey days, from a possible, 102 survey days. A total of 921 shoals comprising 57,941 individuals were documented, with a total of 188 search paths. The total area covered was approximately 24,000/cm2. Filtering of the shoal sightings assuming shoal movement speeds of 0.8 and 2.1 m/s, eliminated (60.5 ± 6.8)% and (46.6 ± 6.9)% of shoals, respectively, reducing the total number of shoals from 921, to 313 and 225, respectively. This corresponds to (76.1±4.8)% and (67.9 ± 5.1)% of individuals, reducing the total number of individuals from 50,313 to 21,520 and 20,141 for the 0.8 and 2.1 speed filters, respectively. 1996 survey In 1996, surveying was conducted between July 6-October 27. The number of ob-servers each day varied between 1-8, covering a total search effort of 97,837 km, during a total of 59 survey days, from a possible, 113 survey days. A total of 527 shoals comprising 43,696 individuals were documented, with a total of 178 search paths. The total area covered was approximately 21,600fcm2. 5.2 Analysis of aerial survey data (1994-96) 354 Filtering of the shoal sightings assuming shoal movement speeds of 0.8m/s and 2.1m/s, eliminated (71.6 ± 5.6)% and (50.0 ± 7.0)% of shoals, respectively, reducing the total number of shoals from 527, to 306 and 196, respectively. This corresponds to (81.5 ± 4.5)% and (66.1 ± 6.0)% of individuals, reducing the total number of individuals from 43,696 to 32,993 and 25,523 for the 0.8 and 2.1 speed filters, respectively. Figure 5.135 shows the spotter survey effort spatial coverage for each survey year (1994-96). Estimates of survey area were calculated by intersecting all replicate survey effort (i.e., search paths) and imposing a geographic grid containing blocks with a res-olution of ten minutes of latitude and longitude. Each block in the grid had an area of ~ 343fcm2. A substantial change in survey effort between year 1994 and subsequent surveys in years 1995-96 was evident. Survey coverage between years 1995 and 1996 were relatively consistent. In each case, spatial gaps in the allocation of search effort within the survey region appeared within a sub-region delineated approximately between (66.0 - 68.5)° longitude and (41.0 - 43.5)° latitude. Survey abundance index Survey effort, Us, termed sightings-per-unit-effort (SPUE) was calculated on the ba-sis of the spotter-pilot search paths and was proportional to population density, assuming survey area is proportional to survey effort. The proportionality between effort and den-sity is a survey calibration coefficient, denoted qs. This coefficient was decomposed into separate calibration coefficients relating to different survey factors. Population density of fish in an area is the mean number of fish per unit area. Survey effort, population density and mean number of individuals for the full survey area (As) are denoted Us, Ds, and Ns respectively. Survey effort in a survey sub-region, r, having Ur. The density within sub-regions is Dr. Survey effort and population density for the full survey area was related to similar measures within sub-regions according to, 5.2 Analysis of aerial survey data (1994-96) 355 SPUE = US = qsDs = q, (5.136) The relations in Equation 5.136 are often used in standardizing fishing effort and for calculating fishery-dependent indices of abundance on the basis of calibrated fish catchability. In fishery-independent surveying, survey calibration coefficients replace the fish catchability coefficient representing the assumed proportionality between C P U E and mean population abundance. The distinction between fishery-dependent and indepen-dent methods relates to estimation of survey calibration coefficients, survey encounter rate and shoal size, whereby independent approaches can yield more precise estimates of abundance, reducing the reliance on an accurate knowledge of fishing dynamics. S P U E can be expressed in terms of shoal sightings or the total number of individuals sighted from shoal size estimates. Un-calibrated S P U E for shoals and individuals within sub-regions (blocks) of size approximately (5x5)km=25 km2, is shown in Figures (5.136-5.138) for each survey year. Survey calibration Calibration of S P U E involved filtering the sightings based on B F T movement to min-imize multiple or repeated sightings of the same shoal, and correction for the probability of being sighted near the ocean surface. The probability of surface sighting was based on the probability depth distribution of their movement. Depth-correction or calibration takes into account shoals and individuals that are below the depth-range of detection for aerial surveys. The survey calibration coefficient, qs as the proportionality between S P U E and mean abundance is separated into a correction in the horizontal xy plane parallel to the ocean 5.2 Analysis of aerial survey data (1994-96) 356 Figure 5.135: Survey effort spatial coverage for spotter-directed surveys of B F T in the G O M , 1994-96. 5.2 Analysis of aerial survey data (1994-96) 357 Figure 5.136: Survey spatial sightings-per-unit-effort (SPUE) with interpolated grid, 1994. 5.2 Analysis of aerial survey data (1994-96) 358 Figure 5.137: Same as Figure 5.136 for year 1995. 5.2 Analysis of aerial survey data (1994-96) Figure 5.138: Same as Figure 5.136 for year 1996. 5.2 Analysis of aerial survey data (1994-96) 360 surface, and depth z plane as, n - l Qs = (qxyqvQz) = [-rjlv \\ / p(ml, m2)dz (5.137) Spatial interpolation of the SPUE for each survey was performed, profiling the spatial trends of S P U E across the survey area. Daily and monthly variation in the S P U E index was calculated (Figures 5.139 and 5.142), corresponding to the ^=2.1 m/s movement filter and no correction for depth, qz—l. Each Figure shows S P U E in units of (indi-viduals/1.8km) (i.e. not S P U E from spatial interpolation grid, but linear measure of survey effort). Sub-figures (A)-(C) show S P U E in units of number of shoals per 1.8km. The increased range of variability in S P U E for individuals is associated with variation in shoal size. Across survey days for each year, S P U E showed considerable variability, even after filtering for mean movement speed. For all years, S P U E ranged within (0-200) individuals/1.8km, omitting outliers. Daily, serial correlation in the survey index The presence of serial correlation in the daily S P U E index was examined by fitting the cumulative observed distribution of SPUE (individuals/1.8km) to a sigmoidal functional form with first-order autocorrelated error. The following fitting function was used, having five parameters (SPUE(t0),t0,a,b,c). The error term was assumed to follow a first-order autoregressive process, denoted as AR(1), with correlation coefficient p, as, In Equations 5.138 and 5.139, t is the independent variable of time, where t refers to survey day. Figure 5.140 shows the results of fitting the cumulative S P U E series for each a (5.138) et = pet-i + ut, - 1 < p < 1 (5.139) 5.2 Analysis of aerial survey data (1994-96) 361 survey year without error, and sub-figures providing the cumulative distribution of S P U E as shoals/1.8km across survey days. Figure 5.141 shows the resulting fit thast includes the error autoregressive process taking place with varying lags (i.e. serial correlation of daily SPUE) . The sub-figures show plots of the residuals obtained from the fitting procedure. For 1994 and 1995, daily S P U E was found to be correlated at a temporal scale of eight survey days, while for 1996, correlation was detected at the scale of one survey day. Calculation of the Durbin-Watson test statistic (DW) under the null hypothesis of no serial correlation in S P U E were outside the range of critical values (dr,,du) at the 95% confidence level. For all survey years, the null hypothesis was accepted confirming that correlation in S P U E would not significantly bias abundance estimation. Statistical results are summarized in Table 5.58. With no significant serial correlation between daily SPUE, the results provided assurance that distinct shoals varying in size are sighted by observers between each survey day for all survey years. Monthly and annual variation in the survey index Monthly variation in S P U E for the months of July-October showed high variability in the S P U E index at the monthly scale. The large confidence intervals calculated for the month of October in the 1994-95 survey years was due to an insufficient number of survey days. The corresponding statistics of monthly S P U E for each survey year are listed in Table 5.59 for both the 0.8m/s and 2.1m/s speed filters. Depth calibration of survey data The results of an analysis of observed fish movements from hydroacoustic tracking data yielded probability depth distributions associated with alternative movement be-haviours of B F T : foraging ( m i movement mode) and travel behaviour ( m 2 movement mode) (see Chapter 2). The probability depth distributions, p(ml) for mode m i , p(m2) 5.2 Analysis of aerial survey data (1994-96) 362 for mode and p(ml, m2) representing pooling of both movement modes, are shown in Figure 5.143. Inspection of these distributions in relation to typical detection depths as-sociated with light-attenuation (aerial and photographic observation) at 10m, and radar (LIDAR) detection at 40m, shows that shifts in the depth distribution or preference of B F T for each movement mode can considerably vary the probability of sighting shoals in aerial surveys. Depth-correction of survey S P U E obtained by fitting the observed distributions to a Weibull probability function and integrating across a range of depth z = (0, zmax) yielded calibration coefficients to enable the correction of S P U E result-ing from the movement of B F T at depth. This function was selected on the basis of its general shape and provided a good approximation to the observed data. In Fig-ure 5.145, results of correcting the spatial S P U E index for depth are shown in sub-figures (A2,B2,C2), with uncorrected SPUE profiles, (A1, B1, C1). Monthly sightings-per-unit-effort (SPUE)(individuals/1.8km)(depth-corrected) were calculated and the results are shown in Table 5.60. Depth-corrected survey relative abundance across years and es-timates of the depth-calibration coefficient, qz for the depth distribution of separable movement modes in B F T movements were calculated, and estimates are listed in Ta-ble 5.61. In this Table, ' V P A Abundance' refers to estimates of their abundance in the West-Atlantic (WR), and were compared to the depth-corrected survey estimates for the G O M sub-region. Figure 5.146 provides the annual trend of the survey relative abun-dance for the G O M (1994-96) corresponding to different movement niters (0,0.8,2.1)m/s. The annual abundance estimates were compared to the age-specific V P A population es-timates for B F T having ages greater than age, a, denoted as a+ within the Western Atlantic management division. 5.2 Analysis of aerial survey data (1994-96) 363 fl V II 3 ^ 08 co S« CO + 3 fl - — cp - d CU CO fl CO S-i w OH CO .!> co i—( fl a A o (H •s A o fl cp PH QJ fl t-l CP I fl fl CP -f l CP SH CP Jl SH SH fl , f l II 1 I co cp O hO H-< fl o Ta CO 43 cp cfl ... _ . . . J T 1 1 r Jul Aug Sep Oct Month Figure 5.142: Monthly S P U E versus elapsed time (days) 1994-96. 2.1 m/s filter, no depth calibration. 5.2 Analysis of aerial survey data (1994-96) 368 Table 5.59: Monthly sightings-per-unit-effort (SPUE)(individuals/1.8km), for movement July Year no filter 0.8 m/s 2.1 m/s 1994 139.1 ± 119.8 93.6 ± 76.3 60.4 ± 5 1 . 5 1995 59.5 ± 45.5 78.2 ± 3 0 . 1 51.7 ± 16.3 1996 88.1 ± 5 3 . 8 106.6 ± 5 2 . 2 91.8 ± 5 0 . 2 August no filter 0.8 m/s 2.1 m/s 1994 32.8 ± 3 1 . 8 22.0 ± 18.9 16.9 ± 15.1 1995 78.2 ± 3 0 . 1 51.7 ± 16.3 45.8 ± 15.7 1996 106.6 ± 5 2 . 2 91.8 ± 5 0 . 2 73.6 ± 39.3 September no filter 0.8 m/s 2.1 m/s 1994 28.6 ± 2 4 . 1 21.6 ± 19.6 14.5 ± 12.4 1995 43.5 ± 2 9 . 1 30.2 ± 18.4 27.4 ± 16.8 1996 71.9 ± 4 3 . 0 71.9 ± 4 3 . 0 55.6 ± 4 4 . 1 October no filter 0.8 m/s 2.1 m/s 1994 117.5 ±110 .1 98.5 ± 102.3 93.6 ± 110.2 1995 677.7 ± 927.4 341.0 ± 4 6 1 . 9 371.3 ± 503.9 1996 125.2 ± 2 8 . 7 104.3 ± 9.4 84.8 ± 8.6 5.2 Analysis of aerial survey data (1994-96) 369 1 . . . . . . P(m,) ? • 1 • c... P(m2) P(m,,m2) - Z 7^ Photographic (10m) / i \\ /LID AR (40m) .1 • ' \\ l l - / •' I \\ . . . . reference depths / • ' ° • f ' i \"^ S^*^ 0 10 20 30 40 50 60 70 80 Depth (m) Figure 5.143: Probability density distribution of B F T movement depth from hydroa-coustic tracking 1996-97. Distributions corresponding to movement modes m i , m 2 , and all movements from the observations are shown. Detection depths for photographic (light attenuation) as 10m, and radar (LIDAR) surface observation, 40m are indicated. 0.30 0.25 H predicted 5% C.I. 95% C.l. © observed Depth(m) Figure 5.144: Observed depth distribution (hydroacoustic observations) fitted to a Weibull probability density function used to obtain integrated correction factors for sur-face S P U E indices in the spotter-aerial surveys. Figure 5.145: Depth-corrected spatial survey S P U E index (individuals/185km), 1994-96. 5.2 Analysis of aerial survey data (1994-96) Table 5.60: Same as Table 5.59 but with depth correction/calibration. July Year no filter 0.8 m/s 2.1 m/s 1994 303.0 ± 179.1 203.9 ± 120.5 131.6 ± 7 7 . 8 0 1995 129.6 ± 99.00 55.8 ± 30.7 47.8 ± 23.4 1996 191.9 ± 117.1 156.1 ± 8 9 . 9 125.8 ± 6 8 . 7 August no filter 0.8 m/s 2.1 m/s 1994 71.5 ± 6 9 . 3 0 48.0 ± 4 1 . 2 36.9 ± 33.0 1995 170.4 ± 65.60 112.7 ± 3 5 . 5 0 99.7 ± 3 4 . 1 1996 232.1 ± 113.7 200.0 ± 109.3 160.3 ± 8 5 . 6 September no filter 0.8 m/s 2.1 m/s 1994 76.1 ± 5 5 . 1 57.4 ± 4 5 . 2 38.6 ± 28.4 1995 94.8 ± 63.3 65.8 ± 40.2 59.7 ± 36.5 1996 156.6 ± 93.60 133.1 ± 9 4 . 9 0 121.0 ± 9 5 . 9 0 October no filter 0.8 m/s 2.1 m/s 1994 255.9 ± 239.8 214.5 ± 2 2 2 . 7 203.8 ± 240.0 1995 1476 ± 2020 742.7 ± 1006 808.7 ± 1098 1996 272.7 ± 62.60 227.1 ± 20.40 184.7 ± 1 8 . 7 0 5.2 Analysis of aerial survey data (1994-96) c p 8 CP cn O £ 2 43 c p CO (-< cu „, S 2 o CP Q cu CJ -a < > cS 00 CN T - H CN o 00 i-H CO l> T f CM CN r - l I O o o m CN CN CN in o co CD CN O) o m as 00 i— I CN CN CN CO o in C D H H O l 00 CO C D N O CO T f m T f m CN in T f CO OS M N O l t- 00 CO CN CO T f T f m CD 0 ) 0 ) 0 O) O) O) CN fi m t-- OS fi £1 CO CO 00 T - H 00 m T f CO m T f T f m cu fc H ^ — ' cfl CO in CN t~ ~s CN s m 00 T f m m T f m T f CO r—1 T f T f in CM 6 | 41996 1 41712 52858 a> CJ CN fl fi CO T - H OS cS fi CN m -o T - H 00 oo fl 3 T - H CO T - H ter e CO T f t -ter < J cf l -a CO O CN T - H o CN fi 00 O 00 03 s O CO OS fi CM t - CM oq CO T f d Is o CO 00 os 1 i-H CM CO CM fi T - H m CO fi 00 T f 00 CP in T f CD Q s £ o CO CO 00 O •a T - H T f in OS T - H m OS O T - H o T - H fH CP H-3 CN CO I O tv t ~ OS CM 96656 o m T - H 96656 o T - H T - H T - H 96656 in CM T f s 1 9910 1042( 9049 T f m CO Yea OS OS os Yea OS OS OS Yea T - H T - H T - H 5.2 Analysis of aerial survey data (1994-96) 373 depth-corrected, no filter depth corrected, 1.5kt filter depth-corrected, 4.0kt filter VPA 9+ VPA8+ VPA 7+ VPA6+ VPA5+ VPA 4+ Figure 5.146: Top: Survey relative abundance for B F T in the study region (1994-96) for (0,0.8,2.1) m/s movement filters, Bottom: Survey abundance estimates compared to I C C A T - V P A age-specific population estimates for the Western Atlantic management division. 5.2 Analysis of aerial survey data (1994-96) 374 In theory, S P U E is proportional to mean abundance as expressed in Equation 5.136. It was expected that abundance estimates obtained each survey day for a given number of observers may include shoals/individuals sighted in previous survey days. For this reason, summation of abundance estimates across all survey days can cause over-estimation of total abundance. Mean abundance estimates calculated as the product of densities and survey area, as Ns=DaAa, were estimated to be 96,401 (1994), 37,695 (1995), and 30,467 (1996). Population density and total survey area estimates are summarized in Table 5.66 and Table 5.68. Since the total survey area was not surveyed by observers each day, and survey effort was spatially allocated in a non-uniform way in the spotter surveys, the corre-sponding daily, calibrated S P U E may have also under-estimated mean abundance. Total abundance was calculated as the cumulative sum of daily abundance estimates across all survey days, under the assumption that each day, spotter observers detect distinct shoals/individuals, sampling their population without replacement. Statistical support for this assumption was provided by no significant serial correlation in daily S P U E having been detected. The total abundance estimates, Ntotai, calculated under this assumption are 47,810 (1994), 50,315 (1995) and 43,696 (1996). After calibration with respect to both horizontal movement (i.e., filtering), the proportion of the total survey area covered, and correction for movement at depth, the total abundance, E(N0bs), for each survey year was estimated to range between (20,278-47,810) (1994), (20,141-50,315) (1995), and (25,523-43,696) (1996). These calibrated ranges include the mean abundance estimates for the 1995-96 survey years, whereas the mean abundance estimate in 1994 that was calculated to be 96,401 lies outside the calibrated range of total abundance of (20,278-47,810). For 1994, mean abundance may have been over-estimated due to insufficient survey effort cov-erage, which would have also contributed to under-estimation of total abundance based on the assumption of sighting of distinct shoals and individuals in shoals each survey day. 5.2 Analysis of aerial survey data (1994-96) 375 Fitting survey abundance to age-specific VPA predictions The estimated ranges in survey relative abundance were fitted to regional population abundance estimates derived from age-specific V P A abundance estimates for the entire Western Atlantic region. Fitted estimates were obtained under a consideration of spatial mixing between the study region (GOM) and western Atlantic sub-regions arising from the seasonal immigration and emigration of B F T shoals. The time-series of abundance for the three survey years was calculated under these considerations, the annual estimates of total abundance from the surveys was also assumed to differ proportionally to true abundance by a log-normal variable [334] according to the following expression, Q>max / \\ Nt = E = U E NZ* e«, et ~ N(0, a2et) (5.140) \"min \\ O. / As detailed in Chapter 4, the annual abundance for the G O M and W R sub-regions across year, t, is, Nt+l,coM = (1 - VGOM-WR)NT,GOM + TYR-GOMNt,WR (5.141) Nt,WR = (1 - TYR-GOM)Nt,WR + rGOM-WRNT,GOM, Vt ~ W(0,al) (5.142) having the population transfer proportions (refer to Table 4.55): YGOM^WR Y W R - * G O M Y W R ^ G O M Y G O M ~ * W R (5 143) In the equations above, rjt and et are normally-distributed error terms with mean of zero and unit variance. The fitting of survey abundance estimates for the G O M to predicted estimates derived from the Western Atlantic sub-regions has the following parameters and variables, NT+1,GOM = f[Ns,u NGOM, NG°M, a m i n , a m a x , q„ T P , T ^ , eu V t , A], X2S) (5.144) 5.2 Analysis of aerial survey data (1994-96) 376 The sum of squares (RSS) is the net contribution of the following terms: RRS=RRS(S)+RRS(e t)+RRS(?7t), for general year shift r, T-r RSS = A} £ { InN? O M - ln qaN?R }2 (5.145) T-r TGOM t+r (i - it \\ G O M ^ W R )NT,GOM + r, i W K - » G O M A W } t=i T—r T—T for successive annual estimates r = 1, where r is the time-series year-lag, across years, t = (1,...,T). The parameters \\ \\ , \\ 2 S were introduced into the expression for RRS as a pre-specified weighting term to govern how strongly the survey should influence the overall fit of the model. The RRS term was minimized using the Conjugate-Gradient method provided in the Solver analysis package of Microsoft Excel™. Results of fitting the survey abundance estimates, Ns• • -Immigration Emigration 1994 o o o- • - • © Qj^~*rft^~~t 1 1 0 0 100 200 300 (C) . . . <>• • • Immigration Emigration 1995 o o o • 1 1 ^ 9 — 0 0 100 200 300 (D) . . . o - • • Immigration Emigration 1996 ( o o-0 100 200 300 Julian Day (Days at Liberty from Jan 1st.) Figure 5.151: (A) Single-point pop-up satellite tagging returns (number versus time at liberty from Jan 1st.) [26,214,236]. (B)-(D) B F T immigration/emigration rate calibrated annually for years 1994-96 to the predicted population abundance for the G O M region derived from ages 7+ of the VPA-estimated Western Atlantic abundance. 5.2 Analysis of aerial survey data (1994-96) 384 80 70 Gulf of Maine population, spotter survey (observed, 1994-96) Gulf of Maine population, predicted (VPA-derived, ages 7+) western Atlantic population (VPA-derived, ages 7+) Gulf of Maine population predicted (mortality scenario 1) Gulf of Maine population predicted (mortality scenario 2) 100 individuals, B F T shoals formed surface-sheets, with a mean size of Ns = 130.6 ±34.57. Estimates of the precision (CV's) of shoal size were 3.17, 1.66 and 1.80 for survey years (1994-96) respectively. Despite the scaling relationship between the annual number of shoals and mean shoal size in the survey observations, uncertainty in shoal size was considerably large. The seasonal variability in the months of July and October may have been due to the immigration and emigration of shoals, a period when the mixing of shoals is expected to be significantly higher if their movements are constrained in certain directions and within a confined spatial region. 5.2 Analysis of aerial survey data (1994-96) 387 Shoal size (number of individuals) 60 (^B) 40 20 4 0 Jul 0 | i i r T | i i i 200 400 600 800 1000 200 400 600 800 1000 I ' ' 1 ' I ' ' ' ' 1 400 600 800 1000 200 400 600 800 1000 Figure 5.153: Shoal size frequency distribution for, (A) : annual and (B)-(E): monthly observations in survey year 1994. 5.2 Analysis of aerial survey data (1994-96) Shoal size (number of individuals) 200 400 600 800 1000 200 400 600 800 1000 200 400 600 800 1000 200 400 600 800 1000 Figure 5.154: Same as Figure 5.153 for year 1995. 5.2 Analysis of aerial survey data (1994-96) zu -(A) 1996 15 - o 0 observed predicted 10 -oo oo Q 3 OO 5 -i 5 OA O O OO Vm > ooo Voce © 0 -O QD uDdB O O O O O o o oo o o oo oomQctnxnnD)ODp I 1 1 1 1 I 0 200 400 600 800 1000 0 200 400 600 800 1000 Figure 5.155: Same as Figure 5.153 for year 1996. 5.2 Analysis of aerial survey data (1994-96) 390 Figure 5.156: Log-log plots of frequency of shoals fiN versus shoal size, ps, for the 1994-96 surveys, fitted to power-law function with exponential decay after a cut-off shoal size, sc (see Chapter 4) [32-34]. 