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Numerical and functional responses of British Columbia trawlers Lapointe, Michael F. 1989

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Numerical and Functional Responses of British Columbia Trawlers by Michael F. Lapointe B.Sc, University of Maine, 1980 A Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in The Faculty of Graduate Studies (Department of Zoology) We accept this thesis as conforming to the required standard The University of British Columbia September 1989 ©Michael F. Lapointe, 1989 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. Department of ~&&0 i-O The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract Numerical responses were examined in the movement of trawlers among seven aggregate fishing areas off the British Columbia coast. Three hypotheses for movement patterns were tested: (1) Movement follows traditional patterns, (2) Movement equalizes the gross dollar returns to effort (LPE$) in each area, and (3) Movement maintains relative LPE$ in each area. On the interannual time scale, I rejected the Equalize LPE$ hypothesis, but failed to reject either the Traditional Patterns or Maintain Relative LPE$ hypothesis. On the intra-annual time scale, I rejected both LPE$ hypotheses, but was unable to reject the Traditional Patterns hypothesis, although traditional movement patterns were evidently being modified by changes in the timing of LPE$ in all the areas and by fishing regulations in two of the seven areas. Three assumptions of the Equalize LPE$ hypothesis were violated, and accounting for violations (especially assumptions concerning movement and areas specific costs) would result in smaller differences in LPE$ between areas and between years within areas. Numerical responses have implications for anticipating the responses of fishing fleets to changes in regulations and for evaluating changes in the economic benefits of alternative fishing areas. Functional responses were examined in a multispecies fishery in Hecate Strait. Two similar techniques for estimating the abundance of exploited fish populations were compared: (1) Virtual Population Analysis (VP A) and (2) Catch-at-age analysis with auxiliary information (CAGEAN). Estimates for each of three species (Pacific cod, English sole, and rock sole) from both techniques were sensitive to the choice of natural mortality rate but insensitive to the choice of fishing mortality rate. In most cases, abundance and catchability estimates from ii alternative input parameters were highly correlated and had very similar time trends. Similar time trends in estimates were also obtained from the two techniques when the best estimates of input parameters were used. Estimates of abundances, catchabilities and catch per predator from both techniques were used to examine functional responses of trawlers to fish abundance. I compared the fit of three alternative single species functional response models: (1) a linear model (type I), (2) a saturating model (type II), and (3) a generalized equation that could mimic four different responses (types I-IV) depending on its parameter values. The generalized equation predicted sigmoid (type III) functional responses for 11 of the 12 data sets. However, only 5 of the model comparisons were statistically significant; all five indicated that a sigmoid response was most consistent with the data. Single species mortality models Fit most data sets poorly, although in most cases, the form of the mortality curve was consistent with the corresponding functional response. I also fit two alternative multispecies functional response models and examined one mechanism, switching, that could result in type III responses. Multispecies functional response and mortality models often resulted in much better fits than single species models for each species. I failed to detect switching in any of the data sets, although the power of the tests of the switching hypothesis was low in most cases. Type III functional responses and multispecies functional responses have implications for: (1) interpretations of abundance indices based on catch per unit of fishing effort, (2) equilibrium yield vs. abundance or harvest rate relationships and (3) simulation models used to evaluate alternative harvest strategies. iii Table of Contents Abstract ii Table of Contents : iv List of Tables , ix List of Figures xii Acknowledgements xvi General Introduction 1 Numerical Responses 4 Functional Responses 5 Chapter 1: Literature Review of Numerical Responses in Fisheries ...7 Introduction ". ; 7 Models of Fisherman's Choice 8 Applications of Search Theory and Optimization 10 Simulation Models 15 Empirical Studies 20 Chapter 2: Effort Allocation in the British Columbia Trawl Fleet 22 Introduction 22 Data and Methods 24 Description of the Fishery 24 Data Sources and Statistical Tests 26 Description of Areas 30 Results 31 Mobility in the Trawl Fleet 31 Traditional Patterns 34 Hypotheses Using Landed Value Per Unit Effort 41 The Dilemma of Effort vs. LPE$ Correlations 41 Numerical responses of effort to LPE$ 46 Equalize LPE$ 51 Maintain Relative LPE$ 52 Summary 55 Chapter 3: Evaluation of Assumptions 58 iv Introduction 58 All Boats are Equal 59 Data and Methods • 60 Results 66 Equal Costs and Desirabilties of Areas 70 Negligible Movement Costs 74 Perfect Information • 76 Historical Data • 78 Personal Experience 78 Plants and Other Fishermen 79 Summary of Evaluation of Assumptions 87 Chapter 4: Discussion of Effort Distribution Analyses 88 Introduction 88 Other Applications of the Hilborn-Ledbetter Hypotheses 88 Data Problems and Statistical Issues '. 94 Data Problems 94 Statistical Issues 95 General Implications • 96 Chapter 5: Literature Review of Functional Responses in Natural Predators • 98 Introduction 98 Functional Response Equations '. 100 Type I Equations 100 Type II Equations 100 Type III Equations 101 Type IV Equations ...103 A Generalized Equation 105 Instantaneous vs. Exploitation Equations 106 Functional Response Mechanisms 107 Type I and II Mechanisms 108 Type III Mechanisms 109 Predator Learning 110 Prey Patchiness or Prey Refugia 113 v The1 Switching Mechanism . ; 113 The Switching Hypothesis .... .. 114 Mechanisms for Switching 117 Switching and the Functional Response 117 Type IV Mechanisms 119 Functional Responses and Stability 120 Chapter 6: Functional Responses and the Fishery 123 Introduction 123 Type I Responses '. 125 Type II Responses 132 Evidence for Type II Responses 137 Type II Mechanisms in Fisheries 143 Type III Responses 144 Type IV Responses 152 Chapter 7: Estimates of Abundance : 154 Introduction 154 Study Area and Fishery Description 155 Data Preparation and Background Methods 159 Aging Methods 159 Estimating Numbers Landed-at-Age 162 Effort Data 164 Models, Data Requirements and Assumptions 165 Virtual Population Analysis 165 Model Description • VPA 165 Data Requirements - VPA 167 Model Assumptions - VPA 169 Catch-at-age Analysis with Auxiliary Information (CAGEAN) 172 Model Description - CAGEAN 172 Data Requirements - CAGEAN 177 Model Assumptions - CAGEAN 178 Application of Models and Specific Methods 179 Computer Programs and Starting Input Parameters .179 Estimates of Abundances and Catchabilities 181 vi Sensitivity Analyses 182 Results 183 "Best" Estimates - VPA and CAGE AN 183 Sensitivity Analyses - VPA 187 Alternative M Values 187 Alternative Terminal q Values 189 Sensitivity Analyses - CAGEAN 194 Alternative M Values 194 Alternative Terminal F Values 196 Alternative X Values 201 Comparison of Estimates from VPA and CAGEAN 206 Summary of Results and Conclusions 208 Chapter 8: Functional Responses of Trawlers 210 Introduction 210 Data and Methods 212 Data Sets 212 Alternative Single Species Models 214 Parameter Estimation 215 Comparison of Models 216 Calculation of Power 218 Examination of Residuals 221 Multispecies Models ' 222 Tests of the Switching Hypothesis 224 Results 225 Comparison of Single Species Functional Response Models 225 Pacific cod 225 English sole 230 Rock sole 234 Single Species Mortality Curves 237 Pacific cod 237 English sole 241 Rock sole 245 Multispecies Functional Response Models 250 vii Pacific cod • 250 English sole 253 Rock sole 255 Multispecies Mortality Curves 257 Pacific cod 257 English sole 261 Rock sole 264 Tests of the Switching Hypothesis 267 Summary of Results 267 Discussion 270 Outstanding Issues 270 Statistical Issues 270 Biological Issues .". 273 General Implications 277 Implications of Type III Functional Responses 278 Implications of Multispecies Functional Responses 282 Perspectives for Future Research 284 Literature Cited 288 Appendix A 306 viii List of Tables Table 2.1. Price indices for selected groundfish species or groups 29 Table 2.2. Summary of mobility statistics for the trawl fleet 33 Table 2.3. Spearman rank correlations between LPE$ and days fished for ten vessel horsepower classes and seven areas. 44 Table 2.4. Spearman rank correlations between individual vessel LPE and number of trips made by all vessels to the same area during the same week for ten horsepower classes 46 Table 2.5. Kruskal-Wallis nonparametric ANOVA comparing weekly LPE$ in the seven aggregate areas 52 Table 2.6. Summary of Kruskal-Wallis nonparametric ANOVAs comparing RPAs between weeks within each of the seven aggregate areas 55 Table 3.1. Kruskal-Wallis nonparametric ANOVA comparing vessel horsepower in the seven aggregate areas 61 Table 3.2. Product-moment correlation matrix for four vessel attributes 61 Table 3.3. Estimated variable and total costs per days fished for three vessel length categories in 1982 72 Table 3.4. Average standardized LPE$s, steaming times/(days fished) (STPE), and vessel horsepowers for each area and the years 1967-80 76 Table 7.1. Product-moment correlation matrices of abundance estimates obtained from VPA using alternative M values for Pacific cod, English sole and rock sole 189 Table 7.2. Product-moment correlation matrices ofcatchability estimates obtained from VPA using alternative M values for Pacific cod, English sole and rock sole 191 Table 7.3. Product-moment correlation matrices of abundance estimates obtained from CAGEAN using alternative M values for Pacific cod, English sole and rock sole 196 Table 7.4. Product-moment correlation matrices of catchability estimates obtained from CAGEAN using alternative M values for Pacific cod, English sole and rock sole 198 ix Table 7.5. Product-moment correlation matrices of abundance estimates obtained from CAGEAN using alternative X values for Pacific cod, English sole and rock sole. .'. 203 Table 7.6. Product-moment correlation matrices ofcatchability estimates obtained from CAGEAN using alternative X values for Pacific cod, English sole and rock sole 207 Table 8.1. Summary statistics for single species functional response model comparisons for Pacific cod '. . 228 Table 8.2. Summary statistics for single species functional response model comparisons for English sole. 231 Table 8.3. Summary statistics for single species functional response model comparisons for rock sole. 235 Table 8.4. Summary statistics for single species mortality model fits to Pacific cod data sets. 238 Table 8.5. Summary statistics for single species mortality model fits to English sole data sets 242 Table 8.6. Summary statistics for single species mortality model fits to rock sole data sets. 246 Table 8.7. Summary statistics for multispecies functional response model fits to Pacific cod data sets 251 Table 8.8. Summary statistics for multispecies functional response model fits to English sole data sets '. 254 Table 8.9. Summary statistics for multispecies functional response model fits to rock sole data sets 256 Table 8.10. Summary statistics for multispecies mortality model fits to Pacific cod data sets .....258 Table 8.11. Summary statistics for multispecies mortality model fits to English sole data sets 262 Table 8.12. Summary statistics for multispecies mortality model fits to rock sole data sets 265 Table 8.13. Summary of tests of the switching hypothesis 268 Table A l . Estimated numbers landed-at-age of Pacific cod 307 Table A2. Estimated numbers landed-at-age of English sole 308 Table A3. Estimated number landed-at-age of rock sole 309 Table A4. Fishing effort for Pacific cod, English sole, and rock sole 310 Table A5. Age specific net fecundities used in CAGEAN for Pacific cod, English sole, and rock sole 311 Table A6. "Best" estimates of starting parameter values used in VPA and CAGEAN for Pacific cod, English sole, and rock sole .; 311 Table A7. Starting input parameters generated by CAGEAN's routine COHORT for Pacific cod, English sole, and rock sole 312 Table A8. Estimated total numbers-at-age of Pacific cod from VPA 314 Table A9. Estimated total numbers-at-age of English sole from VPA 315 Table A10. Estimated total numbers-at-age of rock sole from VPA 316 Table A l l . Estimated total numbers-at-age of Pacific cod from CAGEAN 317 Table A12. Estimated total numbers-at-age of English sole from CAGEAN .... 318 Table A13. Estimated total numbers-at-age of rock sole from CAGEAN 319 xi List of Figures Figure 2.1. Map of seven areas chosen for analysis and two major landing ports for groundfish on Canada's Pacific coast. 25 Figure 2.2. Contribution of the seven areas to weight landed and effort expended 32 Figure 2.3. Proportion of the total annual effort alloted to each area in the years 1967-81 35 Figure 2.4. Proportion of cumulative weekly effort plots for the years 1975-81 in each area. 36 Figure 2.5. Proportion of cumulative weekly effort plots for the years 1967-74 in each area. 39 Figure 2.6. Spearman rank correlations for weekly effort (days fished) and consecutive pairs of years 1967/68 - 80/81 in each area 40 Figure 2.7. Effort and landed value($)/(days fished) vs. week for area 6, Hecate Strait :. 42 Figure 2.8. Spearman rank correlations for weekly effort and landed value($)/(days fished) in consecutive pairs of years 1967/68 - 80/81 in each area 47 Figure 2.9. Spearman rank correlations for the numerical response of effort one week following landed values($)/(days fished) for the years 1967-81 in each area. 49 Figure 2.10. Effort (days fished) in year i+1 vs. landed value($)/(days fished) in year i with regression lines and equations for each area 50 Figure 2.11. Landed ' value($)/(days fished) vs. week for each area 51 Figure 2.12. Landed value($)/(days fished) vs. year for each area in the years 1967-81 53 Figure 2.13. Weekly ratio to provincial average vs. week for each area 54 Figure 2.14. Ratio to provincial average vs. year for each area and the years 1967-81 56 Figure 3.1. Vessel horsepower vs. year for the years 1967-80 in each area ....62 xii Figure 3.2. Mean relative fishing power vs. horsepower class for each of the seven areas. 67 Figure 3.3. Nominal and standardized landed values($)/(days fished) vs. year for the years 1967-80 in each area 68 Figure 3.4. Ratio to provincial average calculated using standardized landed values($)/(days fished) vs. year for the years 1967-80 in each area 69 Figure 3.5. Steaming times/(days fished) and standardized landed values($)/(days fished) vs. year for each area 77 Figure 3.6. Landed value($)/(days fished) vs. year for each area 80 Figure 3.7. Spearman rank correlations for weekly landed value($)/(days fished) in consecutive pairs of years 1967/68 - 80/81 in each area. 81 Figure 3.8. Landed value($)/(days fished) vs. week for each area 82 Figure 5.1. Four types of functional response curves ....99 Figure 5.2. Tinbergen's (1960) search image hypothesis 110 Figure 5.3. Switching hypothesis expressed as ratios and proportions 116 Figure 5.4. Percent mortality curves for four types of functional responses ... 122 Figure 6.1. Type I functional response, and mortality curves for the fishery 127 Figure 6.2. Comparison of two functional response forms for the fishery. ...... 135 Figure 6.3 Implications of switching hypothesis for the fishery 149 Figure 7.1. Map of principal fishing grounds and major statistical areas in Hecate Strait 156 Figure 7.2. Estimated total numbers landed of Pacific cod, English sole and rock sole 158 Figure 7.3. Schematic diagram of VPA method 168 Figure 7.4. Estimates of total abundance of Pacific cod, English sole and rock sole obtained from VPA and CAGEAN using best estimates of input parameters 184 Figure 7.5. Estimates of catchability coefficients for Pacific cod, English sole and rock sole obtained from VPA and CAGEAN using best estimates of input parameters 186 xiii Figure .7.6. Estimates of total abundance of Pacific cod, English sole and rock sole obtained from VPA using alternative M values. 188 Figure 7.7. Estimates of catchability coefficients for Pacific cod, English sole and rock sole obtained from VPA using alternative M values 190 Figure 7.8. Estimates of total abundance of Pacific cod, English sole and rock sole obtained from VPA using alternative terminal q values 192 Figure 7.9. Estimates of catchability coefficients for Pacific cod, English sole and rock sole obtained from VPA using alternative terminal q values. 193 Figure 7.10. Estimates of total abundance of Pacific cod, English sole and rock sole obtained from CAGEAN using alternative M values 195 Figure 7.11. Estimates of catchability coefficients for Pacific cod, English sole and rock sole obtained from CAGEAN using alternative M values 197 Figure 7.12. Estimates of total abundance of Pacific cod, English sole and rock sole obtained from CAGEAN using alternative F values 199 Figure 7.13. Estimates of catchability coefficients for Pacific cod, English sole and rock sole obtained from CAGEAN using alternative F values 200 Figure 7.14. Estimates of total abundance of Pacific cod, English sole and rock sole obtained from CAGEAN using alternative X values 202 Figure 7.15. Estimates of catchability coefficients for Pacific cod, English sole and rock sole obtained from CAGEAN using alternative X values 205 Figure 8.1. Comparison of alternative single species functional response model fits to Pacific cod data sets 226 Figure 8.2. Comparison of alternative single species functional response models from VPA and CAGEAN fit to data sets for Pacific cod 229 Figure 8.3. Comparison of alternative single species functional response model fits to English sole data sets 232 Figure 8.4. Comparison of alternative single species functional response models from VPA and CAGEAN fit to data sets for English sole. 233 Figure 8.5. Comparison of alternative single species functional response model fits to rock sole data sets 236 xiv Figure 8.6. Comparison of alternative single species functional response models from VPA and CAGEAN fit to data sets for rock sole. 239 Figure 8.7.Comparison of alternative single species mortality model fits to Pacific cod data sets 240 Figure 8.8. Comparison of alternative single species mortality model fits to VPA and CAGEAN data sets for Pacific cod 243 Figure 8.9. Comparison of alternative single species mortality model fits to English sole data sets 244 Figure 8.10. Comparison of alternative single species mortality models from VPA and CAGEAN fit to data sets for English sole 247 Figure 8.11. Comparison of alternative single species mortality model fits to rock sole data sets 248 Figure 8.12. Comparison of alternative single species mortality models from VPA and CAGEAN fit to data sets for rock sole 249 Figure 8.13. Predicted multispecies functional response surface for Pacific cod using the data set from VPA for ages 4-10 252 Figure 8.14. Predicted multispecies functional response surface for English sole using the data set from VPA for ages 6-10 255 Figure 8.15. Predicted multispecies functional response surface for rock sole using the data set from VPA for ages 6-10 259 Figure 8.16. Predicted multispecies mortality response surface for Pacific cod using the data set from VPA for ages 4-10 260 Figure 8.17. Predicted multispecies mortality response surface for English sole using the data set from VPA for ages 6-10 263 Figure 8.18. Predicted multispecies mortality response surface for rock sole using the data set from VPA for ages 6-10 266 Figure 8.19. Predicted switching curves for data set from CAGEAN for cod and rock sole and fully recruited ages (4-10;6-10) ". 269 Figure 8.20. Times series plots of LPE and Catchability (q) vs. abundance for selected data sets for each species 275 Figure 8.21, Schematic plots population growth rate vs. population size for two hypothetical fish populations 280 xv Acknowledgements Many people have provided help and encouragement during this endeavor. First, I'd like to thank Dr. Ray Hilborn for his enthusiasm in getting the project started and for his help in obtaining funding. Interactions with fellow students Don Hall, Wilf Luedke and Peter Millington, all of whom shared in the experiment of the new fisheries master's program that didn't quite make it, were particularly valuable. Brain Moore kindly granted me permission to use the sales slips and vessel attribute data that was provided by Dr. Neil Guppy and Dr. Guppy along with Ed Zyblut and Kenny Lorret provided guidance on its use. Doug March provided information on steaming times. Jeff Fargo, Ray Foucher, and S. Jergen Westrheim provided data on Hecate Strait English sole, rock sole and Pacific cod as well as helpful discussions on what to do with it. S. Jergen Westrheim also kindly invited me on a research cruise on the G.B. Reed so that I could get a feeling for sampling at sea. Nev Venables, Frank Crabbe and Jan Lucas allowed me to tag along while they made their port sampling rounds in Vancouver, Steveston and Prince Rupert. Captain Winn Decker, Howard Decker and Risk permitted me to accompany them on the Double Decker for a fishing trip north of Cape Scott where I learned the true meaning of Ralph Yorque and that trawlers make good salmon packers. The company of Chris Foote, Grant Pogson, Wilf Luedke and Ric Taylor made various field trips particularly enjoyable. Darren Gillis and Dr. Randall Peterman and others at Simon Fraser University provided helpful discussions and special thanks to Randall for his patience. Drs. C. J. Walters, C. C. Lindsey, N. J. Wilimovsky and A. V. Tyler provided helpful constructive criticism on the thesis despite its exhaustive length. The research was supported in part by a Department of Fisheries and xvi Oceans grant to R. Hilborn and a Pacific Biological Station Scholarship. I'd like to thank friends at the Institute and my family for their encouragement and for not asking as the thesis dragged on and finally, and most importantly thanks to Linda Berg for her help, encouragement, and patience. xvii General Introduction Predators can exhibit two types of responses to changes in density of their prey: (1) functional responses and (2) numerical responses (Solomon 1949). Numerical responses describe how the number of predators changes with prey density. The change in the number of prey eaten per predator with changes in prey (or predator) density is the functional response. Numerical responses can be further divided into two types; a short term aggregation of predators in response to changes in prey density in space and/or time, and a longer term reproductive response of predators to changes in abundance of prey. Recent studies have found that fishermen have numerical and functional responses akin to those described by Solomon for natural predators (e.g. Hilborn and Ledbetter 1979; Millington 1984; Peterman et al. 1979; Peterman 1980; Peterman and Steer 1981). These studies, and others, have demonstrated that applying theories for natural predators to fishermen is a viable approach that has implications for fisheries management. The potential importance of studying fishermen as predators was outlined by Hilborn (1985) who argued that fisheries management is largely managing fishermen, and that a failure to understand fishermen as predators was a source of many fisheries management problems. Thus he proposed that more research in fisheries should be focused on the interactions among fish populations, fishermen, processors and markets. Hilborn (1985) focused particular emphasis on the study of fishermen as predators which he called fleet dynamics. He divided fleet dynamics into four components (1) investment and fleet size, (2) effort allocation (3) harvest efficiency, and (4) discarding and by-catch. 1 Investment and fleet size are analogous to growth and reproduction in natural predators (i.e. the reproductive numerical response of Solomon above). The lack of understanding of the processes that motivate fishermen to invest in some years but not in others has resulted in an inability to curb overinvestment which is rampant among fishing fleets throughout the world (e.g. Pearse 1982; Kirby 1982; Meuriot 1986). The net results of overinvestment are the same problems that result when there are too many natural predators: (1) excessive pressure on the prey population (creating a conservation problem for the fisheries manager), and (2) inefficient use of the resource (causing economists concern over "dissipation of potential resource rents"; e.g. Gordon 1954). These problems are particularly exacerbated in the fishery because unlike populations of natural predators that usually decline after overharvesting a resource, fishermen are often aided by "supplemental food" in the form of subsidies, loans and unemployment insurance designed to help fishermen survive when prey are in short supply (e.g. Davis and Thiessen 1986). Furthermore, unlike natural predators which must slowly evolve more efficient capture methods, successful fishermen can buy many devices that immediately increase their capture success (e.g. various electronic fish finders). Hilborn (1985) defined effort allocation as the choice of time, location and species to catch. Understanding effort allocation is important for anticipating short (e.g. weekly) and long term (e.g annual) shifts in effort patterns in response to changes in regulations, markets, or fishing opportunities created by managers (e.g. salmon enhancement; Hilborn and Ledbetter 1979). Harvest efficiency relates catch per predator to abundance (i.e. the functional response), and to vessel characteristics (e.g. length, engine horsepower). The form of the functional response has obvious importance because in most 2 fisheries catch per predator is used as an index of the abundance of fish. Knowledge of the determinants of catching power is useful in standardizing predators (e.g. fishing vessels) so that indices of abundance can be adjusted for changes in catching power over time that occur due to changes in fleet composition (e.g. Robson 1966; Kimura 1981). Fishermen are profit motivated, and because the cost of rejecting prey is usually small, they generally only keep the most valuable species. Thus discarding is a major problem in fisheries. Discarding makes it extremely difficult to assess the impact of fishing on nontarget or by-catch species, because: (1) a significant proportion of discarded fish die, (2) the discarded catch of these species is usually not observed, and (3) estimation of discarded catch is difficult. Thus, the motivation for the thesis came from two principal sources. First, I was curious about the theories of functional and numerical responses of predators, and fishermen provided an interesting case study. Second, examining the responses of fishermen as predators provided an opportunity to consider some practical implications of the theories of natural predators for fisheries assessment and management. Specifically, the thesis focuses on two main topics. The first part investigates the numerical responses of the British Columbia trawl fleets to the relative abundance of prey (measured in dollar value). The second part investigates the functional responses of trawlers to the abundance of three species caught in Hecate Strait, an area of particularly rich fishing grounds located between the Queen Charlotte Islands and British Columbia's northern mainland. The organization of each of these parts is detailed below. y 3 I. N u m e r i c a l R e s p o n s e s The first part consists of Chapters 1-4. Chapter 1 is a literature review of numerical response studies in fisheries. Four alternative approaches are reviewed: (1) models of fisherman's choice, (2) applications of search theory and optimization, (3) simulation models, and (4) empirical examinations of fisheries data sets. Each of these approaches is examined in terms of its strengths and weaknesses. Chapter 2 investigates three alternative hypotheses proposed by Hilborn and Ledbetter (1979) to explain movement patterns of salmon seine fishermen that may describe patterns in the numerical responses of the trawl fleet. Because of the multispecies nature of the trawl fishery, and the assumption that fishermen are profit motivated, I used the landed value of the catch per days fished (LPE$) as a surrogate for fish abundance. The hypotheses examined are (1) movement follows traditional (i.e. historical) patterns, (2) vessels move to equalize LPE$ among alternative fishing areas, and (3) vessels move to maintain relative LPE$ among areas (i.e. some areas will have consistently higher LPE$ than others relative to the coastwide average LPE$ for all areas). The hypotheses were examined on and inter- and intra-annual time scale and a fixed spatial scale that divided the British Columbia coast into 7 fishing areas. Chapter 3 investigates some of the assumptions of the Hilborn and Ledbetter (1979) hypotheses arid considers what effects potential violations of these assumptions may have had on the results of Chapter 2. The evaluation of assumptions varies from drawing on existing literature sources to detailed analyses depending on the available data. 4 Chapter 4 discusses the previous three chapters. The discussion first focuses on a comparison of the results from my study with the results from Hilborn and Ledbetter (1979) and Millington (1984) who examined the Hilborn and Ledbetter (1979) hypotheses using data for salmon gillnet fishermen. Next, I discuss some problems with data sets and statistical issues common to examinations of numerical responses in fisheries are > mentioned. I conclude with some general implications of numerical responses. II. F u n c t i o n a l R e s p o n s e s The second part comprises Chapters 5-8. Chapter 5 reviews the literature for functional response in natural predators, to provide the foundation for examination of functional responses in the fishery. Particular emphasis is placed on four alternative functional response types, their equations, and potential mechanisms Chapter 6 reviews the literature on functional responses in the fisheries. Few fisheries studies endeavored to explicitly examine functional responses, but many studies have implications for, functional responses. In order to draw analogies between natural predators and fishermen, Chapter 6 follows a parallel organization to Chapter 5. Estimates of abundance are required for any examination of functional responses. Thus, Chapter 7 examines two similar methods for estimating the abundance of exploited fish populations from data on the age composition of catches gathered over time. Each technique is reviewed in terms of its data requirements, underlying model and assumptions. The two techniques are then applied to data for three species caught in Hecate Strait, and abundance 5 estimates from the two techniques are compared. In addition, I evaluate the sensitivity of estimates from each technique to alternative input parameters (fishing and natural mortality rates). Chapter 8 examines the functional responses of trawlers to the abundance of Pacific cod, English sole and rock sole, which are the major components of a multispecies trawl fishery in Hecate Strait. Firstly, I fit three alternative functional response models and their corresponding mortality curves to the data from each of the three species. Secondly, I consider two alternative multispecies functional response models in attempt to model the three species system as a whole. Thirdly, I examine one mechanism for functional responses in the fishery (switching; Murdoch 1969). The discussion, covering Chapters 5-8, considers outstanding statistical and biological issues and presents some general implications of the findings for fisheries. 6 Chapter 1: Literature review of Numerical Responses in Fisheries Introduction While studies of numerical responses of animal predators have focused on the implications of responses for the total predator-prey system (eg. Beddington et al. 1976), the application of numerical responses to fishing systems has generally taken a different approach for two reasons. First, the analogous processes to reproductive responses in fisheries are changes in fleet size, and/or investments in improved • catching power. Because of the economic nature of these responses, studies tended to focus on economic rather than biological implications (eg. Clark et al. 1979, Aranson 1984, Charles 1983, Charles and Munro 1985, Lane 1988). Second, estimates of absolute abundance are rarely available in commercial fisheries, and when estimates are available they generally have poor precision. Thus, most applications of numerical responses to fishing systems have focused on short term aggregation responses, and the implications of fleet movement patterns. In this chapter I review some alternative approaches to numerical response studies that have been applied to fisheries. I have grouped my review of alternative approaches into four main categories: (1) models of fisherman's choice, (2) applications of search theory and optimization, (3) simulation models, and (4) empirical studies of particular fisheries. For each category, I first briefly review the technique, then highlight studies that have applied the technique, and finally comment on the technique's strengths and weaknesses. 7 Models of Fisherman's Choice A few studies have applied a method known as logit analysis to the examination and prediction of discrete choice behavior by Fishermen. Logit analysis relates discrete choices to a number of discrete or continuous explanatory variables (Swierzbinski 1985). For example, the discrete choice of Fishing location could be related to the continuous explanatory variable LPE$ (landed value of the catch per unit of Fishing effort), and a discrete explanatory variable such as risk of gear damage (classified as high, medium, or low). Other typical discrete choices made by Fishermen include gear type and target species. More speciFically, the logit model relates the probability of choosing a particular alternative (e.g. area to fish) to the explanatory variables. Swierzbinski (1985) gives the formulation for the two choice logit model as P(j|X.)=exp(a0 + 2bX.)/(l + exp(a0~' + LbX.)) ' (1.1) where P(j|Xp is the probability of choosing the alternative j (e.g. an area) given X^ . (e.g. the areas LPE$ and risk of gear damage), L is the sum over explanatory variables, and a^ and b. are constants to be estimated. For more details on choice models see Maddala (1983), Swierzbinski (1985) and the references therein. Eales and Wilen (1986) used the logit model to predict area choice in the California pink shrimp trawl fishery. Using expected catch and distance from home port as explanatory variables, they were able to predict the location of the First set of the day correctly 90% of the time for 3 aggregate areas, and 55% 8 of the time - for 8 smaller areas. The study demonstrated that fishermen are influenced on short time scales (i.e. daily) by relative abundance information, and the authors suggested economic inefficiencies may result from excessive movement into apparently profitable areas. " The logit model was used to predict fishery choice by New England fishermen by Bockstael and Opaluch (1983) (see also Opaluch and Bockstael 1984). Alternative fishery choices included scallopping from New Bedford, otter trawling from southern New England ports, and shrimping from Maine ports. The explanatory variables included the expected net returns, initial wealth, arid a variable that measured the resistance to changing fisheries. They concluded fishermen respond to economic incentives of expected returns and variability in returns, but only after these incentives surpass a substantial threshold. The strength of the logit model approach is its consideration of the behavior of individual fishermen or fishing units. The economists who have applied the approach argue that it is preferable to approaches that deal with aggregate data because the method is applied at the core of the common property nature of fisheries (i.e. decisions by individuals). One weakness of the logit model is that the explanatory variables such as "expected returns" are not observed, but instead must be estimated using proxies such as average catch in a previous time period. This can be a difficult task since catches in subsequent time periods may not be correlated (e.g. Eales and Wilen 1986). Furthermore, expected catches and profits will depend on information flow, that may depend on many nonquantifiable factors. These problems may limit the situations where the logit model can be applied. Swierzbinski (1985) stated that the logit model is not appropriate when the choices are close substitutes, and discussed the appropriate statistical properties and conditions for applying the logit model. More details are 9 also provided in Bockstael and Opaluch (1983) and Eales and Wilen (1986). Applications of Search Theory and Optimization A second approach to the study of movement dynamics has been to apply aspects of the theory of search to fishermen's decisions. This approach casts the choices facing fishermen, in a formal decision making setting and considers how fishermen should respond given an assumed probability of encountering fish (e.g. at a particular location and/or time) and a specified goal of the fishing enterprise (e.g. maximize catch). Thus, the application of search theory has two basic steps: (1) choosing the empirical distribution for the probability of encountering fish (usually clumps or schools of a uniform size), and (2) . applying decision rules that describe what to do given a probability of encountering fish. The choice of the probability distribution in step 1 depends on the spatial distribution of fish. The Poisson distribution,, is appropriate when fish are distributed randomly, but the Negative Binomial distribution is more appropriate when fish are distributed in patches or schools (Swierzbinski 1985). Decision rules depend on the goals of the fishing enterprise. Applications of search theory to movement studies have generally been concerned with the optimal patterns of effort allocation given particular choices for steps 1 and 2. Shotton's (1973) work provided the foundation for much of the more recent applications of search theory to fishing systems. Shotton was one of the first to consider possible distributions for fish schools. He fit several alternative types of distributions to transect data on albacore tuna1 gathered by research 1 All common names of fishes used follow the American Fisheries Society convention (AFS, 1980). 