5.2 Analysis of aerial survey data (1994-96) 391 ^ 52 (A) 1996 •3 48 44 jo < o a 40 1994 1995 I • 1 • 1 I 40 60 80 100 120 140 160 Shoal Size (number of individuals) 250 Jul Aug Sep Survey month Figure 5.157: (A) Comparison of predicted abundance of B F T in the G O M region with shoal size for 1994-96. (B) Shoal size versus survey year, (C) shoal size versus month in each survey year. 5.2 Analysis of aerial survey data (1994-96) 392 Table 5.63: Estimation of aggregation coefficient from fitting of observed shoal size fre-quency distributions to Weibull distribution function (a,b,c,x0,y0), and transformed parameter for power-law/exponential decay function form. Mean and variance of the number of shoals are used to calculate the aggregation coefficient, k (negative binomial spatial distribution of shoals). Parameter standard errors (SE), and associated 95% con-fidence interval ranges (C.I.) on transformed parameters are provided. Fit Parameters Transformed Parameters Parameter Value SE Parameter Value SE 95% C.I. a 31.39 0.4317 a' 41.68 0.4317 0.8461 b 37.77 0.5720 b' -0.1749 0.01720 0.03371 1994 c 1.175 0.01720 sc 37.77 0.5720 1.121 XQ 12.45 0.5471 d 1.175 0.01720 0.03371 Vo 0.06934 0.02170 - - - -a 18.99 0.4476 a' 23.33 0.4476 0.8772 b 23.88 0.6314 b' -0.09001 0.02110 0.04136 1995 c 1.090 0.02110 sc 23.88 0.6314 1.238 XQ 7.421 0.3930 d 1.090 0.02110 0.04136 Vo 0.1630 0.03390 - - - -a 8.946 0.1473 a' 12.12 0.1473 0.2887 b 32.20 0.6857 b' -0.2087 0.02140 0.04194 1996 c 1.209 0.02140 Sc 32.20 0.6857 1.344 XQ 8.519 0.5691 d 1.209 0.02140 0.04194 Vo 0.0706 0.01530 - - - -Year (n) Aggregation Coefficient, k k±a(k) 1994 (380) 1995 (722) 1996 (497) 107.3 56.13 87.31 339.8 92.94 157.14 0.09970 ±0.005117 0.3671 ±0.01371 0.3098 ± 0.01392 5.2 Analysis of aerial survey data (1994-96) 393 Observed aggregation of shoals Figures (5.158- 5.160) show the observed spatial distribution of B F T shoals sighted in the spotter surveys. The distribution of shoals each month of the surveys are also shown. While the unequal allocation of survey effort biased the observed spatial distributions, the large number of shoals sighted within sub-regions provided empirical evidence of aggregation in the spatial structure of their population, and a population spatially con-centrated according to a Clark Type I concentration profile (i.e., few locations with high densities of fish and progressively more with decreasing densities) [163]. Cluster analysis was applied to the observed annual distributions of shoals (Figure 5.161). No significant correlation between the mean aggregation radius (km) and the number of observed shoals within aggregations was found. Geostatistical analysis was applied to the observed spatial distributions using sta-tistical routines in the Surfer surface mapping software (Golden Software Inc., Golden, Colorado). Equations used in the geostatistical analysis are described below, and were adapted from Chiles and Delfiner [181]. The sample variogram as a measure of statistical spatial autocorrelation of a distri-bution was calculated according to, where Nh is the number of pairs of points separated approximately by the distance lag h [181]. The empirical estimator of the variogram is termed the sample variogram, calculated for discrete values of h and approximating the variogram as a continuous function. For N equally spaced points, xa, in one-dimension and at a distance Ax, the sample variogram at lag h = kAx is, xp—xa~h (5.146) N-k (5.147) 5.2 Analysis of aerial survey data (1994-96) 394 This definition can be generalized to spatial data on a two- and three-dimensional grid producing variogram maps. The interpretation of a sample variogram involves character-ization of its range (termed sill), behaviour near the origin (termed nugget effect), and the rate of increase or decrease as the lag distance increases. Variograms can increase to a limiting range value (sill) defined as the distance at which there is no longer any spatial autocorrelation in the data. Variograms can reveal nested or hierarchical spatial structures, each characterized by their own range or sill. The behaviour of the variogram near the origin is linked to the continuity and spatial regularity of a variable of interest. In the current analysis, the variable of interest was shoal aggregation as spatial autocor-relation in the observed distribution of shoals. Variograms may vary with direction and are termed anisotropic, or when no variation occurs are isotropic [181]. To identify the spatial scale at which B F T shoals are observed to aggregate in the survey data (1994-96), standardized variograms were calculated on the survey spatial distributions. These results are shown in Figure 5.162. Standardization involved re-scaling of the sample variogram so that the sample variance or dispersion around the mean was equal to one. Variance was related to the variogram as, a\\za+k - za) = 2 7 ( (a + k) — a) = 2 7 ( A x ) (5.148) 2ry(Ax) is called the variogram, and 7 ( A x ) is called the semi-variogram, although these terms are both commonly referred to as the variogram. The variogram of a spatial process can be related to a covariance function termed the correlogram which is the correlation coefficient between za and za+k- Spatial covariance and the correlogram show how spatial autocorrelation evolves with the distance lag separation. Assumptions regarding the stationarity in the mean and behavioural properties of variance are used in geostatistical analysis, the fitting of variograms and interpretation of spatial distributions that arise due to underlying coupled processes. Petitgas discusses how geostatistics accounts for spatial autocorrelation in spatial distributions of fish, as opposed to random sampling 5.2 Analysis of aerial survey data (1994-96) 395 theory [305]. Petitgas also discusses geostatistical assumptions, how to model spatial correlation, and summarizes the geostatistical equations for computing the precision of abundance and biomass estimates from acoustic survey data. In addition, Warren has addressed the question of the effects of fish movement during research surveys have on the estimate of abundance of northern cod (Gadus morhua) [413]. This study demonstrated the use of geostatistics to separate the autocorrelation in spatial data into spatial and temporal components. The variograms calculated for the spotter aerial survey data on seasonal shoal distribution show spatial autocorrelation (i.e., the location of shoals are correlated as a function of their separation distance) to a range of approximately 150 km. This distance was associated with the spatial range of their aggregation in the survey observations. Periodicity in the variograms indicates the presence of a nested spatial structure within their aggregations with a peak or maximum correlation range of ~ (40-60)km. The aggregation of shoals across each survey year was consistent in terms of the correlation peaks at the spatial ranges of 40-60km and 150km. These results suggest that the distance between shoals is an important feature of their distribution, and may be dependent on their movement response to environmental factors and aggregated abundances of their prey. The occurrence of a second correlation peak in the variograms was interpreted to be the result of shoal movement (i.e., migratory drift) occurring at smaller spatial scales than the range of their aggregations. Results obtained by Warren from the geostatistical analysis of the observed spatial distribution of northern cod (1985-1992), has previously shown that cod populations show strong spatial structure, with positive spatial corre-lation extending to a range of ~ 111+ km [413]. Variograms of their distribution also show within-aggregation correlation variation, and this variation was used to calibrate cod movement across the years of the study. Warren explains the variation in cod spatial 5.2 Analysis of aerial survey data (1994-96) 396 distribution that was coincident with the recent collapse of the northern cod, by con-sidering the following scenario: Suppose initially that prey species are plentiful in the usual feeding grounds. One would expect spatial distribution to remain fairly stable. It would be strongly aggregated around the more favourable feeding centres and gradually decrease from these centres. This may be coupled with a migratory drift. Suppose that over time the prey become scarce, there would be some aggregation about feeding centres that are reduced in size and possible frequency, but more movement in the population occurs as shoals are forced to spend more time and cover greater distances in searching for their prey [413]. This is a heuristic explanation provided by Warren, and a scenario that is consistent with both results the spatial distribution of cod and the results obtained for B F T . However, social communication and learning in addition to prey abundance may lead to increases in the strength of spatial autocorrelation (i.e. aggregation) and constraints on its range. Survey analysis: estimates, uncertainties and precision The analysis of spotter survey data (1994-96) is summarized in the Tables provided on page 405. The precision of survey transects range with coefficients of variation, C V (0.34-0.55) or (34-55)%. The precision of encountering shoals, estimation of population abundance, population density and shoal size were estimated to be small. This may be attributed to the number of observers in the surveys. In this calculation, the maximum attainable precision in the survey measurement variables was determined by the number of transects replicated over days. These results indicate that while the spotter search paths show a degree of regularity across survey years in targeting aggregation centres, the encounter rate of shoals and the number of shoals sighted in the surveying was highly variable. Nonetheless, these results cannot be interpreted to say that spotter-search 5.2 Analysis of aerial survey data (1994-96) 397 sampling is necessarily imprecise in general, only that a higher total number of spot-ter observers participating in the surveying each year, and a more regular or consistent number of observers surveying each day may provide significantly higher survey measure-ment precision. The variogram results showed that spotter-search sampling was able to resolve spatial autocorrelation structure in the distribution of shoals, and to characterize consistent spatial ranges associated with their aggregation, including migratory drift of their aggregations. The effect of the number of survey transects in the various simulated survey measurement schemes were investigated being the leading variable determining the precision of shoal encounter rate and abundance. The abundance estimation included transect length and width, measurement variation in shoal size and encounter rate. Observed population diffusion and advection The spatial range/scale of movement as ~ (40-60)km was further investigated by integrating results of an analysis of their movement from hydroacoustic tracking, single-point and PSAT tagging. Estimates of diffusion (km2/d) and advection were calcu-lated for these available movement observations within the study region. Figure 5.163 provides the calculated diffusion estimates for ultrasonic telemetry/hydroacoustic track-ing (UT)(n=10), short-term light archival tagging (SLA)(n=6), long-term light archival (LLA)(n=3), single-point pop-up tagging (SPl)(n=43). In addition, diffusion estimates are compared between single-point and light-archival tagging movements. This compar-ison involved extracting the start and end locations from the archival tagging data, that would be obtained if the light-archival observations were instead collected from single-point tagging experiments. This additional comparison was performed to calibrate the measurement error in estimating diffusion based on start and end sample points of their movement, with the error when their movements are sampled more continuously as in 5.2 Analysis of aerial survey data (1994-96) 398 archival tagging experiments. The results of this comparison provided a single-point ap-proximation to the light-archival tagging observations (n—3), denoted as L L A . A correc-tion between single-point and archival tagging diffusion estimates was determined based on a calibration scale between observation means, identified as the L L A - L D M correction in Figure 5.163. The correction factor was estimated as 9.6. Correction of the single-point diffusion estimates using this calibration factor are shown in Figure 5.163. The resulting profile for their population diffusion, formed by integrating the observations of their movements (1996-2000), is shown versus elapsed time at a temporal resolution of one day. This profile revealed that diffusion in the population increases linearly within the temporal scale of < 60 days, and varies about a limiting value at higher temporal scales. Diffusion appeared to lead to variability in B F T seasonal distribution within the study region across a temporal range of ~ 190 days, decreasing thereafter. The scaling of the diffusive component of their movements according to the time of year was also exam-ined. The top sub-figure in 5.164 profiles the calculated diffusion estimates versus time as Julian day. The corrected diffusion estimates are shown in the bottom sub-figure, and link the temporal and spatial range of diffusion when they are seasonally resident in the study region. Diffusion of their population scaled from a mean of 165.58 for movement occurring at scales of < 2 days, (573.56-893.72) for 2 -*-» 3.0 : o fc 2.5 : o O 2.0 : s 1.5 : 10 : o —' 0.5 : p 0.0 : 100 200 Julian Day 300 meso-macro scale: (immigration/emigration) (2 ^ IQ I — ' co CO -tf co CO fjj CU CO o cp co II > co •fl I 'I CO fl CO 5 A -is 1 I hJO PH SH <1 CU bO bo * H o cu u CO fl fl° co fl H J >> SH CO fl co 0 5 CO C O X ) co 3 2 p C D PH fl I PH P PH •l-H P PH CO cu . £ CO SH fl CO CO -H la t3 A 2 'co 6 co cu co P H CO W C O 10 C O o CN o Tf CN CO 00 IO CD T - H T H O CO CN CT co Tf co CT T-H in CN 00 CT CT 00 in CN t v CD O l CN CT CD CT CT CO CN T-I CO CN in t~-^ CD O o r>-co T-l o 00 00 CN Tf CD CN O CN t-co in co o CM Tf m CD CT CO t-; l> CM CO T - H Tf m t- o co\" t-H 00 in T-H 10 ^—-' in CM CN °> t - CD CT CT in T-H CN 00 CO CO CT C N m CN - CN o CN 00 CD m CN . c i n m CD T - l O t-CO l-H CD CO O CN co co CO CO N « « in CN Tf CT CN CN^ m in CT CO CO m in Tf 06 m CN CN CN co CO o CN CO CO CO CO CO CO 00 00 o o Tf CN CM CO < < T J II Tf a II < l - H l - H OH cn M | c 1 C cj I, 2 .SP _M c •2 M . ts sp FH a a cS CD ft P cn T o> SP \" h r o a .H J cn co 00 bb 5.2 Analysis of aerial survey data (1994-96) 409 YFT \\\\\\ ! \\ — i \" 1 1 1 1 1 i. A . co 3 cp g co -JH S cp O K S CP Sft I \" \" CO CO 2 Pi rt 3 CP 3e e co a Cj) cd B co CP a XS rt co ^ c3 co O C G O 2 SH CO J 3 cd cp pQ H co C o pj CP SH X o to ~ 7 cp s — ' ^ rt ^ ' o \"& H \"4J rt &H £ -a ~ (T» H—' CO \" CO CD \" H rt H c5 a CP > e e CO co ^ cp <: • 1 E H PQ CO CP SH o cp a, co co +^ 111 \" 1 CO 2^ ^3 ;rt cd O .rt \"43 +J CO s cp h CO o ~ cd .§1 J rt\"3 -3 CO CO cj C3 . r , B >-< CP Q rt L O \"§ .2 -3 & ^ H § o 3 ^ _S SH ^ a E C O cp ^ 1 - 1 PI «•> • . r H CP L O ^ SH ., o rt ISJO \" r t CP fa 3 3 id - .2 8 a ^ fe CP ° rt CO S e e rt o cd rt CO crt S I rt ^3 H rt o OH co CP SH SH O CP r r t + 2 y^ =0 co C3 co CP CP ^ cB CO CP 1 ^ CO CO 3 e £ 3 3 e 3 CO 3 e e 3 « rt rt crt CP co rt ^ ? • • H CD ?? ^FJ SH CP . 2 £ J C ^ E H ^ 3 fa ' PQ E-i PQ CP cd rt Prt H H .cp O > H co , - Z O ,«3 2 fa co o co co CO i—I t— CNI O) CN i—l lO K . CO L O CM C M 00 -H -H -H -H -H O CSJ CO L O CO CO T - H 00 L O 0 1 n s L O o i - H T f L Q . oo i—I L O CO T H CS| r H CO T f O CSI T-H t— CO T-H t—• CS! CN T-H CM T f CSI 00 C O ^ Csi 2 C7) T f T f ,HH O) 1 - 1 t~ -H \"H -H -H -H L O C O r H O ^ CN od o oo oo 3 o co ^ CSI r H CS| T f CSI 00 00 L O T f L O r H Oi CO b-b- CO OS T f 00 Csi b - r H T f CO H EH EH H E-H fa W fa CQ ^ CO CQ > •< co ^ 8 L O H H S cS O cj « s *H TO a> bo ^ EH EH EH EH r , fa W fa CQ ^ CQ CQ i>i < C O 5.3 Survey measurement schemes 411 5.3 Survey measurement schemes Spatial simulation of five survey sampling schemes was performed: (1) random line-transect, (2) systematic line-transect, (3) systematic stratified transect, (4) adaptive stratified, and (5) spotter-pilot search. A depiction of the simulated survey schemes is shown in Figure 5.166. The results of these simulations provide bootstrap statistical confidence percentiles, and estimates of survey bias and uncertainty. The coefficient of variation (CV), the ratio of variance to mean, as the precision of five different spatial sampling designs, was examined through simulation-based inference. The precision of encountering and sighting shoals, and estimation of abundance by considering the con-tribution of shoal size variability was calculated for each survey scheme. Uncertainty in survey precision was obtained under variation of the spatial aggregation of shoals dis-tributed as negative binomial, and the within-aggregation structure of shoals following a bivariate normal distribution with relative degrees of anisotropy. Anisotropy was ex-pressed in the relative scaling of the spatial range of their aggregations in the x— and y—coordinate directions, instead of being angular dependent about aggregation centres. The survey measurement parameters: survey area A(£ls), width (a;), and length (Ls) dimensions of survey transects, total number of clusters/aggregation centres (as), total number of shoals (/x/v) were fixed in the simulations. Population size was also fixed and shoal sizes were determined from sampling of a fixed shoal size frequency distribu-tion. Table 5.71 provides a summary of the definition of survey model parameters and variables. A simulated survey consisted of a series of transects across the survey area with sam-pling spatially referenced to a spatial grid of the survey area. The survey grid comprises 900 equal-sized sub-regions of (18.52x18.52)km—342.99 km2, with a total survey area of 208,195 km2. The size of the spatial grid blocks characterized the distance interval within which initial starting locations and placement of transects was determined. In 5.3 Survey measurement schemes 412 the random survey scheme, transect placement was determined randomly by sampling within the range of each transect interval/spatial grid block. A n example distribution of transect sampling points from simulation of random transect sampling (scheme A) is shown superimposed on the survey grid (Figure 5.168). In systematic surveying, the interval between transects was fixed across all transects and was initially determined by randomly sampling with a transect interval. Replicate survey simulations for a constant number of observers were performed representing sampling by observers each day of a survey. The simulations were used to compare the precision in shoal encounter and abundance estimation across a range of total number of observers/transects, with associated confidence intervals determined by replicate survey sampling with a fixed number of observers under aggregation and shoal size variations in the spatial distribution of shoals. Figure 5.167 provides an example of a set of selected simulation realizations of B F T shoal distribution shown superimposed on the survey grid. Simulation results are also obtained under variation in the number of sampling strata for the stratified and adaptive stratified survey schemes. The spotter search sampling (scheme E) was simulated using actual flight paths. Figure 5.169 shows an example search path for a single track of a spotter-pilot observer. Spotter-pilot search paths for aerial survey of B F T in the G O M region for the month of July in the 1994 survey are shown in Figure 5.170. 5.3 Survey measurement schemes 413 Q cs cS CJ D, CD Si -a X cj CS CD 3 CD CO a >£, co _jJ FH 13 CD > >> fH CD SK > 13 > CD cj bo fl CO CD G o fl o cS cS O - G - P O X O O O !H FH CD CD X X a s 3 3 C fl \"S3 \"3 o o CD 1 fl cc3 CD W fl o \"El CD FH >> CD & 3 CJ +J 13 6 3 co 13 o X \"S c cS cr3 G CD O g .a „ C Q —I \"fl -2 S o I O Xi 2 4-J co fl b O .SH . 3 cj cj - 3 CD CD PH FH SH CO > 1 -a fl fl CD •I a a f 3 3 - H CO Cj Cj 3 >H >H CT1 5 \"> ^ O O CO CO c a CD CDH J X X CD CD CD -I 3 « 13 3 H J cS CS O PH PH 4^ CO CO s G cS 3 G 3 T 3 G CD CD 1 cu - P - U >> 'a +2 3 bO HO cS ' H P o E3> P x 1 o 3 fH -TH • C G bb .2 PH * H G Ji 2 -° cs a> 53 +J \" O -S P C fl +J . 3 . 3 . 3 +^ T H ^4 03 cj 3 -3 CD S aj J3 PH b O M -g 13 CD 1=1 CJ ts „ ft A co o fl\" § 3 P . O CD c? ft fe FH C6 a> ft fe ° SH C f l PH CJ CD N > 'co • H cj 13 £ J CD C O CD CJ G cS 'C > a cS CD S G .2 ter do CD p. a - — eg C O par; 13 p Xi G C O O >| CD CD ft bO bO P fl fl £ cS cS o SH SH CO a S1 3 _a 13 '3 '-4J s *, a a l CO -p '3 p I g E E - s a a a * H J ^ o •« A X H ^ c < ° ^ cl 3 . a. co H Xi cj CO >> S H IW to W cn 6 v C O CQ \"3 G . P P G cS a CD C O > > 00 -d CD Cfl CO CD Cfl CO I H J P . cS G P 3 X >> CJ S CD 3 'co H \"O cS (-T P -3 -fl ^ CO o G CD 3 CT\" CD CD _N 'co b _ a. t2 CD O G cS I CJ 13 13 '•S S CO FH PH P CO G 5.3 Survey measurement schemes 414 Scheme A: Random Transect i i i Scheme C: Systematic Stratified ] w 1 — • • S • -* rata l [YlWd 1 — > ! s rata 4 i L » rata -> ^ — Scheme E: Spotter-Pilot Search Str; ita 1 StlT ta 2 St a ita 4 ^ t r ; ta 3 V Scheme B: Systematic Transect • i i l i • Scheme D: Adaptive Stratified [\"4 i S rata Strut r \" i S ra ..J a J ^ 4 -rata Figure 5.166: Aer ia l survey schemes for spatial sampling of B F T abundance: (A) ran-dom line-transect, (B) systematic line-transect, (C) systematic stratified transect, (D) adaptive stratified, and (E) spotter-pilot generalized search. 5.3 Survey measurement schemes 415 200 300 400 500 Xflcm) 200 300 400 X (km) 500 4O0 300 200 100 -1 0 © % :«Q ® O: : : : : : : : S :: : : : :©:: : p: :: :: : : :£i : 500 400 : » : t : - g | ; : : : : . o-: © : 0 100 200 300 400 X(kin) 100 200 300 400 500 X(k in ) Figure 5.167: Example simulation realizations of B F T shoal distribution on superimposed survey grid across varying levels of aggregation. SOU - * ' • • « • • • * • > * • • mm • •* • »m *B» * • * m • ••• • • 3 0 0 - H - K - : :•:•! M < >:•:•>!-: : • ! - : • : : : : : : : : : : i t M C J M i : ! :< t . : i i s - : c - : : : >! > : . : : 0 I , , r -0 100 200 300 400 500 X ( k m ) Figure 5.168: Example distribution of transect sampling points for random transect scheme A superimposed on survey grid. 5.3 Survey measurement schemes 416 Figure 5.169: Example search path for a single track of a spotter pilot observer. Aerial Surveys for Bluefin Tuna (1994 -1996) Figure 5.170: Spotter-pilot search paths for aerial survey of B F T in the G O M region, July, 1994 (provided by R. Schick, Pelagic GIS program, N E A Q ) . 5.3 Survey measurement schemes 417 Let the survey area covered within a distance a; of a transect line Ls be a, and Pa be the probability of detection unconditional on the position of an observer or shoal. The expected number of shoals detected within a distance to, as yujv = E(n) is then, E[n) = u.N = D-a-Pa (5.150) If us = E(s) is the expected number of shoals detected, then the density of shoals is [51], £>=E(n)-E(s)= nN.j*s c-a- Pa- g0 c-a- Pa • g0 gD is the probability of detection on the transect line. The variance of density was calculated by propagating errors in each variable in Equation 5.151 in quadrature [388] yielding, tHb) = bK\\m^ + °^^\\°2M\\ (5,52) [ \"2 s2 (a-Pa) 9o j This error propagation assumed errors in / j ^ , u.s, Pa, 9o were un-correlated or indepen-dent. The area, a, was expressed in relation to transect width, u>, on either side of the transect line of length Ls, as a = 2uLs, such that the estimated density, D, is, b = J^- (5.153) 1uLPa9o with associated variance re-expressed as, + ^ + ^ + ^ + ^ + ( 5 . 1 5 4 ) { n2 s2 to2 L2 (pfl)2 9 o j Variances in the half-width of transects are set to a2(co)—0 for all survey simulations (i.e., fixed transect width), with variance in the probability of detection unconditional on actual position of a shoal, and ga as the probability of detection of a shoal on the transect line as a2(Pa), cr2(g0) also considered fixed. Shoals located within the transect width and survey grid interval along the transect line were considered to be detected (i.e., sighted) with a fixed detection probability that was independent of shoal size. This assumption is analogous to the assumption of no size bias in shoal detection. 5.3 Survey measurement schemes 418 Equation 5.151 is used in estimating density. For i = (l,...,k) transect lines, the pooled estimate of mean density is, D=J^AJ/£ S (5.155) with variance, a\\D) = —1— y A L i , \\ (5.156) (* - 1) (A - D) 100(1 — 2oc)% confidence intervals (C.I.) are approximated as D ± zayj,a2(D). Shoal encounter rate is the number of shoals detected across the length of a transect, and as with density, is pooled across transect lines. Mean encounter rate is H N / L , with variance, ,2 ^ = pk»>y + .a* AL)y (5157) The precision of the survey schemes were compared by calculating the coefficient of variation (CV) as the ratio of variance to mean, for shoal encounter and abundance across a range of total number of transects. LT denotes the total length of all transects in a simulated survey each of fixed length L S . The equations that were used to calculate C V precision in the number of shoals N are provided below. Mean (u.s) and variance (os) in shoal size were obtained from the shoal size frequency distribution and were used to simulate the negative binomial (i.e., aggregated) spatial distribution of shoals, with variance, as=ris + /x s 2//c, where k is the aggregation coefficient. In the simulations, k was set to range between (0-2.5). The measurement contributions of shoal encounter and shoal size, in Equation 5.155, provide estimates of density. Using the area covered by survey transects and density, estimates of abundance were calculated. Schemes A,B: Random/systematic transect surveys Random transect sampling involved the initial random selection of the position of a transect line, with random sampling also being conducted along each transect line. 5.3 Survey measurement schemes 419 Systematic transect sampling involved initially selecting the transect interval randomly, but with all transects being spaced equally according to the transect interval. Sampling along the transect line in systematic sampling was conducted by initially determining the sampling rate (interval) along a transect line, with all sampling spaced equally at this interval for all transect lines. The precision in shoal encounter for these survey schemes were determined according to, CV(N) = — PN for sampling without replacement, and, 1 o2(N) (Lr-L PN for sampling with replacement. Scheme C: Systematic stratified transect survey (5.158) (5.159) CV(N) = J T h=l 1 whW(N) f LT — L S r>>N,h, r V T \" ( 5 - 1 6 0 ) for h = ( 1 , h s ) sampling strata. ion is the strata proportion/shoal encounters (Nn/N), also termed the sampling fraction for stratum h [185]. Strata should be chosen so that the sampling fraction across strata is homogeneous (i.e., minimal variance), providing greater precision since the total variance across all strata hs, depends only on the size of variances within each stratum, h. Scheme D: Adaptive stratified survey Adaptive stratified sampling involved initial probability selection of grid units (termed primary units), whereby additional grid units were then sampled in the neighbourhood of any primary selected unit, for which the observed value of a variable of interest satisfies a 5.3 Survey measurement schemes 420 specified condition [390] (secondary units). The purpose of adaptive sampling is to take advantage of population characteristics (e.g., such as aggregation in their distribution) to obtain a more precise estimate of population parameters such as shoal encounter, den-sity, and abundance, with a given amount of survey effort. Often the precise location and shape of fish aggregations cannot be predicted before a survey is conducted, so that traditional stratified sampling (as per the systematic stratified scheme, C) are not suffi-cient. In the simulated adaptive sampling scheme, Hk denotes the probability that one or more primary units (i.e., first shoal sightings) are included within a transect sample. This probability, termed an intersection probability, is given by [390], •^ -(\"rM\")-1-^ !-) (5iei) where xk is the number of primary units, N is the total number of grid units, and n is the total number of grid units sampled. Based on primary sightings of shoals by observers, secondary detections are added to the survey sample, based on secondary inclusion probabilities, nkj. The collection of all primary units and all secondary units are termed primary and secondary networks respectively, with index k. The total number of networks is K, and only two networks are considered here so that K—2. The secondary intersection probabilities are determined as, TTfcj = 1-And, N - xk \\ I N - Xj n I \\ n , N — Xk — Xj + Xki \\ I N , *kj = l - Q k - q i - \\ 3 / ] (5-163) n j \\ n where Xkj denotes the number of primary units that intersect the primary and secondary networks k and j respectively. The variables, q k and qj are defined as (1 — 7rfc) and (1 — TXj) respectively. A n indicator variable, Zk is defined to be 1 if one or more primary units that 5.3 Survey measurement schemes 421 intersect network k are included in the initial sample, and zero otherwise. As derived by Thompson [390], a mean estimator £3 for a variable of interest yk is, * = j ^ £ * r ( 5 - 1 6 4 ) k=i The £3 estimator represents a summation over all distinct networks (K=2) in the sample that intersect one or more primary units of an initial sample. In this way, adaptive sampling proceeds by a first-stage of sampling, whereby primary sightings of shoals are made, with a secondary-stage comprising secondary shoal detections for shoals sighted in the neighbourhood of the first shoal detection. This adaptive scheme is two-stage for K = 2 total networks, and multi-staged for K > 2, whereby primary and secondary shoal sightings are determined sequentially. In this way, the sampling stages are combined which distinguishes it from standard multi-stage surveying, where each stage is completed before another stage begins. Statistics of the defined indicator variable are related to the intersection probabilities. The mean of the indicator variable is E(zk) = 7T/t and E(zkZj) = nkj, with variance, a2(zk) = 7^(1 — vr^), and covariance, cov(zk, Zj) = itkj — nicKj, i 3-Here, itkj is termed a joint intersection probability (for K=2, joint probability between the primary and secondary grid networks). With the convention that 7rfcfc = -Kk, the variance in the £3 estimator is, fc=l J = l A n unbiased estimator of the variance involves the indicator variable, Zk, The equations above can be applied within separate survey strata. Thompson shows how the £3 estimators for mean and variance can be referred to separate strata according to a stratified design for an adaptive survey. The total number of units, intersection prob-abilities and variable of interest (e.g., shoal encounter, population density, abundance) 5.3 Survey measurement schemes 422 are all re-expressed in terms of a stratum h, involving this additional subscript (see ref-erence [391] for the corresponding equations). The stratified adaptive survey sampling scheme was simulated, whereby primary and secondary shoal sightings within a given number of strata were calculated based on intersection probabilities across the survey grid under sampling with replacement. Scheme E : Spotter-pilot search Due to the irregular behaviour and observed variability in spotter-pilot searching and spatial sampling, the spotter-search scheme was not formulated as an objective model, but instead involved a simulation using available data on their flight paths from aerial survey experiments. Use of this approach was deemed best to compare with other survey schemes, due to anticipated problems in devising and applying a model that could reliably reproduce and predict the actual spotter pilot search paths. The spotter-search survey scheme was simulated using actual search-paths of spotters for the selected 1994 spotter aerial survey. Sampling was conducted without replacement as per Equation 5.159. The total number of possible observers was fixed in the 1994 survey, with six daily observers. The precision of this survey scheme was simulated under the addition of the spotter observer search paths across survey days. 5.8 Results, summary and future work 423 5 . 4 Results, summary and future work Results The precision of each survey scheme was obtained by sampling a simulated aggre-gated shoal distribution with aggregation coefficient, k, ranging between (0,2.5). The spatial structure of the aggregations were simulated to be 66% anisotropic (a) as the rel-ative extent of aggregations between the survey grid's x— and y— coordinate directions. Simulation results were obtained with a total number of ten possible aggregations. A to-tal of 100 shoals were simulated, with equal transect lengths for all observers each survey day. The width of survey transects was fixed at 0.20 nm=0.37 km on either side of the transect lines, and the simulated shoal size distribution was specified with the parameters (a, b, sc,d) = (100, —1.5,50,1.0). Results of these simulations are shown in Figure 5.171 for the number of shoals encountered within the survey area. The results were generated for a variable number of observers (i.e., transects). The precision of abundance estimates for each survey scheme are shown in Figure 5.172. For all survey schemes, the range of precision in abundance was greater than for shoal encounters as abundance estimation includes uncertainty in shoal size, but no measurement bias due to variation in the ability of observers to sight shoals of different sizes from the transect line. Random transect surveying with and without replacement showed a similar trend of increasing precision as the number of transects increases. The no replacement sampling scheme lead to higher precision (i.e. lower coefficient of variation) after approximately fifty transects are conducted. With replacement, the random scheme became uniform with no further increases in precision up to the total number of transects simulated. The range of precision for shoal encounter and abundance estimation is similar for random sampling. The simulated results show that these schemes should not be used to obtain abundance estimates. The confidence intervals of C V also remain large suggesting that 5.8 Results, summary and future work 424 the effect of aggregation in the distribution of shoals considerably reduced the ability to estimate abundance accurately. In contrast, the systematic survey scheme was shown to be more precise than the random sampling schemes, with rapid increases in precision and a reduced confidence interval range under the simulated effect of shoal aggregation. This result is supported by the predictions of spatial statistics, and the results of a previous study by Fiedler. Fiedler simulated survey transects for model anchovy populations and compared several different survey designs in terms of their precision and efficiency [102]. For approximately ten transects, simulation of this scheme predicts a C V of approximately 0.30 (30%). After twenty transects were completed, the scheme provided a precision of 0.15 (15%), and for thirty transects, CV=0.05 (5%). For surveys performed with a fixed total number of observers in the range (20-30), this scheme is shown to provide sufficient precision in encountering shoals. In estimating abundance, >20 observers are required to achieve 5% precision. For the stratified systematic surveys, four and nine equal-sized strata divided the survey area. The size of the strata depended on the number of strata and total sur-vey area. The boundaries of the strata should be chosen by minimizing the variance of abundance between strata of unequal size. However, equal sized strata were instead used in the calculations as strata boundaries and size allocation were considered independent of simulated spatial distribution. The calculations assuming equal-sized strata therefore provide results on the precision of abundance estimates and shoal encounter rate for an increasing number of strata. Stratified sampling lead to more precise abundance esti-mation, with increasing precision as the number of equal-sized strata increased. The results indicate that the minimum number of observers or minimum number of transects that should be conducted scales with the number of sampling strata. As the number of 5.8 Results, summary and future work 425 strata increased, more rapid decreases in the confidence intervals of precision in abun-dance occurred. This indicates that increasing the number of strata was able to reduce the uncertainty in abundance estimation due to the effect of shoal aggregation. For four strata, and twenty transects, the coefficient of variation is approximately 30%. For a hundred transects, the stratified scheme with four strata increased precision in estimat-ing abundance and attained a value of 5%. For (20-30) observers, a stratified sampling scheme that considers only equal-sized strata does not provide acceptable precision in estimating abundance. This scheme therefore requires devising unequal sizes of strata in proportion to expected abundance, which may determined based on how the presence of shoals is correlated to environmental variables. Nonetheless, the results for equal-sized strata show that a stratified sampling scheme requires a sufficiently large number of observers/transects to provide accurate abundance estimates. This is an important con-sideration, as when the number of transects and observers increases, so does the expected cost of implementing and performing a survey. The greatest increases in precision occur when a high degree of correlation between adjacent transects is present. The simulation results were obtained with the fixed tran-sect width of 0.37 km. From a review of spatial statistics and reported studies (Appendix E), this correlation is expected to decrease if the interval between transects were to in-crease, such that systematic sampling would approach a stratified sampling scheme for appreciably large transect intervals. Likewise, for appreciably small transect intervals, a systematic sampling scheme is expected to approximate random sampling schemes. This means that the precision of systematic surveys relies on an accurate determination of transect interval with precision varying inversely with the distance between transects. The assumed fixed value for transect width lead to distinctive differences in the trend and range of precision for the random, systematic and stratified survey schemes. This in-dicates that the transect width value was appropriate to distinguish differences between 5.8 Results, summary and future work 426 systematic and other survey schemes. Simulations with this value of transect width provided a relatively constant correlation between transects, to ensure that the system-atic scheme was distinguishable and appreciably diverged from approximating either the random or stratified schemes. Results for the adaptive unstratified and stratified sampling schemes for spatial strat-ification with four and nine strata are shown in Figure 5.173. The simulation of this scheme maintained parallel transects within and across strata. The principal difference between this adaptive scheme and the stratified sampling is that it estimates abundance under a two-stage sampling process, whereby shoals that are first sighted direct a sec-ond stage of sampling within its local spatial neighbourhood. Abundance estimates are therefore formed based on primary and secondary intersections or sightings of shoals from the two-stage sampling procedure. This scheme was expected to show a similar trend of increasing precision as the stratified survey scheme, with a reduced number of observers/transects required to achieve equal precision in abundance estimation. The re-sults, however, show that this adaptive scheme had a far higher range of uncertainty than stratified surveying. The increase in precision was more rapid up to thirty transects than the stratified scheme independent of the number of strata. In the case of four strata, with a survey comprising thirty transects, the adaptive scheme can provide necessary precision equal or better than the stratified scheme. For a higher number of strata (i.e., nine strata), abundance estimation was more imprecise than the stratified scheme. This result suggests that unequal sizes of strata dependent on expected population density be used in an adaptive survey scheme. The effect of equal sized strata was more pro-nounced for the adaptive stratified scheme than the standard stratified scheme. Further structuring of the spatial strata for sampling using this scheme are, however, necessary. With unequal sized strata, and a more accurate determination of the probability of first encounter shoals incorporating information on the environmental associations of B F T 5.8 Results, summary and future work 427 shoals, this scheme may provide more precise estimates of abundance than the stratified method. It is expected that these additional considerations could help to stabilize and reduce the uncertainty in estimating abundance, in the case where a large number of transects are used. It is important to note that the total survey area remained fixed for all survey simulations. Further simulations under a variable total survey area, and simulating small or large-scale surveys where the boundary of the area may appreciably bias results, and support the use of an adaptive survey scheme. The adaptive scheme does not require delineation of the boundary for the full survey area, but can be applied under a variable total survey area. In the simulations performed, the strata boundaries within the survey area were specified under the adaptive method within each strata. This scheme, therefore, does not require delineation of the boundaries of strata, and is more applicable to situations where boundaries are difficult to determine, or impractical. The results show that the precision in estimating abundance for an adaptive scheme is less uniform for >30 transects, identified by considerable variation in the mean C V value as the number of transects increases. Also, the confidence intervals for the degree of shoal aggregation remained fairly constant. This indicates that the scheme was not able to reduce the effect of shoal aggregation on the uncertainty of abundance estimates. For the spotter-search survey scheme, the variation in survey precision for an increas-ing number of search paths was similar to the profile obtained for systematic surveying. Estimates of the survey precision in abundance for the (1994-96) survey data, were pre-viously provided in Table 5.64, and C V values range between (4-7)%. These estimates were obtained from the survey data, with the total number of observers less than thir-teen, and include uncertainty in shoal size. The aggregation coefficient, k, associated with these estimates (see Table 5.63) across survey years, range within approximately (0.10-0.37) or (10-37)%. However, predictions shown in Figures 5.171 and 5.172 were obtained assuming uncertainty in shoal size estimation, and no shoal size bias, for a 5.8 Results, summary and future work 428 simulated distribution of shoals with aggregation coefficient, k, in the range (0.00,2.50). The simulation predictions estimate the precision in the abundance estimate to be ap-proximately 0.25 (25%) for five observers. For ^10 observers, precision increases rapidly from a coefficient of variation of approximately (5%) over an increasing number of tran-sects. This prediction agrees with the observed C V values and lies within the range of (4-7)%. While simulation of a spotter search scheme from actual flight paths showed the greatest precision in abundance estimation, large variation in the daily search paths from observational data was estimated to lie in the range (34-55)%. The encounter rate was estimated to be extremely variable with C V values in the range of approximately (128-326)%. While the simulation results indicate that this scheme can yield accurate abundance estimates, the considerable variation in encounter rate make it difficult to infer whether this search scheme is, in general, reliable. This is because large variation in encountering shoals as flight paths change, can lead to large changes in the precision of abundance estimates. While such variation can also serve to make abundance estimates more precise. Precision therefore depends greatly on where shoals aggregate and the ability of spotter search paths to target where these aggregations form. The comparison of C V estimates of the precision in abundance for the actual (ob-served) and simulated (predicted) also showed that the effect of shoal aggregation on the precision of spotter-search is negligible, compared to other survey schemes. This is because the spotter-search scheme involves targeting shoal aggregations. Shoal size showed a decreasing trend between the years 1994-95, and a subsequent increasing trend between 1995 and 1996. This trend was also evident across months, within each survey year through July to October. The largest variability in shoal size observed was in the months of July and October for all survey years, with the months of August and September showing similar mean and variance in their distribution. Mea-surement bias in estimation of the size of shoals due to variation in structure and shape 5.8 Results, summary and future work 429 (i.e., not observer bias) was 7.3%. This value was estimated by comparing visual and automated shoal size estimates. Shoal size bias for different structural formations, in order of decreasing bias is: 11.6% (soldier), 11.5% (dome/packed-dome), 8.5% (mixed), 7.2% (ball), 3.0% (oriented), 1.8% (surface-sheet), and negligible bias (cartwheel). For each survey year, the size, of shoals and 95% confidence interval ranges of its uncertainty are: (73.1,141) in 1994, (49.3,62.9) in 1995, and (73.5,101) in 1996. These observed or uncorrected shoal size range estimates were similar to shoal size estimates calculated in an analysis of B F T shoal structure of observed shoal formations that also showed overlap in the estimated ranges between formation types. For shoal with mean shoal sizes of < 100 individuals, soldier formations were estimated to have smallest mean sizes (Ns = 11.59 ± 1.66), and ball formations had the largest sizes (Ns = 84.25 ± 11.32). For shoal sizes > 100 individuals, B F T shoals formed surface-sheets, with a mean size of Ns = 130.6 ± 34.57. Estimates of the precision (CV's) of shoal size were 3.17, 1.66 and 1.80 for survey years (1994-96) respectively. Despite the scaling relationship between the annual number of shoals and mean shoal size in the survey observations, uncertainty in shoal size was considerably large. The seasonal variability in the months of July and October may have been due to the immigration and emigration of shoals, a period when the mixing of shoals is expected to be significantly higher if their movements are constrained in certain directions and within a confined spatial region. Similarly, the variability in shoal size may have also been biased by interaction effects, whereby individuals mix between shoals when the seasonal population of B F T is expected to exhibit more stability due to reductions in the proportion of shoals immigrating or emigrating. Measurement bias and uncertainty in shoal size, if considered in the simulations, could decrease the predicted precision in abundance. The C V values from the simulations would predict higher C V values, and in the case of ten observers, C V would increase 5.8 Results, summary and future work 430 above the predicted value of 5%. The precision estimated as (4-5)% for the observed distribution of shoals was associated with a smaller range of aggregation coefficient than used in the simulation, where k ranges between (0.31-0.37). 5.8 Results, summary and future work 431 > U (A) Random no replacement with replacement > U (B) Systematic no replacement with replacement 20 40 60 80 Number of transects 100 20 40 60 80 Number of transects 100 (C-9) Stratified systematic 9 strata 20 40 60 Number of transects 20 40 60 Number of transects > O 20 40 60 80 Number of transects 100 Figure 5.171: Simulation results for survey schemes: Precision (coefficient of variation, CV) for shoal encounter/sightings versus the number of transects. 5.8 Results, summary and future work 432 20 40 60 Number of transects 100 (B) Systematic no replacement with replacement 60 80 100 20 40 Number of transects Number of transects Number of transects 0 20 40 60 80 100 Number of transects Figure 5.172: Simulation results for survey schemes: Precision (coefficient of variation, CV) for abundance estimation versus the number of transects. 