10 vessels using troll gear (from Craig and Graham 1961). Of the five distributions he considered, the Negative binomial distribution fit the data best. Shotton's second contribution was the application of Bayes' theorem to make predictions about how expected fish density changed in relation to fishing strategies. The premise for Bayes' theorem is to make expected fish density at any given time depend on two factors: 1. prior sample results (e.g. preseason sampling such as the albacore tuna transects), and 2. current catches as the fishing season progresses. More specifically, Bayes' theorem states P(X|x.) oc P(x.|X)P(X) (1.2) where X is the population density over the fishing grounds, x. is the sample results or catches, P(X|xp is the probability of X given the sample data (called the posterior probability), P(xJX) is the likelihood of obtaining the sample data if X is the actual fish density, and P(X) is the prior probability of X being the fish density (from Lindley 1965, cited in Shotton 1973). Shotton applied the Bayes' formulation to a situation where the objective of the fishing decision was to minimize the loss due to changing grounds unnecessarily. The loss due to changing grounds, L c , was given by L = k(tX + tiX-i//)) (1.3) c t where k is a proportionality constant, t is the time to change grounds, X is the actual fish density on the present ground, tf is the time to "make a trip" on the alternative ground, and \p is the expected catch rate on the alternative ground. Given the formula for expected loss, the albacore tuna data, and an example of two alternative grounds, he derived formulae for determining when it was best to change grounds; Shotton also applied search theory to the British Columbia trawl fishery, that he characterized as having grounds where fish are either practically absent, or present in acceptable quantities. He determined the sequence in which three fishing grounds should be fished under two conditions. In the first case he ignored seasonal changes and assumed the probabilities of encountering fish and time to determine presence or absence were equal in each of the areas. The best strategy was the one with the lowest possible costs (time to cover the fishing ground plus steaming times to and from port). In the second case he allowed for different probabilities of encountering fish and assumed the time to detect presence or absence was proportional to the relative size of each ground and gave the necessary conditions (relative probabilities of encounter) for each sequence to be best in terms of minimizing the relative loss. Other studies also used Shotton's approach. For example, Mangel and Clark (1983) applied search theory to two similar problems. The first considered the optimal allocation of search effort over two fishing ground where the current abundance of fish was assumed to have some known (prior) probability distribution (as in step 1 above). The second problem considered the optimal allocation of vessels over time to one ground. In each problem, new information on catch rates during the season affected the expected probabilities of encountering fish in subsequent time periods, and dynamic programming was used to solve for the optimal decisions. The authors assumed an objective of maximization of the total net seasonal revenue equal to the total value of the catch minus the costs of movement between grounds. 12 The conclusions of the first part of the study were: (1) information obtained from sampling of fish stock can lead to significant increases in expected catches, (2) the amount of information obtained depends on the initial allocation of vessels, and (3) the exact allocation may not be critical, provided some searching takes place on each ground (Mangel and Clark 1983). The second part of the study compared the strategies, of complete cooperation and complete competition among vessels. The cooperative strategy followed the optimal allocation determined from the model, while the competitive strategy allocated all vessels to fish in the first period. The competitive strategy performed progressively worse as the total number of vessels in the fishery increased (Mangel and Clark 1983). There are two major strengths of the search theory applications. First, the probability of encounter is associated explicitly with an assumed spatial distribution of fish. Second the use of Bayes' theorem has some intuitive appeal because it mimics processes that occur in real life fisheries: formulation of prior expectations and updating as more information becomes available (e.g. catches). Prior probabilities can be based on preseason surveys, or on the judgement of skippers if data are lacking. Updating of these preseason expectations occurs through catches obtained as the season progresses. It is unlikely that fishermen use such a formal process in making their decisions, but the concepts of prior expectations and updating certainly apply. While some aspects of search theory applications have intuitive and theoretical appeal, there are weaknesses of the approach that generally have to do with its applicability to actual fisheries situations. First, search theory applications assume random search behavior by fishermen and distributions of fish schools which have been verified only rarely by empirical data. The former assumption is the most questionable, as fishermen are known to direct their 13 effort to areas of high concentrations, particular species, and even particular age classes within species (e.g. Cook 1984). Second, Shotton's (1973) pioneering work was limited because it stipulated the best solutions for a specific example and for a given set of model parameters. Other authors (e.g. Mangel and Clark 1983) have attempted to overcome this limitation, by using dynamic programming. This technique determines the optimal solution by searching over all "possible" (as determined b}' the authors) states (e.g. fish densities), actions (e.g. areas to fish) and time periods and by calculating the value (e.g net seasonal revenue) of each combination. The problem with dynamic programming is that solutions require searching over a number of outcomes equal to the number of states X number of actions X number of time periods (see Walters 1986 for an introduction to dynamic programming). This computational limitation, known as the "curse of dimensionality", has resulted in studies that lack realism. For example, studies commonly assume: (1) schools of uniform size that are not depleted as the season progresses, (2) perfect information, and (3) homogeneit}' of vessels both in terms of catching power and their economic objectives. Furthermore, studies generally only consider a few alternative actions (e.g. two alternative areas). While some of these assumptions can be relaxed, studies have been limited to either qualitative or very small quantitative extensions (e.g. Mangel and Clark 1983). Since the curse of dimensionality largely results from the lack of accessibility to the very highest speed computers, the limitations imposed by the curse of dimensionality may be overcome in the future as technology progresses. 14 Simulation Models The technique of simulation modeling is in widespread use in biology and resource management. Thus, it is not surprising that a few examinations of movement patterns have used simulation models to test assumptions about possible types of movement patterns and their implications for resource exploitation. Yet considering that fisheries in some parts of the world (e.g. North Sea, NW Atlantic) have been monitored and managed for nearly 70 years, it is a bit surprising that the first fisheries models to included fleet components appeared about fifteen years ago and there have only been a few such studies since. One of the first studies to consider fleet components in a fishery model was Caddy's (1975) examination of the scallop fishery on George's Bank in the NW Atlantic. Caddy compared a spatial model -simulating nonrandom recruitment and harvesting with the standard yield per recruit model of Beverton and Holt (1957). The fleet component of Caddy's model was based on the simple formula for allotting effort shown below. C P U E - ,1 A\ ij CPUEm T where f.. is the effort alloted to subarea ij, CPUE.. is the catch per unit effort in subarea ij, CPUE^, is the total catch per unit effort of all subareas, and f^, is the total effort in all subareas. Since the scheme of proportional effort allocation resulted in lower than observed effort in high effort areas, Caddy 15 modified equation 1.4 to account for the average historical effort and catch per effort distributions. Caddy found overall yield declined more sharply as effort increased beyond the point of maximum sustained yield in the spatial model compared to the yield per recruit model and he suggested an increase in yield could be obtained if effort could be diverted away from heavily fished areas. The suggestion of management of effort on a subarea basis was very different from the more general catch quotas and season length limits commonly used for demersal species. A few papers have modeled the annual numerical responses of fishing fleets as a predator-prey system with Lotka-Volterra equations. The basic form of the Lotka-Volterra equations are: = rx - sxy (1.5) dy dt = sxy - my (1.6) where x and y are the numbers of prey and predators, r is the prey's intrinsic rate of increase, s is the rate of capture of prey per predator, and m is the mortality rate of predators. The different formulations for equation 1.6 which in the case of the fishery refer to the change in numbers of boats, are of particular interest. Gatto et al. (1976) applied a discrete time version of such, equations where equations 1.5 was replaced by the Ricker (1954) 16 stock-recruitment model and equation 1.6 was replaced by: Bt+1 = sBt + ^ (1.7) where B is the number of boats, C is the catch and s and i are survival and investment coefficients of the fleet. After examining the models equilibrium properties, the authors concluded that the model was too crude for practical, quantitative applications. The principal weaknesses of the model were the assumption that investment was a linear function of catch per boat in the same year, and a failure to take alternative fishing locations and species into account. Allen and McGlade (1986) considered the Lotka-Volterra approach as an alternative to simple fishery models such as the surplus production model of Schaefer (1957). They argued that Lotka-Volterra models have not been applied to fisheries because of a basic reluctance of managers to treat fishermen as an active part of the system. The first part of their study used Lotka-Volterra equations in a simulation model of the Nova Scotia haddock fishery. Their model for haddock was similar to equation 1.5, except they included 3 age classes. Their equation for change in boats was: 4r = RY(1 - L m] (L8) where Y in the number of boats, R is the rate of response of effort to profitability, C is the revenue per boat needed to maintain effort, and LPE$ is the total landed value of the 3 age classes per boat. The price also fluctuated depending on the catch relative to the available markets. Initial deterministic runs of the model resulted in damped oscillations in both the numbers of fish and 17 vessels with an equilibrium occurring after 70-80 years. Next they allowed the birth rate (equivalent to r in equation 1.5) to fluctuate randomly about a mean value. The result was irregular oscillations in boats and fish.. The oscillations were amplified by the lagged response of effort to fish abundance. The second part of the study considered a multispecies, multifleet spatial model of Nova Scotial trawl fisheries. The additional components of the model included : (1) catch was determined by a the type II functional response of Holling (1965), and (2) allocation of effort to different zones based on their relative "attractivity". Attractivity was a function of the expected profit, the quality of information and the homogeneity of the fleet. This effort allocation scheme was similar to Caddy's proportion effort allocation with the additional component of information flow between fleets. Thus, in Allen and McGlade's model vessels played the dual role of catching fish and transmitting information to other vessels. This form of effort allocation allowed the authors to examine different strategies among fleets such as the stochast - cartesian comparison (see Chapter 3). Allen and McGlade's efforts represent the most sophisticated attempt to include fleet components in fisheries models to date. Hilborn and Walters (1987) presented some general guidelines for simulating stock and fleet djmamics in fisheries with spatial differences. Their model for fleet dynamics included a method of sequential effort allocation. This scheme involved four steps (see their Table 1; p. 1368): 1. Determine the proportion of effort to be alloted to each spatial area for exploratory fishing. 2. Allocate exploratory effort to determine initial catch rates. 3. Divide the remaining total effort into subunits to be allocated sequentially. 18 4. Allocate one subunit of effort to the area with the highest catch rate, update the abundance and catch rate in that' area and allocate next effort subunit to the area with highest catch rate (and so on, repeating step 4). Like Allen and McGlade (1986), they modeled catch as a type II functional response. The model was applied to six stocks of abalone exploited by divers off Australia. They showed how the relationship between catch rate and abundance can be distorted depending on assumptions about handling times and the relative biomasses of abalone in the different stocks. Simulation modeling of movement dynamics has provided some valuable insights into the importance of incorporating the fleet responses in fisheries models. Perhaps the greatest strength of simulation models was stated best by Hilborn and Walters (1987; p. 1366): "The development of simulations forces explicit statements of assumptions and hypotheses, and encourages explorations of alternative hypotheses." This property makes simulation models very good teachers and tools for asking a variety of "what i f questions. Other strengths of simulation modeling include flexibility, general ease of construction, and the relatively short time lag between formulation of hypotheses and results compared to field studies. The main weakness of the simulation approach is that the explicit statements and hypotheses must be written in the form of equations that are suitable for programming. Unfortunately, not all hypotheses are amenable to quantification and often even quantifiable hypotheses are difficult . to model realistically. 19 For example, Allen and McGlade (1986) caution that Lotka-Volterra models assume fluctuations in abundances are causes by factors intrinsic to the model (e.g. abundance of fish fluctuates in response to fishing); external forces (e.g. the environment) are not supposed to influence the variables. In addition, information flow and differential goals among fishermen are difficult to model realistically, because they involve factors that are difficult to quantify. This inability to capture < realism makes simulation models of little value in prediction; there is always something missing, some interaction unaccounted for, and/or some hypothesis not considered in all. simulation models. Empirical Studies Various authors have examined the' movement responses of vessels in particular fisheries. Peterman et al. 1979 found that weekly effort by Georgia Strait salmon trollers increased following increases in weekly catch per effort (an index of abundance). Hilborn and Ledbetter (1979) found a similar aggregation response in British Columbia salmon seiners, although responses varied between years and vessel mobility classes. Mobile seiners (vessels that fished more than one area) showed a nearly linear response of weekly effort to the landed value per effort in the previous week, while effort by stationary vessels had a saturating response to landed value per effort. Millington (1984) found a similar dichotomy between the responses of mobile and stationary salmon gillnetters. Millington also used landed value per effort as his index of abundance. The use of landed values assumes fishermen are sensitive to revenue rather than physical yield and may respond to differences in prices between species and/or seasonal price changes. 20 Hilborn and Ledbetter (1979) proposed some alternative hypotheses to explain the numerical responses of seiners to changes in relative benefit (measured as landed value/days fished) in various salmon fishing areas off the British Columbia . coast. Millington subsequently tested these hypotheses with data from salmon gillnetters. The , hypotheses are: (1) movement follows traditional patterns independent of recent fish abundance, (2) movement equalizes the gross dollar returns to effort' (LPE$) in alternative fishing areas, or (3) movement maintains relative LPE$ in alternative areas. These hypotheses provide a description of movement patterns in fisheries, and my approach in examining numerical responses was to test these hypotheses with data from the British Columbia trawl fishery. 21 Chapter 2: Effort Allocation in the British Columbia Trawl Fleet Introduction Three alternative hypotheses were proposed by Hilborn and Ledbetter (1979) to explain fleetwide movement patterns of salmon seiners and were subsequently tested by Millington (1984) with data on salmon gillnetters. I tested these hypotheses with data from the trawl fleet for two reasons: (1) the hypotheses provide a good framework for examination of movement patterns in the trawl fleet; and (2) the application of these hypotheses to trawlers provides a comparison to previous work on two segments of the salmon fleet. Brief descriptions of the rationale for each hypothesis, the hypotheses (H) and their predictions (P) are provided below. I. Traditional Patterns The rationale for the traditional patterns hypothesis is that fishermen may specialize on a particular area or set of areas, learn the best times to fish in each location, and develop fixed movement patterns that are independent of short term changes in abundance. H: The fleet moves according to historical patterns. P: The proportion of effort alloted to each area should be constant. II. Equalize LPE$ The Equalize LPE$ hypothesis assumes fishermen are profit motivated, and that vessels will move to areas with high LPE$ (landed value of the catch per unit of fishing effort). Hypothesis II makes four additional assumptions: 1. All boats are equal 2. All areas have equal desirabilities and fishing costs 22 3. Movement costs between areas are zero 4. There is perfect information about the potential benefits in each area. If these assumptions hold or a nearly correct, then vessel movement should result in nearly equal LPE$ in all areas. H: The fleet moves to equalize gross returns to effort (LPE$) in all areas. P: Each areas LPE$ should approach the average LPE$ for all areas. III. Maintain Relative LPE$ The Maintain Relative LPE$ hypothesis was proposed for cases when assumption 2 above is violated. If areas have different desirabilities and costs, vessel movement should result in the LPE$ in each area remaining constant relative to the average LPE$ of all areas. Areas with consistently higher costs should have consistently higher LPE$ and vice versa. Note that assumptions 1, 3, and 4 above also apply to the Maintain Relative LPE$ hypothesis. H: The fleet moves to maintain relative LPE$'s in each area. P: The ratio of each areas LPE$ to the average for all areas should be constant. The effects of violations of the assumptions of the Equalize LPE$ hypothesis on the conclusions of this chapter are examined in chapter 3. 23 Data and Methods Description of the Fishery The B.C. trawl fishery provides a good test case for the Hilborn-Ledbetter hypotheses. Over the last twenty years the trawl fishery has averaged about 100 licensed vessels of which 60-80 actively fished in any given year (For a historical review of the development of the B.C. trawl fishery and trawl gear see Lippa 1967). The principal species landed by the trawl fleet include Pacific cod (Gadus macrocephalus), lingcod (Ophiodon elongatus), Pacific hake (Merluccius productus), walleye pollock (Theragra chalcogramma), various rockfishes (Sebastes spp.), flatfishes (Pleuronectid spp.), sablefish (Anoplopoma fimbria) and dogfish (Squalus acanthias). The total landings of all species have fluctuated around 30,000 metric tons in recent years (Leaman and Stanley 1987). The fishery is regulated by the Offshore Division of the Canadian Department of Fisheries and Oceans (DFO), and the primary method regulation has been catch quotas. The fishery is highly structured in space with fishing occuring on a series of relatively discrete trawling grounds at particular depths and seasons for the various target species (Fig. 2.1). For example, during the summer months fishermen may choose between trawling for Pacific ocean perch (Sebastes alutus) in the deep waters (> 100 fathoms) of Goose Island Gully (Area 5), Pacific cod and flatfish in the shallower waters (30-50 fathoms) on Two Peaks-Butterworth ground in Hecate Strait (area 6), lingcod on Big Bank (Area 1), or Pacific hake in area 3 (Fig. 2.1). The opportunities for movement are virtually unregulated with the exception that some areas may become less attractive when quotas for particular species in those areas have been caught. 24 Figure 2.1. Map of seven areas chosen for analysis and two major landing ports for groundfish on Canada's Pacific coast. Trawl gear consists of a large flattened funnel shaped net attached to the stern of boat by two cables (warps). During the towing process, the front of the net is. forced open laterally by two rectangular otter boards that are attached to each warp. Floats attached to the top of the net opening and weights attached to the bottom open the net vertically. The net is usually towed along the sea 25 bottom and fish are forced down the funnel into the rear of the net (codend). The net is retrieved by a winch or pair of winches. There are many variations in the components of trawl gear, see Lippa (1967) for more detail. Data Sources and Statistical Tests • The analysis is based on DFO saleslips and license data for the years 1967-1981 and 1967-80 respectively. A saleslip is recorded for each landing made by a vessel and includes the following information: landings (lbs.) and values ($) of fish by species (or species group), the location of catch by statistical area, fishing effort (days fished), the company purchasing the fish, and the date landed. The license data are collected annually on vessel registration forms and include measurements of various vessel attributes such as length, tonnage, and horsepower as well as the location of home port, type of vessel ownership, and numbers of licenses held. Thirty-one DFO statistical areas were grouped into seven areas that include the major fishing grounds exploited by the trawl fleet. The question of how effort is distributed among fishing areas can be posed at various spatial and temporal scales. In the analysis that follows, I will consider two temporal scales, between and within years, and the fixed spatial scale as determined by the seven areas. The hypotheses predict how effort responds to changes in potential benefits, and therefore it is important to define the units of measurement for these two factors. The units for effort are days fished , and like Hilborn and Ledbetter (1979) and Millington (1984), I have chosen the landed value($)/days fished (LPE$) as my measure of potential benefits. The choice of dollars/days fished assumes that profit is the primary motivation for fishermen. While the 26 measure has the advantage of lumping different species using the common currency of dollars, it ignores any possible species specialization that may be important to some fisheries in particular areas. Hypotheses II & III predict patterns in the landed value/days fished between areas and across the fifteen years of the data base. A price index was needed to account for the affects of inflation over the period, because the prices of the various species landed in each area may have experienced different amounts of inflation. Although Statistics Canada publishes price indices for fish in British Columbia these indices are affected most by high valued species such as, salmon, herring, and halibut that are not landed by the trawl fleet. To eliminate the affects of.. these species, a price index for groundfish was calculated from the saleslip data. The methods used to calculate price indices were as follows. I used the equations below to calculate (1) a price index CI ) for each species (s) and year sy (y) and (2) a total annual index (1^ ). FsyQsb (2.1) Sy~ *so%b = s ?syQsft (2.2) where P is the ex-vessel price/lb. in dollars, Q is the quantity landed (lbs.) and the subscript b denotes the base year. Note that equation 2.1, the standard equation for a price index, reduces to 27 p (2.3) Each species index, I , is the ratio of the price in the index year to the price in the base year. The annual index, 1^ , is a weighted ratio of prices where the weighting factors are the quantities landed of each species in the base year. Following the procedure of Statistics Canada (1983), I selected the average of three years as my base period (1979-1981). I calculated a price index for 12 species or groups for each year using prices and landed values from the saleslip data for all areas and trawl vessels (Table 2.1). I used the price indices in Table 2.1 and the equation below to correct landed values for inflation. where CLV is the corrected landed value of the catch in dollars, LV is the total landed value of the catch, I is the price index, and the subscripts a, y and s denote the area, year, and species. I applied the species specific price index to account for possible variability in inflation effects between areas with different species. I chose certain non-parametric methods (Spearman rank correlation, Kendall's concordance, and Kruskal-Wallis nonparametric ANOVA) for my statistical analyses. LV (2.4) CLV =1 — ay s I say 28 Table 2.1. Ex-vessel price indices for selected trawl caught groundfish species or groups landed in British Columbia 1967-81. (1979-81 = 1). Year Pcod Icod sabl rock sole Species' ilou turb poll hake skat dogf misc yind^  1967 30 23 32 33 28 52 19 32 0 68 36 21 27 1968 29 25 38 32 27 59 ^0 43 0 52 47 24 28 1969 31 27 39 37 24 37 0 87 0 40 67 24 30 1970 36 34 36 39 27 49 38 104 0 48 30 35 35 1971 38 38 33 38 30 48 0 0 0 47 30 38 32 1972 36 40 35 35 32 41 46 54 0 40 30 35 33 1973 42 54 35 43 45 61 66 59 0 54 60 44 4Z 1974 53 72 37 54 63 63 68 65 0 59 48 27 51 1975 53 45 38 55 58 65 67 74 " 0 62 43 53 51 1976 54 58 49 54 56 65 68 70 0 61 54 48 51 1977 67 80 64 65 65 81 67 71 0 60 71 74 64 1978 83 107 91 80 80 109 86 83 0 73 94 15 73 1979 99 129 93 101 102 93 91 99 0 85 100 69 92 1980 99 81 66 99 97 111 106 98 118 100 100 71 95 1981 100 93 116 99 100 103 104 105 98 107 96 107 101 Species codes: Pcod = Pacific cod; Icod = ling cod; sabl = sablefish; rock includes Pacific ocean perch, & yellowmouth, greenstriped, yelloweyed & other rockfishes; sole includes petrale, butter, dover, english, rex, & rock soles; flou includes Pacific sanddab & starry Bounder, turb = turbot; poll = walleye pollock; hake = Pacific hake; skat = skates; dogf = spiny dogfish; misc includes nonfood fish, octopus, halibut, squid, plankton, eulachons, sturgeon, perch, crab, & other fish. y^ind = annual index. I used Spearman's rank correlation (Siegel 1956; p. 202) instead of product-moment correlation in most cases because of the highly variable nature of catch and effort data. The variability was particularly large on the weekly time scale (e.g. Fig. 2.11), and the use of rank correlations minimized the effect of outliers. Another advantage of using rank correlations was that the Kendall's concordance statistic (W; Siegel 1956; p. 229) provides a measure of the similarity of rankings among all years. Note that the use of product-moment correlations (i.e. in Tables 2.3a, b and 2.4, and Figs. 2.6, 2.8, and 2.9) would not have changed the qualitative results. I performed Kruskal-Wallis nonparametric ANOVA's (Siegel 1956; p. 184: i.e. Tables 2.5, 2.6 and 3.1) rather than parametric ANOVAs because of the skewness and heteroskedasticity of the data. I was able to correct for the skewness by log transformation, but variances among treatments (i.e. areas or weeks) were significantly different (by an F test; Cooley and Lohnes 1971). Parametric one-way ANOVAs also showed significant differences, but the assumption of homogeneity of variance was violated. For further explanations of the specific tests mentioned above see Siegel (1956) and Zar (1974). Description of Areas Figure 2.1 shows the sevens areas chosen for the analysis. They are: (1) Southwest Vancouver Island, (2) Northwest Vancouver Island, (3) Inside Vancouver Island, (4) South Queen Charlotte Sound, (5) North Queen Charlotte Sound, (6) Hecate Strait, and (7) West Queen Charlotte Islands. Each area includes a major fishing ground or aggregate of grounds exploited by the trawl fleet. 30 Also noted on the map are the two major ports for landing groundfish on the coast; Vancouver and Prince Rupert. These two ports accounted for 76-97% of the annual landed weight during the time period. Figure 2.2 shows the contribution of each of the seven areas to the weight landed and effort expended. Most of the effort has been exerted in areas 1, 3 and 6. Areas 1, 5 and 6 contributed most of the landed weight. The weight landed in Area 3 is small in comparison to the effort because area 3 is exploited by a large number of relatively small vessels (Fig. 2.2). Results Mobility in the Trawl Fleet Previous studies of movement patterns (Hilborn & Ledbetter 1979, Millington, 1984) have indicated the importance of distinguishing between nonmobile or stationary vessels (i.e. vessels that fish only one area) and mobile vessels (i.e. vessels that fish two or more areas). This section briefly examines three aspects of mobility in the trawl fleet: (1) the ratio of mobile to stationary vessels, (2) the relative contribution of these vessels to the landed weight, values, and effort expended in each area, and (3) mobility differences between areas. The ratio of mobile to nonmobile vessels in the trawl fleet (all years combined) was 1.3:1 (56%:44%). This ratio fluctuated annually from a minimum of 0.92:1 in 1974 to a maximum of 2.03:1 in 1980, but there was no time 2 trend (r =0.03, P<0.56). More than half of the mobile vessels fished in three or more areas per year. Despite nearly equal numbers of mobile and stationary vessels, mobile vessels accounted for 88% of the landed weight and value and 31 2500 -, 2000-"O QJ SZ cn 1500 -cn ro T D 1000 -l _ o LU 500-i r 1 2 r-16000 T T 3 4 5 Area 6 • in c •12800 P. — I c_ - 9600 QJ X) 6400 QJ T3 C ro - 3200 SZ O l — I QJ Figure 2.2. Contribution of the seven areas to. weight landed (lbs.) and effort expended (days fished). Squares are means and error bars the range over the period 1967-81. 76% of the days fished (Table 2.2). Stationary vessels were generally confined to areas with landing ports (i.e. areas 1,3,6). Sixty-six percent of all stationary vessels over the 15 years fished in area 3, while 18% and 12% fished in areas 6 and 1, respectively. But even in these areas, their contribution to landed weights, values and effort expended was rather small relative to mobile vessels with the exception that stationary vessels accounted for 49% of the effort expended in area 3 (Table 2.2). The ratio of mobile to nonmobile vessels within each area also fluctuated annually, but there were no significant time trends in any of the areas. For the 32 Table 2.2. Summary of mobility statistics for the trawl fleet. Numbers are totals or percentages for the period 1967-81. The column "Mobil, class" is the number of areas fished per year (i.e. 1 = 1 area, 2 = 2 areas, 3+ = 3 or more areas). Each area column contains the totals or percentages for each category and mobility class (e.g. of the 645 vessels that fished area 1 over the fifteen year period, 223 fished 2 areas per year). The column "All areas" contains coastwide totals and percentages for each mobility class (e.g. 642 vessels fished only 1 area per year during the years 1967-81). Category Mobil. Area All class 1 2 3 4 5 6 7 areas No Vessel- 1 80 9 422 5 5 117 4 642 years 2 223 23 200 48 31 138 83 373 3 + 342 222 227 345 275 299 68 457 Tot Vessel- 645 254 849 398 311 554 155 1472 years % Vessel- 1 12 4 50 1 2 21 3 44 years 2 35 9 24 12 10 25 54 25 3 + 53 87 27 87 88 54 44 31 % Landed 1 6 •1 23 0 0 19 9 12 Weight 2 35 3 44 10 11 38 43 31 3 + 59 95 33 89 88 43 48 57 % Landed 1 6 3 28 0 1 18 11 12 Value 2 36 3 44 10 11 38 44 31 3 + 58 94 28 90 88 44 46 56 % Effort 1 10 3 49 1 1 25 4 24 2 38 7 32 10 12 37 51 31 3 + 52 90 19 89 87 38 45 45 33 three areas with significant numbers of stationary vessels, the proportion stationary was not correlated with LPE$. The differences in the number of mobile and stationary vessels between areas will be evident in the movement patterns described in later sections. Traditional Patterns The traditional pattern hypothesis for. between-year effort allocation predicts that the proportion of effort alloted to each area remain constant between years. Figure 2.3 shows the proportion of effort plotted vs year for each area. As noted above, areas close to landing ports (SW Vancouver Island, Inside Vancouver Island, and Hecate Strait) have the greatest proportion of effort. The other four areas rarely receive greater than one-tenth of the total effort in any one year. While there is considerable variability in proportions between years, particularly in high effort areas, the areas have maintained quite consistent rank (Kendall's concordance, W=0.87 (P<0.001)). A" concordance of 1.0 would mean that each area's rank remained constant across all years. This tendency for some areas to have consistently higher effort than others is consistent with the prediction of the traditional patterns hypothesis, but it would also result if some areas are consistently more attractive because they have more fish than others (i.e. the Maintain LPE$ hypothesis). I also looked for traditional patterns of within-year effort allocation by examining plots of cumulative effort vs. week. If effort is distributed according to some historical pattern, then the amount of effort exerted in an area each week should be similar between years, and the cumulative effort plots for an area for different years should overlap each other. The analysis was divided into two time 34 o • A 1 SU Vancouver Island 2 NU Vancouver Island 3 Inside Vancouver Island 4 S Queen Charlotte Sound 5 N Queen Charlotte Sound 6 Hecate Strait 7 U Queen Charlotte Islands <- .4-, . 3 -a a CD CD-CD . 2 -.1 -r ~ i 1 1 1 1 1 1 1 1 1 1 1 1 r~ 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 Year Figure 2.3. Proportion of the total annual effort alloted to each area in the years 1967-81. periods, 1975-81 versus 1967-74, in order to detect long term changes in the predicted effort distributions. Figure 2.4 shows the proportion of cumulative effort plots for the seven areas for the years 1975-81. The cumulative effort in each week is expressed as a proportion of the total effort in that area and year, to adjust for the fact that annual effort has increased in all areas over the time period and to permit 35 Week Week Figure 2.4. Proportion of cumulative weekly effort plots for the years 1975-81 in each area. The cumulative weekly effort is expressed as a proportion of the total effort in each area and year. comparison of plots between years. Note the degree of overlap in areas 3 & 6 of the plots for different years. If the total annual effort in these areas could be predicted, a curve fit through the . points would predict the weekly effort for 36 either area with reasonable precision. The cumulative effort plots for areas 3 and 6 are consistent with the predictions of the traditional patterns hypothesis. The proportion of cumulative effort plots for Area 1 (SW Vancouver Island) and Area 7 (W Queen Charlotte Islands) show considerable scatter. These areas have been affected by regulations since 1978 that have closed certain grounds for short periods during each year (S.J. Westrheim, Pacific Biological Station, Nanaimo, B.C., pers. comm.). This effect is particularly evident in Area 1 where a closure to protect spawning cod has affected the timing of effort in the early part of each year (and apparently also the total effort as indicated in Fig. 2.3). Areas 2, 4 and 5 are the farthest from ports of landing. All have a similar seasonal pattern in timing of effort with the major activity occurring during the middle part of the year, but each area shows considerable year to year variability in its effort pattern. The variability could be due to weather, as these areas along with area 7 are the most - exposed areas on the the coast. Whatever the underlying cause, the effort distributions in these low effort areas located far from major landing ports, show much less overlap than areas 3 and 6, and are not consistent with the predictions of the traditional patterns hypothesis. What is different about areas 3 and 6? Some differences have already been presented. Both areas are allotted large proportions of the total effort (Fig. 2.3 ), and each contain one of the two landing ports for groundfish (Fig. 2.1). Areas 3 and 6 also contain the largest proportion of stationary vessels (Table 2.2). These factors may have resulted in the overlapping relationships for areas 3 & 6, when compared to other areas. To examine long term changes in the 37 effort " patterns I plotted the same curves for the earlier time period (1967-74) in the data base (Fig. 2.5). , For area 3, the main difference between years is the length of the winter fishery. For Area 6, the patterns between years are similar within the periods 72-74 and 68-71, but different between these periods. The primary difference between these two periods is an early season (weeks 8-16) drop in effort that only occurs in the latter period. The cumulative effort plots for area 1, the third area with significant numbers of stationary vessels, show more overlap during • the period unaffected by the early season closure. So far, I have just presented a graphical examination of the within-year traditional patterns hypothesis. To quantify the amount of overlap in the cumulative effort plots for each area over the entire time period (1967-81), I performed Spearman rank correlations on the weekly effort alloted to each area for each pair of years. Figure 2.6 shows Spearman rank correlations (rg) plotted for consecutive pairs of years for each area. _The Kendall's concordance values (W) for each area recorded on each plot are equivalent to an average rank correlation between all pairs of years. Area 3 had the highest consistency of within year effort pattern because of the large number of stationary vessels in area 3. Areas 4 and 5 have the next highest concordances, despite a predominance of mobile vessels. Area 6, that had overlapping plots for the 1975-81 period has a rather low consistency of rank for all years combined. In conclusion, given the high concordance in rank of the proportion of effort allotted to each or areas over time, I cannot reject the traditional patterns hypothesis on the time. scale of interannual variation. Within season patterns show much more variability between years, suggesting that traditional patterns 38 Week Week Figure 2.5. Proportion of cumulative weekly effort plots for the years 1967-74 in each area. The cumulative weekly effort is expressed as a proportion of the total effort in each area and year. are modified by other factors. The next sections examine how temporal patterns in effort may be modified by numerical responses of effort to LPE$. 39 o ro QJ o • c CO ro ro QJ ca CO • Days fished .5--.5-67-68 80-81 -.5 5 N Queen Charlotte Sound * * * p • • » 1 1 1 1 1 T 1 1 1 1 1 1 1 1 • U=0.54 6 Hecate Strait s B _ S * , f ^ i ^ x * f 1 1 1 1 1 1 1 1 1 1 1 1 1 1 • U=0.36 7 W Queen Charlotte Islands - > » s • W=0.16 67-68 Year 73-74 Year 80-81 Figure 2.6. Spearman rank correlations (r ) for weekly effort (days fished) and s consecutive pairs of years 1967/68 - 80/81 in each area. Stars (*) indicate significance at P<.05. Kendall's Concordance (W) values are recorded on each plot. All concordances were significant at P<.001. 40 Hypotheses Using Landed Value Per Unit Effort Assuming fishermen are profit motivated, I would expect the pattern of landed value/effort (LPE$) to influence the choice of fishing location and affect the timing of effort in each area. Before numerical responses are explored in more detail, the next section examines a general problem with correlations between effort and LPE$. The Dilemma of Effort vs. LPE$ Correlations Although it is logical to expect increased benefits in an area to attract more effort to the same area (i.e. a positive numerical response), providing supporting evidence is difficult. The main problem is that covariation of effort and LPE$ may be expressed in two different ways. When the measure of benefit is LPE$, it may be impossible to separate the cause from the effect. Two alternative explanations for covariation in effort are: 1. Increases in LPE$ resulting from increased effort 2. Numerical reponses of effort to increased LPE$ Both these explanations must be examined to determine the cause of correlations between effort and LPE$. For example, both effort and LPE$ in Hecate Strait have the same seasonal, pattern (Fig. 2.7), but it is not clear which factor is the cause and which is the effect. 41 125-100 OJ cn cn >. re 75-o u_ U -50-25-cn nt i i n *\t\ —r-—i 1 1 1 1 r-16 ' 24 32 40 52 7500-O J at 6000 ro 4500-1 ^ 3000 -| ro > •o -D 1500 -I a ro if T — r u i — r 16 — i 1 1 — 24 32 Week -r~ 40 in a i 52 Figure 2.7. Effort and landed value($)/(days fished) vs. week for area 6, Hecate Strait. Squares are medians and error bars mark the interdecile range for the years 1967-81. 42 Increased effort could produce increased LPE$ in two ways; either directly through learning and increased efficiency or indirectly through facilitation (e.g. using the positions of other vessels as an aid in locating fish). Therefore, explanantion 1 may be restated as two questions: la. Is individual vessel LPE$ increased when vessels fish more days? (a learning response) lb. Is vessel LPE$ increased when more vessels fish in the same area? (a facilitation response) To examine question la, I calculated Spearman rank correlations, r , between days fished per trip and LPE$ for all trips made to each area in the years 1967-81 (Table 2.3a). The coefficients are broken by vessel horsepower class to account for any possible confounding affects of vessel size (i.e. larger vessels fish more days and have higher LPE$ than smaller vessels). Half of the correlations are not significant at P<0.05. Thirty-one of possible 70 horsepower class-area combinations show significant positive correlations indicating LPE$ increases as the number of days fished per trip increases. These correlations suggest that longer trips are associated with increased efficiency perhaps due to some within-trip learning by the skipper and crew. However, most of the significant positive correlations are for the smaller horsepower classes (i.e. 1-4) and these positive correlations may be an artifact of the effort measure. Many of the smaller vessels make short trips of 1-4 days. The actual number of hours trawled on a one day trip is less than the hours trawled on the second and third days of longer trips due to time spent steaming to and from port. Therefore, effort measured as days fished overestimates actual trawling time for one day trips relative to longer trips. Thus, LPE$ is underestimated for the day trips relative to the longer trips, and that would result in the positive 43 Table 2.3a. Spearman rank correlations (r ) between L P E $ and days s fished for ten vessel horsepower classes and seven areas. Numbers in parentheses indicate horsepower range included in each class. A l l years have been pooled. Correlations significant at P<0.05 are in bold type. Two dashes (--) indicate too few observations to calculate correlation coefficient. Horsepower class 1 2 3 Area 4 5 6 7 1(0-99) .42 .86 .06 .20 - . 0 8 .39 - . 1 7 2(100-199) .34 .28 .21 .21 .48 - . 0 7 - . 2 7 3(200-299) .35 .24 .04 .25 .51 - . 1 7 .39 4(300-399) .18 .37 .22 .28 .30 - . 1 6 .21 5(400-499) .51 - . 0 9 .08 .19 .19 - . 0 2 - . 0 4 6(500-599) .05 .62 - . 1 2 .40 .19 - . 1 1 .29 7(600-699) .19 .27 .47 .13 .08 .17 .14 8(700-799) .20 - . 0 1 - . 0 2 .20 .19 .13 .01 9(800-899) .09 .24 - . 2 2 .20 .32 .04 - . 7 9 10O899) .57 .16 « .12 - . 01 1.0 — Table 2.3b. Spearman rank correlations (r^) between L P E $ and days fished for ten vessel horsepower classes and seven areas. As in Table 2.3a except trips with 1 days fished were excluded from the analysis. Horsepower class 1 2 3 Area •4 5 6 7 1(0-99) .39 __ - . 1 2 .09 - . 2 2 - . 0 5 - . 2 3 2(100-199) .13 .12 .06 - . 1 4 .24 - . 2 1 - . 3 2 3(200-299) .27 .27 - . 0 5 .02 .16 - . 2 7 .10 4(300-399) .14 .38 .09 .02 - . 0 4 - . 2 4 .14 5(400-499) .38 - . 0 7 - . 1 6 .15 - . 0 8 - . 2 0 - . 3 2 6(500-599) .04 - . 2 0 - . 4 7 .16 .12 - . 1 9 .01 7(600-699) .01 .35 .35 - . 0 1 - . 0 5 .01 - . 2 2 8(700-799) .13 .04 - . 0 4 .26 .08 .01 - . 2 5 9(800-899) .03 .21 - . 1 5 .17 .32 - . 0 7 - . 8 6 10O899) .35 - . 5 2 « - . 2 3 .22 1.0 — correlations shown in Table 2.3a. To investigate the possible impact of this bias on my results, I recalculated the correlations excluding the 1 day trips (Table 2.3b). The reduction in the number of significant positive correlations by more than two thirds (from 31 to 10) indicates that the apparent increased efficiency of longer trips was probably an artifact of the effort measure. Negative correlations in either table may be due to the fact that boats with high LPE$ will fill their holds quicker and make shorter trips than vessels with lower LPE$. Furthermore, LPE$ values for various trip lengths (days fished) may not represent a random sample of skippers. More successful skippers are probably overrepresented in the shorter trips with higher LPE$ and vice versa. I was unable to test for a skipper effect because data on the skipper were not collected and recorded for the 1967-81 time period. Evidence against facilitation effects is shown in Table 2.4. Individual vessel LPE$ does not increase with increases in the number of vessels fishing in area. In fact, 6 of 10 horsepower classes show significant negative correlations, perhaps due to competition for preferred trawling sites. Correlations for landed weight per days fished (LPEW) show the same pattern. Therefore, the negative correlations for LPE$ are not due to increased supplies of fish depressing prices. The lack of a local supply effect on prices is not surprising, because most of the groundfish landed in British Columbia are exported and price is not determined by auction. I have refuted two possible explanations for positive correlations between effort and LPE$ flearning and facilitation) that might be confused with evidence of numerical responses to perceived abundance/price changes. The next section examines the numerical responses of effort to LPE$. 45 Table 2.4. Spearman rank correlations (r ) between individual vessel s LPE and number of trips made by all vessels to the same areas during the same week for ten horsepower classes. Correlations are presented for both landed value per effort (LPE$) and landed weight per effort (LPEW). All areas and years have been pooled. Horsepower LPE$ LPEW class n r P(r |p =0) r s s' s s s' s 1 962 -0.10 0.002 -0.17 0.0001 2 5434 -0.34 0.0001 -0.27 0.0001 3 4655 -0.18 0.0001 -0.13 0.0001 4 2968 -0.12 0.0001 -0.10 0.0001 5 943 -0.02 0.56 -0.06 0.08 6 749 0.01 0.78 0.01 0.77 7 641 -0.08 0.03 -0.09 0.03 8 910 -0.04 0.25 -0.04 0.17 9 422 0.16 0.001 0.13 0.007 10 61 -0.20 0.15 -0.20 0.13 Numerical Responses of Effort to LPE$ If the timing of LPE$ is affecting the timing of effort I would expect weekly effort patterns to be more predictable between years when the weekly LPE$ are more predictable and vice versa. Figure 2.8 shows the rg for effort from Figure 2.6 plotted with the r for weekly LPE$ plotted for consecutive pairs of years for each area. The rg values for effort arid LPE$ follow the same yearly trend in most areas with two notable exceptions. Area 3, Inside Vancouver Island, has a very predictable effort pattern but poorly predictable LPE$ pattern. The large number of small stationary vessels that fish in area 3 46 c ro ro ro QJ d CO .5-1-.5-0 -.5 .5-1 SU Vancouver Island • U=0.32 U=0.39 3 Inside Vancouver Island • « • • U=D.11 U-0.68 4 S Queen Char J o u t Sound I I I I I I • U'Q.AA I I I I I I • U=0.51 -i 1 1 1 1 1 1 1 1 1 1 1—l r 67-68 73 74 Year 80-81 • Landed Value($)/days fished • Days fished -.5 5 N Queen Charlotte Sound • U=0.4I U=0.54 6 Hecale Strait • U-0.54 U-0.36 7 U Quern Charlotte Islands Year Figure 2.8. Spearman rank correlations (r ) for weekly effort and landed s value($)/(daysfished) in consecutive pairs of years 1967/68 - 80/81 in each area. Stars (*) above solid squares and below open squares indicate significance at P<.05. Kendall's Concordance (W) values are recorded on each plot. Concordances are significant at P<.001, and P<.0001 for weekly effort and landed value($)/(days fished) respectively. are apparently willing to accept unpredictable LPE$ in exchange for a more regular pattern of effort. The correlations for area 1, SW Vancouver Island, 47 follow the same trend for part of the time period, but the LPE$ pattern is less predictable in the later years. The close correspondence between effort, and LPE$ correlations depicted in Figure 2.8 could be due to a numerical response. If so, I would expect higher effort in an area to follow weeks with high LPE$. Figure 2.9 is a plot of the values for the numerical response of effort one week after the LPE$. The majority of correlations are positive indicating that effort increases following weeks with high LPE$. Areas with the most consistent pattern of LPE$ also show consistent numerical responses between years. Areas 4 and 5 have the most consistent numerical response (highest concordance). These two areas are located farthest from the major landing ports where the cost of steaming to and from port, is greatest. The larger steaming costs increase the importance of fishing these two areas only when LPE$ is high enough to offset costs. I examined the data for evidence of interannual numerical responses, and found mixed results. Regressions of effort in year (i+1) vs. LPE$ in year i had significant positive slopes for areas 3,5 and 7, but slopes were not significantly different from zero in the remaining areas (Fig. 2.10). The positive annual numerical response for area 3 is interesting considering the lack of a intra-annual response (Fig. 2.9). However, the positive relationships for areas 3, 5, and 7 should be interpreted with caution because there were significant positive time trends in both variates in each of these areas. Regressions of residuals from these time trends were not significant for any of the areas. Correlations were found between intra-annual effort and LPE$ patterns,associated with a positive numerical response particularly in two of the seven areas. I also found interannual numerical responses in three of the seven 48 g 1 ™ .5 OJ i_ i- 0 o u 1! 1 c 03 ro OJ a. CO .5-.5 1 SU Vancouver Island * * * » 1 1 1 1 1 • U=0.35 2 NU Vancouver Island t i i i i r i p i • l l I l - I U=0.32 3 Inside Vancouver Island s > 1 1 i 1 \ 1 1 rl 1 I I * I 1 1 1 1 • • N?y • U>0.21 4 S Queen Charlotte Sound * V* » " » * ^m—•—B—BN.* > ^ l l 1 l l l l l l 1 1 1 1 1 1 1 1 1 l 1 1 1 1 1 l • - ! 1 1 1 1 1 1 1 1 1 1— 1 1 1 1 1 U=0.42 - i — r — i — i — i - 1 67 69 /I 73 75 77 79 81 Year Year Figure 2.9. Spearman rank correlations (rg) for the numerical response of effort (days fished) one week following landed values($)/(days fished) for the years 1967-81 in each area. Stars (*) indicate significance at P<.05. Kendall's Concordance (W) values are recorded on each plot. All concordances are significant at P<.0001. areas, although the evidence is confounded by time trends in effort and LPE$. The next two sections inspect these patterns in relation to the two remaining hypotheses that involve responses to LPE$. 49 2000 1500 1000 -| 500 0 1 SU Vancouver Island Y=.27X<550. rJ=.22 NS 600. 450-fe 300^ QJ >-600 1200 1800 2400 3000 2 NU Vancouver Island -o QJ SZ cn 150-1 0 Y--.04X*258. r -.03 NS (Jl ro Q 3500 2625-1750-875-0 600 1200 1800 2400 3000 3 Inside Vancouver Island Y-2.8X-I86. r -.28 * • 0 1000 750-500-250 0 . 200 400 600 800 4 S Queen Charlotte Sound Y=-.05X*406. rV02 NS 1000 700 1400 2100 2800 3500 1000 ^ N Oueen Charlotte Sound Y-.28X-439. r J-.52 » Q • 750-500 250-| 0 0 1000 • 2000 3000 4000 5000 350D-, 6 H t c a , t s , r a l t Y-.D7X-M2S6. r'-.OI NS a 2625-1 1750 875-1 0 • a a ~3? ~a~ CP.D 0 500 375-250-125 0 800 1600 2400 3200 4000 7 U Oueen Chariotie Islands Y-.10X-75. r*-.57 » • 13 1000 2000 3000 4000 5000 Landed value/days fished year i Figure 2.10. Effort (days fished) in year i-fl vs. landed value($)/(days fished) year i with regression lines and equations for each area. NS, * indicate regression slopes not significantly or significantly (P< .05) different than zero, respectively. 50 7500 56254 3750 1875 "P 5000 J= ^ 3750 u_ 2500 1250 ro <5 2680 ™ 2010 TD 1340 OJ XD CZ 670 ro _ J 0 6000' 4500-3000-1500 • 0-I SU Vancouver Island i 1—i—i I i—r 2 NU Vancouver Island 3 Inside Vancouver Island m 111 ~ i i i i j ~ 4 S Queen CharJotie Sound 8 16 24 32 Week 40 52 8 16 24 32 Week Figure 2.11. Landed value($)/(days fished) vs. week for each area. Squares are medians and error bars the interdecile range for the years 1967-81. Equalize LPE$ The Equalize LPE$ hypothesis predicts that boats move to make the LPE$ in each area tend toward the average LPE$ of all areas. The within-year LPE$ Table 2.5. Kruskal-Wallis nonparametric ANOVA comparing weekly LPE$ in the seven aggregate areas. All years (1967-81) have been pooled. H : The areas have the same weekly LPE$s. Area n avg.rank . 1 780 3088 2 780 2032 3 780 2386 4 780 2694 5 780 2828 6 780 3799 7 780 2286 Total n 5460 Kruskal-Wallis H Statistic = 654.1, df=6, P<0.0001 patterns are shown in Figure 2.11. There are significant differences in weekly LPE$ between areas (Table 2.5) indicating that vessel movement has not resulted in equal LPE$ for all areas. Figure 2.12 plots-the inter annual LPE$ pattern for each area. The concordance in rank (W=0.67;P< 0.0001) indicates areas have not tended toward the same average LPE$. These tests clearly reject the hypothesis that vessel movements have equalized LPE$ in the seven areas on either a within or between season time scale. Maintain Relative LPE$ A reinspection of Figures 2.11 and 2.12 shows that while each area does not tend towards the same average, some areas have consistently higher LPE$ than others. This observation is confirmed by the concordance measure for 52 ~5 5000 -, co 3750 -ro - a <O 2500-QJ ZD —I CD > TD QJ TD CZ 1250-• o ? A 1 SU Vancouver Island 2 NU Vancouver Island 3 Inside Vancouver Island 4 S Oueen Charlotte Sound 5 N Queen Charlotte Sound 6 Hecate Strait 7 U Queen Charlotte Islands ~~l I 1 1 1 1 1 1 1 1 1 1 1 1 T" 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 Year Figure 2.12. Landed value($)/(days fished) vs. year for each area in the years 1967-81. between year LPE$. If each area has its own unique time invariant desirabilities and costs, then vessel movement should maintain each areas relative LPE$. The weekly ratios of each area's LPE$ to provincial average LPE$ (RPA) are plotted in Figure 2.13. There are significant differences between weekly RPAs within 5 of the 7 areas indicating that the weekly RPAs are have not remained constant within years (Table 2.6). The RPAs in Area 5 show marginally significant differences. The lack of significant differences in area 7 is probably due to the 53 5.0 2.5 0.0 QJ 5.0 -c n ro L_ QJ > ro 2.5-"ro , i ^ » CJ c: — i 0.0 -> o 5.0 -a o -t-1 2.5-o — i ro or 0.0 -5.0 -2.5-0.0 I SU Vancouver Island - i — i — i — i — r — i — i — i — i — i — i — * i — r 2 NU Vancouver Island ll T I I I I I I 't '7 1 1 I""" f 3 Inside Vancouver Island — i — I — I — I — I — i ' i ' — i — i — i — i — i — r - i — I — I — I — i ' t ' — i — i — i — i — i — r 4 S Quern Charlotte Sound I 1 ' 1 . 1 I I 1 I I I I 0 8 16 24 32 40 52 5.0 2.5-0.0 5.0 2.5 0.0 5.0 2.5 0.0 5 N Queen Charlotte Sound 1 i i i — i — i i i i i i i i 6 Hecate Strait i i i — i — I 1 1 — i — i — i — i — i i 7 U Quern Char 1 one Islands — r - * - i — i — i — i — i — i — i— i — r -0 8 16 24 32 40 52 Ueek Week Figure 2.13. Weekly ratio to provincial average vs. week for each area. Squares are medians and error bars the interdecile range for the year 1967-81. Latter weeks lack rpa values or error bars because effort was either not present or present for only a few years. large variability in weekly RPA for this area (Fig. 2.13). 54 Table 2.6. Summary of Kruskal-Wallis nonparametric ANOVAs comparing RPAs between weeks within each of the seven aggregate areas. All years (1967-81) have been pooled. H^: The weeks have the same RPA. Area H statistic df P 1 83.7 51 0.003 2 72.5 47 0.010 3 103.2 51 0.0001 4 111.4 48 0.0001 5 61.3 48 0.094 6 91.3 51 0.0006 7 41.8 51 0.82 Figure 2.14 shows the annual ratio to provincial average plotted for each area. The concordance (W=0.67;P<0.0001) indicates the annual RPAs have been fairly constant in rank among years. Summary This chapter examined the allocation of fishing effort by the trawl fleet to seven aggregate fishing areas located off the British Columbia coast. Three hypotheses proposed by Hilborn and Ledbetter(1979) were tested: I. Movement follows traditional patterns, II. Movement equalizes the gross dollar returns to effort (LPE$) among areas, and HI. Movement maintains relative LPE$ in each area. The major conclusions of this chapter are : 55 QJ 01 3.0 -, ro i_ QJ > CE ro —i u cz > o t_ Q_ 2.0 1.0 -ro 0.0 o 1 SU Vancouver Island • 2 NU Vancouver Island • 3 Iralde Vancouver Island * 4 S Queen Charlotte Sound o 5 N Queen Charlotte Sound T 6 Hecaie Strait A 7 ll Queen Charlotte Islands i—i 1 1 ' i 1 1—i 1—i—i—r~—r—i r -67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 Year Figure 2.14. Ratio to provincial average vs. year for each area and the years 1967-81. 1. The proportion of effort allotted to each of seven aggregate areas maintains concordance in rank among years which is consistent with the traditional patterns and maintain relative LPE$ hypotheses. 2. There is a traditional component to the within-year timing of effort, but effort timing is modified by changes in the timing of LPE$ in all areas and by regulation in two of the seven areas studied. 3. Vessel movement has not equalized the LPE$ in the areas. 4. The ratios of each areas LPE$ to the average LPE$ of all areas have not remained constant but do show concordance in rank across years which is consistent with the maintain relative LPE$ hypothesis. 56 None of the hypotheses examined explain the observed within-year pattern of effort allocation. However two hypotheses, Traditional patterns and Maintain Relative LPE$, are consistent with the among-year pattern. The latter hypothesis, however, still has some problems. Relative LPE$s may be maintained by movement patterns, but also could result from differences in vessel size between areas. Furthermore, the rejection of the Equalize LPE$ hypothesis may have resulted because differences in costs were not taken into account. Equalization of net LPE$, not maintenance of relative gross LPE$, may be the underlying explanation. The next chapter examines the assumptions of the Equalize LPE$ hypothesis in more detail. 57 Chapter 3: Evaluation of Assumptions Introduction This chapter examines the effects of the assumptions of the Equalize LPE$ hypothesis (Ch. 2 ;p. 22-23) on the conclusions of chapter 2. The examinations for each assumption vary from detailed analyses to discussion depending on available data. Assumptions 1, 3 and 4 apply to both the Equalize LPE$ and Maintain Relative LPE$ hypotheses. Thus, investigations of assumptions 1, 3 and 4 consider. primarily whether violations may have resulted in the false rejection of the Equalize LPE$ hypothesis and secondarily whether violations may have given false support to the Maintain Relative LPE$ hypothesis. Assumption 2 is relaxed in the Maintain Relative LPE$ hypothesis; thus violations of assumption 2 are considered primarily as possible causes of false rejection of the Equalize LPE$ hypothesis. " Section one applies a technique called effort standardization to data from the trawl fleet in order to remove possible violations of the assumption that all boats are equal. Section two considers violations of the assumption of equal costs and desirabilities for all areas. Unfortunately, detailed data on area specific costs were unavailable, so section 2 first considers what evidence have been presented thus far that either supports or cast doubt on the assumption that areas are equally desirable. Next, section 2 presents some estimates of differences in costs between vessel size classes and their implications for the assumption of equal area specific costs. 58 Section 3 uses estimates of steaming times between areas as indices of movement costs to investigate possible violations of the assumption that movement costs are negligible. Section 4 examines the quality of information and the likely extent of information exchange for four sources of information available to the trawl fleet. A literature review of information exchange in other fisheries is used to draw analogies with and predict the likely amount of information flow in the trawl fleet. All Boats are Equal My examination of LPE$ and effort correlations in chapter 2 were subdivided by horsepower class to eliminate possible confounding effects of vessel size. Vessel horsepower differences between areas may have caused some of the LPE$ differences shown in chapter 2. I used a method known as effort standardization to examine the effects of relaxing the all boats are equal assumption on tests of the Equalize LPE$ and Maintain Relative hypotheses. More specifically, I consider whether differences in catching power among vessels fishing in different areas may have caused the LPE$ differences that resulted in the rejection of the Equalize LPE$ hypothesis on the interannual time scale and/or the concordance in the rank of annual RPA's that was consistent with the Maintain Relative LPE$ hypothesis. Effort was standardized using vessel horsepower because horsepower has been indentified as a major determinant of fishing power in trawlers by Gulland 59 (1956a), Beverton & Holt (1957), Houghton (1977) and in British Columbia trawlers by Ketchen & Thomson (1958), Ketchen & Forrester (1966), Kimura (1981), Stocker & Fournier (1984), and Westrheim & Foucher (1985a). This technique has been widely applied to trawl Fisheries (see Westrheim & Foucher 1985a for a review). The above studies standardized effort to account for differences in landed weighted per unit effort among different sized vessels. Landed values were first used in fishing power studies by Carlson (1975). Standardization of effort using landed values may yield different results from standardization using landed weight, if for example, smaller vessels landed smaller weights of higher valued species. However, the Equalize and Maintain Relative LPE$ hypotheses compare LPE$ among areas and years, not landed weight per effort. Therefore, I standardized effort to account for differences in LPE$ among vessels. Before proceeding with the process of trying to correct for differences in vessel horsepower among areas, I review evidence concerning the magnitude of differences among areas. Vessel horsepower has increased in each of the areas over the years 1967-80, and vessel horsepower varies between the seven areas (Fig. 3.1). Table 3.1 presents the results of a Kruskal-Wallis nonparametric ANOVA that indicates that the horsepower differences between areas shown in Figure 3.1 are statistically significant. Similar differences can be shown in other attributes such as length and gross or net tonnage because all are highly correlated with horsepower (Table 3.2). Data and Methods DFO saleslip and license data described in chapter 2 were used for the analysis. The main data requirements for the method described below were 60 Table 3.1. Kruskal-Wallis nonparametric ANOVA comparing vessel horsepower in the seven aggregate areas. Observations are included for each trip made by a vessel to an area. All years (1967-80) have been pooled. H : Areas have the same vessel horsepower. Area n avg.rank 1 3270 9607 2 789 9935 3 6786 5700 4 1456 11946 5 1129 . 14001 6 4531 11780 7 651 13362 Total n 18612 Kruskal-Wallis H Statistic = 5622.26, P<0.0001 Table 3.2. Product-moment correlation matrix for four vessel attributes. Observations are included for each vessel that fished during the time period 1967-80. r 0.05 is the the value of the correlation coefficient necessary for significance at P=0.05. Attribute r value Length ,1.0 Netton1 0.86 1.0 Groston1 0.86 0.97 1.0 Hrspwr2 0.81 0.89 0.90 1.0 Length Netton Groston Hrspwr n=1326, df=1324, r 0.05 = 0.054 1 Netton = Net tonnage, Groston = Gross tonnage; both are volume measures. 1 ton = 100 ft . 2 . Hrspwr = Horsepower of the motor 61 1200-800-600-300 0 1700' 900-l_ OJ g 600-1 CL OJ 300 LTl l_ 5 " _,1200 OJ 3 300 OJ > 600 300-1 1200 900 600 300 0 1 SU Vvcuvxr Island II I 1 I I I I 1 ! I 1 I 2 HU V r a m r Ulwi H l i ' l - i — i — i — i — i — i — i — i — i — i — i — i — i — r 3 Imldr Vaxamr Ultni • 11.111 H 1111H -i—i i i i i i i i—i—i—• i 4 S Owen Charlntr Saund II i i i i i i i i i i i i i i 67 63 71 73 75 77 79 t200-900. 600. 300' 0' 1200 900 600-300-0 1200 900 600 300 0 S N Ourm Oarlolle Sound — i — i — i — i — i — i — I — i — r -6 Hccrtr Strait IMP 11 i i i i i i i i i 7 U Ourm Charlmu l a l n k 1.1 l.l ' » ' I 1 i i i i i i i i i i i—i i i 67 89 71 73 75 77 79 Year Year Figure 3.1. Vessel horsepower vs. year for the years 1967-80 in each area. Squares represent means and error bars the interdecile range. horsepower, landed value of the catch, and effort by vessel and area. Westrheim and Foucher (1985a) standardized effort for major statistical areas exploited by the British Columbia trawl fleet. My analysis follows their approach with two exceptions: (1) I standardized effort based on landed value 62 without qualification level rather than landed weight with an 80% qualification level (i.e. Westrheim arid Foucher (1985a) included only those trips in which the target species or group constituted 80% or more of the total landed weight), and (2) My space-time stratification is defined by the seven areas and years respectively (see Westrheim and Foucher 1985a, p. 1620 for their area-time strata). Westrheim & Foucher (1985a) used the model of Gulland (1956, p. 5) for deriving estimates of relative fishing power: Y..=P.Xt..XD.. Xa.Xe.. (3.1) • y i v y J y where Y=catch of the vessel, P = vessel's absolute fishing power, t= time spent fishing, D=density of fish, a—vulnerability of the fish, e = a random error term, and the subscripts i and j refer to the vessel and area/time period respectively. Landed weight per effort, (LPUE=Y . It .), of a "standard" vessel is substituted sj s/' for a. X D. when these two factors are unknown. J J Relative fishing power (RFP) is then calculated as: R F P - ^ y (3-2) R F P r LPUE . Although Westrheim & Foucher (1985a) used the same underlying model as Gulland, their application methods differed. Gulland used a log-transformed version of equation 3.1 and applied it to individual vessels in restricted area-time cells. Westrheim and Foucher (1985a) calculate the RFP of vessel horsepower classes for selected species groups and broad area-time cells. My analysis involves 63 large aggregate areas and years, so I chose to follow Westrheim and Foucher's (1985a) method. Substituting LPE$ for LPUE, and landed value for landings, the method has four steps (see Westrheim & Foucher 1985a, p. 1623): 1. Compile landed values, nominal effort and LPE$ by horsepower class, area, and year. 2. For each area select a standard horsepower class based on the number of years it fished, cumulative value of landings, cumulative days, fished, and cumulative number of trips made. 3. Calculate mean RFP for each horsepower class as: LPE$.. l y -RFP.=— L i n. i LPE$ . (3.3) sj where RFP= the mean relative fishing power, n = the number of years of records, LPE$ = the landed value per days fished, and i and j are the horsepower class and year, respectively. 4. Regress RFP on horsepower class to correct for horsepower classes with small values of n. and recalibrate to the standard horsepower class: RFPu{.=o + 6 (horsepower class i) (3.4a) RFPu. ,„ ... RFPc = l- ( 3 , 4 b ) i RFPu s where RFPu = uncalibrated estimate of relative fishing power, RFPc = calibrated estimate of relative fishing power, a and b are regression parameters and i and s denote horsepower class and standard respectively. 64 Westrheim and Foucher (1985a) arbitrarily excluded from the regression any horsepower classes that fished only a few years. I performed weighted regression using all the data, where the regression weights were the number of years that each horsepower class fished. Horsepower classes that fished only a few years were included, but were given less weight in the regression in favor of horsepower classes that fished during more of the time series. Using the predicted RFPs from the regressions, the standardized effort is determined for each horsepower class, area, and year as: SE.,.=NE.,.XRFP., (3.5) ikj ikj ik where SE = standardized effort, NE = nominal effort, and i,k, and j denote the horsepower class, area, and year, respectively. The standardized effort for each area and year is: SE,.=Z SE.,. (3.6), where all symbols are as above. Note from equation 3.5 that horsepower classes greater than the standard had RFP's greater than 1.0 and greater standardized effort than nominal effort, while for horsepower classes less than the standard, RFP's were less than 1.0, and the converse was true. Thus areas with larger vessels had lower standardized LPE$ than nominal LPE$ and vice versa for areas with smaller vessels (see below). 65 Results Figure 3.2 shows the regressions of step 4 for each of the seven areas. . Horsepower class accounted for 30-90% of the within area variation in RFP. The poor fits in Areas 2 and 7 are probably due to the low effort exerted annually in these areas that results in greater variability in the LPUE values for different horsepower classes. The effect of using standardized rather than nominal effort on the LPE$ values for each area is shown in Figure 3.3. Area 5 showed the greatest difference between nominal and standardized LPE$. As expected, the standardized LPE$ is less than the nominal LPE$ in most areas, except notably area 3 where standardized LPE$ is greater because most of the effort in area 3 is exerted by small horsepower classes with relative fishing power less than 1.0. I also re-examined the annual numerical responses of Figure 2.10 using the standardized effort and LPE$ values. Only the regression for area 7 retained 2 its significance (r =0.44, P<0.01). However, as before, the relationship is confounded by time trends in the standardized variates and the slope of the 2 regression of the time trend residuals was not significant (r =0.18, P< 0.16). How does effort standardization affect how well the data agree with the two LPE$ hypotheses? The concordance measure for the standardized LPE$'s shown in Figure 3.3 is only slightly less than the concordance for nominal LPE$ (0.61 versus 0.67; cf. Fig. 2.12). Thus it is unlikely that vessel catching power differences resulted in the false rejection of the Equalize LPE$ hypothesis in Chapter 2. The standardized RPA's are shown in Figure 3.4. Effort standardization reduces the between-area differences in RPA, but again the 66 5.0 2.5 0.0 8.0 4.0 Q_ L L cn CZ 0.0 ro Or114.0 7.0 0.0 4.0-1 2.0 0.0 1 SU Vancouver Island YOIX-U2 r?=.89 U^B^ 1 3 9 14 J*^\\ —i 1 1 r 1 1 2 NU Vancouver Island Y=.35X03 r*=.58 -i 1 1 r "i 1 1 1 1 1 1 r 3 Inside Vancouver Island Y-.69X-.40 r -.79 I • i —i 1 1 1 1 1 4 S Oueen Charlotte Sound Y-.24X04 r J-.86 13 13 -i 1 1 1 1 1 1 ——i 1 r 1 2 3 4 5 6 7 8 9 10 4.0 2.0 -0.0 4.0 -2.0 0.0 4.0 2.0 0.0 5 N Oueen Charlotte Sound Y-.08Xt.42 r ?-.76 —i 1 1 r-6 Hecate Strai t Y-.15Xt.60 r ?-.74 —i 1 1 P 1 1 1 1 i r-7 U Oueen Charlotte Islands Y-.08X*.89 r'-.32 1 2 3 4 5 6 7 8 9 10 Horsepower, class Figure 3.2. Mean relative fishing power vs. horsepower class for each of the 2 seven areas. Regression lines, their equations and their r values are shown for each area. Numbers beside squares are the number of years that each horsepower class fished in the respective areas. 67 4000 3000 2000 1000 -D,DOO [ 3000 20D0 1000 -LT) ro W u OJ 2000 ro 1500 •O 1000 QJ T J C 500 ro _ J 0 • 4000 3000 2000-1000 -0 1 SU Vancouver Island of rf rf rf H D D • - 1 — i — i — i — i — i — i — i — i — i — i — i — i — i — r 2 NU Vancouver Island • LD Q • ° • • • 1 - ° • e? r j „ " 1 1 - i 1 1 1 1 1 1 i 1—i 1—i—i r r 3 InsJdf V a n c o u v e r Island ™ .._ _ ^ m m rn rn • • • D D • • —i 1 1 1 1 1 1—1 1—i 1 1 1 1 r 4 S Queen Charlotte Sound • m • CD O * * B 1'ra • • • • • ° ° • - i — r — i — i — i — i — i — i — i — i — i — i '• i — i — r 67 69 71 73 75 77 79 81 Year • Noninal Effort • Standardized Effort 5000 3750 2500 1250 0 5000-3750-2500 1250-0 6000 -4500 3000 -1500 -0 5 N Queen Char lone Sound •f " « . rn ° ° » • % m m a a * a D "CD CD —i 1 1—i 1 1 1 1 1 1 1 1 1 1—r 6 Hecair Strait ~ i — i — i — i — i — i — i — i — i — i — i i — i — i — r 7 U Ourrn Char J one Islands — ! — i — i — i — i — i — i — i — i — i — i — i — i — i — r 67 69 71 73 75 77 79 81 Year Figure 3.3. Nominal and standardized landed values($)/(days fished) vs. year for the years 1967-80 in each area. concordance in rank was only affected slightly, changing from 0.67 to 0.61. While fishing power differences exist between vessels fishing in different areas, taking account of them has only slightly reduced the differences in LPE$ between areas and has not changed the qualitative results. However, the standardization of effort is directed at removing vessel size effects on indices of gross LPE$, 68 QJ c n 3.0 r o L _ QJ > CE —i (TJ —I 2.0 -U d —1 > o i Q _ 1.0 -o O —1 H—1 ro c r 0.0 • 1 SU Vancouver Island • 2 NU Vancouver Island • 3 Inside Vancouver Island • 4 S Queen Charlotte Sound o 5 N Queen Charlotte Sound V 6 Hecate Strait A 7 U Queen Charlotte Islands 67 68 69 70 71 72 73 74 75 76 77 78 79 80 Y e a r Figure 3.4. Ratio to provincial average calculated using standardized landed values($)/(days fished) vs. year for the years 1967-80 in each area. and does not account for effects of vessel size on other factors such as fishing costs or overall mobility. The possible effects of fishing costs will be considered in the next section. 