5.8 Results, summary and future work 433 Adaptive stratified no strata io- ; 1 1 i 1 0 20 40 60 80 100 2 1 0 12 10 8 6 4 2 0 Adaptive stratified 4 strata 1 1 1 1 0 20 40 60 80 100 Adaptive stratified 9 strata — i 1 1 1 — 0 20 40 60 80 100 Number of transects Figure 5.173: Simulation results for adaptive survey schemes: Precision (coefficient of variation, CV) for abundance estimation versus the number of transects for varying spatial stratification. 5.8 Results, summary and future work 434 Summary The key results from the research work in this chapter are summarized as follows: A n analysis of survey observations of B F T provided estimates of their seasonal abun-dance within the G O M for a three-year period (1994-96) to be (47810,50315,43696) in-dividuals. Filtering of survey data for movement bias in both the horizontal (2.1 m/s) led to reductions in the estimated number of shoals encounter and seasonal abundance. Movement bias due to double or multiple-counting of shoals by observers assuming that fish move at speeds of 0.8 m/s and 2.1 m/s reduced the number of shoals encountered by 87.8% and 48% respectively. These same filters reduced abundance estimates by 78.1% and 64.4% respectively. Tracking observations (N=10) (1995-97) provide estimates of hor-izontal movement speeds in the mean range of (1.91-2.25)m/s, and support the assumed speed filter applied to minimize movement bias in survey observations. Using information from hydroacoustic tracking observations, abundance estimates corrected for bias due to vertical movement increased by 50%. The bias introduced by movement in the vertical dimension (the movement of fish at depth) in abundance estimates derived from aerial surveys, therefore, as 50%, is approximately equal to magnitude of bias due to horizontal movement (48-65)%. Considerable nonstationarity in mean abundance index (SPUE) and its uncertainty was observed in survey observations of their seasonal population in the study region (1994-96). The cumulative increase in the estimated number of shoals and abundance sighted during these surveys each day, and unfiltered for movement bias within the daily time-scale, showed no significant correlation. A weak correlation at a time-scale of eight days was detected for two of the survey years. This trend is not significant, but does indicate that the seasonal immigration of shoals can appreciably bias abundance estimates, and that the process of immigration of fish into the region consists of immigration waves as a weak cyclic trend. The timing of when surveying begins and the duration of surveys 5.8 Results, summary and future work 435 for estimating the regional abundance should include a consideration of the immigration process of fish movement into the region. Regional abundance estimates for (1994-96) were statistically compared to abun-dance estimates in the Western Atlantic (western Atlantic stock management zone as-sumed in international stock assessments, ICCAT) . This comparison predicts that the seasonal abundance in the study region each year consists of approximately 16% of the total abundance in the western Atlantic, with 81.5% of fish transferring annually into the region each year, and 16.5% leaving the region. The comparison of stock assessment and survey time-series of abundance for the three-year period infer that the population in the G O M region consists primarily of fish of age seven years or older. The bias and uncertainty in survey abundance estimates due to annual variation in the rate of immigration and emigration of fish for the study region was determined from single-point satellite tagging observations. Inclusion of the annual immigration and emigration bias in the statistical comparison of western Atlantic and seasonal abundance for the G O M region, supports a time-series mortality scenario for the regional population during the years of the surveys having a constant natural and annual variation in fishing mortality of 7^=0.14, and Fyr=(0.519,0.356,0.615) for years (1994,95,96) respectively. No clear relationship between annual estimates of seasonal population size and mean size of shoals was found on the basis of survey observations. Within years, however, the seasonal population shows that the number of shoals and the mean size of shoals scales to each other in a consistent way, according to a power-law function with exponential decay in shoal size at a maximum shoal size threshold. Fitting of this function to the survey observations identified that this relationship can be used to predict the number of shoals and mean size of shoals independent of population size. The sightings of shoals on the basis of annually observed survey distributions (1994-96) yield aggregation coefficients of (0.10 — 0.37) ± 0.01. For this three year period, the 5.8 Results, summary and future work 436 degree of aggregation was observed to change by a factor of approximately 27%. The spatial extent or radius of shoal aggregations distinguished by the presence of spatial autocorrelation in the observed locations of shoals sighted in the surveys was estimated to be 150 km. This value compares with estimates of the radius of Northern cod aggregations of approximately 111 km [177,347, 361,413]. No significant correlation between the mean radius of B F T shoal aggregations and the number of shoals in the aggregation zones was found. Geostatistical analysis yielded a secondary peak in the spatial autocorrelated structure of the observed shoal distributions from survey data. This peak identifies that the location of shoals was correlated across a spatial range of (40-60)km. Hydroacoustic tracking, single-point and light-archival tagging movement observa-tions were used to estimate the degree of random variation in the distribution of shoals. This measure provided an empirical estimate of the bias in estimating regional abundance due to the formation of shoal aggregations and is estimated as the mean diffusion from the movement observations, with a value of 45.95 km2/day. Comparing this estimate with the spatial range (40-60)km of a secondary peak in the spatial autocorrelation of shoals, lead to the explanation that the spatial structure of shoals in aggregations varied randomly due to the movement (i.e., diffusion) of shoals over a daily time period, as they move with respect to the observed aggregation centres. The secondary peak in the spatial autocorrelation of shoal distribution was due to migratory drift. Mean estimates of advection in observed movements of B F T also confirm that advection can be expected to contribute more to variability in the spatial distribution of shoals at a characteristic length or spatial scale equal and greater than 150 km. The spatial extent of autocor-relation in the location of shoals as 150 km was found to be approximately the same scale at which shoals move in a directed way (i.e., the scale at which advection in B F T movements dominates over diffusion). A comparison of advection estimates, using a lim-ited set of results for other tuna species reported in the literature, indicate that bluefin, 5.8 Results, summary and future work 437 bigeye and skipjack tuna move randomly and then in a directed way across a similar range of spatial scale, while albacore and yellowfm tuna show more advective movements at corresponding spatial scales. Changes in when, where and why tuna shoals may ag-gregate that depend on characteristics of their environment likely lead to differences in the spatial structure of shoal distributions. Despite differences in spatial structure, total population size, and range of distribution, the results obtained indicate that the move-ments of bluefin, bigeye and skipjack shoals are directed between aggregation regions that generate approximately the same magnitude of spatial variation in abundance. From simulation of shoal distribution and aggregation, and different survey measure-ment schemes, results determined that random sampling schemes do not yield sufficient accuracy in abundance estimation. The effect of aggregation in the distribution of shoals considerably reduces the ability to estimate abundance accurately in random sampling schemes. For twenty observers, the random sampling had a precision, C V value of ap-proximately 3.0. However, for surveys performed with a fixed total number of observers in the range (20-30), a systematic transect sampling scheme can provide sufficient precision in encountering shoals, and >20 observers are required to achieve 5% precision. The minimum number of observers or minimum number of transects, conducted in a stratified systematic survey, should be chosen to scale with the number of sampling strata. As the number of strata increases, more rapid decreases in the confidence intervals of precision in abundance can be obtained. This indicates that increasing the number of strata was able to reduce the uncertainty in abundance estimation due to the effect of shoal aggregation. For four strata, and twenty transects, the coefficient of variation is approximately 30%. For a hundred transects, the stratified scheme with four strata lead to an increased precision in estimating abundance attaining a value of 5%. For (20-30) observers, a stratified sampling scheme that considered only equal-sized strata did not provide acceptable precision in estimating abundance. This scheme, therefore, 5.8 Results, summary and future work 438 requires devising unequal sizes of strata in proportion to expected abundance, that may established by relating the observed presence of shoals with environmental variables. When adaptive stratified sampling is used, in the case of four equal-sized strata, and thirty transects, the adaptive scheme provided necessary precision equal or better than the stratified scheme. For a higher number of strata (i.e., nine strata), abundance estimation was more imprecise than the stratified scheme. This result suggests that unequal sizes of strata dependent on expected population density should be devised in an adaptive survey scheme. The effect of equal sized strata was more pronounced for the adaptive stratified scheme than the standard stratified scheme. Simulation of a spotter search scheme from actual flight paths showed the greatest precision in abundance estimation, and large variation in the daily search paths from observational data was estimated in the range (34-55)%. The encounter rate of observers sighting shoals was, however, extremely variable. While the simulation results indicate that this scheme can yield accurate abundance estimates, the considerable variation in encounter rate with shoals derived from the observational data on shoal sightings make it difficult to infer whether this search scheme is reliable. This is because large variation in encountering shoals as flight paths change, can lead to large changes in the precision of abundance estimates. However, such variation can also serve to make abundance estimates more precise. Precision therefore depends greatly on where shoals aggregate, and the ability of spotter search paths to target where these aggregations form. 5.8 Results, summary and future work 439 Future work Future research goals are summarized as follows: 1. Perform additional survey simulations to investigate the effects of: (i) varying the total area surveyed, (i) unequal size proportions and number of strata, (iii) varying the width of transects and its effect on correlation in sightings between transect lines. 2. Perform survey simulations coupled to the spatially-explicit individual based model (SIBM) developed for B F T to investigate changes in spatial distribution and abun-dance resulting from individual and shoal movement, interaction, environmental and climate variability. Using a spatially-explicit model, survey measurement pre-cision can be compared under variation in total population size, total number of shoals, and maximum ranges of shoal size (shoal size frequency distribution). 3. Run simulations using a spatially-explicit population model and survey schemes will allow the estimation of survey measurement and uncertainty to be examined in terms of the effects of movement at depth and survey shoal size bias (transect/search path width) 4. Devise different combinations of the survey schemes could be simulated to inves-tigate the application of multi-staged surveying to estimating regional abundance. Brown has proposed that estimation of abundance for bluefin could be improved if a two-phase stratified sampling design is used, a scheme that has been previously applied to aerial surveying of southern bluefin tuna (SBT) in the Great Australian Bight [48,75]. This design involves sampling according to a line-transect survey scheme, factoring the estimation of abundance into the two steps: the mean and variance estimation of shoal size and the number of shoals in the survey region. 5.8 Results, summary and future work 440 5. Continue aerial surveying to record observer altitude, sea-state conditions and esti-mates of observer-shoal sighting distances to provide empirical estimates of survey bias in shoal size estimation. 6. Continue aerial surveying to examine the survey precision of shoal encounter and abundance estimates with > 30 number of daily observers (varying according to the survey implemented), and a greater number of replicate transects conducted by observers within each survey day. A n analysis of replicate transects/search paths will provide data to estimate survey observer bias. 7. Devise experimental stratified aerial surveys to provide a larger set of data to examine the spatial distribution of shoals in relation to environmental variables and to examine and run simulations on the effects of un-equal sized sampling strata. 8. Continue aerial surveying across a longer number of years to obtain a regional abundance time-series >3 years to compare with model and survey measurement schemes time-series predictions of regional abundance. Chapter 6 Summary and Conclusions The purpose of the thesis research was to conduct an investigation on the movement, spatial aggregation and distribution, shoal size and structure, and behaviour of bluefin tuna, and how these factors affect the measurement bias and estimation uncertainty of population abundance. Methods applied in the analysis of movement, spatial distribution and image data include: interpolation of movement data, Lomb spectral analysis, statistical bootstrap simulation, Kalman filtering, geostatistical and geographic information system (GIS) methods. Two new methods were developed as part of the investigation. A new supervised automated digital image analysis system (SAIA) was developed for the three-dimensional analysis of fish shoal structure and behaviour. In addition, \\ 2 statistics and principal component analysis analyses were applied. A spatially-explicit, individual based popula-tion model was formulated to examine the interaction between B F T movement, shoaling behaviour and changes in environment, and how these contribute to variation in B F T aggregation, distribution and abundance. The results of the investigation are summarized in the following sections. A list of recommendations and future goals of this research work is provided in the summary sections of Chapters 4 and 5. 441 6.0 Summary and Conclusions 442 6.1 Regional population abundance Estimation A n analysis of survey observations of bluefin tuna provided estimates of their seasonal abundance within the study region for a three-year period (1994-96) as (47810,50315,43696) individuals. The effect of aggregation on the distribution of shoals reduces the ability to esti-mate abundance accurately in random sampling schemes. For 20 observers, the random sampling has a C V value of approximately 3.0. However, for surveys performed with a limited/fixed total number of observers (i.e. 20-30), a systematic transect sampling scheme provides sufficient precision in encountering shoals, and >20 observers are re-quired to achieve 5% precision. The minimum number of observers or number of transects conducted in a stratified systematic survey should be chosen to scale with the number of sampling strata. As the number of strata increases, more rapid decreases in the confidence intervals of precision in abundance can be obtained. This indicates that increasing the number of strata reduces the uncertainty in abundance estimation due to the effect of shoal aggregation. For four strata, and twenty transects, the coefficient of variation is approximately 30%. For a hundred transects, the stratified scheme with four strata leads to an increased precision in estimating abundance attaining a value of 5%. For twenty-thirty observers, a strati-fied sampling scheme that considers only equal-sized strata does not provide acceptable precision in estimating abundance. This scheme therefore requires devising unequal sizes of strata in proportion to expected abundance and population density, which may be related to biotic and abiotic variables. When adaptive stratified sampling, four equal-sized strata, and thirty transects pro-vides necessary precision equal or better than the stratified scheme. For a higher number 6.0 Summary and Conclusions 443 of strata (nine strata), abundance estimation is more imprecise than the stratified scheme. The effect of equal sized strata is more pronounced for the adaptive stratified scheme than the standard stratified scheme. Simulation of a spotter search scheme from actual flight paths shows the greatest pre-cision in abundance estimation, but variation in the daily search paths from observational data ranges from 34-55%. The encounter rate of observers sighting shoals is extremely variable. While the simulation results indicate that this scheme can yield accurate abun-dance estimates, the considerable variation in encounter rate with shoals questions the reliability of this search scheme. Large variation in encountering shoals as flight paths change can lead to changes in the precision of abundance estimates. However, such vari-ation can also make abundance estimates more precise. Precision therefore depends on where shoals aggregate and the ability of spotter search paths to target aggregations. Bias and uncertainty Filtering of survey data for movement bias in the horizontal led to reductions in the estimated number of shoals encounter and seasonal abundance. Movement bias due to double or multiple-counting of shoals by observers assuming that fish move at speeds of 0.8 m/s and 2.1 m/s reduced the number of shoals encountered by 87.8% and 48%, respectively. These same filters reduced abundance estimates by 78.1% and 64.4% re-spectively. 6.2 Movement: immigration and emigration Bluefin tuna enter the Gulf of Maine/northwestern Atlantic study region during June and July, and leave the region during September and October. Considerable non-stationarity in mean abundance index (SPUE) and its uncertainty is observed in spotter 6.0 Summary and Conclusions 444 survey results. The cumulative increase in the estimated number of shoals and abun-dance sighted during daily surveys unfiltered for movement bias, show no significant correlation. A weak correlation at a time-scale of eight days was detected for two of the survey years. This trend is not significant, however, and indicates that immigration of shoals can appreciably bias abundance estimates, and that entry into the region consists of immigration waves with a weak cyclic trend. The timing and the duration of surveys for estimating the regional abundance should include a consideration of the immigration of fish into the region. Estimation Regional abundance estimates from aerial survey data were compared to abundance estimates in the western Atlantic. This comparison predicts that the seasonal abundance in the study region consists of approximately 16% of the total abundance in the western Atlantic, with an annual transfer rate of 81.5% into the region, and 16.5% leaving the region. Bias and uncertainty The bias and uncertainty in survey estimation due to annual variation of B F T im-migration and emigration rates for the study region was determined from single-point satellite tagging observations. Inclusion of the annual immigration and emigration bias in the statistical comparison of western Atlantic and seasonal abundance yields a time-series mortality scenario for the regional population with a natural and fishing mortality of M y r =0.14, and ^=(0.519,0.356,0.615) for years 1994-96 respectively. 6.0 Summary and Conclusions 445 6.3 Spatial aggregation and distribution Estimation Bluefin tuna aerial survey results yielded aggregation coefficients of (0.10 — 0.37) ± 0.01. For the three year survey, the degree of aggregation changed by approximately 27%. The spatial extent of shoals distinguished by the presence of spatial autocorrelation in the observed spatial distribution of shoals sighted in the surveys is estimated to be 150 km. This value compares with estimates of the radius of Northern cod aggregations of approximately 111 km [177,347,361,413]. No relationship between the mean radius of B F T shoals and the number of shoals in aggregation zones was found. Geostatistical analysis yielded a secondary peak in the spatial autocorrelated structure of the observed shoal distributions. This peak identifies that the location of shoals is correlated across a spatial range of 40-60km. Bias and uncertainty Hydroacoustic tracking, single-point and light-archival tagging movement observa-tions were used to estimate the degree of random variation in the distribution of shoals. This measure provides an estimate of the bias in regional abundance due to the formation of shoal aggregations and is estimated as the mean diffusion from the movement obser-vations of 45.95 km2/day. The contribution to measurement bias due to non-random, persistent movement is estimated to be 7.96 km/day. Comparing the estimate of mean diffusion with the spatial range 40-60km of a secondary peak in the spatial autocorrelation of shoals, suggests that the spatial structure of shoals in aggregations varies randomly due to the movement of shoals over a daily time period, as they move with respect to the observed aggregation centres. The secondary peak in the spatial autocorrelation of shoal distribution is, therefore, due to migratory drift. Mean estimates of advection in 6.0 Summary and Conclusions 446 observed movements also confirm that advection contributes more to variability in the spatial distribution of shoals at a spatial scale ^150 km. The spatial extent of autocorre-lation in the location of shoals as 150 km is approximately the same scale at which shoals move in a directed way. A comparison of advection estimates, using a limited set of results for other tuna species, indicate that bluefin, bigeye and skipjack tuna move both randomly and in a biased way across similar spatial scale, while albacore and yellowfin tuna show more advective movements. Changes in aggregation patterns that depend on environmental shifts can produce differences in the spatial structure of shoals. De-spite differences in spatial structure, population size and distribution, the movements of bluefin, bigeye and skipjack shoals directed between aggregation regions may generate approximately the same magnitude of spatial variation in abundance. 6.4 Shoal size and structure The image analysis method developed in this thesis was used to calibrate, measure, sort, compare and archive information on fish shoal structure and behaviour. The scheme integrates a user interface for supervising automated measurements and conducting the-oretical extrapolative calculations of shoal shape and structure. The method was tested and applied to 463 digitally-processed aerial survey photographic observations of B F T shoals for a three-year period. Testing of the method was performed using observations of the first survey year (N=160, 1994), with a complete analysis performed for observa-tions from subsequent years (1995-96). The aerial surveys were conducted by fish-spotter observers and the New England Aquarium (NEAQ). The digital image database consists of shoals detected in July-October in the Gulf of Maine. The components of the image analysis scheme are: (1) data initialization (data archiving and database management, spatial and intensity image calibration), (2) pre-processing steps (user environment settings, image area-of-interest delineation and image 6.0 Summary and Conclusions 447 histogram output), (3) individual object identification (intensity filtering and threshold-ing, manual image measurement, (4) post-processing (automated object signature de-tection/rejection, object measurement calculations and object statistics, shoal structural statistical discrimination and classification, (5) three-dimensional visualization of output shoal structure for real-time simulation, and (6) archiving of results. The SAIA scheme, applied to the survey observations of bluefin tuna shoals, enables efficient and detailed calculation of shoal structure. Future improvements could include the use of a fully automated digital image capture system (hardware and software and image database algorithm extensions and algorithms) to permit rapid retrieval, efficient archiving, and analysis of large sets of observational data. The automated analysis of shoal structure, together with future direct observations of fish shoaling dynamics, provide a method to calibrate shoal size estimates, shape and structure observations in fisheries surveys using aerial, acoustic or radar technologies. The shoal structure results obtained from the SAIA image analysis method can also be used to compare large sets of observations of shoal structure to theoretical models of individual fish movement and interaction, and fish shoal structure and collective emergent behaviour. Estimation I used reduced x2-statistics to compare the relative precision of visual and automated method for identifying individuals in shoals, and shoal structure pattern-corrected esti-mates of shoal size. Calculated visual, automated and corrected estimates of shoal size Ns were not significantly different (p < 0.05) across survey years. The shoal observations were further sorted into seven characteristic formation types. (A) cartwheel, (B) surface-sheet, (C) dome, (D) soldier, (E) mixed, (F) ball and (G) ori-ented. xV^Z-st^tistics shoal size estimates for each formation type indicate an increased 6.0 Summary and Conclusions 448 precision of the automated procedure in comparison to visual estimation for the four of the seven formations: surface-sheet, soldier, mixed, ball formations. With the exception of the soldier formation, all of these formations comprise a substantially large number of individuals for which the automated image analysis method is able to reduce bias in visual shoal size estimation. Separation distance between first nearest-neighbours (NND) ranged from ~ (0.1 — 2.0) B L . The smallest NND's in this range occur in the cartwheel and ball formations, with the largest occurring in the surface-sheet and oriented formations. This range can be compared to a mean three-dimensional NND of 0.9 B L as a reasonable assumption for a typical fish shoal [319]. As a function of shoal size, N N D in their shoals increases to a maximum and then decreases. Shoaling individuals have between one to six first nearest-neighbours (NNDcrit = 1.5BL). The surface-sheet and ball formations have between one to three neighbours. Approximate modal values of preferred bearing angles between neighbouring individuals are 30°, 90° and 120°, indicating that individuals prefer to be positioned horizontally directly beside each other, in a range of (30 — 60)° from this horizontal position. Bluefin tuna show highly polarized shoal structure. Differences in the mean polar-ization of their shoal formations is evident, with shoal orientation angles varying in a range of (5 - 30)°. The mean packing density of bluefin tuna shoals across all formation types was (0.80±0.072) BL~3. Systematic measurement uncertainty introduced on packing density associated with the fixed shoal depth assumed in the intensity calibration of the images was estimated as Sp = ±0.0267 BL~3. For both random and systematic measurement uncertainty, shoal packing density is (0.80 ±0.098) BL~3 or a range of (0.71-0.90) BL'3. Pitcher and Partridge conducted an extensive analysis of data on cruising shoals of saithe, herring and cod [319]. Packing density estimated for saithe (Pollachius virens) and 6.0 Summary and Conclusions 449 herring (Clupea harengus) laboratory shoals were 0.71 BL'3 and 1.43 BL'3 respectively [319]. They indicate that on the basis of the results of their experiments and evidence from other work, packing density, approximated as 1.0 BL'3, is supported for fish when they shoal, although volume may increase (packing density decrease) in more loosely organized shoals. Estimates of packing density from laboratory experiments can be compared to those from observations of shoals in the wild using acoustics instruments. Misund [254], for observations of saithe and herring shoals in the wild, reports that packing densities are generally one order of magnitude lower than the prediction of 1.0 BL'3 of Pitcher and Partridge [319]. Packing density, estimated for bluefin tuna shoals as (0.80 ± 0.098) BL'3 lies within the estimated range in laboratory experiments. Comparing with the approximation pre-dicted by Pitcher and Partridge of ps — 1.0 BL'3 suggests that packing density estimates obtained from the image analysis under-estimate true packing density. Some other sys-tematic bias contributing to measurement uncertainty, in addition to error introduced from extrapolating shoal depth in the analysis, may not have been considered. However, packing density estimates for shoals in the wild obtained by Misund were less than those estimated in the laboratory. Under this consideration, packing density estimated for B F T shoals in the wild obtained in this analysis is also less than available laboratory estimates (saithe, herring, cod), and given the differences in laboratory and wild measurements, compares fairly well to the approximation of ps = 1.0 BL'3. The most realistic theoretical aggregative packing model predicts a volume per fish of 0.60 BL3, or packing density of 0.84 BL'3. Theoretical optimum packing of spheres yields a volume per fish of 0.64 BL3 (i.e., packing density of 0.78 BL'3) [319]. Mean packing density for B F T was estimated to be greater than these theoretical estimates. The maximum packing density for B F T estimated from the observations is (0.94±0.561) BL'3 at the 95% confidence level, corresponding to a range of (0.3740 — 1.4956) BL~3. This 6.0 Summary and Conclusions 450 estimate indicates that shoals have packing densities close to the theoretical optimum. As summarized by Misund [254], if individuals pack denser at greater speeds or higher levels of arousal, then this could cause variation in packing density between regions within a shoal [291, 319]. Separate from the scaling assumptions of ellipsoidal shoal structure and volume, relatively small changes in nearest-neighbour distances (NND's) may create large changes in packing density. Such variation may be especially apparent in free-swimming shoals due to variation in shoal speed and density as shoals change their orientation directed, encounter food patches, or respond to predators. Results also show that even without a consideration of individual velocities and turning angle variations, bluefin tuna maintain rigid shoal shape and structure. For example, for all formation types, the number of edge individuals increases exponentially to a maximum of 25-30, independent of shoal shape and size. This indicates that individual tuna on the edge of the shoal is well-determined and their first neighbours are then positioned with respect to the edge individuals at preferred distances, number of nearest-neighbours, and bearing angles. The internal arrangement of individuals in shoals is determined by the number of edge individuals. The well-defined formations together with the association of each formation type with a specific range of shoal size (total number of individuals), indicate that the restriction of a number of edge individuals constrains the overall shape and internal structure of their shoals. Preferred ranges in the numbers of first nearest-neighbours, their separation distance, and relative bearing angles may be related to sensory lateral line perception, movement efficiency in relation to reductions in hydrodynamic drag and increased propulsion, the need to maintain an angular range of vision for both independent and inter-fish coordi-nated movement, maintain hydrostatic stability due to positive and negative buoyancy (swim-bladders), or in relation to the need for an efficient, regulated supply of water (oxy-gen) passing through their gills (or mouth) in relation to dissolved oxygen concentration 6.0 Summary and Conclusions 451 and direction of water flow [1, 2, 57, 290, 307,409,410,414, 416,418,419]. Bias and uncertainty No clear relationship between estimates of population size and mean size of shoals was found in results presented in Chapter 5. However, within years the number and size of shoals scales to each other according to a power-law function with exponential decay in shoal size at a maximum shoal size threshold. The relationship can be used to predict the number of shoals and mean size of shoals independent of population size. Further survey-ing of the population is required to determine whether relationships between population size, number of shoals and shoal size exists. Shoal size shows a decreasing trend between 1994 and 1995, and an increasing trend between 1995 and 1996, across months within each survey year (July-October). However, the month of October was sampled only for one to two weeks. The largest variability in shoal size observed was in July and October for all survey years, with August and September showing similar mean and variance in their distribution. Significant differences in the time-of-day frequency of occurrence between their soldier and ball formations (jp < 0.001) were found. These formations have the smallest (soldier, Ns = 11.6 ± 1.66) and largest (ball, Ns — 84.25 ± 11.32) observed mean shoal sizes, where mean shoal size is < 100 individuals. For shoal sizes > 100 individuals, shoals form surface-sheets. This indicates that shoal size is a driving factor influencing alterations in their shoal formation and vice versa. Shoal structures are dictated by internal and external shoal structural variables. Each formation is characterized by variation in shoal length, width, depth, nearest-neighbour distance (NND) and bearing angle between neighbours (BA). Measurement bias in estimation of the size of shoals due to variation in structure and shape is 7.3%. This value is estimated by comparing visual and automated/corrected 6.0 Summary and Conclusions 452 shoal size estimates. Shoal size bias for different structural formations, in order of de-creasing bias is: 11.6% (soldier), 11.5% (dome/packed-dome), 8.5% (mixed), 7.2% (ball), 3.0% (oriented), 1.8% (surface-sheet), and negligible bias (cartwheel). For each survey year, 95% confidence interval ranges of uncertainty in shoal size are; (73.1,141) in 1994, (49.3,62.9) in 1995, and (73.5,101) in 1996. Overlap in estimated ranges across formation types, however, does exist. For shoals with mean shoal size is < 100 individuals, soldier formations have the smallest mean sizes (Ns = 11.6 ± 1.66), and ball formations have largest sizes (N3 = 84.25 ± 11.32). For shoal sizes > 100 individuals, their shoals form surface-sheets, with a mean size of Ns = 130.6 ±34.57. Estimates of the precision of shoal size are 3.17, 1.66 and 1.80 for survey years (1994-96) respectively. Despite the scaling relationship between the annual number of shoals and mean shoal size uncertainty in shoal size is large. Measurement bias and uncertainty in shoal size, if considered in the simulations, would decrease the predicted precision in abundance. For example, the C V values from the current simulations would predict higher C V values, and in the case of 10 observers, C V would increase above the predicted value of 5%. The precision estimated as 4-5% for the observed distribution of shoals is associated with a smaller range of aggregation coefficient than used in the simulation, where k ranges between (0.31-0.37). Principal component analysis using to a set of seven variables relating to shape and internal structure explains approximately 70% of the total variance observed in their shoals. Scatter plots of the first two principal components (PCI: shoal shape, PC2: internal structure) can be used to develop a signature test in the automated analysis of shoal images from aerial surveys. By combining a signature test within the SAIA scheme, signatures for the different formations can be used in determining threshold settings for NNDcrit, and the best pattern-matching threshold, Rcru for comparing individual 6.0 Summary and Conclusions 453 positions between visual and automated shoal size estimation. The extension of the SAIA analysis to include automated feedback of P C A analysis results allows for the procedure to recognize shoal formations (shape and structure) effectively from static digital images. This addition would make it possible to automate analysis of time-lapsed observational data of fish shoaling dynamics. In this way, the automated analysis method would track real-time changes in individual movement velocities and turning angles, correlations between individual movement parameters, and shoal shape. The results obtained would provide a basis for comparing real-time observations of shoaling dynamics with theoretical models of individual and shoaling behaviour. 6.5 Movement: foraging, short and long-range searching Individual tuna travelling in shoals exhibit intrinsic movement characteristics ac-cording to how vary their travel speed, and change their orientation. As they travel, they alter their speed in relation to orientation. When they forage and move within a restricted area, correlation between changes in speed and orientation are strong, and when they travel, searching for food over larger distances, coupling between variation in their orientation and speed is reduced. Estimation Two-dimensional interpolation of geoposition estimates was used to identify distri-bution characteristics for turning angle and move-speed. The turning angle of observed movements of individual tuna is distributed according to unimodal and bimodal forms of the circular-normal, von Mises distribution. Move-speeds follow the form of a gamma-distribution. Turning angle and move-speeds are in the range of (2.0 ± 0.46)m/s and (23.3 ± 2.71)°, respectively (N=10). Lomb spectral analysis of the movement observa-tions identified two separable movement modes rai,m2 by re-sampling the data across 6.0 Summary and Conclusions 454 a range of sampling rates. Tuna in mode mi (intensive search, foraging) travelled at a speed of (1.2 ± 0.44) m/s, with turning angle, (17.0 ± 1.1)°, and a higher rate of turning than in mode m 2 (extensive search, travel), with the range of speed, (2.3 ± 0.71) m/s, turning angle, (21.0 ± 1.87)°. Individual tuna appear to move very differently when foraging on food that is avail-able locally, or searching other regions for food. These aspects are distinguished as movement modes and represent coupling of the processes of movement and behaviour. During dusk and dawn, bluefin tuna consistently dive rapidly across depth. A n expla-nation for this diving activity is not clear. However, these dive events separate the two movement modes and indicate that when they dive deeply, they switch between modes of movement. Bluefin tuna respond to changes in their environment [27,40,174, 345]. They respond to changes in temperature gradients and the local abundance of prey, however they do also respond to water flow and changes in bathymetry. Comparison with data shows that their movement response has two main components: sensitivity (ability to detect gradients) and the distance they are from the largest local environmental gradients. These movement responses are made over short distances, and it is not clear whether they may also have the ability to detect and respond over larger spatial ranges to changes in their environment. They may possess an ability to learn and remember spatial habitats where prey are abundant, where the water is warmest, where the depth of the water column is sufficiently small that can provide improved prey visibility, or where water flow is reduced and helps to minimize energy exerted to counteract hydrodynamic drag. Variability in the 2D distributions of movement parameters is best resolved or ex-plained using three-dimensional interpolation that combines information on horizontal and vertical movement forming space trajectories. Bootstrap simulation is a useful tech-nique for statistically extrapolating data with a small sample size (N=13), to compare 6.0 Summary and Conclusions 455 movement observations with the predictions of different theoretical models. Theoretical random walk models are able to predict their movement, but rely on strong assumptions on the degree of intrinsic correlation and external bias that is present. Comparison of observed movements for 48 hour periods, show that tuna move between local foraging regions. Correlated and biased random walk models predict lower and upper limits on displacement and spatial movement range over time. These attributes lead to variability in movement observations, as the degree of external bias depends on the spatial scaling between where they are located at any point in time and areas that attract them. The analysis of further observations may link external movement bias and changes in local prey abundance. Such a link may also occur between movement bias and their diet (changes in preferred prey). Bias and uncertainty The process of movement observed for tuna is biased by behaviour and environmental factors, with the environment exerting a larger bias. This external bias can also be termed process bias and is distinguished from estimates of bias when movement is sampled, termed measurement bias. Observations of bluefin tuna movement over 30 day periods, as with daily movements, show considerable process bias. Kalman filtering is a method to examine these observations and requires use of movement and measurement models. Results show that applying this method to interpret geoposition data derived from PSAT tags, is significantly biased in its measurement by a distance of 237.5 km. Uncertainty in estimating the location of tuna from light-archival data, independent of the theoretical assumptions, is estimated as (0.855-12.5)km2. Tracking observations (N=10) provide estimates of movement speeds in the range of (1.91-2.25)m/s, and support the assumed speed filter applied to minimize movement bias in the spotter survey observations. Using information from hydroacoustic tracking 6.0 Summary and Conclusions 456 observations, abundance estimates corrected for bias due to vertical movement increased by 50%. The bias introduced by movement in the vertical dimension (the movement of fish at depth) in abundance estimates derived from aerial surveys, therefore, as 50%, is approximately equal in magnitude of bias due to horizontal movement (48-65)%. While sampling and measurement of population abundance must consider how fish move, the spatial structure of their environment and preferred habitats is very impor-tant. Habitat spatial locations and characteristics are required to provide: (1) reliable interpretation and improved prediction of their movement under the assumptions of bi-ased and correlated random walk models, (2) to spatially stratify surveys used to mea-sure actual abundance in relation to expected population densities, and (3) to orient survey sampling transects or search paths so that they are not aligned parallel in the primary movement direction of tuna. Movement observations can be used to identify how transects should be oriented in adaptive stratified survey schemes, as individual movement observations can be compared with random walk predictions to identify the direction in which sampled movements are most biased. The orientation of transects can then be chosen perpendicular to this direction. This approach, requires consider-able sampling of movements across a survey region. Alongside such experiments, aerial surveys identify spatial regions that attract tuna, and movement observations collected near the boundaries of these regions where tuna aggregate, may identify habitat connec-tivity [19,23,24,47,92,101,109,192,206,237,259,263]. In relation to local population densities, the proportions of tuna that move between identified habitat regions can be estimated and used to identify their spatial re-distribution. 6.6 Interaction of individuals and shoals In Chapter 4, I formulated an integrated model to describe how movement, shoal structure and population abundance are coupled by formulating a spatial, individual 6.0 Summary and Conclusions 457 based (SIBM) model for bluefin tuna. The model is formulated on the basis of results of separate analyses of their movements and shoal structure. The model considers individ-ual and shoal movement and interaction, shoal size and structure. The model provides a logical description of how these considerations are coupled to the formation of aggrega-tions of shoals and is developed to predict the distribution of bluefin tuna shoals in the study region. Spatial simulations of this model allow a comparison of different survey measurement schemes in estimating population abundance and its uncertainty. More comprehensive validation tests and simulations to generate model predictions in relation to key parameters are required. Process tests of the model presented in this thesis lead to the following interpretations of how individual tuna and shoals interact within their environment. 1. Simulation test cases reveal that simulated and observed trajectories at the non-random and random limits both show movement. mode characteristics based on turning rate and move-speed variation. 2. Model predictions show that the assumptions of random variation in turning angle and move-speed does not prevent shoals from emigrating from the study region, but that as process variance in turning angle and move-speed increases, the rate of emigration (and immigration) decreases. 3. Under the assumptions of a correlated random walk model, immigration and emi-gration movement is associated with a high degree of correlation in turning angle and move-speed. 4. Model process tests show that the directionality of the simulated movements of fish is maintained under harmonic turning angle and move-speed varying in the range of observed movements. Under the assumptions of the model, the alteration of directionality in movement is directly associated with fish responding to changes 6.0 Summary and Conclusions 458 in their environment. The model predicts that directionality and external bias are separable mechanisms that couple together. These two mechanisms, when coupled together, can produce large variation in movement predictions, similar to patterns seen in observations. 5. As the degree of randomness in the variation in turning angle increases, predicted movements show the movement of fish becomes concentrated in space, where the duration of their residency increases, with more directed movement taking place between such regions. 6. As the degree of correlation in turning angle increases under random variation, the number of zones within which their movements become concentrated increases, with more directed movement as they move between zones. 7. Random variation in move-speed and turning angle increases the spatial extent of regions where movement is concentrated. 