69 Equal Costs and Desirabilties of Areas The assumption of equal costs and desirabilities is relaxed in the Maintain relative LPE$ hypothesis, but the assumption is essential to the Equalize LPE$ hypothesis to account for costs and noneconomic factors that may affect a skipper's choice of one area over another. This section first considers whether results presented thus far suggest that areas have equal desirabilities. Next, it presents some estimates of differences in costs between vessel size classes and their implications for the equal cost1 assumption. Desirability is an ambiguous term, but in the fisheries context, factors that affect the desirability of areas include (1) protection from or exposure to weather, (2) distance to home port, (3) personal preference based on past experience, (4) risk of gear damage, (5) available species and markets (6) depth distribution of available species and other intangibles that are not accounted for by the LPE$ benefit measure. I have presented some evidence that suggests areas differ in their desirabilties. One example is the high concordance in rank of the proportion of effort alloted to each area annually. The concordance for effort is higher than for RPA, which coupled with the weak annual numerical responses (Fig. 2.10) suggests that the fleet is not simply allotting their effort strictly in response to some constant relative benefit. Other factors not included in the benefit measure may be important (i.e. desirabilities and costs). In addition, a large proportion of total coastwide effort is consistently exerted in area 3 (Fig. 2.2), despite an unpredictable LPE$ pattern, and higher 70 and more predictable LPE$s in adjacent areas (Fig. 2.8). Area 3 does have attractions: it is close to Vancouver, its waters are sheltered from major storms by Vancouver island, and its proximity to markets lets fishermen have a more regular work schedule. This latter quality is apparent in the the very consistent intra-annual pattern of effort in area 3 (Fig. 2.8). However, the apparent desirability of area 3 to smaller vessels may be dictated by undesirable qualities or the operational dangers of the other areas. Small vessels are limited to periods of relatively calm weather on the adjacent offshore areas (i.e. areas 1,5) and large steaming distances preclude trips to the other areas (e.g. areas 4-7). The revenue from a filled hold on a small vessel might not offset the fuel costs for travelling such distances. The second part of the assumption is equal fishing costs of the areas. The most significant and consistent fishing cost (excluding movement costs and unpredictable costs due to gear damage or equipment breakdown) is the cost of fuel for trawling and movement within areas. If the number of hours trawled is proportional to the number of days fished, my gross LPE$ values for each area should be roughly proportional to the Net LPE$. However, equalizing gross LPE$ is only equivalent to equalizing net LPE$ under the assumption of equal fishing costs in the areas. Therefore, if my rejection of the Equalize LPE$ hypothesis was due to violations of the equal area specific cost assumption, I must show differences in the area specific costs that are proportional to gross LPE$. One factor that could result in costs that are proportional to gross LPE$ is the difference in vessel horsepower among areas shown in the previous section. Padilla (1986) presented a table of costs per vessels-ton-day for three length categories derived from a sample of 59 B.C. trawlers taken in 1982. His figures are hard to interpret however, because length and tonnage are highly Table 3.3. Estimated variable and total costs per days fished for * three vessel length categories in 1982. Length(ft) Variable^ Total cost($)/day cost($)/day <64 1067 1447 65-84 2336 3244 >85 5129 6311 * Modified from Padilla (1986). 1 Includes crew share, provisions, fuel, oil, and grease. correlated (Table 3.2), and he did not present the raw data. In an effort to reinterpret his data, I used the 1980 attribute data to calculate the mean gross tonnage for each of his vessel length categories and multiplied these values by his cost figures to obtain costs per day fished for each length category (Table 3.3). Although these figures are approximations, they clearly show both total and variable costs increase with vessel size. Thus, it is likely that including vessels' fishing costs would reduce differences in net landed value per day fishing between areas, because of vessel size differences between areas. In the case of effort standardization this reduction in LPE$ differences between areas was slight. Therefore, it is unlikely that accounting for costs due to vessel size differences would result in a failure to reject the Equalize LPE$ hypothesis. 72 Other differences between areas such as trawling depth, local currents and type of bottom could affect the relative costs of trawling per hour. There are insufficient data to predict what effect these factors have on the relative magnitude of area specific costs. Movement costs within areas and costs of searching are probably related to the size of the areas and the proportions of bottom that are trawlable. The costs of movement within areas are probably small relative to the cost of trawling, because: (1) most trawling grounds are well known to most trawlers, (2) navigational aids such as Loran provide accurate and repeatable location information and (3) searching is often combined with trawling to offset costs. However, the ratio of trawling cost and searching cost undoubtably varies considerably with conditions such as abundance and availability of fish, target species, bottom type, and experience of the skipper. In conclusion, areas probably differ in fishing costs, because of differences in many of the above factors. The equalize LPE$ hypothesis is most sensitive to violations of, the assumption of equal desirabilties and costs but violations should not effect the Maintain Relative LPE$ hypothesis unless the relative area-specific costs and desirabilities vary over time. However, the evidence presented is insufficient to predict the quantitative effects of accounting for fishing costs in each area. Qualitatively, accounting for differences in costs due to vessel size differences among areas, would reduce differences in net landed value per day fishing between areas which would be consistent with an Equalize Net LPE$ hypothesis. 73 Negligible movement costs Given the large aggregate areas and the existence of only two major landing ports at opposite ends of the province, movement costs are substantial, not negligible. Furthermore, costs of movement within the areas are probably small in comparison to the costs of steaming to and from port. This section first considers the impact of movement costs on movement patterns. The second part presents estimates of steaming times (a gross measure of movement costs) and their implications. I was unable to include a detailed analysis of the effects of movement costs for lack of data. For Vancouver based vessels, the cost of steaming to the northern areas is much greater, so the major decision is whether to fish the Areas 1 or 2 on the west coast of Vancouver Island or the northern areas (4-7). Given that most of the fleet fishes out of Vancouver, I expect movement costs to force vessels to be more selective in their choice of when to fish the northern areas as compared to the southern areas. The pattern found in the consistency of numerical responses supports this assertion. Two areas in Queen Charlotte Sound had the most consistent numerical responses, while area 3 had the least consistent numerical response and areas 1,2 and 6 were intermediate (Fig. 2.9). The less consistent numerical response in area 6 compared to areas 4 and 5 may result from the greater number of stationary vessels based in Prince Rupert that fish in area 6 (Table 2.2). The low consistency of area 7, is probably due to the low amount effort exerted in 74 this area (Fig. 2.2), and the areas exposure to greater extremes of weather. It is apparent that movement costs are different between areas and that these differences have had some effect on the observed vessel movement patterns. To examine the effects of movement costs, I estimated the total steaming times from home port to each area (and back) for all vessels each year. The steaming times are based on the distances from each port to each of the areas assuming a steaming speed of 10 knots (D. March, B.C. Trawlers' Association, pers. comm.). I then divided these total times by the number of days fished in each area. If movement costs are responsible for the rejection of the Equalize LPE$ hypothesis, then steaming times/(days fished) (STPE) should be proportional to the LPE$ in each area. On the average, areas with higher gross LPE$ also have greater STPE and higher proportions of larger vessels (Table 3.4). While there are exceptions (e.g. areas 2 and 6 have relatively higher and lower steaming times for their LPE$, respectively) to this general trend, Table J3.4 suggests that the inclusion of movement costs would tend to reduce differences in net economic attractiveness among areas. Similarly, on an annual basis, LPE$'s are positively correlated with STPE's in 6 of the 7 areas (Fig. 3.5). If movement costs were included annual differences in LPE$ between years in each area would also be reduced. The correlations between LPE$ and STPE in Figure 3 are not perfect (i.e. only 4 of the positive correlations are significant (P<0.1)), but STPE is only a crude measure of movement costs. I have estimated steaming times to the major ground in each area, or to the center of areas with several grounds (e.g. Area 6, Hecate Strait), but the areas are quite large so actual steaming times might vary around these "average" estimates. Also, cost of steaming varies with vessel 75 Table 3.4. Average standardized LPE$'s, steaming times/(days fished) (STPE), and vessel horsepowers for each area and the years 1967-80. Area LPE$ STPE (hrs) Horsepower 1 2 3 4 5 6 7 1427 1685 756 1699 2225 2091 2098 5.8 14.8 4.2 11.6 13.2 11.5 17.1 309 312 194 402 503 367 482 size, the amount of fish aboard, and may have even varied with vessel size over the years as newer, more fuel efficient vessels entered the fishery. Despite these factors, the qualitative effect of including movement costs is to reduce differences in LPE$ between areas, and reduce differences in LPE$ between years within areas. The evaluation of movement costs does not support either LPE$ hypothesis over the other. Nevertheless, Table 3.4 and Figure 3.5 suggest that movement costs are an important consideration in the trawl fleet, and that equalization or maintenance of relative Net LPE$ may be the net result of movement patterns. Perfect Information Both the LPE$ hypotheses rely on vessel movements in response to perceived benefits to either equalize or maintain relative LPE$. The amount and 76 15000 • 11250 -7500-3750 0 -—; 7500 gS62S l JZ 3750-OJ 1875 cn 0 c —I 7500 EE ro QJ 5625-1 U7 3750 ro 1875-0 7500 5625-3750 1875 0 1 SU Vancouver Island r=-.16 • • 1 » . D • • • a • " • • u m a a • • • - i — i — i — i — i — i — i — i — i — i — i — r ~ 2 NU Vancouver Island r—.39 D O * < ? 1 . • ' " • • D • • "T 1 1 1 1 I I 1 3 Inside Vancouver Island r--.06 • m • • • • -i—i—i—i—i—i—i—i—i A S Oueen Charlotlf Sound • • • r.-.48 • • CD • • • • • a —i • — i — i — i — i — i — i — i — i — i — i — i — i — r 67 69 71 73 75 77 79 8t Year • Steaming time • Landed valuel$)/days fished (Standardized) 15000 11250 7500 3750-0 32000-24000 16000 8000 0 7500 5625-3750-1875 0 5 N Outen Charloite Sound r-.26 D D m D * - i — i — i — i — i — i — i — i — i — i 6 Hecate StraJi r--.26 • • • • • D a - i — i — i — i — i — i — i — i — i — i — i — i — i — i — r -7 U Queen Charlotte Islands r-.68 ° m • • • • • • " • • • • Q CO cn CO a a 67 69 71 1 1 1 1 1 1 73 75 77 Year Figure 3.5. Steaming times/(days fished) and standardized landed values($)/(days fished) vs. year for each area. Product moment correlation coefficients (r values) are shown for each area. Two stars (**), one star (*), or NS indicate significance at P<.05, P<0.1, or not signficant respectively. Note: Scale for landed value($)/(days fished) is the same as Figure 3.3. quality of information affects fishermen's perceptions and presumably influences their choice of fishing location. There are several sources for such information and I will briefly discuss four of these below. 77 1. Historical data 2. Personal experience of the skipper or crew 3. Processing plants 4. Other fishermen Historical data Some of the trawling grounds off British Columbia have been exploited since 1912, and the Fisheries Research Board of Canada has kept records of trawlers since 1944 (Lippa 1967). Reports of monthly landing by species and statistical area are available from university libraries. The locations of many of the traditional fishing grounds are clearly marked on most nautical charts. Reports of exploratory trawl surveys conducted by the Department of Fisheries and Oceans are commonly sent to participating and/or interested fishermen. I am not certain of the extent to which fishermen use historical data, but it is likely that from a combination of the historical sources and conversations with willing oldtimers, an inexperienced trawler could gain general knowledge on seasonal distributions of species in each of the areas. However, these sources do not provide the timely information concerning catch rates in specific areas needed to respond to within and between trip changes in abundance. Personal Experience Personal experience is perhaps the most trusted source of information because it does not depend on the opinions of others. Personal experience may 78 be a particularly valuable source of information on general seasonal patterns of availability in particular areas, but it is limited to the experience of the skipper and possibly crew members if they have worked previously on other vessels. Personal experience cannot provide within-year information of catch rates in areas other than those fished by the given skipper, and therefore as a sole source of information it would likely result in quite restricted movement patterns for most vessels. Plants and Other Fishermen Of the four information sources, plants and other fishermen have the greatest potential to provide the timely information on catch rates needed to choose the "best" fishing location. Before discussing these information sources, it is relevant to consider: What is perfect information from the individual vessel perspective? Figures 3.6 (cf. Fig. 2.8), 3.7 (cf. Fig 3.4 open squares), and .3.8 attempt to quantify the information that would be available in a scenario where each vessel knew the LPE$ obtained and area fished by every other vessel. On both the among season (Fig. 3.6), and within season (Fig. 3.7) time scale, this complete information sharing scenario provides some clues about which areas are on average better than others and the relative predictability of seasonal LPE$'s in each area. The value of the perfect state depends on the ability of the skipper to discern these patterns from the rather variable individual observations (Figs. 3.6, 3.8). The spatial areas chosen by skippers (ie. fishing grounds) are smaller than these larger aggregate areas, but these figures at least present a rough idea of the level of information available on the spatial and temporal scales over 79 4000 3000 2000 1000 P 4000 QJ JZ ^ 3000 ^ 2000 _ g 1000 -w 0 Q J 2000 ZD — I ^ 1500 TD looo QJ T D C 500 ro _ l 0 4000-3000 2000 -1000 0 1 SU Vancouver Island — i 1 r T 1 1 1— 7 N V V a n c o u v e r Island - i — i — i — i — i — i — i — i — r -3 Inside V a n c o u v e r Island 444*44444444444 ~ i 1 1 — I — I r 4 S Queen Charlotte Sound I! I I I' 1 1 1" 1 1 1 1 I 1 67 69 71 73 75 Year 77 79 81 5000 3750 2500 1250 -I 0 5000-3750-2500 1250 0 6000-4500 3000-15U0-0 S N Queen Charlotte Sound I ! I I I I "I I I 1 1 6 Hecate S t r a i i Hut ~l 1 l 1 1— ~ l 1 1 1 1 1 1 1 r — 7 U Ouren Chariot it Islands i^ ,tl((i"i4-i i i — i — r — i — i — i i i i i i 67 69 71 73 75 77 79 81 Year Figure 3.6. Landed value($)/(days fished) vs. year for each area. Squares are medians over vessels and error bars the quartiles for the years 1967-81. which fishermen make major decisions. Deviations from the perfect information state depend on the amount and quality of information exchanged either between processors and fishermen, or among fishermen. Some indication of the amount and quality of information exchanged comes from anthropological studies. 80 c o ro QJ 1_ l _ O <_) LZ ro c ro ro QJ a. CO 1--.5 1-.5-.5-1--.5 1 SU Vancouver Island • U=0.28 2 NU Vancouver Island • U=0.29 3 Inside Vancouver Island +-v-• U-0.09 4 S Oueen Charlolte Sound • U-0.41 - i — i — i — i — i — i — i — i — i 1 — r — T — i — r 67-68 73-74 Year 80-81 .5-.5--.5 5 N Queen Charlolte Sound I I I I I I I 1 I I I • U=0.41 6 Hecate Straii I I I I—I I I I I I I I I I • U=0.52 7 U Queen Charlotte Islands 67-68 80-81 Figure 3.7. Spearman rank correlations (rg) for weekly landed value($)/(days fished) (medians over vessels) in consecutive pairs of years 1967/68 - 80/81 in each area. Stars (*) indicate significance at P<.05. Kendall's Concordance(W) values are recorded on each plot. Concordances are significant at P<.01. Fishing operations are often vertically integrated with company owned boats directed to some extent by plant managers (e.g. Andersen 1972, 1973). In 81 15000 -11250 -7500-3750 0 • T D 6000-QJ J= -Ol 4500-•—1 U - -UD 3000--ro 1500 -T D \ 0 -QJ 2000-ID —1 -ro 1500-> T D 1000 -QJ T D a 500-ro _ J 0 10000 -I 7500 5000-1 2500 0 1 SU Vancouver Island hi 2 NU Vancouver Island •I »1—T I — r * — I "!• • t ••••»•• 3 Inside Vancouver Island kk "T——\ 1 1 1 T Hi ~i 1 1 1 r~ 4 S Owen Charlotte Sound T 1 1* 1*1 f-—1 1- i"f •• 6 16 24 32 40 52 Week 27000 -| 20250 13500 6750 0 • 27000 20250-I 13500 6750-I 0 20000 -15000 -10000 -5000 0 5 N Queen Charlotte Sound A • ? • — r * - i — t — r " — i — i — i — » t — i — i " 6 Hecate Slralt i 1 1 i 1 1 1 i i i 1 * • !• i "I 1 1 1 1 1 1 1 * * *•! 7 U Queen Charlotte Islands • -ll' 1 0 8 16 24 32 40 Week 52 Figure 3.8. Landed value($)/(days fished) vs. week for each area. Squares are medians over individual vessels and error bars the quartiles for the years 1967-81. such cases, there is cooperation between the fleet and plant managers. Plants provide advice about species and areas in exchange for catch estimates (called 82 hails) from skippers. Some plants may monitor catch hails and plot the information on charts available to all skippers who fish for the plant (Andersen 1972). The utility of catch hails and locations depends on the accuracy of the catch hails. Andersen (1972) found the accuracy of catch hails for Newfoundland trawlers was inversely related to the number of days left in the trip because skippers were reluctant to give away any information that might jeopardize their current trip. The British Columbia trawl fishery has some vertically integrated plant/vessel operations. There are also fishermen cooperatives where vessels are often operator owned. In these cases, fishermen have more freedom, but there may be less information sharing with plants. I expect that the amount of information exchange between plants and fisherman in terms of location and species would be greater in vertically integrated operations. In any case, plant-vessel information is limited to vessels with the same plant and therefore still only represents a subset of the available information. Information sharing among fishermen has been the subject of much anthropological study. Orbach (1977) found most tuna seiners were involved in some communication system. Seiners formed "code groups" that shared information concerning the location of tuna schools using elaborate codes to prevent non-group members from stealing information. Information groups were formed for a combination of economic, social, and status purposes, but the main reason was probably economic because the tuna range over 5,000,000 square miles and information sharing considerably reduces searching costs. Similarly, Gatewood (1984) found Alaskan salmon seinermen formed small groups that shared information as they searched for salmon just in advance of fishery openings. 83 Stuster (1978) suggested differences in information sharing varied with the type of fishing strategy. He thought "trapper fishermen" (e.g. lobster, crab trappers, and set line fishermen) were the least likely to share information, while "hunter" fishermen (eg. trollers, harpooners, seiners) that actively pursue migratory species were most likely to share information. He found trollers for albacore were in the hunter category and they formed code groups. Stuster placed California trawlers in the trapper category and gave anectodal accounts of how they practice deception, and work alone to protect the location of "hot" spots. Stiles (1972) found Newfoundland longliners practiced "radio camouflage" by avoiding specific statements of results or intentions. Based on observations of Newfoundland deep sea trawlers, Andersen (1972) concluded that much of the information exchanged between skippers was purposely distorted and deceptive. Distortions included underestimation of catch by as much as 50% when fishing was good, and overstatement of the catch when fishing was poor to trick competing skippers into steaming to a new position in the expectation of better fishing. Deceptions included misinformation concerning bottom snags, fishing depth and location, and radio silence to preserve exclusive rights to a particularly good location. Andersen (1972) also identified certain non-deceptive transactions including discussions of trawling strategy among 2-3 trawlers from the same plant when they were fishing in the same area but at some distance from other vessels, and "donations" by successful skippers of good fishing locations after another skipper had a run of bad luck. Andersen (1972) identified four key factors that were responsible for the deceptive patterns of information exchange among Newfoundland trawl skippers. The first was the unequal distribution of catch values between skippers both within and between plants. Each plant had poor fishermen and top fishermen or 84 "highliners", and the highliners always received the best boats and crew from the plant. The second factor, was the occurrence of bad breaks (e.g. equipment breakdown, accidents) that were more frequent on the older vessels and that made older vessels less desirable to skippers and crew. The third factor was uncertainty in hunting a common property resource. Although skippers can at times detect their prey with electronic devices such as echo sounders, results of efforts are never certain until the trawl is retrieved. The uncertainty was compounded by competition from other domestic trawlers and foreign fleets and it accentuated the importance of local knowledge. Andersen argued these three factors alone did not lead to the misinformation he observed. The most important factor contributing to the deception was the co-adventure nature and vertical integration of the fishing operations. The co-adventureship occurred on two levels; between plants and skippers and between skipper and crew. At the first level there were formal and informal contracts that stated that the plant provided gear, crew and determined skippers on their vessels. This resulted in rewards of better vessels and crew for successful skippers. Skippers were responsible for the crew, choice of fishing area, and within trip storage of fish. Crew were paid on a share system based on a proportion of the revenue from each trip's catch. It was the system of co-adventureship that acted in combination with the other three factors to make it in each skipper's economic interest not to exchange any information that might jeopardize his catch relative to others. The British Columbia trawl fishery has a similar organization to the Newfoundland trawl fishery, and while the plants and vessels may not be vertically integrated to the extent mentioned above (e.g. especially the fishermen's cooperatives), the pattern of rewards to successful skippers can occur from both 85 free movement of crew between vessels, and from highliners' greater abilities to purchase new vessels or upgrade equipment on existing ones. Thus, incentives for disinformation rather than information sharing exist, and my personal experience with British Columbia trawl skippers indicates that radio silence and catch underestimation are not unique to Newfoundland trawlers. I have presented anecdotal evidence supporting violation of the perfect information hypothesis, but I cannot quantify the affect of violations on the LPE$ hypotheses. One recent study, that modeled information exchange in a fishing system, may shed some light on the expected effects of disinformation on movement patterns. Allen and McGlade (1986) considered movement and information exchange in a model that they applied to Nova Scotian trawl fisheries. The model considered two groups of fishermen: stochasts and cartesians. Stochasts were characterized as high risk takers who fished according to some personal or random scheme and who were only weakly influenced by information from others. Cartesians were unwilling to take risks, and fished only in areas promising the best known returns. One scenario compared the two groups under conditions of no information sharing between groups. When cartesians withheld information from stochasts, there was little effect on the movement patterns or general success of each group, but if the situation was reversed, the cartesians perished. In another scenario, weak mutual exchange of information was found to be beneficial to the fleet as a whole. While in reality there is undoubtably a continuum of strategies between the two extremes, the study suggests that a poor information state favors vessels with greater movement, and as long as there is some weak level of information exchange, vessels will discover and move to areas of high catch rates. 86 Summary of Evaluation of Assumptions The major results of the evaluation of assumptions are: 1. Accounting for vessel size differences slightly reduces differences in area LPE$'s. 2. Inclusion of area specific costs would reduce differences among area LPE$'s because of difference in vessels size between areas and in costs between vessel sizes. 3. Accounting for movement costs would reduce average differences in LPE$ between areas, and differences in LPE$ between years within each area. 4. Disinformation is probably common in the trawl fishery, but fleet responses to catch rates may depend only on a few highly mobile vessels and weak information exchange. The results indicate that assumptions 1-3 of the Equalize LPE$ hypothesis have been violated and that accounting for violations (especially differences in costs) would result in smaller differences in LPE$ between areas and between years within areas. The evaluation of assumptions does not provide unequivocal support for either LPE$ hypothesis, but instead points to the importance of including costs in movement studies. The evaluation also suggests that equalization or maintenance of relative net LPE$ may be the underlying result of and impetus for movement patterns. However, a more detailed accounting of costs is required to distinguish between these two alternatives. 87 Chapter 4: Discussion of Effort Distribution Analyses Introduction Section one of this discussion compares my findings with Hilborn and Ledbetter (1979) and Millington (1984), and attempts to draw some generalizations from these three movement studies. Section two discusses some data and statistical problems common to the three studies and to fisheries data sets in general. Section three considers some general implications of movement pattern studies. Other Applications of the Hilborn-Ledbetter Hypotheses My comparison will focus on five aspects of the fleets in an attempt to draw some generalizations between fishery characteristics and movement patterns. The five factors are: 1. number of vessels 2. availability of information on catch rates and/or abundance to to fishermen 3. ratio of mobile to non-mobile vessels 4. type of management regulations 5. movement hypotheses considered "best" Hilborn & Ledbetter's (1979) study on the salmon seine fishery included about 500 vessels. Information on abundances is provided to those vessels in the form of preseason forecasts published by the Department of Fisheries & Oceans, and area specific catch rates are available from the large number of vessels fishing each week. The ratio of mobile vessels (defined as vessels fishing more 88 than one of their eight aggregate areas) to stationary vessels varied between 5:1 in 1973 and 12:1 in 1976. Management regulations consisted of short area/time openings of 2-4 days over a season that lasts from May to November (Hilborn and Ledbetter 1979). Hsu (unpublished data, cited in Hilborn and Ledbetter 1979) suggested that seiners moved in traditional patterns, but Hilborn and Ledbetter did not explicitly test the traditional patterns hypothesis. They dismissed the existence of traditional patterns based on differences in numerical responses between years. However, I found that movement in traditional patterns by trawlers does not necessarily imply similarity in numerical responses between years (e.g. Area 3, Figs. 2.7 & 2.9). Their study clearly rejected the Equalize LPE$ hypothesis on both a between and within year time scale. They concluded that differences in LPE$ between areas were due to a combination of economic and relative noneconomic desirability, and boat movement tended to maintain annual ratios of each areas LPE$ to provincial average LPE$ (i.e. RPAs, the Maintain Relative LPE$ hypothesis). They supported their conclusion with an example where salmon enhancement resulted in increased total stock available in one area, and these fish attracted more vessels that maintained the area's relative LPE$. They did not consider the Maintain Relative LPE$ hypothesis on a weekly time scale and their study involved four years. The salmon gillnet fishery examined by Millington (1984) consisted of about 2500 vessels, some of which also participated in the salmon troll fishery. Gillnetters have access to the same preseason forecasts as seiners, and their larger numbers may mean better inseason catch rate information. Most of the stationary boats fished in one area (Area 29, Fraser river; for a map of statistical areas see Millington 1984 p. 23-24), where they were outnumbered by mobile vessels by a 2:1 ratio. In all of Millington's other areas combined, the 89 mobility ratio averaged about 20:1 over' the three years examined. Unfortunately, because Millington considered smaller areas, his ratios are not comparable with those of Hilborn and Ledbetter or my study. Without a re-analysis of Millington's data and ignoring the special case of area 29, I estimate a comparable ratio for aggregate areas would be bounded by the estimates for area 29 and the other areas, and is probably about 4:1. Management regulations in the gillnet fishery are similar to those in the seine fishery, but openings tend to be shorter (1-2 days) because gillnetters fish more in river mouth areas where salmon are more vulnerable. Millington's (1984) test of the Traditional patterns hypothesis was a comparison of the probabilities of vessels fishing particular area pairs. He concluded that the nearly constant probabilities for adjacent areas were due to geographical proximity and that the interannual differences of probabilities for area pairs that were separated by larger distances were due more to changes in strengths of salmon runs. He did not examine within year effort distributions because he thought his time series was too short, and the reasons (e.g. regulations) for unexpected changes in these effort distributions would have been hard to isolate. Like Hilborn and Ledbetter (1979), Millington (1984) clearly rejected the Equalize LPE$ hypothesis. Annual RPAs did not maintain consistent rank between years, but overall ranks appeared to be related to movement costs (i.e. remote areas had the highest RPAs and areas close to major ports had the smallest RPAs consistent with the Maintain Relative LPE$ hypothesis). Millington (1984) attempted to forecast weekly boat numbers based on the Equalize LPE$ and Maintain Relative LPE$ hypotheses and found neither model 90 performed much better than the historical mean effort level except in one centrally located area (area 8). He also found the median LPE$ of the first and last week each boat fished (that he called the starting and stopping LPE$s) were nearly equal in two of the three years studied and concluded that similar economic criteria were used by the fleet as a whole in determining when to start and stop fishing each season. However, distributions of starting and stopping LPE$ for individual vessels differed indicating that starting and stopping times were modified by traditional patterns (i.e. some vessels likely started and stopped in the same weeks each year regardless the LPE$). Millington's (1984) study essentially found none of the hypotheses explained movement patterns very well, although based on his attempted forecasts, historical patterns were better predictors of within year patterns than methods based on either of the LPE$ hypotheses. As Millington (1984) acknowledged, his between year comparisons might have been improved if he had been able to include years that covered a full life cycle of salmon abundance (especially for sockeye salmon). The number of vessels fishing in the trawl fishery averaged about 100 over the study period. The ratio of mobile to nonmobile vessels was 1.3:1, but mobile vessels accounted for most of the landed catch, value and effort expended (Table 2.2). As in Millington's (1984) study, stationary vessels were particularly prevalent in one area (area 3). The lack of preseason forecasts, small vessel numbers and low mobility result in a much poorer information state than in either of the salmon fisheries. In comparison to the salmon fishery, the trawl fishery was virtually unregulated over the study period, and what regulations were in effect (i.e. catch 91 quotas) were not strictly enforced (because they were not enforcible within the Fisheries Act at that time). However, there were some isolated seasonal closures to protect spawning aggregations of various species, particularly in areas 1 and 3. Plants have also sporadically imposed miscellaneous regulations, such as trip quotas on certain species or minimum sizes (for economical filleting), but these are largely undocumented and it is difficult to evaluate their impact. The trawl fishery essentially operated year-round in a generally unrestricted regulatory environment. Unlike the two salmon studies, I failed to reject the traditional patterns hypothesis on the annual time scale, and on the within season time scale for one area (area 3). However, my methods for examining traditional patterns differed from both of the previous studies. Like both salmon studies, I clearly rejected the equalize LPE$ hypothesis. And finally, like Hilborn and Ledbetter, but unlike Millington, I failed to reject the maintain RPA hypothesis on an annual time scale, but was able to reject it on a weekly time scale. My study examined evidence for possible violations of the assumptions of the Equalize LPE$ hypothesis. The examination suggested my rejection of the Equalize LPE$ hypothesis may have been due to violations of its assumptions. Differences in LPE$ between areas were partly due to differences in vessel sizes (and their associated catching power and costs) between areas, and differences in movement costs to each area. However, the examination did not unequivocally support either LPE$ hypothesis, but instead suggested that equalization or maintenance of relative net LPE$ might be the result of movement patterns. The examination fell short of a complete accounting, of the effects of fishing costs, but it revealed the importance of including costs, when possible, in movement studies. 