8. The effects of correlation in turning angle and move-speed are most evident when these parameters are assumed to vary randomly. 9. Correlation of turning angle and move-speed, when both parameters vary randomly, increases the degree that fish will either move directly away, or will move directly towards a given region. 10. Model movement predictions show that the alteration in the correlation of either turning angle or move-speed lead to very different patterns, scaling with the vari-ance of these parameters. 11. New predictions are generated on the basis of two separable movement modes for foraging and travel. Process testing shows that movements can be simulated ac-cording to these two modes that are distinguished by differences in two parameters: 6.0 Summary and Conclusions 459 the range and frequency of variation in turning angle and move-speed, and correla-tion between these parameters. Process testing shows that the movements of fish can still become concentrated. 12. A new explanation of how shoals move over time is formulated: shoals in a popula-tion move according to two modes. They must interact (move so that they intersect other shoals) in order to change their movement mode, as a function of the sizes of interacting shoals. The only time this is not necessary is during times of the day (dusk/dawn) when mode-switching occurs. This formulation is predicted by process testing of the model. Process tests demonstrate that modal variations may explain the link between individual decision making, changes in shoal size, and predicts that shoals must move to regions where other shoals are aggregated. 13. New predictions estimating the degree of shoal interaction and mixing that are needed to produce changes in the frequency distribution of shoals and to match observations are generated by the model. Process tests show that the model can describe and simulate individual and shoal movement, behaviour, and interaction population processes. Together, these processes produce shoal size frequency dis-tributions associated with high fusion and high fission. Model predictions indicate that both fusion and fission must take place to produce distributions that match observational data. When shoal fusion dominates, larger shoals form. However, for a fixed population size, the number of smaller shoals is also predicted to increase. This same effect is evident when shoal fission dominates over fusion. Process tests predict that the exchange of fish between shoals is regulated by individual decision making as an explanation of the power-law scaling in the number of shoals at a given size in survey data. 14. Model predictions identify locations of shoal aggregation that correspond closely to 6.0 Summary and Conclusions 460 those seen in aerial survey observations. The predicted locations do not all match the observed locations in any specific survey year, but across the three survey years, the predictions and observations are consistent. 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Appendix A Abbreviations and Notation A . l Abbreviations List of important abbreviations and relevant pages Abbrev i a t i on Exp lana t ion Page A I Agent-based algorithm 4 A C F Autocorrelation function 49 A V H R R Advanced Very High Resolution Radiometer 296 B F T North Atlantic Bluefin Tuna (Thunnus thynnus) 1 B L Body-length 10 B M S Y Population biomass at M S Y 18 C A Cellular automata 4 C.I.,CI Confidence Interval 93 C P U E Catch per unit Effort 19 E A Evolutionary algorithm 4 E C N A S A P North America Strategic Assessment Project 288 F D Finite-differencing 4 F E M Finite-element mesh 285 FIESTA Fish Estimation, Schooling and Tuna Abundance Model 251 G A Genetic algorithm 4 GIS Geographic Information System 4 G O M Gulf of Maine 36 GPS Global positioning system 36 G T Game-theoretic model 4 ICCAT International Commission for the Conservation of Atlantic Tunas 124 JLS Join, leave and stay decisions 115 kt knots: 1 kt = 1 nm/h = 0.514 m/s 352 KS Kolmogorov-Smirnov test 302 L A T Latitude 84 L I D A R Light-detection and Ranging 349 L G T Longitude 84 495 Appendix A. Abbreviations and Notation 496 List of important abbreviations and relevant pages Abbreviation Explanation Page M A R M A P Marine Resources Monitoring, Assessment & Prediction Program 290 M S Y Maximum Sustainable Yield 18 nm Nautical miles: 1 nm = 1.852 km 352 N W O Northwestern Atlantic Ocean 14 N E A Q New England Aquarium 20 N M F S U.S. National Marine Fisheries Service 288 O D E Ordinary differential equation 4 P C A Principal component analysis 117 P D E Partial differential equation 4 pdf Probability density function 75 P M N N Pattern-matching neural network 4 PSAT Pop-up satellite archival tag 11 SAR Synthetic Aperture Radar 21 SIBM Spatially-explicit individual-based model 2 S M R Standard metabolic rate 557 S P U E Sightings per unit effort 344 SST Sea-surface temperature 3 S T M State-transition matrix 551 V B Visual Basic programming language 118 Y / R Yield-per-recruit method 18 Appendix B Chapter 2: Background, Derivations, Extended Results B . l Move-speed and turning angle distributions 42.5 42.4 42.3 42.2 42.1 42.0 41.9 41.8 9602 (W=159kg.,o=150-200) start\" 'T ~w *\\ end V -70.5 -70.0 -69.5 Figure B.174: Left: Observed movement path of individual 9602 in 2D with estimated weight, W, and belonging to a shoal of size, a. Right: Frequency distributions of move-turning angle and move-speed, with frequencies scaled between (0,1). 497 Appendix B. Chapter 2: Background, Derivations, Extended Results 42.45 42.40 42.35 42.30 A 42.25 42.20 42.15 H 42.10 1 I.O H 0 1 2 3 4 5 6 cp(rad) Figure B.175: Same as Figure B.174 for observed movement path, 9603. Figure B.176: Same as Figure B.174 for observed movement path, 9604a. Appendix B. Chapter 2: Background, Derivations, Extended Results 41.30 41.15 H 1 1 1 1 r-.18 -69.16 -69.14 -69.12 -69.10 -69.08 -69.( v (m/s) 4 5 Figure B.177: Same as Figure B.174 for observed movement path, 9604b. 42.40 42.35 H 42.30 42.25 42.20 H 42.15 -70.5 -70.0 -69.5 Figure B.178: Same as Figure B.174 for observed movement path, 9701. Appendix B. Chapter 2: Background, Derivations, Extended Results Figure B.180: Same as Figure B.174 for observed movement path, 9703. Appendix B. Chapter 2: Background, Derivations, Extended Results Figure B.182: Same as Figure B.174 for observed movement path, 9705. Appendix B. Chapter 2: Background, Derivations, Extended Results B.2 Spectral identification of movement modes Appendix B. Chapter 2: Background, Derivations, Extended Results 503 9602 1.0 0.8 H 0.6 s CL, 0.4 0.2 0.0 —

s Figure B.185: Same as Figure B.183 for observed movement path 9604a. Appendix B. Chapter 2: Background, Derivations, Extended Results 506 Appendix B. Chapter 2: Background, Derivations, Extended Results Figure B.187: Same as Figure B.183 for observed movement path 9701. Appendix B. Chapter 2: Background, Derivations, Extended Results Figure B.188: Same as Figure B.183 for observed movement path 9702 Appendix B. Chapter 2: Background, Derivations, Extended Results Appendix B. Chapter 2: Background, Derivations, Extended Results 510 9704 0 20 40 60 80 100 120 140 160 180 11* 12 0.0 0.2 0.4 0.6 200 0.0 0.1 0.2 0.3 0.4 Figure B.190: Same as Figure B.183 for observed movement path 9704. Appendix B. Chapter 2: Background, Derivations, Extended Results Figure B.191: Same as Figure B.183 for observed movement path 9705. Appendix B. Chapter 2: Background, Derivations, Extended Results B.3 Space trajectories Figure B.192: (Top to Bottom): (a) 3D interpolated observed movement trajectory (9602), (b) time-series profiles of vertical inclination (polar), 9(n) (rad), horizontal di-rectional (azimuthal), (n) (rad), turning, 200 lIllllllliTfrru 0 1 2 3 NND (BL) Figure C.201: Same as Figure 3.63 for year 1996. Appendix C. Chapter 3: Background, Derivations, Extended Results 120 80 40 (B) Surface-sheet NND (BL) 2 3 NND (BL) 240 200 H 160 120 80 40 H 0 0 1 2 3 NND (BL) (F) Ball Figure C.202: Same as Figure 3.64 for year 1996. Appendix C. Chapter 3: Background, Derivations, Extended Results 525 C.1.2 Frequency of nearest neighbours Appendix C. Chapter 3: Background, Derivations, Extended Results 500 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 2 0 15 1 0 5 (G) 100-199 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours Figure C.203: Same as Figure 3.65 for year 1996. Appendix C. Chapter 3: Background, Derivations, Extended Results 527 40 0 5 10 15 20 25 Frequency of Nearest-Neighbours 60 50 40 H 30 20 -10 - | 0 n (C) Packed-dome 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours -(B) Surface-sheet r-i n - J l r f l 41 n i l n 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours 0 5 10 15 20 25 Frequency of Nearest-Neighbours Figure C.204: Same as Figure 3.66 for year 1996. Appendix C. Chapter 3: Background, Derivations, Extended Results 528 C.1.3 Bearing angle between nearest-neighbours (BA) Appendix C. Chapter 3: Background, Derivations, Extended Results 30 60 90 120 150 180 BA(°) 40 30 20 10 (B) 10-19 i r 30 60 90 120 150 180 BAO 40 30 -| 20 10 (C) 20-29 0 in l i b i 1 1 1 1 r 30 60 90 120 150 180 BAO 30 60 90 120 150 180 BAO 14 12 10 8 6 4 H 2 0 0 (E) 40-49 30 60 90 120 150 180 BAO 30 60 90 120 150 180 BAO \"1 i i 30 60 90 120 150 180 ~ r 30 60 90 120 150 180 BAO BAO Figure C.205: Same as Figure 3.67 for year 1996. Appendix C. Chapter 3: Background, Derivations, Extended Results 530 60 90 120 150 180 B A O 180 B A O 30 60 i i 90 120 150 180 B A O 80 60 40 20 (E) Mixed 1 1 y . . . . y • • • • y 0 30 60 90 120 150 180 B A O 80 60 40 20 0 \\ - i 1 i 1 r 30 60 90 120 150 180 30 B A O (F)Ball 60 90 120 150 180 B A O 30 60 90 120 150 180 B A O Figure C.206: Same as Figure 3.68 for year 1996. Appendix C. Chapter 3: Background, Derivations, Extended Results C.1.4 Shoal polarization Appendix C. Chapter 3: Background, Derivations, Extended Results 120 100 CFS-0 -100 -50 0 50 100 Figure C.208: Same as Figure 3.71 for year 1996. Appendix C. Chapter 3: Background, Derivations, Extended Results 534 C.2 Convex hull refinement of ellipsoidal shoal structure (A) Cartwheel Formation 2 H C/3 V„ (10 3BL 3) pq 2 1 H V h (103 BL 3 ) Figure C.209: Comparison of ellipsoidal surface area (SAS) and volume (Va) and refine-ment of surface area (SAh) and volume (Vh) with fitted convex hul l for B F T Cartwheel formations. Appendix C. Chapter 3: Background, Derivations, Extended Results 535 (C) Dome Formation Figure C.210: Same as Figure C.209 for dome formation. Appendix C Chapter 3: Background, Derivations, Extended Results 536 (D) Soldier Formation 1(a) 0 2 4 6 8 1 0 1 2 V S ( 1 0 3 B L 3 ) 0 2 4 6 8 1 0 1 2 V H ( 1 0 3 B L 3 ) Figure C.211: Same as Figure C.209 for soldier formation. Appendix C. Chapter 3: Background, Derivations, Extended Results 537 Figure C.212: Same as Figure C.209 for mixed formation. Appendix C. Chapter 3: Background, Derivations, Extended Results 538 (F) Bal l Formation 0 2 4 6 8 10 V h (103 BL 3 ) Figure C.213: Same as Figure C.209 for ball formation. Appendix C. Chapter 3: Background, Derivations, Extended Results 539 Figure C.214: Same as Figure C.209 for oriented formation. Appendix D Chapter 4: Background, Derivations, Extended Results D . l Seasonal population This section provides a summary of the Virtual Population Analysis (VPA) method used in the population assessment of B F T [334, 342]. A n equation for estimating regional population abundance based on annual transfer, natural and fishing mortality rates be-tween different spatial regions is provided. The SIBM model uses this equation to specify the size of their seasonal population. A fish population is comprised of individuals of different ages termed age-classes, year-classes or cohorts. The abundance of each age-class decreases over time due to mortality. Renewal of abundance or recruitment is provided by spawning and birth and recruitment of individuals into younger age-classes. The following derivations assumes that there is no recruitment of new fish and all parameters are constant for all individuals during a time period. A n age-structured population model follows each age or year-class through its life rather than following a population over time. As time passes, fish age-classes age. Let Na be the number of individuals in the 0-age class and Nt, the number in the tth ages-class respectively. Let Z be the instantaneous total mortality, initially assumed to be constant for all individuals and ages. For populations subjected to harvesting, total mortality, Z , is composed of mortality due to both harvesting and natural causes. Let F and M be the instantaneous fishing and natural mortality, so that Z = (F + M). The abundance of each age or year-class decreases as it ages due 540 Appendix D. Chapter 4: Background, Derivations, Extended Results 541 to mortality. A simple deterministic differential equation for each age-class, describing a population subjected to these mortality impacts, is dJ^S.'= -ZN(t) = —(F + M)N(t) (D.167) dt where the term, FN, represents the rate of removals due to fishing, and, MN, is the rate of removals due to natural causes, and t represents age or year class. The solution to this equation is, N(t) = N0e~zt = N0e-{F+M» (D.168) Equation D.167 can also be re-expressed as, dt \\dt dt J y 1 in terms of the rate of catch, C = FN (removal due to fishing), and death rate, D = MN, due to natural causes. Integrating Equation D.169 over individuals of age 0 to r yields, C T JdC = JFN0e~Ztdt (D.170) o o The solution of this equation is known as the Baranov catch equation [334], C = | i V 0 ( l - e~ZT) (D.171) The Baranov equation can be further simplified by considering average age-class size over a period of time. The average abundance over age classes t\\ to t { (D.193) Q(s, s') = 2/3s splitting Appendix D: Chapter 4: Background, Derivations, Extended Results 548 For case (3): shoal fusion: small shoals do not remain for long durations, with larger shoals forming, and fission is independent of shoal size (i.e., is constant), whereby influences favouring shoal splitting affect all groups uniformly; 1 A(s, s') = aa(s') = -% merging a(s),a(s') = - = * < s s (D.194) s { fl(s, s') = f splitting For all the cases, as indicated by Gueron and Levin, regardless of initial conditions, shoal size converges to a limit [3,136], where the total number of shoals, G, and population size, P, are related by, l imG(t) = G(t) = J — (D.195) t-»oo V Ci The stationary distribution of shoal size can be shown to be uniquely determined by, f(s) = 2-e~2(£> (D.196) a or for a general functional form of the kernel a(s), f(s) = £ (-^) e~Xs (D.197) a \\a{s)J oo where A «- P = 2 - / - ^ - e - A z d z (D.198) a J a(z) o D.3 Lagrangian equations A system is defined as an assemblage of inter-related quantitites/entities considered as a whole. If the attributes of a system change with time, a system is termed dynamic, whereby the partial or full evolution of a dynamic system over time is called a process. The form of any system of higher-order ordinary differential equations can be reduced to an equivalent system of first-order equations. The states (x(t)) of a dynamic system, in generalized notation, have time-varying characteristics, ^ - = ±(t) = f(t,x(t),u(t)) (D.199) Appendix D. Chapter 4: Background, Derivations, Extended Results 549 In general, a dynamic system is then formed by n system equations of n dependent variables {xi | 1 < i < n), and r known inputs {«; | 1 < i < r}, describing the evolution of the system states with respect to the independent variable t (time), xi = fi(t,xi,x2,x3, • • • ,xn,UUU2,U3, • • • ,ur) %2 = f2{t,Xl,X2,X3, - • • ,Xn,Ui,U2,U3, - • • ,Ur) (D.200) as, Xn = fn{t, Xi,X2,X3, ••• ,Xn,Ui,U2,U3, ••• ,Ur) For linear dynamic systems, the n set of ODE's can be expressed in vector notation d x(t) = - x ( i ) = F(t)x(t) + C(t)u(t) (D.201) where the elements and components of system matrices and vectors are functions of time, -fn(t) / i a ( * ) Mt) • hiit) Mt) • • Mt) F(t) = M(t) Mt) Mt) • • Mt) Jnl(t) fn2(t) Mt) • fnn(t) --cu(t) cu(t) Mt) • • C i r ( t ) \" C2l(*) C22(t) Mt) • • Mt) C(t) = c3i{t) c32(t) Mt) • • Mt) -Cnl{t) Cn2(t) Mt) • cnr(t) . (D.202) (D.203) ur(t) f (D.204) u(t) = (u1(t) u2(t) u3(t) • This shows that a model description of a dynamic system can contain a matrix F(t) as a dynamic coefficient matrix, with matrix elements termed, dynamic coefficients. The matrix, C(t), is termed the input coupling matrix, with input coupling coefficients. The vector of inputs (Equation D.204) is termed the system's input vector. Appendix D. Chapter 4: Background, Derivations, Extended Results 550 Equation D.199 describes a system continuously in time because it is defined with respect to an independent variable, t varying continuously over a real-valued interval t £ (t0,tf). Difference equations are the discrete-time representations of continuous differential equations. The continuous time-interval is separated into ordered times, tk, where t0 < h < h < • • • 4 _ i < tk < tk+i <••• (D.205) The following discrete version of Equation D.199 is then, x(tk+1) = f{x{tk),tk,tk+i) (D.206) Each state is therefore representable as a function of previous states. For discrete systems having uniform time-intervals, At, each discrete time, tk is then, tk = kAt (D.207) Forward or backward differences between states x(tk) represent the evolution of dy-namic systems. A forward difference x(tk+i) — x(tk) of a state variable can be expressed as: (1) a function of all ip independent variables, or (2) in terms of the forward value x(tk+i) as a function 0 of the independent variables, with the addition of the previous state value x(tk) also as an independent variable [127], x(tk+i) ~ x{tk) = ip(tk,x(tk),u(tk)) x(tk+i) = (t){tk,x{tk),u{tk)) 4>(tk,x(tk),u(tk)) = x(tk) + ijj(tk,x(tk),u(tk)) In the case of linear systems (Equation D.201), the dependence between system states and inputs can be further represented by matrices replacing the functions 4> and ip, x(tk+1) - x(tk) = y{tk)x(tk) + C(tk)u(tk) Appendix D. Chapter 4: Background, Derivations, Extended Results 551 x(tk+i) = &(tk)x(tk)+ C(tk)u(tk) $(t f c) = I + *(**) Here I is the identity matrix. In discrete equations, the matrix $ is called the state-transition matrix (STM), and C is termed the input coupling matrix. Differential equations can be classed as homogeneous or non-homogeneous, used to describe whether the system equations are homogenous with respect to dependent variables. The state-transition matrix $ is a solution of the homogeneous part of a general non-homogeneous differential equation. In this way, the first term in Equation D.201 is • the homogeneous part of the system, where the contribution of both terms, comprise, in general, a non-homogeneous differential equation. D.4 Adaptive step-size Runge-Kutta integration Numerical integration in the SIBM model uses a fifth-order Runge-Kutta scheme, • r i = hf(xk,yk) r2 = hf(xk + a2h, yk + 6 1 2 n ) r 6 = hf(xk + aeh, yk + b61n + • • • + i W s ) Vk+i = Vk + c i n + c2r2 + c 3 r 3 + c 4 r 4 + c 5 r 5 + c 6 r 6 + 0(h6) The embedded fourth-order formula is then, y*+i = Vn + cjri + c*2r2 + C3Y3 + C4V4 + c*5r5 + c*6r6 + 0(h5) with error estimate, 6 A = (j/fc+i - y*k+1) = Y^(ci ~ c*)ki (D.208) (D.209) Appendix D. Chapter 4: Background, Derivations, Extended Results 552 The relationship between the error estimate A and step-size h. If A 0 is desired accuracy, and a step hi yields an error of A i , then the step achieving the desired accuracy is, A a 2 The difference between A c and A i is monitored in the integration scheme such that adjustments in step-size can be made, and in practice, because the error estimate in Equation D.210 is not exact, but only accurate to fifth order, a safety factor, rs, is used (set a few percent less than one). This replaces Equation D.210 as, The adaptive step-size scheme provides local monitoring of truncation error to ensure accuracy and adjust step-size. For further details of this method see Numerical Recipes [330]. Desired numerical accuracy was fixed at 0.1% for all model simulations. D.5 Movement and behaviour dynamics D.5.1 Movement correlations, modes and mode-switching events The index / in the equations that follow references each of the coordinate directions I = (x,y,z). D.5.2 Move-speed movement mode (7774,777,2) variation Move-speed coupling over time for shoals/individuals in each of the two movement modes ( m i ,777.2) is harmonic. The equations represent a superposition of two independent frequencies (u)a,ub) of move-speed, %(£&) variation, with amplitudes (pa,p0). Movement (D.210) 1 (D.211) Various terms appearing in the model's Lagrangian equations are detailed below. Appendix D. Chapter 4: Background, Derivations, Extended Results 553 modes are characterized with different frequencies and amplitudes. In addition, move-speed coupling is determined by pp during mode-switching events. This term is described in Section 4.5. Move-speed coupling decreases exponentially over time tk for time-lags where pp is specified in Equation D.242. D . 5 . 3 Move-angle operator The move-angle operator Ai translates the direction of shoal/individual movement over time according to changes in the polar and azimuthal orientation angles (dive and turning angle)(f%, iptj), Successive Euler angle rotations from Cartesian coordinates S(x,y,z) —> S'{x',y', z'), under a counter-clockwise rotation about the z axis (xy plane) by an angle ip (i.e., clockwise rotation by angle -S\" is given by, A(6, sin 9', sin((9 — 7r/2) —» cos 9' A(tk,9\\tk),p (grid layer, p), and scales with move-speed, Vij(tk). Approximation of this function relates turning rate only to local environmental gradients, neglecting changes in turning rate related to move-speed. The zooplankton grid layer is selected to alter the rate of change in move-turning angle. Turning rate and prey abundance contribute to external bias in individual/shoal movements that can lead to both the formation and break-up of shoal aggregations over time. Furthermore, with turning rate being spatially dependent, shoals/individuals begin attracted to certain spatial regions and aggregating there, are then re-directed by variation of turning angle in response to prey abundance. The angles of movement are defined by the following functions; Vertical, inclination angle correlation: 9(tk) = A e e - ^ f l f o - i ) (D.221) Turning angle correlation: p(tk) — M scales within the same range as predation risk, / / ( r^ , Wi). For an individual i joining a shoal j, containing (sij — 1) other individuals, each individual receives a proportion of the total energy/prey resources available to the shoal, denoted K I , S . The parameter K T ' S scales the foraging benefits offered by a shoal of size with respect to the proportion of benefits that can be provided to each individual. / f f a , W i ) - $ P ( * f c ) - M (D.224) Appendix D. Chapter 4: Background, Derivations, Extended Results 558 The minimum value of foraging benefit provided to a shoaling individual K * ' 7 ' 5 , or available to a joining individual, is the proportion K 7 , 5 that equalizes the benefit of joining a shoal or foraging alone. By setting to/- — wf, the minimum value, K * ' J ' S is, ...,.s _ (nsl'%(tk) - l»s - n'} ^ -T 7?