92 Some generalizations can be drawn from these three examinations of movement patterns in fisheries. First, tightly regulated fisheries, with short periods of fish availability and better information on catch rates favor greater fleet movement and numerical responses. From the fishermen's perspective, tight regulations and short periods of fish availability exacerbate the common property nature of fisheries. Fishermen are forced to make quick decisions that may affect the success of the whole season. For example, the salmon seine fishery has legal fishery openings that last only a few hours, and most salmon runs are only available in any one area for a few weeks. Thus, the seine fleet has "evolved" to include large vessels that are capable of greater movements. Second, fisheries with year round seasons are more likely to have traditional fleet movement patterns. In these types of fisheries, there is less reliance on a particular trip to make the season profitable and therefore fishermen can work at their own pace to a greater extent than is possible in the salmon fisheries. In some cases, traditional patterns may result from low mobility and be seemingly unrelated to patterns in resource availability (e.g. Fig. 2.8, area 3). However, traditional patterns may also result from high mobility, if movement is tightly linked to consistent numerical responses and repeatable patterns of fish availability among years (e.g. Figs. 2.8 & 2.9, areas 4 & 5). Third, costs are important considerations in movement studies. For example, including fishing costs was important in the trawl fleet because the fleet is a heterogenous group with different size vessels fishing different areas and because vessels may only land their catch at ports located at opposite ends of the province. Similarly, given that Hilborn and Ledbetter (1979) failed to reject the Maintain Relative LPE$ hypothesis, the assumptions of equal fishing costs and negligible movement costs may have also been violated for the salmon 93 seine fleet, despite the fact that the fleet is more homogenous, and landing areas (in the form of small plants or mobile packing vessels) are available in every area with a major seine opening. Data Problems and Statistical Issues Data Problems The three studies on different segments of B.C. fisheries have all used the same basic source of data: Department of Fisheries and Oceans saleslips. While this database contains a wealth of information, it is not ideally suited for movement studies. For example, in examining numerical responses, all three studies chose the basic unit of one week for comparisons between effort and LPE$. However, the saleslip data only indicate the date landed, so weeks refer to the week of reported landing. Daily catch or hail information would be better because the time scale is more similar to the time scale of information flow in the fisheries (i.e. fishermen may learn of catch rates and respond within a trip instead of only between trips). Unfortunately, accurate daily catch data are rarely available, which makes testing the affects of different levels of temporal aggregation difficult. Determining the appropriate spatial scales for analysis is also problematic for this data set because information from saleslips is used to monitor catch quotas. When the quota is either nearly caught or completely caught, there is an incentive for skippers to misreport the location. In the case of the trawl fleet, the misreporting is probably isolated to trips involving particular species or groups (e.g. Pacific ocean perch and other rockfish), but to my knowledge the 94 extent of misreporting has not been evaluated. Nonetheless, researchers interested in movement studies are faced with a trade-off between selecting areas large enough to have some confidence in the accuracy of the information, but small enough to capture the movement dynamics. In the case of the trawl fisher}-, a more reliable data base from port sampler interviews exists, but its continued existence depends, justifiably, on the confidentiality of the information. For a general discussion of data aggregation and other problems with fisheries data sets see Gates (1984). Statistical Issues Even with accurate reporting, fisheries data usually present several problems that make application of the normal statistical recipes inappropriate. LPE$ observations are not usually distributed normally (see Swierzbinski 1985 for a discussion of typical distributions). Furthermore, while several observations of LPE$ for a given time period are often available, these usually represent different vessels with different catching power and therefore they often cannot be treated as replicates because vessel size differences may confound the ability to detect differences between areas. Thus data are usually aggregated (i.e. 2C/2E), which leaves one observation per test cell. One example of lack of replication is the test of the RPA hypothesis on the annual time scale. For statistical purposes, the null hypothesis is Hq: There is no significant difference between years in the ratio to provincial average of each area. However, with only one observation per cell (i.e. year; Figs. 2.14, 3.4), the null hypothesis cannot be tested in the normal fashion. Therefore, previous authors 95 and this study have attempted indirect methods of testing the RPA hypothesis. Hilborn & Ledbetter (1979) constructed a two-way ANOVA to test for differences in landed value per boat week where one effect was the area and the other was the year. They found' very significant differences between areas and years. However, their method does not test the RPA null hypothesis because significant differences between areas and years could result even if RPA did not remain constant among years. Millington (1984) tested for differences in RPA between years by calculating rank correlations between years of the RPA's across areas. He found the correlations were not significant, which indicated the ranking of area RPA's differed between years. The Kendall's concordance measure I reported (i.e. for Figures 2.14 & 3.4) is analogous to an average of Millington's rank correlations calculated for each possible pair of years. But again the concordance measure, while quantifying the agreement in ranks of RPA between years, falls short of a true test of the null hypothesis. General Implications Understanding movement patterns has implications for (1) predicting the responses of fleets to changes in regulations, and (2) evaluating changes in the economic performance of fishing fleets and areas. The primary regulatory tool that has been used to manage the British Columbia trawl fishery an other trawl fisheries is a catch quota. In recent years in the British Columbia trawl fishery, quotas for some species (e.g. various rockfishes) have been allocated on a quarterly or trip basis. While these regulatory measures are usually effective in achieving stock conservation, they often result in economic inefficiencies because more trips are required to land a 96 given quantity of fish and "scrambles" for shares of the quota occur more frequently. Economists have suggested alternative regulatory tools (e.g. transferable individual vessel quotas, area licensing, royalties on landings or effort) which can in theory result in more efficient harvesting and greater net ecomomic benefits from the resource (e.g. Moloney and Pearse 1979; Clark 1980). Implementation of these schemes will require an understanding of vessel movement patterns. For example, if a tax is to be used to shift effort among species and/or areas, understanding the potential numerical response (either positive or negative) is essential. An understanding of traditional movement patterns would be important to evaluating area licensing proposals. The techniques applied to evaluate the movement hypotheses would be useful for examining economic measures of fleet performance on a coastwide or area specific basis. LPE$ whether absolute or relative (i.e. Ratio to some average LPE$) provides a much more meaningful measure of changes in the economic state of the fishery over time than the commonly used aggregate measures such as landed value of the catch, because LPE$ measure value relative to inputs (i.e. effort). LPE$ can also be used to compare alternative areas over time. The standardization of effort based on dollar value of the catch as suggested by Carlson (1975) and that I applied to the trawl fleet in Chapter 3, is an important step in calculating LPE$ indices, especially for heterogenous fleets such as the trawl fleet. 97 Chapter 5: Literature Review of Functional Responses in Natural Predators Introduction Solomon (1949) defined the functional response as the relationship between the number of attacks or consumption rate of a predator, and the prey or predator density. Four forms of the functional response relating the number of attacks per predator per unit time to prey density were proposed by Holling (1959a, 1965) (Fig. 5.1). The type I response is characterized by a linear rise in the number of attacks as prey density increases followed by a plateau in attacks at higher prey densities. The number of attacks in a type II response also rises to a plateau, but the form is curvilinear with a decelerating slope. The type III response is sigmoid. The type IV response is similar to a type II response, except that the number of attacks actually decreases at high prey densities. The remainder of this chapter is divided into three main sections. Section 1 presents equations for the four main forms of functional responses (Fig. 5.1) and a generalized equation which is able to describe all four forms depending on its parameter values. There are three primary reasons why I present the equation(s) for each functional response. First the equations provide a precise definition of each functional response. Second, it is helpful to refer to the equations when considering mechanisms for the functional response. Third, my review of functional respones in fisheries (chapter 6) will occasionally refer to these equations to draw analogies between processes in natural predation and fishing. 98 Prey density Figure 5.1. Four types of functional response curves. Section 2 reviews the mechanisms for each type of functional response. The switching mechanism for type III responses is discussed in particular detail because it has been commonly found in natural predators and it is particularly relevant to multispecies fisheries. Section 3 briefly considers functional responses in the context of the stability of predator-prey interactions. Only the type III response is shown to be potentially stabilizing. Chapter 6 reviews several studies that have found type II responses in fisheries; this form is potentially destabilizing to fish populations (e.g. Clark 1974). 99 Functional Response Equations Type I Equations The type I response can be described by two equations (Hassell 1976): NA = cTN when N < N (5.1a) NA = aTN when N & N (5.1b) x x where NA is the number of prey attacked per predator, a is the rate of effective search (the area or volume searched per unit time; Holling 1965,1966), T is the time prey are exposed to the predator, N is the prey density and N^ is the threshold prey density above which the number of attacks is constant (Fig. 5.1). Type II Equations The most commonly used equation for the type II response is Holling's (1959b) disc equation. Holling's studies of the functional response dissected predation into its principle component processes (Holling 1959a,1961). He classified the components of predation as either basic (universal) or subsidiary (not always present). Holling's three basic components were the rate of successful search (a), the time prey were exposed to predation (T) and the time predators spent handling each prey (t^). Subsidiary components included hunger, learning by the predator, inhibition by the prey, facilitation and interference between predators, and avoidance learning by the prey. Holling (1959b) assumed that the rate of 100 encounter of predators and prey is proportional to the prey density (NA = rxT N) and the time spent searching (T = T—t, XNA). These assumptions lead s S i t to the disc equation: NA = _ £ ™ _ (5.2) l + at,N h In the disc equation the rate of effective search, a, determines how fast the type II curve approaches the plateau, and the maximum number of attacks is equal to T/t^ (Fig 5.1). Note that the disc equation is algebraically identical to the Michaelis-Menten equation for enzyme kinetics (Michaelis and Men ten 1913). A second equation for the Type II response is attributed to Gause (1934 cited in Taylor 1984). NA = A (l-e"aN) (5.3) max where ^m a x *s the maximum number of prey attacked per predator (i.e. the asymptote for the type II response in Fig. 5.1) and all other terms are as described above. Type III Equations Two main types of equations have been proposed for the type III response. Both equations hypothesize an increase in the rate of effective search with prey density in order to produce the accelerating portion of the sigmoid response (Fig. 5.1). 101 Hassell et al. (1977) found that the rate of effective search (a), increased with prey density in an form analogous to a type II response, for some predators that exhibited a type III functional response. _ uN rr A \ ° ~ l+W (5-4a) Substituting equation 5.4a for a into the disc equation yields the type III response of Hassell et al. (1977). NA = (5.4b) l + vN + ut,N h The second expression for the type III response is attributed to Murdoch and Oaten (1975) and Real (1977). Real's derivation follows directly from the disc equation. First, Real rearranged the disc equation (eq. 5.2) by dividing the numerator and denominator by aN yielding N A = t A + l / ( c N ) • . ( 5 - 5 a ) Next, he noted that in this form the number of prey attacked, NA', is the total time available for foraging (T) divided by the total time necessary to search for, capture and handle one prey (t^  is the time spent handling each prey, and the time necessary to locate one prey (tj) is equal to l/(aN)). Real hypothesized that in the type in response, t^  would be a decreasing function of the number of encounters as predators gain experience, or t = - ^ r (5.5b) 1 a N n 102 where n is a parameter measuring the number of encounters a predator must have with prey in order to reach maximum efficiency. Substituting this expression back into equation 5.5a yields the second expression for the sigmoid response. NA = — — (5.5c) l + at,Nn ti Note that equation 5.5c also assumes that a increases with prey density, because Tl-1 an identical equation would result if aN were substituted for a in the disc equation. Thus, equation 5.5c yields a sigmoid response for n>l and a type II response if n=l (Real 1977). Type IV Equations Tostowaryk (1972) modified the disc equation to generate a type IV response. Again the derivation starts with equation 5.2. First, note that equation 5.2 may be rewritten as NA = a(T-t^NA)N . (5.6a) Tostowaryk observed that group defense by sawfly larvae reduced the attack rate by an Hemipteran predator at high prey densities resulting in a type IV response. To incorporate this observation into the disc equation, he hypothesized that group defense acted to increase handling times. He found that the increase in handling time was proportional to the square of the larval density and modified the disc equation as shown below: 103 NA = o(T-(t/i+cN2)NA)N (5.6b) 2 where cN represents the added time needed for handling due to group defense. Solving 5.6b for NA yields an equation for a type IV response. NA = 5- (5.6c) l + at,N+acN h Alternatively, various equations from nonpredator-prey literature could be adapted to describe a type IV functional response. One example, is the Ricker reproduction curve shown below (Ricker 1954, 1975). R = Sea"fcS (5.7a) where R is the number of offspring and S is the number of parents. Substituting NA for R and N for S yields an expression that generates a dome shaped curve. NA = N e ° '6 N (5.7b) In this form, e° (analogous to the rate of effective search above) determines how steeply the response rises at low prey density and lib represents the prey density where the number of attacks is maximum (i.e. the prey density corresponding to the peak of the type IV curve in Fig. 5.1). 104 I A Generalized Equation Finally, Fujii et al. (1978) derived one equation that can produce all four functional response types. Fujii et al. follow Hassell et al. (1977) in assuming that the rate of effective search varies with prey density, but Fujii et al. use the expression shown below instead of equation 5.4a: cN a = de (5.8a) where d and c are constants. Substituting this expression for a into the disc equation (5.5a) yields Fujii et al.'s "descriptive" equation: T N A = = , , „ T X T , ,m (5.8b) t^+ l/(dNexp(cN)) Equation 5.8b has the following properties (Fujii et al. 1978): 1. A type I response is generated when c = dXt^ 2 . 2. A type II response results when c = 0 and t,>0, or when n c and t^ > 0 and c < dXt^. 3. A type III response occurs when c > dXt,. n 4. A type IV response is produced when c < 0. Thus, if equation 5.8b is fit to a given data set, the estimated parameter values can be used to judge which form of functional response best describes the data. Note, however that merely determining the "best fit" parameter set does 2 Technically, the Type I response requires two equations (i.e. eqs. 5.1 a & b), but equation 5.8b does produce a linear rise with a decelerating approach to a plateau under the conditions specified in 1. 105 not provide any definitive clues as to the mechanisms that are causing the observed form of the response. Also, if parameter confidence intervals are taken into account, it is likely that more than one form of functional response may be equally consistent with the data. Nevertheless, equation 5.8b provides a consistent method to test for, or model alternative functional response forms. Instantaneous vs. Exploitation Equations All of the equations presented thus far are based on experimental settings where prey can be replaced as they are depleted by predation. These equations are called instantaneous equations because they do not account for prey depletion by the predator. Instantaneous equations only apply technically to cases where either: 1. prey are replaced as they are depleted (e.g. during the course of an experiment), 2. the overall depletion of prey during the total time period of exposure is small, or 3. predators search systematically (i.e. they do not return to previously depleted areas; Rogers 1972; Hassel et al. 1976). Thus, instantaneous equations apply to cases that are unaffected by "diminishing returns" (Royama 1971). To be of some value in modeling predator-prey systems, these instantaneous equations must be incorporated into exploitation equations. Exploitation models of predation account for prey depletion by integrating over the time during which prey depletion occurs (Royama 1971). The general form for the rate of change in prey density (o!N/dt; for the case of depletion by predation only) is given by = -NAP (5.9) 106 where NA is the instantaneous functional response equation, and P is the number of predators (Royama 1971). The exploitation equation is determined by first integrating equation 5.9 between the limits N = N^ (the initial prey density) and N = N0~NE, and t=0 and t=T, and then solving the result for NE (the number eaten taking into account prey depletion; Royama 1971; Fujii et al. 1978). For the disc equation (eq. 5.2), the resulting exploitation equation is NE = N0( l -e~o ( F r - t/ tN E )) (5.10) where are symbols are as in equation 5.2 and NE and P are defined above. Equations 5.9 and 5.10 assume that predators do not interfere with each other. For more complicated exploitation models that account for interference see Beddington (1975) and Royama (1971). Equations for the other functional response forms can be derived in the same fashion. However for type HI responses, exploitation equations can be quite complicated (e.g. eq. 36; Fujii et al. 1978). Exploitation equations are appropriate for use in population models to determine deaths due to predation. The next section considers some properties of predators and their prey that have been proposed as mechanisms for the four types of functional responses. Functional Response Mechanisms In his original work, Holling (1959a, 1965) found type III responses by small mammals predators to the density of pine sawfly cocoons, and a type II response by preying mantis to the density of fly adults or larvae. Other authors 107 have since found all four forms in a variety of vertebrate and invertebrate predators (See Murdoch and Oaten 1975, Fujii et al. 1978, and Taylor 1984 for reviews). These studies and those of Holling (1959a&b, 1965, 1966) proposed various mechanisms that could lead to the four types of functional responses that have been documented. Type I and II Mechanisms First, the plateaus in response types I-ITI, and the peak in type IV, result from limited time available for foraging relative to handling (i.e. T and t^ in above equations 5.1a & b, 5.2, 5.4b, 5.5c, 5.6c and 5.8b) and/or satiation of the predator (Holling 1965). The type I response results if a predator has a constant time spent searching until satiation (Taylor 1984), and this form has been found for some filter feeding zooplankton species (Murdoch and Oaten 1975). Two mechanisms are commonly proposed for the type II response (reviewed in Taylor 1984). The first mechanism is a continuous decrease in searching rate or percent of time spent searching as the gut is filled, arising primarily from the effects of hunger upon the motivation to hunt (Gause 1934 and Ivlev 1961, cited in Taylor 1984; resulting in equation 5.3). The second mechanism is a reduction in searching time available due an increased proportion of the total time spent handling as prey density increases (Holling 1959b; resulting in the disc equation, 5.2). It is likely that both mechanisms operate in many cases. These two mechanisms also cause the type III response to decelerate to an asymptote. 108 T y p e III M e c h a n i s m s T h e two p r e v i o u s sections h a v e discussed the m e c h a n i s m s for the deceleration and a s y m p t o t e of the t y p e III response. T h e u n d e r l y i n g cause for the a c c e l e r a t i n g portion of the type III response is a n i n c r e a s e i n the rate of effective s e a r c h (a) as p r e y d e n s i t y i n c r e a s e s f r o m low to i n t e r m e d i a t e levels ( H o l l i n g 1965; M u r d o c h a n d O a t e n 1975; e.g. v i a equations 5.4a or 5.5b). S e v e r a l m e c h a n i s m s have been proposed t h a t could r e s u l t i n i n c r e a s e d s e a r c h r a t e and/or efficiency. A l t e r n a t i v e m e c h a n i s m s for t y p e III f u n c t i o n a l responses were reviewed b y M u r d o c h a n d O a t e n (1975). T h e y considered t h a t the f o r m of the f u n c t i o n a l response was not only d e t e r m i n e d b y the b e h a v i o r a l options a v a i l a b l e to the predator, but also the c h a r a c t e r i s t i c s of the p r e y (e.g. density, d i s t r i b u t i o n i n space a n d time). M u r d o c h a n d O a t e n (1975) outlined three p r i n c i p l e m e c h a n i s m s for the t y p e III response. 1. T h e predator is s t i m u l a t e d to spend a g r e a t e r p r o p o r t i o n of time h u n t i n g and/or h u n t i n g efficiency i n c r e a s e s t h r o u g h l e a r n i n g w h e n p r e y abundance increases. 2. T h e p r e y occur i n patches of di f f e r e n t densities a n d the p r e d a t o r s feed more often i n the h i g h d e n s i t y p a t c h e s and/or a l i m i t e d n u m b e r o f s p a t i a l p r e y r e f u g i a are a v a i l a b l e a n d p r e y i n r e f u g i a are less v u l n e r a b l e to attack. 3. S w i t c h i n g b y the predator between less a b u n d a n t a n d more a b u n d a n t p r e y (by v a r i o u s m echanisms, see below). 109 Predator Learning Studies of parasite-host systems (analogous to predator-prey systems in many respects) have found that parasites may be stimulated to spend a increasing proportion of their time hunting as host density increases (Burnett 1964, Takahashi 1968). However most examples of mechanism 1 operating in predator-prey systems have involved some form of predator learning. In fact, predator learning was the first mechanism suggested for the type III response (Tinbergen 1960, Holling 1965). Tinbergen's (1960) concept of a predator search image is one example of predator learning that dominated the early literature on Type III functional response mechanisms. Tinbergen (1960) observed a sigmoid relationship between the proportion of a given prey species (caterpillars) in the diet and the density of the same species available to the predator (great tits); a straight line relationship would be expected from random search (Fig. 5.2). To explain this observation, Tinbergen suggested that the birds learned to concentrate on certain characteristics of the prey (size, shape, color, etc.), and he called this learned response the development of a specific searching image. Thus, when species i was abundant, birds focused on species i and overlooked less abundant prey resulting in a diet with larger proportions of species i (Fig. 5.2). When the abundance of species i decreased, Tinbergen hypothesized that the birds abandoned their specific search image for species i and acquired a search image for a more abundant prey type resulting in lower proportions of species i in the diet than expected from random search. The plateau in the curve at high densities was attributed to the need for a varied diet (Fig. 5.2). 110 Figure 5.2. Tinbergen's (1960) search image hypothesis (from Hassell 1976). Holling (1965) saw analogies between his work with small mammal predators feeding on sawfly cocoons and Tinbergen's work on birds. Holling (1959a, 1965) observed a type III functional response and suggested that one logical component of predation that could result in the accelerating portion of the response was a predator learning to associate stimuli with the prey. He supported this hypothesis with experiments in which naive predators were exposed to a constant density of prey, and he found that the attack rate increased with the time predators were exposed to the prey. He also found an extinction of the learning response (i.e. forgetting). Next, he developed a simulation model that included learning, and it generated a type III response. Despite the fact the predator was provided with an alternate food, he did not suggest a strict search image explanation for his observations, perhaps because the alternate food (dog biscuits) was rather unpalatable in comparison to the sawfly cocoons. I l l While the ability of predators to develop visual cues for and learn to associate stimuli with particular prey has not been questioned, the idea of search image formation and learning as the likely mechanisms for Tinbergen's and Holling's observations has been criticized. Murdoch and Oaten (1975) extensively reviewed Tinbergen's search image hypothesis and concluded that it had been subject to misinterpretation in the literature. Below are two major points of clarification they provided concerning the search image hypothesis: 1. A sigmoid relationship between the proportion of prey in the diet and the abundance available does not necessarily imply compensatory mortality on the given prey during the accelating part of the curve (i.e. does not imply a type III functional response). Inferences concerning the shape of the functional response can be drawn from these plots only if the absolute predation rate is provided or the total number of all prey species in the diet is constant. 2. Tinbergen's verbal account of the, search image hypothesis, and ; subsequent analysis by Mook et al. (1960), actually suggest that search image formation implies a change in behavior in time, as well as, or instead of with prey density. Therefore, to test the hypothesis, it is necessary to look at attack rates through time. Conditions where the development of a search image will lead to a type III functional response are reviewed further in Murdoch et al. (1975). Critics of Tinbergen's (1960) search image hypothesis and Holling's (1965) learning explanation have suggested that both authors' observations could be better explained by mechanisms 2 and 3. These mechanisms are reviewed in the 112 next two sections. Prey Patchiness or Prey Refugia Mechanism 2 is presumably widespread among predators (Murdoch and Oaten 1975) and it has been modeled by Hassell and May (1974). Royama (1970) first suggested mechanism 2 as an alternative explanation for Tinbergen's (1960) observations. Royama (1970) showed that if the proportion of time a predator spends in patches containing a given prey type increases as the density of that prey type within patches increased, the proportion of prey types in the diet would vary relative to the absolute number available in a manner consistent with the search image hypothesis. Furthermore, Royama's explanation results in a type III functional response and it does not require any learning on the part of the predator. When two or more prey types are involved, Royama's mechanism may result in the observations consistent with switching (see below). The Switching Mechanism In addition to clarifying Tinbergen's search image hypothesis, Murdoch and Oaten (1975) suggested an alternative explanation to learning for Holling's (1959a,1965) observed type HI functional response in deer mice. They suggested that the observations could be explained if the mice tended to concentrate on the feeding pattern that was more rewarding. Since cocoons were preferred to dog biscuits, the reinforcement from a cocoon meal should be greater than for a dog biscuit meal and so they expected searching would persist for cocoons even when cocoons were rare. However, when cocoons were sufficiently rare, Murdoch and 113 Oaten hypothesized that the mouse switched to biscuits. A similar mechanism has been proposed for predation by guppies (Murdoch et al. 1975). Taylor (1984) proposed a similar alternative to Holling's learning hypothesis, except Taylor suggested that if cocoon density was sufficiently low, the deer mice mightly simply choose not to search for cocoons and dig them out of the sand, because the activity was unprofitable. Thus, Taylor suggested that an increase in time spent searching (relative to the total time exposed) by mice as cocoon density increased would result in a type IH response. This "switching off' mechanism of Taylor (1984) has been proposed for a variety of zooplankton species (e.g. Adams and Steele 1966; Parsons et al. 1969; Steele and Henderson 1977). Murdoch and Oaten (1975) also compared the search image hypothesis to the switching hypothesis (Murdoch 1969). This comparison is briefly summarized in the next section. The Switching Hypothesis The switching hypothesis relates the proportion of two prey types in the diet to the proportion of the types available (Murdoch 1969). This hypothesis is distinct from the search image hypothesis that relates the proportion of a prey item in the diet the that item's absolute density available (Murdoch and Oaten 1975; cf. Figs. 5.2 and 5.3b). Murdoch (1969) defined switching in the case of two prey as a changing preference such that the ratio of two prey types in the diet would increase at a rate disproportionate to the change in their relative availability. He 114 defined the null case (no switching) by the equation = c (5.11) where and Dg are the numbers of prey types 1 and 2 in the diet and and Ng are the numbers available to the predator. The parameter c is a constant that measures preference, such that when c=l prey types have equal preference, when c<l prey type 2 is preferred and vice versa for c>l (Murdoch 1969). Thus, the switching hypothesis defines preference as the ratio of the numbers of two prey types in the diet when equal numbers of' the each prey type are available to the predator. Note that D is the same as NA in the functional response equations, and c is o-^la^ (see below). In the absence of switching, c is expected to be constant, and the relationship expressed in equation 5.11 would be a straight line (Fig. 5.3 A; Murdoch 1969, Murdoch et al. 1975). If switching occurs, the ratio of prey items in the diet should increase faster than linearly as the ratio of prey items available increases (i.e. c is a increasing function of N^ /N^ ; Fig 5.3 A). The switching model is often expressed in terms of the proportion of a prey item in the diet (i.e. D^ /(D^ -t-D^ )) relative to the proportion of that prey item available (i.e. N^AN^+N^)). When expressed in this way, the null hypothesis is still a straight line and the curve for switching is sigmoid (Fig. 5.3 B; see Murdoch and Oaten 1975). Several studies have tested the switching hypothesis, and the principal conclusions from the tests were summarized by Murdoch and Oaten (1975; also see Murdoch et al. 1975). Generally, studies failed to reject the null hypothesis (i.e. no switching) under two conditions: either when preference at equality (i.e. 115 Proportion Available ( N < / [ N 1 +N 2]) Figure 5.3. A. Switching hypothesis expressed as ratios. Switching curve was drawn using equation 5.16 with \^/\^= l .B. Switching hypothesis expressed as proportions. Switching curve was drawn using equation 27 in Murdoch and Oaten (1975; p. 62). 116 •when N^=Ng) was consistently strong (i.e. c >> 1) or consistently weak among individual predators (i.e. c < < 1). Switching was found in cases when preference at equality was weak (i.e. c << 1) averaged over all predators, but strong and highly variable among individuals. Mechanisms for Switching Various mechanisms have been proposed to explain the occurence of switching. Murdoch et al. (1975) listed four common mechanisms: 1. Prey types are found in different areas, and predator spends longer in an area as the relative reward rate there increases. 2. Rate of detection or recognition depends on recent experience. 3. Rate of acceptance/rejection of each species depends on recent experience. 4. Rate of successful attacks on each species depends on recent experience. Examples of studies that attributed switching to each of the above mechanisms were presented in Murdoch et al. (1975) and Murdoch (1977). "Flocking" (i.e. individual predator diets influenced by those of predators near them) was suggested as a switching mechanism by Murdoch and Oaten (1975). Switching and the Functional Response In order to include switching in functional response models, it was necessary to extend the models to include more than one species. Murdoch (1973) 117 generalized Holling's disc equation to k prey species as shown below. NA. = ^ l + Za.t .N. • . i h i where all symbols are the same as in equation 5.2 and the subscripts refer to prey species. Equation 5.12 is called the multispecies disc equation (MSDE). Murdoch and Oaten (1975) show that if the MSDE is expressed for two prey species, (as in equations 5.13a & b below) and the equation for the attack rate on one prey is divided by the attack rate for the other, the result is the switching null hypothesis (equation 5.14 below): N A ; = a l N l T <5-13a> N A 2 a2N2T (5-13b) 1 + althl™l+a2th2N2 NA a N where NA and o ^ ° 2 a r e e Q u ' v a l e n t to D and c of equation 5.11. Murdoch and Oaten (1975) incorporated switching into the MSDE, by assuming the rate of effective search for each prey species increased linearly with the proportion of the species available. They let a. — XP. where P. = N7(N 7 +N 9 ) (Note: in their notation, X. = cP. because they used X. to denote the rate of effective i i i J i search). With these substitutions, the functional response model for switching in the two prey case is NA. = \W. ( 5-15) 1 + hthlPlNl + hth2I'2N2 118 Equation 5.15 results in a switching hypothesis of NA X N 2 -wt~ = y - ' -V (5-16) N A 2 *2 N 2 2 Note that the ratio of the two prey items in the diet (i.e. NA^/NA^) increases faster than linearly with the ratio available, as expected if switching occurs (e.g. Fig. 5.3a). For a more detailed method of including switching in the functional response see Oaten and Murdoch (1975b). The MSDE will generally result in a type II response for both prey species (Murdoch 1973; Murdoch and Oaten 1975). When switching is included (as in eq. 5.16), a type III response results for both prey species if handling times are equal among prey species and the total density is fixed, but in general a type III response occurs in one species (often the preferred prey; Murdoch and Oaten 1975). In summary, when the diet of the predator is composed of more than one prey species, various mechanisms may result in switching in the predator. When switching occurs, it is a common mechanism for the type III response. However, the occurence of switching does not necessarily lead to a type III response. The generation of a type III response depends on the extent to which increases in the rate of effective search due to switching (that causes an acceleration in the response) are counteracted by increased total handling time (that causes the response to decelerate). 119 Type IV mechanisms Holling (1965) found that when he included learning in his simulation model of the vertebrate functional response, a type IV response sometimes resulted for distasteful prey. He thought that this response could result in the field if predators developed specific nonsearch images (i.e. the reverse of Tinbergen's 1960 concept). Subsequent studies have attributed type IV responses to three main mechanisms: predator confusion, predator disturbance by prey, and group defense in the prey. The group defense mechanism was proposed by Tostowaryk (1972) and is included in the functional response equation 5.6c above. Confusion of the predator was first proposed by Welty (1934). He found that several Daphnia present in the immediate field of vision of a goldfish offered conflicting stimuli that blocked the feeding response. Mori and Chant (1966) documented the disturbance mechanism. They found that a predacious mite was disturbed by contacts with other prey at - high prey density, and would consequently abandon attacks. For other examples of type IV responses see Fujii et al. (1978). Functional Responses and Stability Much of predator-prey literature concerns the ability of predators to stabilize prey populations. Thus, many investigations of functional responses have commented on the implications of the form of the functional response for stability (e.g. Holling 1965; Murdoch and Oaten 1975; Oaten and Murdoch 1975a, b; Hassell 1978; Taylor 1984; Kuno 1987). Oaten and Murdoch (1975a) pointed out the importance in defining stability because of the various types of stability that 120 exist in mathematical models and the difficulty in defining stability in real communities. Holling (1965) and Murdoch and Oaten (1975) others have emphasized that it is actually the form of the total response (functional and numerical) that is relevant to stability. With these caveats, and for the purposes of this brief summary, I will define stability as damped fluctuations about some average ("equilibrium") value and consider which forms of the functional response can be potentially stabilizing to prey populations. Holling (1965) stated that a functional response could be stabilizing if the number of attacks resulted in increased percent mortality as prey density increased. To determine which functional responses cause compensatory mortality, the equations are expressed as percent mortality for each type (Fig. 5.4). Thus, only the type HI response can potentially stabilize prey populations. Types I, II and IV are potentially destabilizing, particularly if the prey have a depensatory reproduction curve (e.g. Clark 1974). Oaten and Murdoch (1975a) considered two measures of the stabilizing effect of functional responses. First, they considered whether the response causes compensatory mortality, which occurs when f(N) > f(N)/N (5.17) where f(N) is the first derivative of the functional response equation (f(N)) with respect to prey density (N). Equation 5.17 holds when an increase or decrease in the prey population results in a greater than proportionate increase or decrease in predation by a single predator. Their first measure of the stabilizing effect was the range of prey abundances (Nm) over which compensatory mortality occurs. The second measure was f(N ) that provides a measure of the range of 7 7 1 ' parameter values for a given predator prey model over which the functional 121 0 Prey density Figure 5.4. Percent mortality curves for the four types of functional responses shown in Figure 5.1. Only the type III response, results in compensatory mortality over a range of prey densities and is potentially stabilizing. response will yield stability (see Oaten and Murdoch 1975 a,b; Levin 1977; and Oaten and Murdoch 1977 for more detail). Oaten and Murdoch (1975b) examined the implications of switching for stability. They concluded that switching is not always stabilizing, because switching does not always result in a type III functional response. For an examination of the implications of incorporating functional responses into various predator-prey models see Bazykin (1974), and Kuno (1987). For an analysis of fisheries stability implications see Jones and Walters (1976) and Walters (1986). 