*M ) (D'226) wf(tk) is estimated with « 7 ' 5 = K * , ! ' S , SO that foraging benefit between individuals and shoals considers the minimum foraging benefit a shoal can provide. Further empirical information on how foraging benefits differs between individuals in shoals of different sizes, Sj, is required for B F T to more accurately characterize ft7,5. For two interacting shoals (j, j'), I assume that individual i in shoal j, and individuals i' in shoal j' decide to join, leave or stay (JLS) in their shoals according to a comparison of net reproductive output - termed 'fitness'. JLS decision join w{j{tk) < wf,(tk) stay wjjiU) > wf,(tk) (D.227) [ leave w7-(£ fc) < wf(tk) The join decision involves a comparison of whether the fitness trade-off for an individ-ual i in shoal j is less than the trade-off state for the same individual i in the interacting shoal f. The stay decision compares the fitness trade-off of individual i with shoal f. If the trade-off is less if the individual were in shoal f, then it stays in its shoal j. The leave decision compares the fitness trade-off of individual i within its shoal j. If the individual has a higher fitness than its shoal, it leaves. Hence, shoal sizes are adjusted in the model according to individual JLS decisions over time. Appendix D. Chapter 4: Background, Derivations, Extended Results 559 Individual movement mode alteration The alteration of the movement mode of an individual within a shoal at time tk is represented as a self-organizing process. General features of self-organizing processes, and application of such a scheme for shoal movement dynamics infers the following [155, 183,270,341], • evolution of a shoal into an organized form in the absence of external constraints • shoals can move from a large region of state-space (refer to definition of in Appendix E3) into a persistent smaller one (attractor) under the control of the shoal itself • translates move-speed and turning angle correlations between local movement rules of individuals in a shoal A self-similar return or contraction mapping function f is defined as: if f is a contraction mapping then 3k G [0,1], such that || f(x) - f(y) \\\\<\\\\ x - y \\\\, \\/x,y G Rn G h Here n is the spatial dimension of space R. The mapping function in the model is, f(x,mi:j(tk)) = h(mij(tk - r m ) ,my(t f c - ( r m - 1)), • • • ,mi:i(t - 2),mi:i(t - 1)) • rriij (tk) (D.228) In the model, rriij takes two assignments mi and m 2 movement modes. Functions involving rriij modes require the use a binary operator, denoted B . This binary operator, is defined to operate on the set movement mode and returns either [0,1]. If an individual i is in a shoal j and is in a movement mode at time tk then, B(m^(£fc)) yields values [0,1]. This operator is denoted as B m i such that if an individual is in movement mode mi at time tk, then B m i ( m i j ( 4 ) = 1, B m 2 (m i : , (£ f c ) = 1. Similarly, if an individual is in movement mode m 2 , with operator, B m 2 , then Bm 2(mjj(tfc) = 1, Bmi(m i : )(£fc) = 1. In this way, values between [0,1] resulting from operating B m l ' m 2 on movement mode state Appendix D. Chapter 4: Background, Derivations, Extended Results 560 mij, can be related in functional form, where the inverse operation B ~ l m 2 re-assigns the movement mode state (mi ,m 2 ] , based on the binary state. The time-lag in movement mode alteration is r m and encompasses indirect factors affecting movement in terms of an individual's movement mode memory. The parameter Tm adjusts the number of previous states in movement mode mij(t) influencing its time-transition at tfc. If m individuals are in the ith individual's view then the time-transition of movement mode is, m-ijitk) = B l(mij(tk)) n tk — l = -. — r YI YI [ B m i ( \" V j ( c r ) ) - B m 2 ( m ^ ( a ) ) ] + Bml(mij(tk-i)) + Bm2(mi:j(tk)) The inner summation involves comparison of individual movement mode states across the previous a times from tk-i back in time by time-lag r m . Summation is then performed across all movement mode states in time across m nearest-neighbours [i! = 1 , n ) , in the ith individual's sensing range. The number of neighbours involved in this identification process can be set to vary with shoal movement variance/fluctuation (by assuming a maximum distance/range (NND) for individual interaction extending to the nth nearest-neighbour), or alternatively, can depend on a fixed m number of nearest-neighbours. In the model the first approach is used, so that fluctuations in N N D resulting from individual attraction/repulsion are taken into account. Movement mode states for a shoal j represents averaging of all states of its member individuals, where i = 1 to shoal 1 S j mj(tk) = B-'imjitk)) = ~J2 [BmiKjfaO) + Bm 2(rnr,(4))] (D.229) Appendix D. Chapter 4: Background, Derivations, Extended Results 561 The time-lag, r m , may be a characteristic of various shoaling structural formations, dependent on such factors as shoal velocity. Whether observed formations arise from vari-ation in Tm remains to be determined. Equation D.228 is termed a self-similar mapping or contraction function arising from individual movement interactions within shoals assumed to be controlled by incomplete identification of each individual in sensing or responding to the movement tendencies and alterations of their nth order nearest-neighbours (NNS). In the model, the formulation and use of a contraction mapping is explained simply as a function that describes the behavioural alteration whereby individuals adopt one of the two assumed movement modes. The model's contraction function, on the basis of successive interactions that occur between shoaling individuals, leads them to adopt the same mode of movement over time. The degree to which the movement modes of individuals traveling in shoals con-tract over time depends on their ability to maintain a nearest-neighbour distance below the threshold, NNDcru. Separation distances between individuals changes according to the move-speed and turning angle harmonic and correlation characteristics, but also neighbour attraction and repulsion. In this sense, individuals mimic the movement of their neighbours, by identifying their positions and velocities. Gunji and Kusunoki, have demonstrated that the global, emergent properties of fish shoals may result from local, individual interactions following an incomplete or indefinite identification of its neigh-bours [138,139]. Nearest-neighbour distance has also been shown to characterize spatial relationships in populations [69], suggesting that maintenance of neighbour separation distance is not only important for individuals to sense and respond to each other, but also a factor in how individuals/shoals interact and alter their movement behaviour. Appendix D. Chapter 4: Background, Derivations, Extended Results 562 Individual mode-switching events Events in time whereby individuals switch between moving according to either the mi and m2 movement modes are determined by the mode-switching function, Between the times taw>i,tsw<2 that mode-switching events are scheduled to occur, the movement mode of an individual at time tk switches from the previous mode at time tk-i-At times tsw(tk)} Sj(tk-\\) = sf(tk-i) [mj(tk),mj(tk)] Sj(tk-i) > Sj>(tk-i) (D.234) Appendix D. Chapter 4: Background, Derivations, Extended Results 564 Shoal mode-switching events Two mode-switching events for shoals are assumed to occur at the same time as individual-mode switching events (see Equation D.227). Switching-events are scheduled to occur at dusk and dawn each day at t — (tSWii, tsw^)- At tk = tSWti or tk = tsw>2 times, mj(tk) = 3,V?'. 1 m,j{tk-\\) = 2, tk 7^ taw,i,tSWi2 m^tk) = \\ 2 rrijitk-i) = l,tk=£ tswA,tSWt2 (D-235) 3 rrij(tk-i) = 1, 2, tk = tSWf\\,tsw%2 D . 5 . 6 Neighbour individuals: attraction and repulsion Nearest-neighbour attraction and repulsion are described by the following functions, Fi = Filmijit^^gimijitk)), Xgirriijitk)), Ra, Rr] (D.236) For 0 < NNDu. < Rr nu\\ ) V^ iC**)) + (*i(*fc) - a*i(*fc-i)) l = x ri\\tk) = \\ {V.161) Xgirriijitk)) + (yij(tk) - yij(tfc-i)) I = y For Rr < NNDui < Ra T,u\\ ) - w » ( ^ i ( * * ) ) [ c o s ( ^ m « ( * * ) ) + (^(*0-^j(*fc-i))] l = x m o o c ^ Pl\\tk) — \\ [\\J.Z6b) -wff(mij-(tfc))[sin(?T77iij(tib)) + (yij{tk) - yij(tk-i))] I = y The above equations used to describe variation in the attraction/repulsion between individuals are oscillatory and refer to individuals having nearest-neighbour distances, x' = NNDiti> < Ra, where Ra is their attraction range. Alternate forms for attrac-tion/repulsion between nearest-neighbours consider normalized/Gaussian forms (Ka, Kr) Appendix D. Chapter 4: Background, Derivations, Extended Results 565 as attraction/repulsion potentials [37,260,275]. -x' N Rr < - ^ a The range of repulsion is assumed to be less than the range of attraction in the model (Rr < Ra)-D .5 .7 Movement response to environment and prey Ei = Et[ep, Xp, y(rp, %), 0p] (D.239) Considering the z-component of movement being dependent upon position (depth), z(t), and environmental variables, &p(t) (p = l,...,m), according to the chain-rule, the rate of change in movement depth is, d , ^ , XN dz(t) dz(t) d$Jt) 4 . v p over contributions from each pth environmental variable. Movement in relation to envi-ronmental gradients (dz(t)/d&p) is modelled according to the separation distance between the position of a shoal or individual and their local environment. They respond with an intrinsic movement sensitivity to environmental gradients. Sensitivity and distance sep-aration represent intrinsic and extrinsic considerations respectively, characterizing how individuals perceive and respond within their local ocean environment. Change in move-ment position over time with respect to environmental gradients is, = ep(l - e^r*) ( 1 + (3)2* ) (D.241) dz(t) Appendix D. Chapter 4: Background, Derivations, Extended Results 566 This equation has the key flexible form of response to environmental gradients involv-ing the sensitivity parameters (e\\,ep, ...,em), time-lags {T\\,TP, . . . , r m ) , and time-averaged separation-distances yp (y±, yp,ym). The leading term in the above expression is its key function from which additional higher-order terms are formed as a polynomial series expansion. The term dz(t)/dt is represented in relation to a switching-function, p0(t). Rapid diurnal movement at depth corresponds to a mode-switching event observed in the B F T hydroacoustic movement observations coinciding with daily times of dusk and dawn. In association with the characterization of B F T movement according to the two movement modes, a mode-switching function is defined to occur at times tk coinciding with time of day events, { 0 rriiAtk) = 1,2 (D.242) Sgn(z(tk) - %(tk))z(tk) mij(tk) = 3 Truncating Equation D.241 to first-order, and substituting the mode-switching function p0 and into Equation D.240 yields, *(«•»,(«))_ (1_e(-^r»)JM) p.243) dt dt The movement contribution term Ei(tk) for each coordinate direction with respect to environment gradients appearing in Equations 4.121 then becomes, tk+l Ed t k where yp is defined by, o + l , N «(**)= / JE^ 1 - ^ \" 2 ^ \"P))da (D,244) = 7 - Z — / \\ z(tk - a) - %(tk - a) \\ da (D.245) tk—Tp Appendix D. Chapter 4: Background, Derivations, Extended Results 567 as the time-average separation-distance between the position of an individual or shoal to a local environmental gradient in the model. In general, for a gradient //(tfc — tk-i), the equation for the time average of separation distance is, tk 5?(tfc,tfe-i) = T — - . f (u{tk) - u(tk^))du (D.246) tk — t f c - i J t t — i Appendix D. Chapter 4: Background, Derivations, Extended Results D.6 Multi-Layered spatial environment D.6.1 Observed environmental association of shoals Appendix D. Chapter 4: Background, Derivations, Extended Results 569 July 1995 0.1 0.2 0.3 0.4 Flow magnitude (m/s) 0.5 1.0 1.5 2.0 Mean concentration (ug/1) 2.5 • — i 1 r -50 -100 -150 -200 -250 -300 -350 , a c 120 -o 3 o-2 80 -u. 40 -o -0.0 0.5 1.0 1.5 2.0 2.5 Mean concentration (ug/1) 30 25 20 15 10 5 0 (e) Bathymetry Ui llhi 0 -50 -100 -150 -200 -250 -300 -350 Depth (m) 50 -, 40 -(J c 30 -o 3 er 2 20 -u. 10 -0 , u c 60 -u 3 req 40 -u-20 -0 -0 200 400 600 800 1000 1200 2 2 Mean concentration (N x 10 Im ) Figure D . 2 1 7 : Same as Figure D . 2 1 5 for September, 1995. Appendix D. Chapter 4: Background, Derivations, Extended Results October 1995 0.1 0.2 0.3 0.4 Flow magnitude (m/s) o c 13 0.5 0.0 0.5 1.0 1.5 2.0 2.5 Mean concentration (ug/1) 0 -50 -100 -150 -200 -250 -300 -350 Depth (m) o c o 0 5 10 15 20 25 30 Mean sea-surface temperature (°C) 0 200 400 600 800 1000 1200 2 2 Mean concentration (N x 10 /m ) Figure D.218: Same as Figure D.215 for October, 1995. Appendix D. Chapter 4: Background, Derivations, Extended Results 573 July 1996 0.1 0.2 0.3 0.4 Flow magnitude (m/s) 0.5 c Mean sea-surface temperature ( C) Mean concentration (ug/1) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 2 2 Mean concentration (N x 10 Im ) 50 -100 -150 -200 -250 -300 -350 Depth (m) Figure D.219: Frequency distributions of environmental variables (Current flow veloc-ity, SST, chlorophyll-a concentration, zooplankton abundance, and bathymetry for BFT shoals observed in spotter-surveying (July, 1996). Appendix D. Chapter 4: Background, Derivations, Extended Results 574 August 1996 o 2 a 0.1 0.2 0.3 0.4 Flow magnitude (m/s) 0.5 1.0 Mean concentration (u.g/1) 0.5 -50 -100 -150 -200 -250 -300 -350 Depth (m) Mean sea-surface temperature ( C) 200 400 600 800 1000 1200 2 2 Mean concentration (N x 10 Ira ) Figure D.220: Same as Figure D.219 for August, 1996. Appendix D. Chapter 4: Background, Derivations, Extended Results September 1996 0.1 0.2 0.3 0.4 Flow magnitude (m/s) 60 I 40 3 5\" 20 H 0 (c) Chlorophyll-A 0.0 0.5 1.0 1.5 2.0 Mean concentration (ug/1) 0.5 2.5 0 -50 -100 -150 -200 -250 -300 -350 Depth (m) 0 5 10 15 20 25 30 Mean sea-surface temperature (°C) 0 200 400 600 800 1000 1200 2 2 Mean concentration (N x 10 /m ) Figure D.221: Same as Figure D.219 for September, 1996. Appendix D. Chapter 4: Background, Derivations, Extended Results October 1996 F l o w magnitude (m/s) o c \"CP o3 4-3 CO 4 ^ CO CP +J CP 43 4 J bO fl 1 43 t-< o 03 O CO •J3 S3 to O +3 +J o3 fl 4-= x> co :t3 t - i fc! O co - ; g CO co J3 + 3 CO CP -t-s CP 4-= CP fl 1 4-3 4 2 03 4 3 O t - i cu 43 4 ^ CP CO M o3 T3 t - i 4 3 fl a A - S CP •a* I « o CP > t H CP CO 4 3 O as 32 43 +^ fl .23 03 03 +j 4-> CO i H CP 4-3 4-3 CP CP CP M 4 J O CP P Q, r - H £ 4 1 EH £ 43 fl 03 CT1 > CP O O £ O O O.o P. M M ' i CD O O O O O O O O d o d o o CD oo o CC l O N T f n< iq iq o d d d V V 8 q ~ V v v r - l i—I o o S o o g o o 0 -V V s. LO CD T f T f 00 co ^ o> T f £r l>- . CN CO . Tt< 5 CN N 2 CN CN O l ° . « O M v S S S O 00 LO O 00 CO 00 CO CO 00 C D o o o o o o o o o g §3 o o o o q jg £j d d d d o 0 - 0 ' oT T f ' cJ\" 55 S3 H N ^ ^ S M Cf) CN M CO N ™ ™ d d d d d ° ° o o o o o o o o o o o T f O O O O O Q iv' O \" O T ^ OJ\" 22 H W O l CD CO y CO LO LO C D t> \"i d d d d d ° o o o o o q p p p p d d d d d CN rH OJ LO O 00 CO OJ O O T f i>- iv oq os d d d d d r H O o t -d p CO CN OJ CN § P V v, CN OJ LO LO fT. rH d d «-; o oo o o V ° \\ , LO O o LO CO OJ , CD •i LO O O r< O O P P d d LO' O t-H LO CN ec3 40 r H C N C O T f T f r H C N C O T f O O O O O O O O O C O C D C D C D C O N N S N 0 1 0 ) 0 ) 0 ) 0 ) 0 3 0 0 3 0 ) Appendix D. Chapter 4: Background, Derivations, Extended Results 580 o a 3 a* a 3 s •3 u 1.0 0.8 0.6 0.4 ] 0.2 0.0 r rrt; rf i\"\\ I i l i t N 9601 9602 9603 9604a 9604b 9701 9702 9703 9704 9705 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Flow Magnitude (m/s) 0.1 0.2 0.3 0.4 Flow Gradient (m/s/Aj.) 0.5 Figure D.224: Same as Figure D.223 for water flow velocity (m/s). A range of gradi-ent response curves corresponding to Equation D.241 are shown for ep = (1.6 — 3.3), Xp = (0.004 - 0.200), 0P=1. Appendix D. Chapter 4: Background, Derivations, Extended Results I S-i co 1 SH co t -Q CO a3 a3 C O T t Q jo o o o o o o o o o o q q o o o o o q LO o d d d d d d d d o V V V V V V V V V O l T - H T-H Tf T - H rH 00 t - i>- c o LO 00 00 00 oo LO Tf Tf o q CO CO O l O l t - LO LO t> o d d d d d d d d ,—v —v ,—.. —, .—v ,—v ,—^ T - H T - H T - H rH rH rH rH o o o O O O O q q o O O O q o d d d d d d d o V 1 O) ocf Tf C\" t> o d d d d d d d T - H T - H T - H rH rH rH rH o o o O O O O o o o O O O O CO d d d d d d d o t— V V V V V V V i O l O l o> Tf 00 oo CO rH t- 00 T - H LO t -q q d d d „—v ,—s T - H T - H o O o o CO d d o CD t i i i i i i O l T - H o ' o T - H LO Tf d d T - H o q CM d o CO i i i i i i > O l o 00 d ^! cS - D o T-H CN CO Tf Tf rH CM CO Tf CS o O o o o O O o O CO co CO CO CO t - t -H O l O l o> O l O l O l O l O l Appendix D. Chapter 4: Background, Derivations, Extended Results 582 100 200 300 Depth (m) 400 o > o 100 200 300 400 Depth (m) 2.0 Jr \\W \\ Bathymetric Gradient (m/Ay) Figure D.225: Same as Figure D.223 for bathymetry. A range of gradient response curves corresponding to Equation D.241 are shown for ep = (0.0001), Xp = (0.0005 - 0.0600), 0P=1. Appendix D. Chapter 4: Background, Derivations, Extended Results CO a S-l co r— Q CP I CO c3 cd CO L O b-Q CD 0 ..—^ — .-—~ T - H O l O l O l LO r H O l o r H o O l O l O l CO o O l CN o o O l O l O l cq o O l T f o 9705 d d d d d d d d d V 9705 V *—' ~-—' — —y V • — 9705 o r H O l O l b - T - H 789( LO CO r H r H LO CN b - 00 789( CO LO 00 b - b - 00 789( d d d d d d d d d T-H CO LO CO o o o o o o o o o o o p p p o d d d d d d d V V V V V V V i O l CO O l co T f o T f T f oo CN 00 b - co O l r H o CN cq b - cq LO O l b - b -d d d d d d d d T - H co~ ioT co\" o r H i(0.435) CO CO T - H co LO o i(0.435) CO o b -o d V (0.4 (0.4 (0.4 (0.8 •o>) i(0.435) i O l 683( CN r H o 00 T f O 683( 00 b - 00 LO b - O J 00 683( O l O l O l O l O l d d d d d d © — v , s T—t o o? ST oT oo\" r H O l O l O l CO O CN o o O) O l O l cq O d d d d d 0> V — s — ••—^ — ' 0> 1 • • 01 683( o r H b - o 944( 683( LO 00 T - H 00 b -oq i-H o> 944( d d d d O d — v ,—^ ..—.. . .— v T - H r H r H T - H o o O o p p O p T—t d d d d «o o b - V, «o c 1 i i O l CN CO •Tf ' co' co O O) O l b - oq co b - O l 00 d d d d d . .— v T - H o? co' 00 o O l co t-o O l CO CO T f d d d d o V 1 1 • CO O l 683( CO 00 CN O CN O l 683( co O l CO d d d d - — * ^ , , T - H O l O l O p O l O l 03 O l O l T f d d d o V i • 1 1 i i CO cn 683( LO b -00 T f 683( CN 00 d d d ^ v r H O O 999) CO o CO d V ©, i , i , , , , 01 j0.683( 0.808 ,—s T - H o p CN d o CO V i i i i 1 1 i i O l |0.652( J4 03 r Q 13 r H CN CO T f T f r H CN CO T f O O o O o o o o o CO CO CO CO CO b - b - b - b -O l O l O l O l O l O l O l O l O l Appendix D. Chapter 4: Background, Derivations, Extended Results 584 0.0 0.4 0.8 1.2 1.6 Chlorophyll-A Concentration (p.g/1) o _o > o 4) s o a. E o V N 1.2 0.8 i 0.4 0.0 \\ v Chlorophyll-A Gradient (10\"3) Figure D.226: Same as Figure D.223 for chlorophyll-a (ChlA) concentration (pg/l). A range of gradient response curves corresponding to Equation D.241 are shown for ep = (1.6 - 3.3), Xp = (0.004 - 0.200), 0P=1. Appendix D. Chapter 4: Background, Derivations, Extended Results o O O O o o O o O o O O q o o O o O LO d d d d d d d d d o V V V V V V V V V cn CT) V V V V V V i i 1 cn 00 CN CO CN CN I V o CT) rH o LO LO CD Tf Tf CD CO 00 d d d d d d T—1 rH rH rH rH O O O O O o O q q q r-H d d d d d O I V V V V V V i i i 1 cn 0.367 0.659 0.918 0.678 0.732 — ^ — ^ , — S . — S rH T—1 rH rH O O O O O o O O Tf d d d d o V V V V i i i i 1 CO cn 10.475 0.535 0.947 0.329 . — . , — S , — , rH rH rH O O o 03 Tf O O o d d d O V V V i i i i i 1 CD cn 0.332 0.396 0.803 — V „—.. rH rH O O O q CO d d o CD V V i i i i i i 1 cn |0.947( 0.662( — ^ rH o q IM d O CO V i i i i i i i 1 cn 0.535( ee x> o rH CN CO Tf Tf rH CN CO Tf cs O o o o o O o o o CD CD CD CO CO I V I V I V I V CT) CT) CT) CT) CT) CT) CT) CT) CT) Appendix D. Chapter 4: Background, Derivations, Extended Results 586 1.0 H . 2 Zooplankton Concentration (10 N/m ) Zooplankton Gradient (104 N/m 2/^) Figure D.227: Same as Figure D.223 for zooplankton concentration (iV/10ra 2). A range of gradient response curves corresponding to Equation D.241 are shown for ep = (1.6 — 3.3), A p = (0.004 - 0.200), /? p =l. Appendix D. Chapter 4: Background, Derivations, Extended Results 587 L O o b -o O cn cn CT _ _ CT CT CT ° P CT CT CT V S ° ° o d V « op 53 CT o o o o o CT co rr CT 2 CT CO P \" o V O M O H o 3 3 £ CO ^ ^ T f <= O O o o o d V O i — i i—i i — i r r C T r - 1 o O O O O S L O O L O C N O O O P T f O C N o V o o o o o o o 0 0 CO CN CT 0 0 CT CN CO CO L O b -c5 c5 d d CN T f L O C N C O CO C N L O C O b -b -CN o o o o CO CO O T f T f T f T f © P . P . P . v P d co T - I C N x a i ' c S ' o C N O C N L O X 2 C O L O b - T f 0 0 QQ [I. CO d d P o o o o CT CT CT CT CT CT O O P . CT CT CT P P . ° d d d ° ° V 2 - S S V V CN T? - * d P CT O CO O O CO L O b- b -CT L O O O O O O o p p p q d> ci c5 d> d> \\/ v, ST b^\" b - T f \" T f ' T f L O r H b - b -CO T f CO CT CT c5 d ci ci c3 T f o CO CT O O O O p o p p d d d d C O C N r H C O L O 0 0 b - r H b- CT CT C N d d d d _ CT CT P g g ?n CN r H S ?P b -o o P CT CO o3 OS - D CN CO T f T f r H CN CO T f o o o o o o o o C O C O C O C O b - b - b - b -CTCTCTCTCTCTCTCT Appendix E Chapter 5: Background, Derivations, Extended Results E . l A review of spatial statistics in survey design Surveying of a population allows objects of interest to remain undetected within the field of observation. This forms an important distinction to census approaches which aim to enumerate all possible objects. Given the detection of n objects, survey meth-ods estimate how many of them are within the sampled area over time. Selection of a sampling design must consider aspects of the object detection: visibility, sightability, unit/object definition, object size and characteristics [94,323]. Object visibility relates to environmental conditions (e.g., weather and sea-state) and time of day and is the degree to which an object within the detection range of an observer can be detected relative to optimal conditions. Sightability can be defined as the degree to which objects can be detected under optimal visibility conditions. Sightability is dependent upon the distance between an observer or detection instrument to an object varying in both the horizontal and vertical (depth) spatial dimensions, and time of day. Sightability is biased due to a varying ability of different observers to detect objects, being independent of the various factors believed to determine whether an object is sighted or not. Visibility and sighta-bility are closely related factors, and reducing bias in each of these factors can involve the analysis of survey data under replicate sampling experiments [343]. Primary units or objects in fish surveys may be defined as shoals or individuals. When individuals are organized into shoals, shoals become the primary unit of a survey. 