122 Chapter 6: Funct ional Responses and the Fishery Introduction Fishing activities include most processes found in animal predation (e.g. search, pursuit, handling and capture), yet it was not until this decade that the functional response concepts of Holling (1959a and b,1965) were first applied to a fishery (Bienssen 1979, Peterman 1980)3. The relatively recent application of Holling's ideas to fisheries is especially surprising given that fisheries analysts and managers commonly assume that the catch per unit of fishing effort (CPUE; i.e. the number attacked per predator (vessel) per unit time) is proportional to abundance (i.e. a type I response) and the alternative forms of the functional responses have very different implications for the relationship between CPUE and abundance (e.g. Fig. 5.1). One factor contributing to the late application of Holling's ideas to fisheries is that fishing effort data are seldom collected or reported in terms of their components. A common effort measure is days fished without mention of the hours per day spent travelling, searching, capturing, and handling, even when some of these time components are recorded in fishermen's logbooks. Furthermore, it is usually recommended that the best component of effort for use in CPUE indices is the amount of time the gear is actually fishing (e.g. hours trawling; Gulland 1955; Beverton and Holt 1957), therefore it has generally been 3 Peterman's study was the first to formally apply Holling's concepts, however the recognition that fishing effort consisted of various components occured much earlier (e.g. Gulland 1955; Beverton and Holt 1957), and Paloheimo and Dickie (1964) were the first to propose a catch equation of type II form, but they did not recognize its relationship to Holling's work and they were unable to find suitable data sets for its application. Bienssen also apparently independently derived Holling disc equation and applied it to an Australian abalone fishery. 123 considered uneccessary to examine the other components. In fact, most studies to date have simply hypothesized how the components of fishing effort might vary given an "observed" form of the functional response. Despite the relatively recent application of functional responses to fisheries, many studies have recognized the potential biases in CPUE as an index of abundance that result from measures of fishing effort that either aggregate all components (e.g. days fished) or include only one component (e.g. hours trawled; e.g. Gulland 1964; Garrod 1964; Paloheimo and Dickie 1964; Radovich 1976). It is important to note that while CPUE is commonly used to mean catch per unit of fishing effort, the use of LPUE or LPE (i.e. landings per unit of effort) is more accurate because in many fisheries a portion of the catch is * discarded at sea. In this review, I use the abbreviation used by the cited authors, but in Chapters 7 and 8, I use LPE, since unknown quantities of groundfish are discarded by the trawl fleet. The next sections consider the functional response models of fisheries, their assumptions, and the consequences of violations of the assumptions in more detail. Section 1 discusses the type I functional response and its assumptions in the fisheries context. The assumption of a type I response (with a very high limit on attack rate) is implicit in most theories of fish population dynamics. Section 2 reviews some studies that have questioned validity of assuming a type I functional response, many of which have provided evidence for type II responses in fisheries for a variety of fishes. Section 2 also reviews potential mechanisms for type II responses in fisheries and the management consequences of incorrectly assuming a type I response when the true response is of type II form. Sections 3 and 4 consider the possibility of and potential mechanisms for 124 type III and type IV responses in fisheries. Type I Responses Most fisheries models assume a type I functional response. This assumption is expressed in the Catch Equation Ct = qft\ (6.1) where C is the number of fish caught in year t, f is the fishing effort (e.g. number of vessel-hours trawling), N is the average number of fish available, and q, the catchability coefficient, is defined the as the proportion of the population caught by one unit of fishing effort (Baranov 1918; Ricker 1940). Equation 6.1 states that CPUE (C/f ) is proportional to the average abundance N where q is the proportionality constant. The term average abundance is used because fishing and natural mortality will deplete the population over the course of a fishing season. Ricker (1944) showed that is related to the initial abundance at the start of the season (N )^ by Nf = N0(l-e'Zt)/Zt (6.2) where is the total instantaneous mortality rate and e is the base of natural logarithms (Z =^ —In(total annual survival rate)). Note that may be expressed as the sum of the instantaneous rates of fishing mortality (F^ ) and natural mortality (M f ;i.e. Z^ = F f + M ( ) and that fishing mortality is given by Ff = An alternative expression for the catch equation due to Palohiemo 125 and Dickie (1964) is: Ct = qrpt • (6.3) where D^ is the average density of fish at time t (i.e. /A, where A is the total area occupied by the population), and q' = ca. In this form c is a parameter that measures the efficiency of the gear over the area a swept by one unit of gear for a fixed time period (e.g. by a trawl net over one standard fishing period). The two forms of the catch equation (i.e. eqs. 6.1 & 6.3) are equivalent if the area of the stock (A) remains constant at all levels of abundance. Thus, the catch equation descibes a type I response as depicted in Figure 6.1 A and it implies a constant percent mortality per predator (i.e. constant catchability coefficient) at all levels of fish population density (Fig. 6.1 B). Note that the catch equation only predicts the solid lines of Figures 6.1 A & B, but in practice it is recognized that at high levels of abundance CPUE may level off and percent mortality may decline as indicated by the dotted lines. Gear saturation, limited hold capacities and boat limits imposed by fish processing plants are among the mechanisms that have been proposed for this leveling of catch rate at high abundance levels (Radovich 1976; Rothschild 1977). The principal assumptions of the catch equation as given in Ricker (1940) and summarized in Paloheimo and Dickie (1964) are: 1. Effort units, f, operate independently and do not compete with each other. 2. Catchability (q) is constant. 126 Figure 6.1. Type I functional response (A) and mortality (B) curves for the fishery. Solid lines are predicted by equation 6.1. Dashed lines are not predicted by equation 6.1, but are believed to be part of type I responses in fisheries. Violations of assumption 1 are undoubtedly related to the level of abundance as well as the level of effort. Radovich (1976) found there was a greater reliance of vessels on communication with other vessels (i.e. less independence) when abundance was low in the California sardine purse seine 127 fishery. In principle, it would be possible to measure the added benefit of cooperation by comparing the CPUE of vessels fishing independently and cooperatively on similar densities of fish. However, in practice these experimental comparisons would be very difficult to set up (see Paloheimo and Dickie 1964 and Rothschild 1977 for theoretical models of cooperation). The second part of assumption 1 concerns competition. Rothschild (1977) identified two types of competition: competition among vessels for fish (interference) and competition among fish for space in or on the gear (saturation). As with cooperation, if the extent of interference could be measured, CPUE indices could be adjusted accordingly. However, an alternative model that includes interference would be more appropriate (e.g. Watt 1959; Rogers and Hassell 1974; Beddington 1975; Rothschild 1977). The effects of saturation depend on the length of time the gear is fished. If catch and effort data are available for subperiods, the choice of short time periods should minimize the effect of saturation. However, if saturation is continuous (i.e. mimicing satiation), a type II response is more appropriate (e.g. Gulland 1955; Rothschild 1977). Fishery scientists have recognized that many factors affect the catchability coefficient. Gulland (1955) listed four factors: 1. the fishing power of a vessel for a type of fish. 2. the vulnerability of that type of fish. 3. the aggregation of fishing units on the fish. 4. the concentration of fishing units on the fish. Obviously, it is an impossible task to control for all these factors that affect catchability. Thus, in practice, attempts to meet the constant catchability assumption have usually centered on careful selection of the effort units (i.e. f ) 128 such that effort is proportional to fishing mortality (i.e. F=qf ). Garrod (1964) and Gulland (1964) have reviewed the problem of determining effective fishing effort (see also Robson 1966). Changes in fishing power usually act to increase catchabilty over time. Effort standardization (e.g. Gulland 1956a) may be used to account for catchability changes that result from changes in fishing power. Vulnerability refers to the degree to which fish are exposed and susceptible to the fishing gear. Exposure varies with the temporal and spatial distribution of fish in relation to fishing. Susceptibilty depends on physical and behavioral characterstics of the fish in relation to the gear (e.g. the girth of a fish in relation to the mesh size of a gillnet). Vulnerability is usually quantified in terms of relative catchability. For example, the vulnerability of fish of various ages may be expressed as the ratio of the catchability of fish at each age to the catchability of fish at a reference age. Vulnerability changes resulting from physical changes in the gear that directly affect susceptibility (e.g. changes in mesh size) may be accounted for by procedures analogous to effort standardization. However, changes in the vulnerability of fish resulting from changes in exposure often result in seasonal changes in catchability (e.g. Garrod 1964). Accounting for seasonal changes in exposure of a population is only possible under certain circumstances (see below). Changes in exposure may result from changes in the level of aggregation and concentration. Gulland (1955) defined the aggregation of fishing effort as the ratio of the density of fish in the immediate area of the gear to the average density in the general area where the gear is working. He defined concentration of fishing effort as the ratio of the average density in this general area to the 129 average density of the whole accessible stock. Changes in aggregation or concentration of effort may cause both annual and seasonal variations in catchability. If cyclical or seasonal variation in vulnerability is similar between years, changes in seasonal effort patterns (i.e. concentration and aggregation) may be accounted for by taking a weighted average CPUE among subperiods and/or subareas where the weights are the sizes of the subareas and the durations of the subperiods (Gulland 1955; 1964). However, if seasonal variations in vulnerability differ among years, Garrod (1964) suggested using the CPUE from a subperiod(s) during which the average catchability is constant or nearly constant among years. Given the many factors that affect catchability, it is relevant to consider the general conditions necessary to meet the constant catchability assumption. Paloheimo and Dickie (1964) reviewed intuitive models for the distributions of fish and fishermen that would meet the constant ..catchability assumption. Baranov's (1918) intuitive model was a population of immobile fish that was uniformly distributed over its range. Areas fished by each vessel were of areas fished during previous sets (e.g. trawls) and independent of areas fished by other vessels. Ricker's (1940) intuitive model included a very mobile fish population that quickly repopulated locally depleted areas. Under such circumstances, the conditions for uniform exploitation and independence of effort would be met even with stationary gear as long as the gear did not directly compete or affect fish redistribution. Alternatively, Ricker hypothesized that if mobile vessels were widely dispersed over an area occupied by a moderately mobile fish population, average catchability determined from non-uniformly and uniformly distributed populations would be nearly the same. These intuitive models clearly appeared to be special 130 cases, and therefore Paloheimo and Dickie (1964) examined the implications of fish distribution in more detail. As an alternative to the assumption of uniform distribution of fish, they proposed that fish were distributed in clumps or schools of a uniform size. Next, they derived an alternative catch equation based on the division of fishing effort into fishing time (time spent actually capturing a school) and time spent searching for a school. Their alternative catch equation was where C is the catch, g is the proportion of fish in a school of radius rthat are caught, D is the overall density of fish in the whole area containing n schools, T is the time required to exploit a school, and t is the total operation time spent both exploiting and searching. Next, Paloheimo and Dickie (1964) compared their catch equation (i.e. eq. 6.4) with the classical equation (i.e. eq. 6.3). They concluded that CPUE indices based on C/f (i.e. catch per fishing time) would be misleading if they were applied to fisheries that involved search for clumps or schools of fish. In such cases the apparent catchability coefficient may be expected to be inversely proportional to the overall density. Paloheimo and Dickie (1964) suggested catch per operation time as an alternative to catch per unit fishing time. However, the authors stated that understanding this alternative index would depend on knowledge of the size of schools and the density within schools as well as the overall density, and detailed information on school characteristics is rarely available. 131 In summary, the applicability of the type I functional response model to fisheries is highly dependent on the degree to which the model assumptions fit the data. The constant catchability assumption can be met in some cases by careful selection of the effort measure. However, some assumptions cannot be accounted for by standardization the components of the type I response, but may be accounted for in alternative models. The next section considers the type II response. Type II Responses Paloheimo and Dickie (1964) provided a theoretical basis for a type II functional response by fishermen. Although it was not recognized by the authors, Paloheimo and Dickie's model (eq. 6.4) is identical to Holling's 1959b disc equation (eq. 5.2) with a —2rg, T=t, and t^ = Tlgn. Paloheimo and Dickie clearly demonstrated the implications of a type n response for CPUE as an index of abundance. The most important implications -are that CPUE will not be proportional to abundance and the apparent catchability coefficient (q) will be inversely related to abundance. A study by Beinssen (1979) is a second example of a fisheries researcher apparently independently (he cites neither Holling nor Paloheimo and Dickie) deriving an expression equivalent to the disc equation. Beinssen (1979) modeled CPUE of divers harvesting abalone off the SW coast of Australia. He observed that total diving time (T) was comprised of search time (S) and handling time (H) and he postulated that total handling time was proportional to the number of abalone caught (i.e. H = hC; where h, the average time spent handling each abalone, is equivalent to t, in the disc equation; eq. 5.2). Next, he hypothesized 132 that catch per unit of searching time rather that catch per unit of total diving time should be proportional to the density of abalone (D), because searching time was directly related to the process of locating abalone. Thus, he assumed that C/S = rD (6.5) where the parameter r, defined as fishing power, has units of area covered per time spent searching, and is equivalent to the rate of effective search a in the disc equation (eq. 5.2). Substituting T-hC for S in equation 6.5 yields C/(T-hC) = rD (6.6) Unlike Paloheimo and Dickie (1964), Beinssen (1979) was able to estimate the parameters (r,h) for his model by using data from three separate experiments where he had mark-recapture estimates of D, and estimates of T and C for research divers. He rearranged equation 6.6 to yield T/C = h + l/(rD) (6.7) 2 and estimated h (5.1 seconds per abalone) and r (1196 m per hour spent searching) by regression. Note that equation 6.6 can also be rearranged to yield C = TTrhTT (6-8) which is identical to the disc equation (eq. 5.2) with o=r and t^=h. Thus, Beinssen (1979) demonstrated that a type H functional response equation can 133 result from a type I functional response equation (i.e. eq. 6.3) in fisheries which meet the following three conditions: 1. total fishing time is comprised of time spent handling and time spent searching, 2. catch per unit search time rather than catch per unit total time is proportional to density, and 3. total handling time is proportional to total numbers caught. The potential management consequences of assuming a type I response when the true response is type II were outlined by MacCall (1976) and Ulltang (1976,1980). The first consequence of incorrectly assuming a type I response is that the manager will underestimate the decline in the population, particularly at low abundance levels (i.e. < N^, Fig. 6.2 A). The situation is further exacerbated by the second consequence, which is underestimation of the fishing mortality caused by one unit of fishing effort (i.e. the catchability coefficient) at low stock sizes (i.e. q^ > q ,^ Fig. 6.2 B). Thus, the manager will be overly optimistic and will be less likely to take the stringent measures (i.e. reductions in effort or allowable catches) that may be required to prevent overfishing (e.g. MacCall 1976; Ulltang 1976; Sinclair et al. 1985). The extent that managers may be misled depends on the degree of nonlinearity in the type II response. Under certain conditions, a type II response may lead to stock collapse (e.g. Clark 1974). See also Fox (1974) and Condrey (1984) for the implications of density dependent catchability for surplus production models (i.e. logistic models). The first evidence for type II responses in fisheries appeared about ten years after Paloheimo and Dickie's (1964) original model. One misconception about Paloheimo and Dickie's model was that knowledge of the size of schools and density of fish within schools was required in order to test the model empirically. Thus, the delay may have been due to an inability of researchers to view some of Paloheimo and Dickie's rarely measured components (e.g. school size 134 NII NI Abundance (N) Figure 6.2. Comparison of two functional response forms for the fishery. Given the same observed CPUE, the type I response overestimates abundance (A) and underestimates the catchability coefficient (B) relative to the type II response. and density within schools) as parameters rather than variables. A simpler model proposed by Fox (1974) has been widely applied in studies of functional responses in fisheries: 135 q = aN® (6.9) where q is the catchability coefficient, N is the abundance of fish and a, and /3 are parameters to be estimated. The relationship of Fox's (1974) model to functional response models is evident if the right hand side of equation 6.9 is sustituted for q in equation 6.1 and the resulting expression is solved for C/f yielding Equation 6.10 was derived independently by MacCall (1976) and Ulltang (1976). Thus, while Fox's (1974) model is for density dependent catchability, it applies also to functional responses because when /3 = 0, q is constant and a type I response is indicated, for -1 < 0 < 0, q is inversely proportional to N and a type II response results and for 0 >0, q is an increasing function of N and C/f versus N mimics the accelerating portion of a type III response. Thus, finding that —1< |3 < 0 in equation 6.9 is the same as detecting a type II response (assuming f measures the total time (T) prey are exposed to the predator; e.g. Fig. 6.2). Estimates of a and /3 are obtained by regression using log-transformed versions of equations 6.9 and 6.10. Thus, for equation 6.9 the regression equation is C/f = aN 0 + 1 (6.10) ln(q) = ln(a) + 01n(N) (6.11) Similarly, for equation 6.10 the regression equation is 136 ln(C//) = ln(a) + (0-rl)ln(N) (6.12) Many of the studies I review below as evidence for type II functional responses in fisheries used equations 6.9-6.12 to test for density dependent catchability. Evidence for Type II Responses Based on Paloheimo and Dickie's model, the most likely situation to expect a type II response is in a fishery for schooling fish, where a large proportion of the fishing effort is spent searching for schools. Thus, it is not surprising that some of^the first empirical tests of a type II response came from fisheries for various clupeoids (e.g. herring, anchovy) that show strong schooling behavior. Not coincidently, these initial tests were made when several clupeoid stocks around the world were either near collapse or had already collapsed. For the theoretical mechanisms that may cause stock collapse in clupeoids see Clark (1974) and Ulltang (1980) and for a review of clupeoid stocks and their management see Murphy (1977) and various papers in Saville (1980). Fox (1974) did not apply equation 6.9 by itself, but he incorporated the equation into a surplus production model that he fit to two species. He found 0 = —0.3 for Pacific sardine {Sardinops sagax caeruled) and 0 = 0.4 for Pacific yellowfin tuna (Thunnus albacores). Before Fox proposed his model, Schaaf and Huntsman (1972) had suggested that the catchability coefficient was inversely related to abundance in Atlantic menhaden (Brevoortia tyrannus). Later, Schaaf (1975) provided quantitative evidence when he fit Fox's model and found 0 = -0.74. 137 MacCall (1976) fit equation 6.11 to cohort analysis estimates (method of Tomlinson 1970) of catchability and abundance of Pacific sardine, and he found /J = —0.61. Ulltang (1976) also used equation 6.11 to estimate the extent of density dependent catchability in the Norwegian herring (Clupea harengus) and he found 0 = -1.376. However, Ulltang (1976) thought the value of 0 = -1.375 was probably an overestimate because when /3 < — 1, CPUE decreases with increasing abundance. These studies led Radovich (1976,1979) to question the use of CPUE as an index of abundance in fisheries for pelagic schooling fishes. Pope (1980) compared the fit of a density dependent catchability model (i.e. a type II response) with a density independent model using data from several North Sea herring fisheries, and he found that the density dependent model was more probable and fit most data sets better. The collapse of the world's largest fishery (i.e. Peruvian anchoveta; Engraulis ringens) may also be partly attributed to density dependent catchability (e.g. Murphy 1972; Paulik 1971; Csirke 1980; Beverton 1983). More recently, Winters and Wheeler (1985) found that the catchability coefficient was inversely related to stock area for a number of northwest Atlantic herring populations. The primary mechanism was that fishing effort was able to operate more efficiently as the area occupied by the stock declined. The relationship held across stocks that occupied different sized areas and within stocks when stock area declined with abundance. The apparent decrease in stock area with declines in abundance found by Winters and Wheeler casts doubt on the type I response assumptions of the catch equation because most applicatons assume that stock area is constant (i.e. equation 6.1 is used instead of equation 6.3). 138 Not all studies of density dependent catchability have involved clupeiods. Some studies have involved trawl fisheries for various groundfish. For example, Pope and Garrod (1975) and Garrod (1977) found that the catchability coefficient in two separate trawl fisheries for Arcto-Norwegian cod (Gadus morhua) was inversely related to stock biomass, although neither study provided estimates of p\ Garrod (1977) also found that average catchability for three stocks of north Atlantic cod was inversely related to stock area. Houghton and Flatman (1981) found /3 = -0.522 when they fit equation 6.11 to catchability and biomass estimates from another northeast Atlantic cod stock that was exploited by a trawl fishery in the west-central North Sea. The relationship was not statistically significant when numbers were used instead of biomass. Archibald et al. (1983) hypothesized that catchabilty of Pacific ocean perch (Sebastes alutus) increased with decreasing abundance because the time spent searching increased with decreasing abundance and these changes in search time were not accounted for by the effort measure (i.e. hours trawled). Angelsen and Olsen (1987) considered that catchability would be expected to vary with effort and density in their study of the Lofoten cod fishery. The effect of effort was expected because the fishery consisted of various stationary gear types (e.g. hand lines, long lines, gill nets and Danish seines) that became very dense during the height of the season and thus CPUE decreased due to interference and saturation effects (e.g. Rothschild 1977). Therefore, Angelsen and Olsen suggested a modification of the density dependent catchability model (eq. 6.9) as shown below. q = aEaN^ (6.13) or for equation 6.10 139 C = aEa+V + 1 (6.14) In separate regressions on ln(q) on ln(E) and ln(q) on ln(N), they found that q was inversely related to effort and abundance for both longline and gillnet gear types (i.e. for longlines a = —1.967, and /3 = —0.852 and for gillnets a = — 1.660, and 0 = —0.826). They combined the coefficents from the separate regressions by substituting the equations for q vs. E and q vs. N into the 2 2 equation q = (C/EN) and solving for C. The resulting catch equations were CL = 9.697 X103N°-574E0-0165 (6.15) CQ = 2.969 X103N°-5 S 7E0 J 7 0 (6.16) where and are the catches by longlines and gillnets. Angelsen and Olsen's (1987) study was unique because abundance estimates came from acoustic surveys. In addition, the surveys were detailed enough in one year to provide an estimate of within-year changes in availability. Thus, the equations above are the first estimates of density dependent and effort dependent catchability on a intra-annual time scale. The study does have a few problems, however. The first problem is that Angelsen and Olsen (1987) applied the relative availability curve estimated in one year to total abundance estimates from 10 years, and then used the resulting within-year abundance estimates for all years in one regression. Thus, the X variate values (intra-annual abundances) for different years are not independent and would constitute psuedoreplicates (Hurlbert 1984). Therefore, the only data appropriate to include in statistical tests of within year density or effort effects on catchability are the data for the year 140 that the relative availability surveys were conducted. The second problem concerns Angelsen and Olsen's (1987) method of combining the constants for effort and abundance; it ignores potential interactions between the two effects. Since the effects of effort and abundance undoubtably interact to affect catch and q, it would be better to estimate the constants simultaneously by fitting equation 6.13 or 6.14. However, in practice it may be very hard to estimate the relative amount that catchability is affected by effort and abundance since these two factors commonly covary because of numerical responses. Under such circumstances, the estimates of a and 0 would be confounded (i.e. highly negatively correlated). The result of this confounding would be that two hypotheses, one that assumes large effects of density (i.e. large /3) and small effects of effort (i.e. small a) and the other that assumes large effects of effort (i.e. large a) and small density effects (i.e. small 0) would be equally consistent with the data. A few studies have examined density dependent catchability in fisheries for anadromous species. The advantage of these studies is that estimates of abundance are often from direct counts of adults that return to freshwater to spawn. These estimates may be more reliable than abundance estimates for marine fishes that are based on a variety of stock reconstruction models (e.g. Tomlinson 1970, Pope 1972; also see Shardlow et al. 1985 for a different opinion). For example, Peterman (1980) found that the catchability coefficient in various native Indian food fisheries was inversely related to the ' abundance of various Fraser river sockeye salmon (Oncorhynchus nerka) stocks. Peterman was the first to draw direct analogies between Holling's (1959b) functional responses and predation by fishermen. Thus, Peterman characterized native Indian food fishermen as type n predators. One drawback of Peterman's (1980) study was 141 that the effort measure he used, number of permits, does easily relate to the total time prey are exposed to the predator (T). Peterman and Steer (1981) extended Peterman's (1980) work when they examined the functional responses of sportfishermen to the abundance of chinook salmon (Oncorhynchus tshawytscha). The authors found that the catchability coefficient was inversely related to abundance and that a type II response was indicated (see also Shardlow et al. 1985). Peterman (1980) and Peterman and Steer (1981) fit both the Fox (1974) model (i.e. eqs. 6.9-6.12) and Holling's (1959b) disc equation to the salmon fishery data. In another study involving an anadromous species, Crecco and Savoy (1985) examined density dependent catchability in a commercial drift gillnet fishery in the Connecticutt river for American shad (Alosa sapidissima). Using Fox's (1974) model, they found 0 = -0.743 (eq. 6.9) and /J = -0.724 (eq. 6.10). The authors suggested that the primary mechanisms for density dependent catchability were strong schooling behavior during upriver migration by the shad coupled with nonrandom search behavior by shad fishermen. Crecco and Savoy (1985) also drew analogies with Holling's (1965) work and they suggested that another possible cause of density dependent catchability was increased handling times (emptying and resetting the nets) at high shad abundance levels. Two studies examined functional responses in invertebrate fisheries. Beinssen (1979) found that r=1196 and h = 5 (equal to o and t^  in the disc equation; eq. 5.2) and thus he found that abalone divers were type H predators. Hilborn and Walters (1987) modeled an abalone fishery using a type II functional response and sequential movement of effort to the most desirable area first (i.e. the numerical response) and found that a saturating relationship between CPUE and abundance resulted from simulations. 142 Type II Mechanisms in Fisheries Athough Fox's (1974) model has been useful in describing the relationship between catchability and abundance, it has not provided any clues as to the possible mechanisms that cause density dependent catchability. Thus, most studies have only speculated on the mechanisms based on more detailed theoretical studies (e.g. Paloheimo and Dickie 1964; Rothschild 1977). Four main mechanisms have been cited as potential causes of type II responses in fisheries. 1. Gear saturation or limits to searching capacity. 2. Decrease in searching time available because of an increase in the proportion of time spent handling as abundance increases. 3. The areas searched by a unit of nominal effort represents a larger proportion of the area occupied by a clumped population as abundance decreases, because the area of the stock decreases with decreasing abundance (because of schooling by the fish) and/or because of nonrandom search by fishermen. 4. Fishing effort is directed at a subset of the total stock (i.e. particular age or size classes) that represents a large proportion of the total numbers and/or biomass and the proportion of the population composed of this subset increases as abundance decreases. Saturation is analogous to satiation in natural predators. Gulland (1955) and Rothschild (1977) have provided theoretical models of saturation. Mechanism 1 is a possible cause of a type II response in Angelsen and Oslen's (1987) study of an inshore cod fishery and was cited as a potential cause in Peterman's (1980) Indian food fishery study. 143 Mechanism 2 follows directly from Holling's (1959b) disc equation (i.e. e.q. 5.2). Density dependent searching time was considered as the primary type II mechanism by Beinssen (1979) and Archibald et al. (1983). Peterman (1980), Crecco and Savoy (1985) and Hilborn and Walters (1987), considered mechanism 2 as one of a few possible alternative causes of type H responses in their studies. Mechanism 3 was first suggested by Paloheimo and Dickie (1964). Most studies of density dependent catchability in clupeiods have cited mechanism 3 as the likely cause (e.g. Fox 1974; MacCall 1976; Radovich 1976; Ulltang 1976; Pope 1980; Winters and Wheeler 1985). Mechanism 3 has also been suggested as a possible cause of density dependent catchability in cod fisheries by Pope and Garrod (1975), in a shad fishery by Crecco and Savoy (1985), and in abalone fisheries by Hilborn and Walters (1987). Crecco and Savoy (1985) demonstrated that when overall shad abundance was low, fishermen targetted on the peak periods of migration and on areas of high local abundance. Mechanism 4 was suggested as a possible cause of density dependent catchability in a North Sea cod fishery by Houghton and Flatman (1981). The ability of fishermen to direct their effort at age classes with high biomass has also been found in North Sea fisheries for other species (e.g. Cook 1984; Cook and Armstrong 1985). The next section considers type HI responses in fisheries. Type III Responses A few studies have attempted to detect a type HI response in fisheries. Peterman (1980) fit Fujii et al.'s (1978) generalized equation (i.e. eq. 5.8b) to data for the Indian food fishery, but he did not find parameter values (i.e. c > 144 dXt,) that were consistent with a type III response. Peterman and Steer (1981) n also rejected the type III response in their analysis of chinook salmon sport fisheries. However, their rejection of the type ni response appears to have been based on a visual inspection of the data. The authors suggested that a type III response could occur in fisheries if fish density was low and the area occupied by the fish was very large relative to the searching ability of fishermen (i.e. conditions where learning might act to increase the rate of effective search), but these conditions did not apply to the sport fisheries they examined which were either confined to rivers or to a small inlet. Despite the lack of empirical evidence for type IH responses in fisheries, the mechanisms for type III responses in natural predators reviewed in chapter 5 can also be expected to apply to fishermen. The three principle mechanisms for type in responses listed in chapter 5 were: 1. The predator is stimulated to spend a greater proportion of time hunting and/or hunting efficiency increases through learning when prey abundance increases. 2. The prey occur in patches of different densities and the predators feed more often in the high density patches and/or a limited number of spatial prey refugia are available and prey in refugia are less vulnerable to attack. 3. Switching by the predator between less abundant and more abundant prey (by various mechanisms, see below). Mechanism 1 is probably the least likely mechanism to apply to fishermen. Undoubtedly, fishing is a learning process and learning might result in an increase in the rate of effective search. However, the circumstances in which to expect learning in fisheries (i.e. newly developing fisheries) are also likely to 145 be associated with high stock levels (i.e. virgin biomasses) and under such conditions any increase in the rate of effective search caused by learning would be offset by increased handling times. Furthermore, cases which have examined fisheries that have depleted populations (e.g. the clupeiod examples above), have not detected a decrease in catchability at low stock levels that would be predicted by a type III response (Fig. 5.4). Fishermen are known to direct their effort at areas of high density (e.g. Saville and Bailey 1980; Cook and Armstrong 1985) and even at high biomass age or size classes of a stock (e.g. Cook 1984). This by itself would counteract type III effects. However, exploited fish populations are likely , to have several possible refuge areas where they could escape from predation (e.g. areas of untrawlable bottom, areas closed by regulations). Thus mechanisms 2. is very likely to operate in fisheries in spite of effort direction at concentrations. Many of the world's fisheries involve more than one species. Thus, of the three type III mechanisms, the switching mechanism is probably the most likely of the above three mechanisms to apply to fishermen. Larkin (1979) stated: ..."man as a predator has an arsenal of technology with which he can be an accomplished switcher, and with modern processing and marketing techniques it may well be possible to be even more flexible in the future". Given the likely applicability of switching to fisheries it is worth exploring in more detail the switching hypothesis as it applies to fishermen. In considering switching hypotheses for fisheries, it is helpful to recall how the null hypothesis for switching in natural predators follows directly from the 146 multispecies disc equation (MSDE; i.e. equations 5.12-5.14). Recall that the switching null hypothesis was NA^/NA^a^N^a2^2 ^'e' ec*' However, it is commonly assumed that fishermen are profit motivated (e.g. Clark 1976), and if so, the switching hypothesis presented for natural predators should be modified to account for the economic values of the species involved. The MSDE can easily be modified to calculate the attack rate (catch rate) in dollars by multiplying the catch rate in numbers (CR = C/f) by the price per fish. Thus, for the two fish species case: y _ a^N^T (6.17a) 1 + althlNl+a2th2N2 V2 = °2P2N2T ( 6-1 7 b ) 1 + althlKl+a2th2N2 where and Vg are the catch rate in dollars (i.e. V^.