588 Appendix E. Chapter 5: Background, Derivations, Extended Results 589 Shoal size estimation and uncertainty must therefore be considered in the estimation of population abundance and density. Shoal size estimates are biased by as a function of the distance between an observer and a shoal, and may also be further biased by the observable characteristics of shoal structure, in the case where shoals, being three-dimensional objects are observed across only two spatial dimensions (e.g., aerial surveys), and when observers estimate shoal size under a qualification of shoal structure. If the primary survey unit of observation comprises a collection of individuals, in the case of surveys of shoaling fish populations, estimation and variability in the number of individuals in a shoal is required. Variation in group or shoal size, increases the coefficient of variation (i.e., reduces precision) in the density estimation based on individual counts [94]. In addition to increased variance in abundance or density estimates, the detection of shoals from a observation line or point is biased by shoal size, whereby larger shoals have a higher probability of being detected than smaller shoals as a function of increasing distance from an observer [51]. The recording of object sightings, detection distances with respect to survey observers, and the use of aerial photography can provide important data that can be analyzed to reduce the bias in estimates of shoal size. The fundamental problem with line and point transect sampling theory is that abundance estimation is increasingly sensitive to the absolute sample size as population size increases, and are not dependent on the fraction of a population sampled. The necessary size of a sample (e.g., number of transects) is therefore an important factor in survey design for reliably estimating population abundance. For populations comprising groups of different sizes that are aggregated in space and distributed across a large high coefficient of variation (low precision) in abun-dance makes it difficult to design an effective and reliable sampling scheme. Aggregation of shoals leads to further complications in the estimating of both survey uncertainties and biases under any survey design, as visibility (i.e., sea-state), sightability, object definition, Appendix E. Chapter 5: Background, Derivations, Extended Results 590 size and characteristics may both be correlated and/or vary differently within aggrega-tions, in contrast regions where aggregations are not present. The factors believed to govern the formation of aggregations that include climate, oceanographic, habitat, social behaviour and movement, predator-prey trophic interactions become increasingly impor-tant in the design of surveys where populations are aggregated, with a greater degree of possible interaction as a result of being closer together [303,304]. Furthermore, as a result of a varying interaction probability between shoals distributed in space within and between aggregation structures, survey encounter rate and shoal size may be dependent due to the movement, behaviour and size of shoals being dependent strongly dependent on their interaction (density-dependent interaction). The assumption of an underlying random distribution of objects when populations are aggregated is not valid. Traditional survey schemes establish spatial plots or quadrants at random (e.g., cir-cular, square or rectangular) using finite estimation of density as the number of object counts per unit of area [146]. The random allocation of sampling within these plots is necessary for unbiased estimation of population density, as the population being sam-pled is considered to be distributed according to an unknown underlying spatial process. Systematic and cluster sampling ensures homogeneous coverage of the area in which a population is distributed and permits the random allocation of sampling starting loca-tions. Jolly and Hampton articulate that no distributional assumptions are required to derive unbiased estimates of mean and variance in traditional/conventional sampling methods [184,185]. When population structure is spatially aggregated, stratified survey sampling, where greater sampling effort is allocated to quadrants proportional to population density, can be used [352,357]. However, as a result of aggregation, survey spatial data contain a large proportion of zero counts, imposing the requirement for a theoretical consideration Appendix E. Chapter 5: Background, Derivations, Extended Results 591 of how a population is distributed in space. The statistical A-distribution used is lognor-mal and provides an allowance for zero or null observations has been applied to fish and plankton surveys, by Pennington, acoustic surveys by MacLennan and MacKenzie, and spotter-pilot aerial surveys of northern anchovy (Engraulis mordax) [211,228,298-301]. Similarly, use of a theoretical negative-binomial distribution is commonly applied in the analysis, modelling and simulation of aggregated populations [302,389]. Under distri-butional assumptions, the statistical analysis of survey data can involve a consideration of regions containing no or a small probability of sightings to provide more reliable and efficient abundance estimation [211,299,301]. Sampling designs for large-scale surveys of populations containing units of very dif-ferent sizes, typically involve a consideration of population size stratification, ratio es-timation, and survey effort allocation. A key question is how to sample an aggregated population and allocate survey effort while designing and delineating survey sub-regions on the basis of different sighting probabilities. Brewer details a combined estimation and selection scheme for large-scale surveys, and discusses several classes of robust sampling designs [39]. A robust survey scheme based on unequal probability sampling is advan-tageous in large-scale surveys by removing the requirement for allocation and design of spatial stratification based on population size [59,152,210]. Existing survey methods that consider population stratification and unequal sampling probabilities are adaptive, iterative schemes that assign sampling intensities in a given quadrant to previous density estimates derived by imposing a different size and geometry of sampling quadrants [390-392]. Aerial surveys typically consist of simultaneous observers following parallel transects conducting measurements independently. Strip transects in aerial surveys, delineated according to flight transects having a range of detection perpendicular to flight direction (i.e., transect width) are often used to form samples based on an interpolating spatial Appendix E. Chapter 5: Background, Derivations, Extended Results 592 grid for calculating, standardizing or calibration various factors affecting object sightabil-ity. The line-transect method is often preferred over the point-transect method because samples are obtained according to random spatial assignment within an area [332]. More sampling time at low cost is provided in transect sampling compared to the time re-quired to travel between random points in areas. However, point-transect methods are more advantageous if sampling points are allocated to occur randomly along fixed tran-sect lines. It is important that transect direction does not parallel physical or biological features/gradients as such factors are then co-variates introducing systematic bias in abundance estimates [335]. Instead, transects should extend between the boundaries of a study region, of unequal length, and be placed at an effective distance to ensure that objects are not detected on separate transects. For surveys repeated over time for the purposes of examining trends in abundance and population density, the sampling time interval should be large enough so that successive object sightings are not correlated. Moreover, the speed of movement of objects must be considered relative to the spatial and temporal interval of sampling [402]. For the case of stationary objects, no bias is introduced if a single object is repeatedly counted on multiple lines or points. When objects move after detection across sampling lines/points in a short time period, it is critical that abundance estimation involve filtering any potential bias of multiple object counts [343]. For populations that are believed to exhibit smooth spatial trends in the large-scale density over an area, the sampling lines or points can be systematically, rather than, randomly placed. Fiedler has previously simulated survey transects for model anchovy populations, and compared several different survey designs in terms of their precision and efficiency [102]. Three transect survey designs were compared: systematic, random and stratified sys-tematic. Random surveying consists of a series of transects separated by a randomly de-termined distance interval between transect lines, whereas systematic surveying consists Appendix E. Chapter 5: Background, Derivations, Extended Results 593 of a series of transects separated by a constant transect interval. Stratified systematic surveying involves dividing a population into spatial strata, whereby a series of tran-sects simulated within each strata. Simulations of the three survey schemes show that systematic sampling may result in considerable gains or losses in precision compared to random-transect sampling [102]. The greatest increases in precision occur when a high degree of correlation between adjacent transects is present. This correlation decreases as the interval between transects increases, such that systematic sampling approaches a stratified sampling scheme for appreciably large transect intervals, and for appreciably small transect intervals, approaches a random sampling scheme. Therefore, the preci-sion of systematic surveys rely on an accurate determination of transect interval with precision varying inversely with the distance between transects. Precision may be reduced when periodic variation in a population is present and the interval between transects matches the period of variation in abundance. In the case of shoaling populations, where survey abundance involves estimation of shoal size, periodic variation in abundance or spatial autocorrelation is a result of variability in the frequency distribution of shoal size. If the mean and variance in detecting shoals (i.e., shoal encounter rate) is spatially correlated within in different sub-regions of a survey area, stratified systematic surveying is expected to provide more precise abundance estimates by allocating more survey effort in proportion to the variability in abundance. Spatial simulation of surveys verify that the patchy or aggregated distribution of fish shoals is a major source of error in population abundance estimation, and the precision of stratified systematic surveys may not significantly different from unstratified surveys when shoals are randomly distributed. Using a simulation-based approach, Jolly and Hampton have recently demonstrated, however, that in the case of a stratified random transect scheme, precision in abundance estimation was increased by a factor of 2:4 compared with unstratified random transect Appendix E. Chapter 5: Background, Derivations, Extended Results 594 sampling, based on a uniform allocation of survey effort in each strata. The difference between the stratified systematic and stratified random transect designs is based upon whether survey transects are spaced at random or constant interval within each strata. When survey effort as the number of transects were allocated, instead, in proportion to the expected encounter rate within each strata, a further increase of 1.4 compared to uniform allocation in the stratified scheme was achieved with a coefficient of variation in abundance of 16%. These results suggest that for populations with aggregated shoals, stratified systematic schemes may be more precise than stratified random survey designs, dependent upon a reliable determination of encounter rate within spatial strata. Shoal size may vary between strata with size estimates biased as a function of the distance between shoals. When shoals are separated by small distances, the ability of observers to distinguish separate shoals and determining their sizes may become increasingly difficult. Squire has previously estimated the apparent abundances of several pelagic fishes off the California coast from aerial surveys conducted by spotter-pilots [379]. This study provides a number of important variables that affected the statistical accuracy of the spotter-pilot data that were also difficult to evaluate. These are the individual differences in the ability of spotters to: (i) locate surface shoals, (ii) characterize shoals according to the target species, and (iii) estimate the size of shoals. Squire concludes that variation in the estimation of shoal size was the dominant source of survey error across individual spotters. The spotter-search scheme target shoal aggregations - regions where popula-tion abundance is high. The main distinction between spotter and transect sampling is that sampling points are conducted in an non-linear fashion rather than along a transect line. For all survey schemes, navigation error makes it impossible to replicate both lin-ear transects and nonlinear, spotter search paths without error. Considerably variability is presented in the sampling paths of a spotter search scheme. If the search paths are considered to vary according to shoal sightings whereby observers attempt to continually Appendix E. Chapter 5: Background, Derivations, Extended Results 595 direct their search on the basis of first detecting a shoal (i.e., primary sighting probabil-ity), and subsequent search being directed in its spatial vicinity or neighbourhood (i.e., secondary sighting probability), the variability of spotter search sampling may be seen as a highly adaptive scheme for maintain encounter rates and targeting regions of high abundance. Spotter search may be considered a higher-order scheme, resembling the adaptive survey sampling which involves spatial stratification of a survey area according to unequal detection or sighting probabilities. A theoretical comparison of spotter search and adaptive stratified survey sampling schemes must therefore rely on an examination of the variability of sighting/detection probabilities and how this variation is related to the underlying spatial structure of a population. Under theoretical assumptions of population spatial structure and its variability in time, survey measurement errors associated with interpolating observations from spotter search and adaptive survey schemes can be determined. Angulo and coauthors have re-cently investigated the problem of space-time optimal sampling design [9]. The key point of their investigation was to demonstrate how interactions taking place between shoals in populations affects the configuration of possible alternative sampling strategies. This study indicates the importance of formulating specific optimality criteria related to sur-vey designs, and shows how an optimal sampling scheme is decomposed into two parts: (i) the spatial distribution of sampling sites, and (ii) the time frequency between succes-sive samples. The spotter search scheme attempts to vary its effort according to both components: when and where shoals are detected. This scheme relies on efficient match-ing of the dynamics of the underlying space-time stochastic processes that determining population structure, abundance and their variability. The application of geostatistical interpolation and bootstrapping extrapolation tech-niques can be compared to the predictions of spatially-explicit population models. This approach may be the most adaptive method for estimating abundance and its variability, Appendix E. Chapter 5: Background, Derivations, Extended Results 596 even though there is a strong reliance of survey design and measurement sampling on population dynamics. Simmonds and Fryer have considered acoustic survey design and the estimation of mean abundance for spatially autocorrelated populations. They have investigated how the precision of survey schemes in both space and time relate to the objectives of estimating mean and variance in abundance. They examined the how the choice of survey design affects the bias and precision of the sample mean as an estimator of mean abundance. They also investigated different ways to estimate the error variance of the sample mean, using: (i) variance estimates pooled within spatial strata, and (i) geostatistical model estimators of type spherical and exponential. By simulating different survey schemes, the results of their study identify that the best strategy has: (a) a sys-tematic survey with use of a geostatistical variance estimator, when the main objective of a survey is to obtain the most precise estimator of abundance, and (b) a stratified random survey with two transects per strata and a pooled variance estimator, when the objective is to obtain a precise estimator of variance in abundance. A single survey scheme which is able to provide precise estimates of both mean and variance in abundance is conceivable. Such a scheme may be developed by applying appropriate constraints on a spotter search scheme devised on the basis of a knowledge of the underlying population dynamics. Similarly, a two-stage survey scheme that combines use of the same or different sampling methods could also be devised. Brown has proposed a two-phase sampling design could be used, in the case of estimation of B F T abundance. This approach has been applied in aerial surveys of southern bluefin tuna (SBT) in the Great Australian Bight [48,75]. This survey design samples according to a line-transect scheme, factoring the estimation of abundance into the two steps: (1) estimation of mean shoal size and the number of shoals within the survey region, and (2) replicate spatial sampling to determine population variance. Whether a single or multiple-staged survey is applied, repeated surveying of a population is crucial for providing estimates Appendix E. Chapter 5: Background, Derivations, Extended Results 597 of the encounter rate of shoals and to estimate variance in abundance. Image analysis techniques of photographic and acoustic data can be used to improve the estimation of mean and variance of shoal size and associated shoal detection and characteristics. Appendix F Curriculum Vitae B I O G R A P H I C A L N O T E S Born: September, 21 s t-, 1973. Toronto, Canada Academic Studies: M.Sc. Astrophysics, University of Calgary, Alberta, Canada, 1997 B.Sc. (Honours), Mathematics and Physics, University of Guelph, Ontario, Canada, 1995 B.Sc. (Year 1), Biology, University of Waterloo, Ontario, Canada, 1991 G R A D U A T E STUDIES Field of Study: Fisheries Science Courses Description Instructors R M E S 501 Perspectives in Resources and Environment Dr. L . M . Lavkulich R M E S 502 Graduate Seminar in Resources and Environment Dr. L . M . Lavkulich R M E S 500B Resource and Environmental Workshop Dr(s). H.E. Schreier & K . J . Hall FISH 504 Quantitative Analysis in Fisheries I (audit) Dr(s). T . J . Pitcher & D. Pauly FISH 505 Quantitative Analysis in Fisheries II (audit) Dr. C.J . Walters FISH 506 Critical Issues in Fisheries Development: (Modules): World Fisheries - Status and Prospects Dr(s). T .J . Pitcher and D. Pauly Ecopath Ecosystem Modelling Dr. D. Pauly 598 Appendix F. Curriculum Vitae 599 Sustainable Fisheries Certification - International Protocol Dr. T . J . Sproul The Applicability of Sociological Analysis to Fisheries Research Dr. R. Matthews A W A R D S 2001 Student Scholarship Conference Award, 52nd. International Tuna Conference. Lake Arrowhead, C A , U.S.A. 2000-2001 University Graduate Research Fellowship Award, U . B . C , Vancouver, B.C. , Canada. 2000 Student Scholarship Conference Award, 51st. International Tuna Conference. Lake Arrowhead, C A , U.S.A. 1. Newlands, N . and Lutcavage, M . From individuals to local population densities: North Atlantic Bluefin Tuna (Thunnus thynnus). J.R. Sibert and J.L. Nielsen (eds.), Electronic Tagging and Tracking in Marine Fisheries, Kluwer Academic Publishers, The Netherlands, pp. 421-441. (2001) 2. Newlands, N . Learning from Uncertainty: Population monitoring, modelling and simulation of Atlantic Bluefin Tuna (Thunnus thynnus). 5th Annual Workshop of the Institute for Resources and the Environment: Addressing the Knowledge Gap in Water and Energy: Linking Local and Global Communities, 19th February, pp. 18-38. (2001) 3. Pauly, D., Beattie, A . , Bundy, A . , Newlands, N . , Power, M . and Wallace, S. Not Just Fish: Value of Marine Ecosystems on the Atlantic and Pacific Coasts, In: Just Fish: Ethics and Canadian Marine Fisheries. Coward, H. , Ommer, R. and P U B L I C A T I O N S Appendix F. Curriculum Vitae 600 Pitcher, T. (eds.) Institute of Social and Economic Research (ICER Press), Memo-rial University of Newfoundland, St. John's, Newfoundland, Canada, pp. 34-46. (2000) 4. Power, M.D. and Newlands, N . A report on historical, human-induced changes in Newfoundland's Fisheries Ecosystem. Ecosystem Approaches for Fisheries Man-agement, University of Alaska Sea Grant, AK-SG-99-01, Fairbanks, pp. 391-404. (1999) 5. Lutcavage, M . and Newlands, N . A strategic framework for fishery-independent assessment of bluefin tuna. Int. Comm. Conserv. Atlantic Tunas Coll. Vol. Sci. X L I X (4). pp. 1-4. (1999) 6. Mackinson, S., Vasconcellos, M . and Newlands, N . A new approach to stock assess-ment: model-free estimation of stock-recruitment relationships using fuzzy logic. Can. J. Fish. Aquat. Sci. 56(4), pp. 686-699. (1998) 7. Mackinson, S. and Newlands, N . Using local and scientific knowledge to predict distribution and structure of herring shoals. ICES-CM-1998/J:11, Copenhagen (Denmark), 18 pp. (1998) 8. Newlands, N . John Murray's Expedition of the Indian Ocean, International Fish Database, FishBase 98 C D - R O M . I C L A R M , Manila, Philippines. (1998) 9. Newlands, N . Asymmetric emission model of E X O 2030+375. Proceedings of the 11th. Canadian Conference on Black-holes and Relativistic Astrophysics, Univer-sity of Calgary, Calgary, Alberta, Canada. (1997) Appendix F. Curriculum Vitae 601 P R E S E N T A T I O N S 1. N . Newlands, Spatially-explicit individual-based modelling of bluefin tuna, integrat-ing new data from acoustic tracking, satellite tagging and aerial survey experiments PIMS-MITACS Seminar on Mathematical Biology, U . B . C , 2002. 2. N . Newlands, Relative abundance estimation under alternative spatial sampling strategies (FISH 500 seminar), Fisheries Centre, U . B . C , 2001. 3. N . Newlands, Aerial surveying of Atlantic bluefin tuna (Thunnus thynnus), Gulf of Maine: relative abundance estimation under alternative spatial sampling strategies (talk), 52nd International Tuna Conference, Lake Arrowhead, C A , U.S.A, 2001. 4. N . Newlands and M . Healey, Where have all the salmon gone? An environmental sensor array (ESA) with global application (talk), Workshop Presentation/Research Proposal to the Canadian Centre for Innovation in Research: Integrated Monitoring of Salmon Migrational Movements with short and long-range hydrophone transduc-ers and active archival tagging using M E M S technology. Green's College, U . B . C , 2001. 5. N . Newlands, Approaches in population spatial dynamics modelling and simulation: Case of highly migratory, schooling fish populations (invited lecture), Dr. Solange Brault, Department of Biology, University of Massachusetts Boston, Boston, M A , U.S.A, 2001. 6. N . Newlands, Learning from uncertainty: population monitoring, modelling and simulation of Atlantic bluefin tuna (Thunnus thynnus) (talk), 5th- Annual Work-shop of the Institute for Resources and the Environment (IRE): Addressing the Knowledge Gap in Water and Energy: Linking Local and Global Communities, 2001. Appendix F. Curriculum Vitae 602 7. N . Newlands, Stalking the last tuna: monitoring, modelling and simulation of At-lantic bluefin tuna (Thunnus thynnus) ( R M E S 502 seminar), Department of Re-sources and the Environment, U . B . C . , 2001. 8. N . Newlands, From individuals to local population densities: North Atlantic Bluefin Tuna (Thunnus thynnus) in the Gulf of Maine/Northwestern Atlantic, (talk), Sym-posium on Tagging & Tracking Marine Fish wi th Electronic Devices, Honolulu, Hawaii, U . S . A , 2000. 9. N . Newlands, The meso-scale distribution of fish shoals: modelling shoaling pattern using heuristic rules under heterogeneous habitat structure (seminar), Department of Zoology, U . B . C . , 1999. 10. N . Newlands, Spatial-temporal dynamics of bluefin tuna (talk), Fisheries Centre, U . B . C . , 1999. 11. N . Newlands and S. Mackinson, Generating the mesoscale pattern of fish shoals (FISH 500 seminar), Fisheries Centre, U . B . C . , 1999. 12. N . Newlands et al, Mini-symposium on modelling of marine resources ( R M E S 502 seminar), 1998. 13. N . Newlands, The role of shoaling behaviour in the spatial dynamics of range col-lapse in fisheries (F ISH 500 seminar), Fisheries Centre, U . B . C . , 1998. 14. N . Newlands and S. Mackinson, Using heuristics to model the structure and distri-bution of herring shoals (F ISH 500 seminar), Fisheries Centre. U . B . C . , 1998. "@en ; edm:hasType "Thesis/Dissertation"@en ; vivo:dateIssued "2002-11"@en ; edm:isShownAt "10.14288/1.0074854"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Resource Management and Environmental Studies"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Shoaling dynamics and abundance estimation : Atlantic bluefin tuna (Thunnus thynnus)"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/13501"@en .