— (CR^.); and p^, p^ are the price per fish in dollars (i.e. average weight per fish (kg) X price per kg)). Since a profit maximizing fishermen is interested in the values VI and V2, the switching null hypothesis (i.e. no switching) for the fishery implies: V c p N V- = inrw- (6-18) 2 °2P2 1N2 Thus, given a^a^, equation 6.18 hypothesizes that in the absence of switching, the ratio of the catch rates in dollars of two species is proportional to their relative values and to their ratio of their numbers available in the environment (Fig 6.3a). A mixed species fishery which is completely nonselective with respect to species and without discards might fit this simple null model. 147 However, if fishermen are profit motivated, and if there is a physical way to select which species to catch, it would be better to catch the higher priced species in disproportionately larger quantities relative to the lower price species (i.e. fishermen should switch), if there is a physical way to select which Thus, I would expect a switching fishermen to increase his attack rate for the higher valued species. For example, switching for the case of the fishery could be incorporated into the MSDE in a way analogous to Murdoch and Oaten's derivation for natural predators (i.e. eqs. 5.15 & 5.16), except I will assume that rate of effective search for each species increases linearly with the proportion of the value of the species available (i.e. p.N•)• Thus, I let o. = XP . where P. = p .^N7(p^N^ H-p^N )^ and with these substitutions, the functional response model for switching in the two prey case is n j } _ XP.N.T (6.19) Kjti. — I l l l Equation 6.15 results in a switching hypothesis of C R i \ P i V C R2 X2 p 2 2 N 2 2 (6.20) Note that in the case of the fishery, the catch rate depends on both abundance and price of each species. Thus, a switching fishermen is expected to increase the relative catch rate (i.e. CR^ . also V.) of the more valuable and/or more abundant species faster than linearly with changes in its relative abundance (Fig. 6.3 A). The switching curves are sigmoid and the null model is a straight line if plots are drawn in terms of the relative proportions caught and available (Fig. 6.3 B). 148 Switching 3. Proportion Available (N / [N +N ]) Figure 6.3. A. Implications of switching hypothesis for the fishery. Straight lines are for the ratios of catch rates in dollars for the case when p^ = Pg (solid line) and when p^ = 2p^ (dashed line). Curves are for switching. B. Switching hypothesis in fishery when catch rates and numbers available are expressed as proportions. Straight line denotes no switching, upper curve is for switching when p = 2pg, and lower curve is for switching when p = p . 149 There are two primary mechanisms available for switching by fishermen. In cases where nonselective gear is used and species distributions overlap, fishermen may simply discard less valuable or otherwise undesirable species (i.e. switching mechanism 3 for natural predators; chapter 5). Alternatively, fishermen may attempt to target on particular species by fishing with selective gear or by spending a greater proportion of their time searching areas where more valuable species are concentrated (i.e. switching mechanism 1 for natural predators; chapter 5). The other two mechanisms for switching in natural predators (rate of detection dependent on recent experience, rate of successful attacks dependent on recent experience) could also apply to the fishery. For example, trawlers commonly use echo sounders to detect the presence of fish on or near the bottom. Some trawl captains claim they can tell the type of fish from a echo soundings and this ability would clearly depend on experience. The rate of successful attacks might also depend on recent experience because skippers must learn how to use the gear most effectively in the particular areas where different species are distributed (e.g. trawling on soft vs. hard bottom, flat vs. sloped bottom, etc.), and under various tidal conditions. However, these latter two mechanisms are probably much less common causes of switching in the fishery than the two primary mechanisms. From examining equation 6.17, and considering the many variables that might affect fishermen's decisions, it is obvious that the form of the switching curve for any particular case will be affected by many factors. There is a void of research in this area, but one approach which seems promising is the application of optimal foraging theory (e.g. Stephens and Krebs 1986). The "prey" and "patch" models described by Stephens and Krebs (1986) are applicable to the discarding and targeting mechanisms above and could be used to determine 150 switching rules (i.e. what proportion of each species to take, and when to switch fishing grounds), and switching curves resulting from optimal foraging strategies (D. Gillis, Biological Sciences, Simon Fraser University, Burnaby, B.C. pers. comm.). However, in order to provide meaningful results however, these studies should modify the simple models to take account of fishing costs, risk, and state dependent foraging (e.g. the decision to discard may depend how much space in available in the vessel hold). Finally, it is worth considering what factors peculiar to fisheries may affect the likelihood of detecting type HI responses in existing data sets. In "single" species fisheries, the likely mechanisms for switching are learning, spatial refugia and other factors that cause an increase in the rate of effective search as fish density increases from low to moderate levels, or vice versa. The learning mechanism seems unlikely to apply to fisheries for the reasons mentioned above. Also, it does not appear that the rate of effective search declines in clupeiod fisheries when the population drops to low levels (several studies reviewed above found type H responses). In fact, the rate of effective search appears to increase even when the stocks have been depleted to very low levels. As these fisheries develop, the increases in fishing power due to improved gear technology and searching ability combined with the strong schooling behavior of the fish more than compensate for the decrease in the rate of effective search expected from decreased abundance. Clupeiod fishermen may simply act like type H predators. However, type II responses also appear to describe fisheries data from nonclupeiod fisheries (e.g. Garrod 1977). Garrod's (1977) Arcto-Norwegian cod stock is actually part of a multispecies fishery where type HI functional responses are more likely. Management may have kept abundance levels above 151 those where the rate of effective search declines. Alternatively, in fisheries where fishermen select target species by choosing particular gear types or areas, fishermen may simply switch to a different target species when their rate of effective search declines. Thus, in contrast with data from laboratory experiments with natural predators, fisheries data sets may not contain the range of observations over which the accelerating part of the type III response is generated. In unselective fisheries, fishermen may continue to catch species at low abundance levels, but unknown quantities of discarding may confound any attempts to discern the form of the functional response. As Murdoch and Oaten (1975) pointed out, the occurrence of switching does not necessarily imply a type III response. T y p e I V R e s p o n s e s The three main mechanisms for type IV responses in natural predators were reviewed in chapter 5: predator confusion by prey, predator disturbance by pre}' and group defense in prey. It is unlikely that any of these mechanisms operate fisheries. In fact, it ironic that schooling behavior that may act to confuse natural fish predators increases the vulnerability of fish to predation by man (e.g. Clark 1974; Clark and Mangel 1979). Paloheimo and Dickie (1964) found conditions where their model gave a response similar to a type IV response. Their model predicted a decrease in the rate of attack as school radius increased (NA; their C(t)/t) if the overall density of fish was constant and density within schools decreased as school radius increased (their Fig. 1 mislabelled as Fig. 2 in the paper). However, in order for these conditions to generate a type IV response to overall density, the additional 152 condition of school radius increasing with overall density must also be present. . Type II responses combined with a positive numerical response and competition could also result in apparent type IV responses. The mechanism in this case is analogous to predator disturbance by prey, except predators would be disturbing each other (i.e. interference competition). Thus, it is important to recognize that with most fisheries data sets apparent functional responses include interactions among fishermen; they are not analogous to results from experiments with natural predators where individual predators are used and the effects of competition or facilitation are thus excluded. There is another mechanism for type IV responses in fisheries: fishermen are not all alike, and the poorer ones may only fish when (or where) fish are unusually abundant or valuable. In this case, the average catch/fishermen can easily decline as abundance increases, though for any single fishermen the response is monotonic increasing. Thus, it is important to distinguish between individual behavior and performance versus average per capita behavior of the aggregate fisherman population (C. J. Walters, Resource Ecology, Univ. of British Columbia, pers. comm.). 153 Chapter 7: Estimates of Abundance Introduction Any examination of functional responses requires estimates of prey abundance and percent mortality per predator (i.e. the catchability coefficient). However, obtaining these .estimates for marine Fisheries is problematic because direct estimates of fish populations are rarely available. Therefore, fisheries scientists have developed a variety of indirect methods for estimating demographic statistics. In this chapter, I first describe two related techniques that are used to estimate abundances and catchabilities of exploited fish populations from data on the age compositions of catches over time. Each technique is described in terms of the underlying mathematical model, data requirements and key assumptions. I apply these methods to three fish species that are the principal components of a multispecies trawl fishery in Hecate Strait. Next, I examine the sensitivity of estimates obtained from each technique to alternative assumed input parameters (i.e. fishing and natural mortality rates), and finally I compare the estimates obtained from the two techniques. Estimates of abundances and catchabilities obtained from the techniques are used in Chapter 8 which examines the functional responses of trawlers in Hecate Strait. A. • I „> The first technique, Virtual Population Analysis (VPA;Fry 1949)" was developed by Jones(1964), Murphy(1965), and Gulland(1965,1983). The second technique, Catch-at-AGE ANalysis with auxiliary information (CAGEAN; Deriso et al. 1985), is one of a variety of techniques (cf. Doubleday 1976; Pope 1977; Paloheimo 1980; Fournier & Archibald 1982; Pope and Shepherd 1982) developed 4 Fry originally coined the term "Virtual population" but his method differs from the techniques commonly referred to as Virtual Population Analysis (e.g. Gulland 1965, 1983) 154 as alternatives to VPA. I will use CAGEAN to denote the Deriso et al.'s (1985) technique (see below) and CAGEAN to denote the computer package that uses the technique. The two techniques have similar data requirements (e.g. estimates of numbers landed-at-age), but differ in some of their assumptions and estimation procedures. CAGEAN has more favorable statistical properties, but because of its recent development has only, been applied to a few species. VPA is a simpler procedure that has been applied to many species. I chose to apply both methods for two reasons: (1) to determine whether the two techniques would provide similar estimates, and (2) to compare the sensitivity of estimates obtained from the two methods to alternative input parameters. Before proceeding with further details on the methods, the next section reviews the study area and the fishery used to examine functional responses. Study Area and Fishery Description My analysis of functional responses focused on the multispecies trawl fishery in Hecate Strait (Fig. 7.1). Landings statistics collected by Department of Fisheries and Oceans are available from various stock assessment documents (e.g. > Westrheim 1980; Stocker 1981; Tyler and McFarlane 1985; Tyler et al. 1986). In addition to these basic statistics, the fishery has also been the subject of recent detailed analyses including a method for allotting effort among species (Westrheim 1983), estimation of relative fishing power of trawlers (Westrheim and Foucher 1985a), and various analyses of age composition data sampled from the landings (Ketchen 1961,1964,1970; Kennedy 1970; Chilton and Beamish 1982; Foucher and Fournier 1982; Foucher et al. 1984). In addition to these studies, Hecate Strait has been the focus of a five-year project that began in 1984 and that has the primary goal of developing an ecological basis for mixed-species 155 Figure 7.1. Map of principal fishing grounds and major statistical areas in Hecate Strait (from Westrheim 1983). 156 fisheries management (Tyler 1986, 1989). Using the wealth of information on this fishery available in the above studies, the description below briefly highlights the principal species involved, and general spatial and temporal patterns in their landings. The principal species caught by the bottom trawl fishery in Hecate Strait are Pacific cod (Gadus macrocephalus), rock sole (Lepidopsetta bilineata) and English sole (Parophrys vetulus) (Westrheim and Foucher 1985b). Annual landings of these species have fluctuated cyclicly (Fig. 7.2). For statistical purposes, Hecate Strait has been divided into two major areas, 5C and 5D, each of which contains a few trawling grounds (Fig. 7.1). Detailed catch information including fishing ground and depth of capture have been collected by port samplers since the 1940's, however uncorrectable encoding errors make data unreliable for the pre-1956 period (Westrheim and Foucher 1985a). Pacific cod and rock sole are caught in both major areas, but the only substantial landings of English sole come from area 5D (Westrheim and Foucher 1985b; Fargo 1985). Area 5D produced 67% of the cumulative landings of cod for the period 1960-81 (Foucher and Westrheim 1984) and 62% of the cumulative landings of rock sole for the period 1956-85 (Fargo 1986). The landings of both rock sole and English sole comprise primarily (>70%) females that are larger and more vulnerable than males (Fargo 1985). Landings for all three species fluctuated seasonally and among grounds. Area 5C landings of cod and rock sole are from Horseshoe and Ole Spot grounds during April-September (cod) and June-October (rock sole), and from Reef Island-Cumshewa ground during January-June (cod) and April-May (rock sole; Fig. 7.1). In area 5D, the principal ground for all three species is Two 157 Figure 7.2. Estimated total numbers landed of Pacific cod (areas 5C + 5D; ages 2-10; 1961-1985), English sole (area 5D; ages 3-17;1956-1979) and rock sole (area 5D; ages 4-15; 1956-1978). Peaks-Butterworth during April-September. A small amount of rock sole is also landed from Shell ground during these months (Fig. 7.1). During October-March, cod and English sole are landed from White Rock-Bonilla ground (Fig. 7.1; Westrheim and Foucher 1985b; Fargo 1985). Principal depths of capture vary among species, seasons and years (Westrheim and Foucher 1985a). For example, the principal depths of capture of cod and English sole were 30-49 fathoms on Two Peak Butterworth ground, except for a few years when cod were caught at 10-19 fathoms. Rock sole have generally been caught in shallower water (10-19) fathoms, but have also had 158 substantial landings at 30-59 fathoms. Data Preparation and Background Methods Aging Methods Both stock assessment methods used require estimates of the numbers of Fish landed-at-age for each of the three species. Therefore, a brief description of the aging techniques used by the scientists at Pacific Biological Station for these three species is warranted. Age composition samples for the three species were collected from commercial landings by port samplers stationed in Vancouver and Prince Rupert. Representative Fishing grounds for age compositions were selected for each species based on the relative quantity of the landings from each ground and other criteria. Cod were aged from length frequencies (see below) and the representative grounds varied quarterly. White Rocks-Bonilla ground was used for quarters I (Jan.-March) and IV (Oct.-Dec.) and Horseshoe and Two Peaks-Butterworth grounds were used for quarters H and i n (April-Sept.; Fig. 7.1; J. Westrheim pers. comm., Pacific Biological Station, Nanaimo, B.C.). Sample sizes taken for length frequencies ranged from 100-300 fish per month during the appropriate quarter(s) for each representative ground (Foucher and Westrheim 1984). English and rock sole are aged from otoliths and Two Peaks-Butterworth ground was selected as the representative ground for both species (Fig. 7.1; Fargo 1985). Annual sample sizes have averaged 611 (range 100-957; no samples for 1972) for English sole and 596 (range 127-1345; no samples for 1965 and 1972) for rock sole (sample sizes are for females only for both 159 species; see below). Of the three species, Pacific cod has been the most difficult to age reliably. Length frequencies and data from tagging experiments were used initially by Ketchen (1961,1964) who later showed that otoliths were of. little use (Ketchen 1970). Concurrently, a technique using scales was described by Kennedy (1970), but in 1978 the use of the scale method was suspended pending validation because it was suspected of being inaccurate especially for older fish (Chilton & Beamish 1982). A method using dorsal fm rays yielded ages that were similar to those obtained from scales for younger fish, however it was thought that ages from fin rays were more reliable for older fish (Chilton & Beamish 1982). This method encountered technical problems however, that made it. difficult to get consistently readable fin rays from port samples. At present the only technique being applied uses length frequencies following the general approach of Schnute & Fournier (1980) adapted for cod as described by Foucher & Fournier (1982) (in particular, see the appendix of their paper for details of the changes). Any approach that uses length frequencies must deal with the problem of overlapping distributions (i.e. ranges of the length frequency distribution that likely contain fast growers from one age class mixed with slow growers of the next oldest age class). Early methods such as Harding (1949) and Cassie (1954) relied on graphical methods of separating out this overlap and assigning proportions to each group. For example, Cassie's (1954) method involves the visual inspection of a plot of cumulative frequencies versus length to identify 2 inflection points. The choice of infection point could then be evaluated using a x test. Schnute & Fournier's (1980) method is not completely objective. It identifies shapes of length frequency distributions that are consistent with simple biological 160 assumptions about growth (i.e. a von Bertalanffy growth curve). Foucher and Fournier (1982) compared the estimated ages obtained from their method (termed the computer method) to those obtained from scales, Cassie's (1954) graphical method, and Ketchen's (1964) tagging experiment. Ages obtained from the graphical and computer method were similar to each other, but different from ages estimated by the scale method. Foucher and Fournier (1982) suggested that these differences were possibly due to errors in scale reading. They reached this conclusion based on an examination of mean lengths-at-age determined by the two methods. Estimates of mean lengths-at-age from the computer method were similar to those determined from tag returns (cf. Foucher et al. 1984). In conclusion, Foucher and Fournier (1982) state that Pacific cod is a fast-growing species with a relatively small number of fished age classes, and therefore it should be an excellent candidate for aging by length frequency analysis. These characteristics of cod should minimize problems introduced by the lumping of older age classes common to length frequency techniques (Schnute & Fournier 1980). Proportions-at-age estimated using this techniques were used in my analysis. In contrast to the problems encountered with cod, flatfish aging has proceeded rather smoothly. Both English sole and rock sole are aged using otoliths. Initially, ages were obtained from surface readings, but techniques using broken and burnt otoliths or thin cross sections were considered to be more accurate for older individuals, particularly for English sole. Preliminary results from age validation experiments using rock sole also suggest that readings from sectioned are more accurate (Fargo and Chilton (1987). Chilton and Beamish 161 (1982) describe the aging techniques in more detail. All the ages for rock sole in my study were obtained from surface readings. The times series of ages for English sole is a mixture of surface and broken and burnt readings. In one experiment, approximately 600 English sole otoliths were aged by both techniques, so that the whole time series could be standardized to one method. I chose to follow the approach of J . Fargo (pers. comm., Pacific Biological Station, Nanaimo, B.C.) and convert all ages to surface ages because these ages make up 20 of the 26 years in the time series, and to maintain consistency between rock sole and English sole aging methods. Estimating Numbers Landed-at-Age I obtained tables of the number of fish landed-at-age for subsamples from the representative grounds for each species (flatfish, J . Fargo, cod, R. Foucher pers. comm. Pacific Biological Station, Nanaimo, B.C). Given these estimates of numbers of fish landed-at-age in samples from the commercial fisher}', the next step was to estimate the total numbers landed-at-age in the appropriate statistical areas. The statistical areas vary for the three species due to differences in stock delineation. In the case of cod, the statistical areas are 5C + 5D (Fig. 7.2) since tagging studies show that movement precludes stock separation within Hecate Strait (Westrheim 1984). The fishery for rock sole in Hecate Strait is considered to be made up of two stocks, one in area 5D the other in area 5C. This judgement is based on abundance trends in landing statistics, length frequency anomalies (Fargo 1985) and preliminary results from a tagging experiment 162 (Harling et al. 1982; Ketchen 1982). Age composition data are only available for the northern stock (i.e. area 5D). Hecate Strait is presumed to contain a single stock of English sole (i.e. in area 5D), based on landing statistics (Fargo 1985) and tagging experiments (Ketchen 1953). In addition to divisions by stock, aging of the two flatfish species is also broken down by sex. Approximately 75% of the fishery for English sole involves females from 4-10 years of age (Fargo 1985). Similarly, approximately 80% of the landed weight of rock sole is comprised of females. Males are generally smaller, and therefore less vulnerable to the gear. Thus, only female English sole and rock sole were considered in the analysis. The exclusion of males is consistent with published stock assessments (Stocker 1981, Fargo 1985). In summary, numbers landed-at-age in samples were inflated to total numbers landed in area 5C & 5D for cod and in area 5D for both rock sole and English sole females. The formula used was N. = (W/w)XnXPj. (7.1) where N is the total landings-at-age i, W is the total weight of landings, w is the weight of fish sampled, n is the number of fish sampled, p is the proportion of fish sampled in age group i. For cod, equation 7.1 was applied to landings on a quarterly basis for each representative ground. For both flatfish species, equation 7.1 was applied to samples from Two Peaks-Butterworth ground. For years when no sample were taken (1972 for English sole, and 1965 and 1972 for rock sole, I used the average p. calculated from the total landings for all years with samples; i.e. the average estimated proportion landed-at-age). All age compositions were extrapolated 163 to total landings for a standard calendar year (i.e. Jan.-Dec). This deviates from past assessments for cod that used a "cod year" (i.e. April-March; cf. Stocker 1981 and Westrheim and Foucher 1985b). I chose the calendar year as my time stratum because it allowed for comparison between assessments of all three species. The estimated numbers landed-at-age for the three species are presented in Appendix A (Tables A1-A3). Effort Data In addition to estimates of numbers landed-at-age, estimates of fishing effort were also needed. I used effort data published in Tyler and Saunders (1987). Since all three species are caught in the same general area, and often in the same trawl haul, it was important that the effort used represented the effort "directed" at the particular species of interest. This is a problem common to multispecies fisheries and the main concern is not to include effort associated with incidental catches of a species that could result in misleading interpretations of stock abundance indices based on LPUE. The effort data I used were allocated to individual species by a method known as the Option 2 Method (Westrheim 1983). The method allots effort to a species based on an examination of the landings for designated time (quarter year), area (fishing ground), and depth (10 fathom intervals) strata. For example, if the depth strata 10-20 fm and 40-50 fm contained the largest and second largest total catches of English sole in quarter 2 (April-June) at Two Peaks-Butterworth ground, then the Option 2 effort for English sole would be the total effort exerted at those depths (toward all species) during quarter 2 on Two Peaks-Butterworth ground. Effort data were also standardized to account for changes in fishing power by 164 Westrheim and Foucher (1985a). The effort data used for each species are presented in Appendix A (Table A4.). Models, Data Requirements and Assumptions Virtual Population Analysis Model Description - VPA The technique of VPA is based on following a group of fish of the same birth year (a cohort) through time. The concept is most easily explained using a simple population model. If a cohort of fish is subject to a short fishery followed by a period of natural survival, then the cohort numbers next year (t+1) are related to the cohort numbers this year (t) by the following balance equation. Nt+i = ( Nr c, ) s (7-2) where N is the number of fish, C is the catch, and s is an annual natural -M survival rate (from Ricker 1975, p.198 with s=e ). Note that equation 7.2 can be solved for N^ yielding N, = ®*t+l)/s + Ct (7-3) Given estimates of N ^ ^ , Cy and s, equation 7.3 is used recursively working backwards through time to solve for the N's in previous years. This is the fundamental basis for VPA. In most of the published literature, this simple concept is hidden in more complicated formulas written with instantaneous rates. The equations below from 165 Gulland (1965 cited in Pope 1972, also see Gulland 1983) are the most widely used. N t+1 (7,4) N (1-e t -F-M) (7.5) C t F. + M where N and C are the abundance and catch in numbers, e is the base of natural logarithms, and F and M are the instantaneous rates of fishing and natural mortality respectively. The subscript t can refer to either year or age within a cohort. Equation 7.5, referred to as the Catch Equation, was first derived by Baranov (1918; cf. eqs. 6.1-6.3). Equations 7.4 and 7.5 are for the more general case when the fishery occurs for a longer period and it is necessary- to account for competing fishing and natural mortality. The main reason for their complicated appearance has -F -M more to do with notation than generality, however. Note that e t of equation 7.4 is the total annual survival rate (from both fishing and natural causes). Therefore, (1—e t^^ ) of equation 7.5 is the total annual death rate, and the ratio, F/(F+M), is the proportion of deaths due to fishing. Equation 7.4 then, states that the numbers next year are the numbers this year times the survival rate, while 7.5 states that the catch is equal to a proportion of the total deaths due to fishing. A third equation used in VPA comes from solving 7.4 for N^ and substituting the result into 7.5. It is 166 Nt+1 = (F,+M)e"F'*M (?-6) ~ Fta-e-FiU) Data Requirements - VPA Three items are required to start the estimation. 1. N^s, the population sizes of the oldest age at which the each cohort was fished. 2. C Q f , an estimate of the catch of each age a for the cohort for a series of years (t) 3. M, the natural mortality rate For the complete cohorts (i.e. cohorts that have reached the oldest fished age (A) by the last year (T) of the catch data, shown as solid ellipses connected by solid lines in Fig. 7.3), VPA uses the catches of the oldest age (A) in each year t (i.e. C^s enclosed by the solid rectangle in Fig 7.3), along with M and the terminal F values (i.e. F^s) in equation 7.5 to calculate the numbers of fish of the oldest age each year (i.e. N^ fs; item 1 above). For incomplete cohorts (i.e. cohorts that have not reached the oldest age by the last year of the catch data shown as open ellipses connected by dashed lines in Fig. 7.3), VPA uses the catches of each age (a) in the last year T (i.e. C^^s enclosed by the dashed rectangle in Fig. 7.3), along with M and the age specific F values (i.e. F^^s) in equation 7.5 to calculate the numbers of fish of the each age in the last year (i.e. Na^s). For both complete cohorts and incomplete cohorts, estimation proceeds back in time through each cohort (i.e. along the diagonal lines in the direction indicated by the arrows in Fig. 7.3) using first equation 7.6 to solve for F T - and then equation 7.4 to solve for N • r and so on. 167 Figure 7.3. Schematic diagram of VPA method. Ellipses represent observations of catches-at-age for 10 years from a hypothetical fish population with 6 fished ages. Open ellipses connected by dashed diagonal lines denote incomplete cohorts and solid ellipses connected by solid diagonal lines denote complete cohorts. For further details see text. Figure 7.3 is modified from Figure 1 in Bradford and Peterman (1989). 168 F ^ may be estimated from F^=qE f , where q is the average catchability coefficient for fully recruited ages, and E f is the annual fishing effort. Numbers caught-at-age are commonly estimated by sampling and aging a portion of the landings as described above. The natural mortality rate, M, is either assumed at some arbitrary value or estimated along with the catchability coefficient by various regression methods (e.g. Gulland 1969; Paloheimo 1980). For incomplete cohorts, estimates of F Q^S are required for each age in the last year. Estimates of F^^s are obtained from ^aT=(la'p^'j' where q ^ is the catchability coefficient for each age a in year T. Estimates of q^j, are obtained from the q^s for each age averaged over complete cohorts after the initial pass of the VPA (see equation 7.14 below). Model Assumptions - VPA There are four principal assumptions of VPA (Walters 1987a). 1. N represents a closed group of fish. There is no immigration or emigration. 2. The odds of dying from either fishing or natural causes are the same at each moment during the year.5 3. M is constant over time and ages. 4. F . is known (i.e. F . = q .EJ for all ages a in the last (most recent) a,t a,t a,t r year t, and for the oldest age in all other years. Migration effects on VPA were investigated by Ulltang (1977). He developed a model to keep track of cohort numbers for the hypothetical case where a cohort continuously migrates from area A to area B at a constant rate. If emigration were included in the estimate of M for area A, VPA could be 5 Assumption 2 applies only to continuous fishery version (i.e. to equations 7.4-7.6, not equations 7.2 and 7.3). 169 used. However, he found no simple way to account for immigration to area B. In practice, the conditions of assumption 1 are approximated by choosing the area of the fishery large enough to minimize or balance additions and losses due to migration (e.g. the choice of areas 5C and 5D for Pacific cod). Ulltang (1977) and Sims (1982) investigated the effects of violations of assumption 2 on N estimates from VPA. Ulltang concluded that an uneven distribution of M or F throughout the year produced negligible errors in VPA. Sims concluded that errors were not severe unless M was large and the fishery was seasonal with heavy exploitation at one end of the year or the other. Walters (1987a) stated that assumption 2 was not too critical; F & M estimates can be viewed as integrals across seasonally varying rates. Violations of assumption 3 are difficult to detect due to confounding with possible changes in F with age and time. Effects of errors in M were investigated by Agger et al. (1973) and Ulltang (1977). They concluded that errors in M were transferred to estimates of N and F. If M was overestimated, then N was overestimated and vice versa, but relative N values from year to year were approximately correct. Overestimates of M result in underestimates of F and vice versa. Trends in true M with time were converted to trends in F estimates from VPA. Bradford and Peterman (1989) showed that incorrect values of M or F caused spurious time trends in estimates obtained from incomplete cohorts. Bradford and Peterman (1989) found that the magnitude of the spurious time trend was inversely related to the ratio of Fj/M. These time trends disappeared however if the F^, values for these incomplete cohorts were calculated from the results of an initial VPA and the VPA was repeated iteratively until starting and ending Fj, values converged (method of Rivard 1983). Lapointe et al. (1989) found that a time series of increasing fishing mortality rates coupled 170 with an incorrect value of M caused spurious time trends in abundance estimates for complete cohorts. The magnitude of the bias depends on the range of cumulative fishing mortality rates among cohorts that were present during the beginning and end of the increasing F period. Given the potential effects of an incorrect choice of M, or time trends in M, Walters (1987a) cautioned users of VPA to only consider relative rather than absolute results. Assumption 4 was investigated by Pope (1972). He showed that incorrect choice of terminal F^, values introduced errors in N^, estimates particularly for the last few years of a catch-at-age data set, but these errors were small when cumulative fishing mortality over the life of the cohort is >2.0. For a cohort that is fished for 10 years, this represents an average annual exploitation rate of > 18%. This property of VPA limits its utility in providing estimates of current abundances because the the current estimates are the sums over the incomplete cohorts that have been fished for only a few years (i.e. low cumulative fishing mortality). Therefore, estimates for recent cohorts are generally most subject to error caused by incorrect terminal Tj, values. Pope (1972) suggests that N^, estimates can be errant for all years if F is low compared to M. When terminal F^, are estimated from Fy,=qE^, . it is important that effort be standardized such that q is constant over the time period used (e.g. Gulland 1956a). Effort standardization is particularly essential to the regression methods used to estimate M and q (e.g. Paloheimo 1980). In addition to these assumptions, errors in aging catch samples, may introduce errors in VPA. Walters (1987a) noted that aging errors mask variations in numbers at a given age across years as fish will be incorrectly assigned from small to large cohorts and vice versa. There is no simple way to correct for aging errors. Walters suggested reallocating the age sample according to various assumptions about the magnitude of errors. Other methods (cf. Fournier and Archibald 1982; Deriso et al. 1985) assume a particular distribution for the aging errors and adjust their calculations accordingly. The method of Deriso et al. (1985) also permits pooling of older ages, where aging errors are more likely to occur particularly when fish are aged from bone structures (e.g. otoliths) or other hard parts (e.g. scales, fin rays). In conclusion, because violations of VPA may result in erroneous estimates of abundances and fishing mortalities, it is important to interpret the results carefully. Possible violations of assumptions 1,2,4 may be evaluated to some extent for individual data sets. Violations of the constant natural mortality assumption are difficult to detect, and dictate the use of only relative results. The effects of aging errors are not addressed by VPA alone, but may be evaluated through simulation given some assumed distribution for the errors. More recent techniques (e.g. Deriso et al. 1985) may be less sensitive to the effects of aging errors. Catch-at-age Analysis with Auxiliary Information (CAGEAN) Model Description - CAGEAN The goal of CAGEAN (Deriso et al. 1985) and its predecessors is the same as that of VPA: estimation of cohort numbers and fishing mortalities. The primary impetus for CAGEAN came from the recognition that VPA estimates more parameters than there are observations. For example, a catch-at-age data set with 20 years and 10 ages would contain 200 observations. The VPA equations using these data contain 221 unknowns (i.e. 200 F values, M, and the N values for the oldest age in each year; 172 N^s). In order for VPA equations to be solved, M and the N values (or F^, values) must be supplied. A second reason for the development of methods such as CAGEAN is that VPA does not provide very reliable estimates of current abundance due to dependence of the most recent estimates on the terminal F^, values. CAGEAN offers two remedies for these problems. The first is to reduce the number. of parameters to be estimated from the catch-at-age data and the second is to use additional data (called auxiliary information) to help define uncertain parameters (e.g. F values) and to increase the number of observations. The resulting model has three components: a catch-at-age component . using the catch equation (eq. 7.5), a component that relates fishing mortality to fishing effort, and a spawner-recruit component that relates the number of recruits (i.e. offspring surviving to first fished age) to the number of spawners (i.e. parents). CAGEAN's first component is aimed at reducing the number of parameters to be estimated from the catch data. Like VPA and most other methods, CAGEAN starts with the catch equation (eq. 7.5), however the form of the catch equation used by CAGEAN comes from combining equations 7.4 and 7.5. Given these two equations, it follows that catch can be related to recruitment abundance (i.e. abundance at first fished age) by „ _T M -F -M) I(-F -M) n _ F N (1-e a,t 'e a,t (7.7) C - a,t r,t a.t F +M a,t where all symbols are the same as in 7.5 except that N^ t is the recruitment abundance, and the subscripts c and t denote age and year. The term £(-F -M) e a,t is the cumulative survival rate from time of first recruitment to time t. For clarity, the notation of equation 7.7 has been simplified from that of Deriso et al. (1985; eq. 7, p. 817). 173 Unfortunately, like VPA, this restatement of the catch equation still has more parameters than observations. The key step to reduce the number of parameters was suggested by Pope ( M S 1974 cited in Doubleday, 1976). Pope's idea was that the fishing mortality rate F at a given age and year could be modeled as the product of a year effect F (due to changes in fishing effort), and an age selectivity effect S q (due to gear selection or partial recruitment). F = F X S (7.8) a, t t a This model reduces the number of fishing mortality parameters from A ages X T years to A + T (A age selectivity effects, and T fishing mortalities). Equation 7.8 is commonly referred to as the separability assumption. CAGEAN assumes at least one of the S Q S = 1 because the age effects can only be measured relative to the total fishing rate (F^ for each year. The separability assumption makes estimation possible. The estimation is accomplished by nonlinear least squares, with the residual sums of squares (SSQ) given by SSQ(catch) = L flogC' -logC f (7.9) a,t a,t a,t where C'a t is the predicted catch from equation 7.7, and C q ^ is the observed catch. Like VPA, CAGEAN uses an assumed M that must be estimated outside the model. Equation 7.9, CAGEAN's first component, is nearly identical to that of Doubleday (1976). CAGEAN diverges from this earlier work in the use of auxiliary information. CAGEAN uses two types of auxiliary information to help with parameter estimates. CAGEAN's second component uses fishing effort to provide more information about the 174 F estimates of equation 7.8. The model for the second component is F = qXE ( Xe vl (7.10) where F^ is the full-recruitment fishing mortality, E f is the fishing effort, q is the catchability coefficient, and ev^ represents lognormal random variability about the relationship. This form of fishing mortality-effort relationship was first used in the catch-at-age estimation scheme of Fournier & Archibald (1982). The sums of squares for the second component is where \j is the ratio of variances (variance of observed logarithm of catch from that predicted by 7.7 divided by the variance" of observed : logarithm of effort). This second component, called an auxiliary sums of squares, is added to the least squares equation 7.9. Deriso et al. (1985) consider X^ to be a weighting term used to adjust the amount of influence of the effort data. A very large value for X^ would result estimates of F^ that are exactly proportional to E f . The third component of CAGEAN was developed to help bound the N estimates of equation 7.7. Again, following Fournier & Archibald (1982), Deriso et al. (1985) assume that recruitment is based on a Ricker (1954) spawner-recruit function with lognormal variability. The model is SSQ(effort) = X ^ (log(Fp-log(qEp)2 (7.11) N = be t a-bSx+v (7.12) r,t+r 175 where N is the recruitment abundance, S is the spawning stock egg production (different from S ;^ the age specific selection coefficient), a and b are stock-recruitment coefficients, is a random variable, and the subscripts rand t denote recruitment age and year. S is the sum over all ages of the N Xe , where e^  is the age specific net fecundity (i.e. percent mature X fecundity). Estimates of age specific net fecundities for each species are presented in Appendix A (Table A5.). Note that the third component does not add any observations; N is estimated simultaneously in equations 7.7 and 7.12 of the r,f model. The sums of squares for the third component is SSQ(spawn) = X 2 £[log(N. ^ J-dogS+a-bsf (7.13) where is the ratio of variances as in X^ except the denominator is the variance of the stock-recruitment relationship. A high value for X^ would result in recruitment abundance estimates that are a deterministic function of spawning stock. Equation 7.13 is added to equations 7.9 and 7.11 to yield the complete CAGEAN model, where the problem is to find the parameters that minimize SSQ(catch) + SSQ(effort) + SSQ(spawn). When all three components are included (i.e. X^,X,g>0), CAGEAN estimates the following parameters: 1. Full-recruit fishing mortality rates (Fj3; i.e. the fishing mortality rates on ages with S^ s = 1) for each year of catch data (a total of T values). 2. Age specific selection coefficients (Scs) for each age as desired (a maximum of A - l values). 3. The abundances for the youngest age class each year for all ages in the first year (a total of T+A- l abundances). 176 4. A catchability coefficient (q). 5. Two stock-recruitment model parameters (a and 6). Thus, CAGEAN estimates a maximum of 2A + 2T-1 parameters from A X T + T observations. Data Requirements - CAGEAN The data requirements for CAGEAN are as follows: 1. Estimates of numbers caught-at-age 2. An estimate of natural mortality (M) 3. A time series of fishing effort 4. Estimates of net fecundities-at-age To start the nonlinear estimation, CAGEAN requires starting guesses for each of the parameters (see above). However, if supplied with an estimate of the terminal fishing mortality rate in the last year, the CAGEAN program will generate its own starting values using a routine called COHORT (Deriso et al. 1985). The CAGEAN program is also designed to permit estimation of parameters using just the first component or any combination of the first component and the other two components. Therefore, estimates can be obtained with data for items 1 and 2 only. However, as noted by Deriso et al. (1985), the earlier work of Doubleday (1976) and Pope (1977) showed that catch-at-age data alone are insufficient to estimate abundance reliably because estimated fishing mortality and estimated abundance are negatively correlated. 177 Model Assumptions - CAGEAN Deriso et al. (1985) list six assumptions for the model (see page 818). 1. Catch is given by Baranov catch equation times a lognormal random variable. 2. Natural mortality, M, is fixed at known value. 3. Fishing mortality is separable into an age-dependent factor (selectivity) and a year-dependent factor (full recruitment fishing mortality). 4. Full recruitment fishing mortality is proportional to fishing effort times a log-normal random variable. 5. Recruitment is given by a Ricker spawner-recruit function times a log-normal random variable. 6. The variance ratios, Xr and X9 are fixed at known values. Assumptions 1 and 2 are shared with VPA. Implicit in assumption 1 are the closed population assumption and "equal odds of mortality" assumption of VPA. Although violations of these first two assumptions have not been investigated in CAGEAN, similar conclusions to those found for VPA could be expected. The procedure of Fournier & Archibald (1982) permits estimation of M, but the accuracy of the estimate is hard to evaluate with actual rather than simulated data. The validity of the separability assumption rests on selectivity-at-age remaining constant over time. If there are known changes in selectivity due to changes in gear, then CAGEAN permits division of the data into separate selectivity estimation periods. Accounting for more subtle changes, for example, 178 due to changing market conditions, would be more difficult. The last three assumptions concern the auxiliary data. The key to meeting assumption 4 is a standardized effort time series, such that is proportional to effort. Note however, that CAGEAN permits variability about equation 7.10, with the amount of variability depending on the choice of X^. As with selectivity, CAGEAN permits the division of the time series into "catchability periods" if changes have occurred that cannot be accounted for by standardization. CAGEAN also permits flexibility in choice of the stock recruitment function in assumption 5, but modifications to the computer program are required. The effect of assumptions 4 and 5 on parameter estimates depends on the values assumed for the variance ratios, X^ and X^. Large values (e.g. 1000) for the variance ratios mean that equations 7.10 and 7.12 hold exactly while small values (e.g. 0.0001) allow large variability to occur about the relationships and greater reliance on the catch data for parameter estimates. Deriso et al. (1985) examined the sensitivity of parameter estimates to X values using data from Pacific halibut and VPA estimates for complete cohorts as a benchmark. They found errors in estimates were reduced when medium or large values were used when compared to errors from estimates using no auxiliary information. Since the choice of X values is arbitrary, experimenting with different values is warranted. Application of Models and Specific Methods Computer Programs and Starting Input Parameters Two computer programs were used for the estimation. VPA estimates were obtained from CATANAL (Walters 1987a) and the program CAGEAN 179 (version 2.0) was used in the . CAGEAN estimates. Both programs were implemented on a IBM compatible microcomputer. CATANAL was obtained from C. Walters, Resource Ecology, Univ. British Columbia, Vancouver, B.C. V6T 1W5 and CAGEAN was obtained from P. Neal, International Pacific Halibut Commission, P.O. Box 95009, Seattle, WA 98145-2009. Starting natural mortality (M) and catchability (q) values for VPA were estimated by CATANAL using the regression technique of Paloheimo (1980). Estimates were highly variable depending on the range of ages included in the regression. I selected M and q estimates that were consistent with published values (Westrheim and Foucher 1985, Fargo 1985, Walters et al. 1982) for lack of any independent reliable estimates. Terminal fishing mortalities (Fj&) for the initial runs of VPA were calculated by CATANAL from Fy,=qE^,. Subsequent VPA iterations use average age specific q's and effort to estimate terminal F values for incomplete cohorts with the same equation. I repeated iterations (2-4 times) until average age specific q's estimated in successive runs converged (i.e. the iterative method of Rivard 1983). This eliminates the possibility of the spurious time trends discussed by Bradford & Peterman (1989). Estimates of M and terminal F used in VPA were also supplied to an initialization routine (i.e. COHORT) in CAGEAN that calculates starting parameter values for the nonlinear estimation. Initial parameter values for each species are shown in Appendix A (Tables A6-A7.). Data for the same ages and years were used in both techniques, but older ages were pooled in CAGEAN to reduce possible affects of aging errors. The ages pooled were ages 8-10 for cod, ages 11-17 for English sole and ages 11-15 for rock sole. Age specific selection coefficients were estimated for ages 2,3 for cod, ages 3,4,5 and 11-17 (as a group; a total of 4 180 S s) for English sole and ages 4,5 and 11-15 (a total of 3 S s) for rock sole. d a For a table of the catch-at-age data and starting parameters for each species, see Appendix A (Tables A1-A7). Estimates of Abundances and Catchabilities Abundance estimates for VPA came directly from the VPA equations. For CAGEAN, abundance estimates for older ages were calculated from equation 7.4 using the estimates of recruitment abundances, fishing mortalities in each year, and the assumed M value. Full-recruit fishing mortalities were multiplied by the S values for all ages with S s < 1.0. a ° a Catchability coefficients for VPA were estimated for each year t using the equation LCtM (7.14) where ZC f , Z N f are the catches and abundances summed over fully recruited ages in year t (i.e. ages 4-10 for cod; 6-10 for English sole and rock sole), and LNt+j is the abundance summed over the first fully recruited age + 1 to the last fully recruited age +1 for each species in year t+1. Thus, VPA can only estimate catchabilities for T- l years because Z N f + ^ is not available for the last year. I used fully recruited ages in the VPA calculations so that the catchability estimates could be compared with those obtained from CAGEAN (see below). CAGEAN's estimates one catchability coefficient that represents the relationship of F^ to E f as determined by equations 7.10, 7.11. In order to estimate annual catchability coefficients, I rearranged equation 7.10 yielding 181 (7.15) Note that dividing F by is equivalent to multiplying the average q estimated by CAGEAN by e€t where are the residuals from equation 7.11. Sensitivity Analyses I estimated the sensitivity of each method to alternative starting parameter values by comparing estimates of abundances and catchability coefficients obtained from the "best" estimates of input parameters ±50% for each species. Thus, I compared VPA estimates obtained from Mg (subscript B denotes the best estimate) with estimates obtained from M^ + 50% and M^-50%, and similarly for q^ and q^±50%. I made the same comparisons for CAGEAN estimates obtained from M^ , M^ + 50%, and Mg-50% and Fg, F^ + 50%, and F^-50%. I used the alternative input parameters, M and F and the rountine COHORT to generated different starting values for the nonlinear estimation. These starting values, along with M were then input into CAGEAN while holding \j and X^ fixed at 0.5. I also investigated the sensitivity of CAGEAN estimates to the choice of X values. I compared estimates obtained from five alternative X value combinations; (1) X r X 2 = 0, (2) X ,^ X 2 = 0.5, (3) X ; , X 2 = 100.0, (4) X^-0, X ^ l . O , and (5) X^  = 1.0, X£ = 0. I selected these five combinations because they represent the following five qualitative uses of the auxiliary data: (1) no auxiliary data, (2) a moderate amount of auxiliary data (combined weight same as catch data), (3) heavy reliance on the auxiliary data, (4) auxiliary spawner-recruit data only, and (5) auxiliary effort data only. In my comparisons of estimates obtained from different X values, I used the 182 M D and F D estimates for each species. Finally, I compared the estimates of abundances and catchabilities obtained from the CAGEAN and VPA. In this comparison, I used the best estimates of the input parameters and X^, X2 = 0.5 because the previous sensitivity analyses within methods gave estimates that were, highly correlated and showed very similar temporal patterns. Results "Best" Estimates - VPA and CAGEAN i Estimates of abundance obtained from VPA and CAGEAN using the best estimates of input parameters are shown in figure 7.4. Estimates from the two methods are very similar and a more detailed comparison is presented below. All three species show cyclical fluctuations in estimated abundances over time (Fig. 7.4). Cod is the most abundant species with a peak abundance in 1964 and a trough in 1970. A second peak in cod abundance is evident in the mid to late 1970's although it is much less pronounced than the earlier peak (Fig 7.4). Based on pre-1960 LPUE data and the fact that cod abundance appears to have increased in recent years (Fargo and Tyler 1989), troughs in cod abundance have also occurred in the late 1950's and early-mid 1980's. Thus, for the 40-50 yrs for which there are data, cod abundance has fluctuated cyclically with a period of 9-11 yrs. The cause of the cyclical trends is unknown, but studies have found that cod recruitment rates are positively correlated with the abundance of herring in Hecate Strait (Walters et al. 1986) and with above average late winter water temperatures (Tyler and Westrheim 1986) and negatively correlated with northward water-mass transport (Tyler and Westrheim 1986). 183 Pac i f i c cod U3 o X CD O o a Z3 •<. "a ~o 61 6'3 ' 65 ' 67 " 69 ' 71 ' 73 ' 75 " ll ' 79 " 81 ' 83 ' 85 Engl ish sole 56 ' 58 ' 60 ' 6'2 ' 64 ' 6'6 ' 6'8 ' 70 ' 72 ' 74 ' 76 ' 78 rock sole 56 " 58 ' 6'0 ' 6'2 ' 6'4 ' 6'6 ' 6'8 ' 70 ' 72 ' 74 ' 76 ' 78 Year Figure 7.4. Estimates of total abundance of Pacific cod (ages 2-10), English sole (females ages 3-17) and rock sole (females ages 4-15 and northern stock only) in Hecate Strait obtained from VPA and CAGEAN using best estimates of input parameters. Note: The legend in the plot for Pacific cod applies to all three species. 184 The abundances of English sole and rock sole females are similar in magnitude and show very similar time trends (Fig. 7.4). Peaks in English sole abundance occurred in the late 1950's, 1965 and 1974, and troughs occurred in 1962 and 1971. Peaks and troughs in rock sole abundance appear to lag about 1 year behind the peaks and troughs in English sole abundance. As with cod, the causes of the cyclical trends in abundance of the two flatfish species are unknown, but recruitment of rock sole is positively correlated with water temperature (Fargo 1985). Patterns in VPA and CAGEAN estimates of catchability coefficients were similar to abundance estimates (Fig. 7.5). Cod catchability coefficients were generally smaller than catchability coefficients for either of the flatfish. Peaks of cod catchability occurred in 1962, 1966 (VPA; 1967 CAGEAN) 1973, and 1983 (CAGEAN) and troughs occurred in 1964 (VPA; 1965 CAGEAN), 1968 (VPA; 1970 CAGEAN), 1974 (VPA), and 1980 (VPA; 1978 CAGEAN; Fig. 7.5). The estimated catchability of English sole shows a major peak in 1960, and minor peaks in 1970 and 1975. Troughs in English sole catchability occurred in 1966 and 1974 (VPA; 1972 CAGEAN) (Fig. 7.5). Rock sole catchability.. estimates fluctuated rather violently in the first two thirds (i.e. 1956-69) of the time series, when peaks occurred in 1958, 1961 and 1967 and troughs occurred in 1959, 1965, 1969 (VPA; 1968 CAGEAN). Fluctuations in estimated catchability of rock sole are damped in the latter part of the time series (i.e. 1970-1977) with peaks in 1970-71, and 1975 and a trough in 1974 (Fig. 7.5). 185 Pacific cod x cr " o C D O o a o " a o 2 -Legend VPA CAGEAN 61 ' 63 ' 65 " 67 ' 69 ' 71 '. 73 ' 75 ' 77 ' 79 ' 81 ' 83 ' 85 English sole 56 ' 58 " 60 ' 6'2 ' 6'4 ' 66 ' 6'8 ' 70 " 72 ' 74 ' 76 ' 78 rock sole 56 ' 5'8 ' 6'0 ' 6'2 ' 64 ' 66 ' 68 '. 70 ' 72 ' 74 ' 76 ' 78 Year Figure 7.5. Estimates of catchability coefficients for Pacific cod (ages 4-10), English sole (ages 6-10) and rock sole (ages 6-10) obtained from VPA and CAGEAN using best estimates of input parameters. Note: The legend in the for Pacific cod applies to all three species. 186 Sensitivity Analyses - VPA Alternative M Values Estimates of abundance for the three species obtained from VPA using Mg, M ^ + 50%, and M^-50% are presented in figure 7.6. Abundance estimates were greater for M ^ + 50% and less for M^-50% than for Mg for all three species (Fig. 7.6). However, within species abundance estimates for all M values were highly correlated (Table 7.1). Thus, while the use of different M values affected the magnitude of abundance estimates, it resulted in similar time trends (Fig. 7.6). Catchability estimates were generally less affected by alternative M values than abundance estimates because catchabilities were estimated for fully recruited ages only, and therefore they were not affected by the abundance estimates for younger ages that are more biased when M is incorrect (Lapointe et al. 1989; Fig. 7.7). Catchability estimates were less for Mg + 50% and greater for M^-50% than for Mg for all three species (Fig. 7.7) because for any given year the abundance and catchability are inversely proportional to each other (i.e. via eq. 7.14). As with abundance estimates, within species estimates of catchability for all M values were highly correlated (Table 7.2). Thus, the use of different M values resulted in similar time trends in the catchability estimates (Fig. 7.7). My results are consistent with past studies which have investigated errors in M (e.g Agger et al. 1973; Bradford and Peterman 1989). However, I did not find different time trends in incomplete cohorts when different M values were used as documented by Bradford and Peterman (1989), because I used iterative VPA (Rivard 1983). 187 Pacific cod O X OJ o d a T J cr _Q O " a 61 63 65 67 69 71 73 75 77 79 81 83 85 English sole M=0.33 M=0.22 M=0.1 56 58 60 62 64 66 68 70 72 74 76 78 Year Figure 7.6. Estimates of total abundance of Pacific cod (ages 2-10), English sole (ages 3-17) and rock sole (ages 4-15) obtained from VPA using alternative M values. M values in legend for each species are Mg+50% (top), Mg (middle) and Mg-50% ft>ottom). 188 Table 7.1. Product-moment correlation matrices of abundance estimates obtained from VPA using alternative M values (M^, M^ + 50%, and MD-50%) for Pacific cod, English sole and rock sole. Note: r 0.05, and r 0.01 are the values of the correlation coefficient necessarj' for significance at P=0.05 and P=0.01, respectively. Pacific cod M value r value 0.68 1.0 1.02 .81 1.0 0.34 .95 .62 1.0 0.68 1.02 0.34 n=25, df=23, r 0.05 = 0.40, r 0.01 = 0.50  English sole M value r value 0.22 1.0 0.33 .92 1.0 0.11 .96 .78 1.0 0.22 0.33 0.11 n = 24, df=22, r 0.05 = 0.40, r 0.01 = 0.52 rock sole M value r value 0.25 1.0 0.375 .98 1.0 0.125 .99 .96 1.0 0.25 0.375 0.125 n = 23, df=21, r 0.05 = 0.41, r 0.01 = 0.53 Alternative Terminal q Values The choice of alternative terminal q values had very little affect on magnitude of estimates of abundances or catchability coefficients for any of the three species (Figs. 7.8, 7.9), and within species estimates from the three 189 Pacific cod X CD o o O SZ o a o 63 65 69 71 73 75 English sole 85 58 60 Year Figure 7.7. Estimates of catchability coefficients for Pacific cod (ages 4-10), English sole (ages 6-10) and rock sole (ages 6-10) obtained from VPA using alternative M values. M values in legend for each species are M g + 50% (top). M f i (middle) and Mg-50% (bottom). 190 Table 7.2. Product-moment correlation matrices of catchability estimates obtained from VPA using alternative M values (Mg, Mg + 50%, and Mg-50%) for Pacific cod, English sole and rock sole. Pacific cod M value r value 0.68 1.0 1.02 .95 1.0 0.34 .97 .86 1.0 0.68 1.02 0.34 n = 24, df=22, r 0.05 = 0.40, r 0.01 = 0.52  English sole M value r value 0.22 1.0 0.33 1.0 1.0 0.11 1.0 .99 1.0 0.22 0.33 0.11 n = 23, df=21, r 0.05 = 0.41, r 0.01 = 0.53  rock sole M value r value 0.25 1.0 0.375 1.0 1.0 0.125 .98 .98 1.0 0.25 0.375 0.125 n = 22, df=20, r 0.05 = 0.42, r 0.01 = 0.54 alternative q values were very highly correlated (all rs>0.99, P<0.01). I found little affect of the choice of alternative q values because these values only enter into the VPA calculations in the estimation of the abundances of the oldest age class in each year (i.e. N s in eq. 7.5) and in almost all years the catches of the oldest age class (i.e. C^ s in eq. 7.5; C^s in Fig. 7.3) are zero for all three species (see Appendix A; Tables A1-A3). Thus, regardless of the choice of terminal q value, the estimate of abundance of the oldest age class will also be 191 Pacific cod 30 • 10 o OJ o c a -a c Z2 a ~o "o 10 8 6 -4-2 0 10. 8. 2 0 q=2.89X10 q=5.79X10 q=8.68X10 -5 69 71 73 75 English sole q=4.09X10 q=6.14X10 56 58 60 62 64 66 68 rock sole 70 72 74 76 78 q=2.59X10"5 q=5.17X10~5 q=7.76X10"5 56 58 60 62 64 66 68 Year 70 72 74 76 78 Figure 7.8. Estimates of total abundance of Pacific cod (ages 2-10), English sole (ages 3-17) and rock sole (ages 4-15) obtained from YPA using alternative terminal q values. The q values in legend for each species are q^ + 50% (top), q-Q (middle) and qg-50% Obottom). 192 Pacific cod x cr CD o o a o o o q=2.89X10 1=5.79X10" 63 65 67 69 71 73 75 77 79 81 83 85 English sole q=2.04X10 q=4.09X10" q=6.14X10" -5 56 ' iB. ' 60 ' 6'2 " 64 ' 66 ' 68 ' 70 ' 12 ' U ' 76 ' 78 rock sole q=2.59X10" q=5.17X10" q=7.76X10" T— 1 T 1" • 1 T 1 1 r r- * " T ' 56 58 60 62 64 66 68 70 72 74 76 78 Year Figure 7.9. Estimates of catchability coefficients for Pacific cod (ages 4-10), English sole (ages 6-10) and rock sole (ages 6-10) obtained from VPA using alternative terminal q values. The q values in legend for each species are q_, + 50% (top), q^ (middle) and q^-50% (bottom). D D D 193 zero. These results are consistent with Pope (1972) who found that VPA abundance estimates are insensitive to the choice of terminal fishing mortality when cumulative fishing mortality on the cohort is large because large cumulative fishing mortality results in few fish surviving to, and being caught in, the oldest age class. Sensitivity Analyses - CAGEAN Alternative M Values The affect of alternative M values on CAGEAN abundance estimates was similar to that found for VPA estimates; abundance estimates were greater for Mg + 50% and less for M^ -50% than for Mg for all three species (Fig. 7.10), and within species estimates for all M values were highly correlated (Table 7.3). Thus, the use of different M values resulted in similar time trends in the CAGEAN abundance estimates (Fig. 7.10). CAGEAN catchability estimates (Fig. 7.11) were generally less affected by alternative M values than abundance estimates because catchability estimates were not affected by abundance estimates for partially recruited younger ages which are more biased when M is incorrect (Lapointe et al. 1989). Catchability estimates were less for M^ + 50% and greater for Mg-50% than for for all three species (Fig. 7.11) because for any given year abundance and fishing mortality are negatively correlated (i.e. via eq. 7.4) and fishing mortality is proportional to catchability (i.e. via eq. 7.15). As with abundance estimates, within species estimates of catchability for all M values were highly correlated (Table 7.4), and different M values resulted in similar time trends in the catchability estimates. 194 90 Pacific cod M=1.02 M=0.68 M=0.34 English sole 56 " 58 ' 60 ' 62 " 64 ' 66 ' 68 ' 7'0 ' 72 ' U ' 76 78 13 rock sole 0 M=0.375 M=0.25 M=0.125 56 58 60 62 64 66 68 70 72 74 76 78 Year Figure 7.10. Estimates of total abundance of Pacific cod (ages 2-10), English sole (ages 3-17) and rock sole (ages 4-15) obtained from C A G E A N using alternative M values. M values in legend for each species are Mg + 50% (top), M f i (middle) and Mg-50% (bottom). 195 Table 7.3. Product-moment correlation matrices of abundance estimates obtained from CAGEAN using alternative M values ( M D , M D + 50%, and ts ts M ^ -50%) for Pacific cod, English sole and rock sole. M value Pacific cod r value 0.68 1.02 0.34 1.0 .91 .96 0.68 1.0 .78 1.02 1.0 0.34 n = 25, df=23, r 0.05 = 0.40, r 0.01 = 0.50 M value English sole r value 0.22 0.33 0.11 1.0 1.0 1.0 0.22 1.0 .99 0.33 1.0 0.11 n = 24, df=22, r 0.05 = 0.40, r 0.01 = 0.51 rock sole M value r value 0.25 1.0 0.375 .95 1.0 0.125 .98 .87 1.0 0.25 0.375 0.125 n = 23, df=21, r 0.05 = 0.41, r 0.01 = 0.53 Alternative Terminal F Values The magnitude of, and trends in, CAGEAN estimates of abundances or catchability coefficients for all the three species were insensitive to the choice of terminal F values (Figs. 7.12, 7.13; all rs>0.99, P<0.01). However, the underlying reason for the lack of effect of alternative F values on CAGEAN estimates is different from that for VPA estimates. Recall that the alternative 196 Pacific cod -4-I X CT " o o o a o o 63 65 67 69 71 73 75 77 79 81 83 85 English sole M=0.11 M=0.22 M=0.33 Figure 7.11. Estimates of catchability coefficients for Pacific cod (ages 4-10), English sole (ages 6-10) and rock sole (ages 6-10) obtained from CAGEAN using alternative M values. M values in legend for each species are Mg + 50% (top), Mfi (middle) and Mg-50% (bottom). 197 Table 7.4. Product-moment correlation matrices of catchability estimates obtained from CAGEAN using alternative M values (M^, Mg + 50%, and Mg-50%) for Pacific cod, English sole and rock sole. Pacific cod M value r value 0.68 1.0 1.02 .90 1.0 0.34 .97 .78 1.0 0.68 1.02 0.34 n = 25, df=23, r 0.05 = 0.40, r 0.01 = 0.50 English sole M value r value 0.22 1.0 0.33 1.0 1.0 0.11 1.0 .98 1.0 0.22 0.33 0.11 n = 24, df=22, r 0.05 = 0.40, r 0.01 = 0.51 rock sole M value r value 0.25 1.0 0.375 .99 1.0 0.125 1.0 .97 1.0 0.25 0.375 0.125 n = 23, df=21, r 0.05 = 0.41, r 0.01 = 0.53 terminal F values were used in. CAGEAN's routine called COHORT which calculated alternative starting parameters for the nonlinear estimation. Thus, CAGEAN estimates were insensitive to the choice of terminal F values because the nonlinear estimation converged on nearly identical parameter estimates over the range of alternative input parameters generated by F^±50%. Thus, unlike VPA, CAGEAN is robust to the choice of terminal F value regardless of the cumulative fishing mortality on cohorts, unless these terminal F values are 198 Pacific cod to O X cu o c o Z3 _Q O "a ~o 61 8 6 4 2 ^ 0 -t- F=0.205 a F=0.41 • F=0.615 63 65 67 69 71 I I 1 — 73 75 77 79 81 83 85 English sole F=0.175 a F=0.35 4* F=0.525 56 58 60 62 64 66 rock sole 68 70 72 74 76 78 F=0.09 F=0.18 F=0.27 Figure 7.12. Estimates of total abundance of Pacific cod (ages 2-10), English sole (ages 3-17) and rock sole (ages 4-15) obtained from CAGEAN using alternative F values. F values in legend for each species are Fg + 50% (top), Fg (middle) and Fg-50% (bottom). 199 Pacific cod 0.5H x cr F=0.205 F=0.41 F=0.615 I I r - T "1 1 I 1 o 61 63 65 67 69 71 73 75 77 79 81 83 85 English sole F=0.175 a F=0.35 •» F=0.525 -4- F=0.09 = F=0.18 F=0.27 Year Figure 7.13. Estimates of catchability coefficients for Pacific cod (ages 4-10), English sole (ages 6-10) and rock sole (ages 6-10) obtained from C A G E A N using alternative F values. F values in legend for each species are F^ + 50% (top), F D (middle) and Fg-50% ftx>ttom). 200 different enough to cause the nonlinear estimation to converge on different local minima. Alternative X Values Cagean estimates of abundances obtained from the five alternative X values combination are shown in figure 7.14. The magnitude of, and trends in abundance estimates were generally insensitive to the range of X value combinations examined with the exception of X^  = X2=100 for all three species and Xjj = Xg = 0 for the two flatfish species (Fig. 7.14; Table 7.5). The use of X^  = X2 = 100 resulted in greatly damped fluctuations in abundance for all three species in comparison to other X value combinations (Fig. 7.14). Abundance estimates obtained when X^  = X2 = 0.5 were intermediate in magnitude for all three species compared to estimates obtained using other X value combinations (Fig. 7.14). Cod abundance estimates were least sensitive to the range of X values explored (Fig. 7.14); estimates from all X combinations were highly correlated; all r s>0.70, P<0.01; Table 7.5). Flatfish abundance estimates were unrealistically high later in the time series when X ^ X ^ O (Fig. 7.14). Furthermore, these estimates were not significantly correlated with estimates from other X value combinations in either species, with the exception that rock sole estimates for \j = X^  = 0 were significantly negatively correlated with estimates obtained when X^  = X,,= 100 (Table 7.5). Intermediate weights on auxiliary data (i.e. X^  = X2 = 0.5, \j = l, X^  —0, or X^  = 0, X^  = 1) yielded estimates that were very highly correlated for both flatfish species (all rs>0.96, P<0.01 for English sole; all rs >0.89, P<0.01 for rock sole; Table 7.5). 201 Pacific cod 56 58 60 62 64 66 68 Rock sole 70 72 74 76 78 Figure 7.14. Estimates of total abundance of Pacific cod (ages 2-10), English sole (ages 3-17) and rock sole (ages 4-15) obtained from CAGEAN using alternative X values. Note: Legend in plot for rock sole applies to all species. 202 Table 7.5. Product-moment correlation matrices of abundance estimates obtained from CAGEAN using alternative X values for Pacific cod, English sole and rock sole. Pacific cod Xvalue r value xrx2=o 1.0 XJ,X2 = 0.5 .97 1.0 XrX2 = 100 .70 .76 1.0 X2 = 0X2 = 1 .72 .86 .73 1.0 Xi = lX2 = 0 1.0 .98 .71 .73 1.0 xrx2=o xrx2=o.5 xrx2=ioo x2=ox2=i x; = ix2=o n = 25, df=23, r 0.05 = 0.40, r 0.01 = 0.50 English sole Xvalue r value xrx2=o 1.0 X r X 2 = 0.5 .05 1.0 xrx2=ioo -.35 .62 1.0 x2=ox2=i .11 .99 .60 1.0 xJ = ix2=o .24 .98 .53 .97 1.0 xrx2=o X r X 2 = 0.5 xrx2=ioo \- ox2=i X7 = ix2=o n = 24, df=22 , r 0.05 = 0.40, r 0.01 = 0.51 rock sole Xvalue r value xrx2=o 1.0 X r X 2 = 0.5 -.24 1.0 x7,x2=ioo -.60 .69 1.0 x2=ox2=i .15 .92 .43 1.0 xi = ix2=o -.27 1.0 .71 .90 1.0 xrx2=o X r X 2 = 0.5 x;,x2=ioo h = ox2=i ix2=o n = 23, df=21, r 0.05 = 0.41, r 0.01 = 0.53 203 The unrealisticly high abundance estimates found for flatfish when no auxiliary data were used are consistent with previous studies (e.g. Doubleday 1976; Pope 1977; Deriso et al. 1985) that showed how catch-at-age data alone give unstable estimates. The problem is particularly acute later in the time series when cohorts are incomplete, because there are fewer catch-at-age observations per cohort. Large X values resulted in damped fluctuations in abundance estimates in all three species for three reasons. First, when X^ is large, CAGEAN predicts that q is constant because a large penalty is attached to effort residuals in equation 7.11. Second, with a constant q, fluctuations in F^ s depend solely on fluctuations in E s^ yet effort has not fluctuated that much for any of the species over the data series. Thus, the Fj3 are relatively constant and moderate for all species (i.e. average F=0.46 ±0.05SE for cod; 0.46 ±0 .06SE for English sole; 0.38 ±0 .06SE for rock sole). Third, when X^ is large, recruitment is a deterministic function of spawning stock because of the large penalty attached to spawner-recruit residuals in equation 7.13 and fluctuations in recruitment due to spawning stock variations are relatively small. Thus, when Xj and X^ are large, CAGEAN generates abundance estimates that are the same as would be predicted from a deterministic population model with moderate exploitation rates. The patterns in the sensitivity of catchability estimates were similar to those found for abundances (Fig- 7.15). Catchability estimates obtained when X^  = X 2 = 0.5 were intermediate in magnitude for all three species compared to estimates for other X value combinations. When X.^ = X , , = 100, estimates of catchability coefficients were nearly constant for all years and species. Constant catchability is expected when X^  is large because under such conditions CAGEAN converges to the constant catchability model of Paloheimo (1980; Deriso et al. 204 Pacific cod ' o X c r cu o cu o o D xz o o O English sole Legend X1=0.X2=0 X1=.5.X2=5 X1=100,X2=100 X1=0,X2=1 X X1=1,X2=0 rock sole -r f <* o 76 78 Figure 7.15. Estimates of catchability coefficients for Pacific cod (ages 4-10), English sole (ages 6-10) and rock sole (ages 6-10) obtained from CAGEAN using alternative X values. Note: Legend in plot for English sole applies to all species. 205 1985). Cod catchability estimates for all X value combinations except the constant catchability model (i.e. X^ = X,, = 100) were highly correlated (all rs>0.81, P<0.01; Table 7.6). The estimates of catchability coefficients for both flatfish species were unrealistically low during the years that abundance estimates were unrealistically high because of the negative correlation between abundance estimates and fishing mortalities (hence q's; Figs. 7.14 and 7.15). However, English sole catchabilities estimated from all X value combinations were highly correlated ( ra0.79, P<0.01; Table 7.6). Correlations among rock sole catchability estimates obtained from different X value combinations were lowest for X^ = X2 = 100 and highest for intermediate X values (all rs> 0.89, P<0.01; Table 7.6). In summary, estimates of abundances and catchabilities were generally not sensitive to the alternative X value combinations when intermediate values were used. Catchability and abundances estimates obtained when Xj^X^ — O.S were intermediate compared to estimates obtained from other X values. Comparison of Estimates from VPA and CAGEAN Estimates of abundances and catchabilities obtained from VPA and CAGEAN using the best parameter estimates were generally very similar (Figs. 7.4 and 7.5). Cod abundance estimates from VPA were highly correlated with CAGEAN abundance estimates (r =0.92, p <0.01). Cod catchability estimates obtained from the two methods were less highly correlated than abundance estimates (r=0.68, P<0.05). Cod catchability estimates from the two techniques were particularly different early and late in the time series (Fig. 7.5). 206 Table 7.6. Product-moment correlation matrices of catchability estimates obtained from CAGEAN using alternative X values for Pacific cod, English sole and rock sole. Pacific cod X value r value xrx2=o 1.0 XJ,X2 = 0.5 .93 1.0 XrX2=100 .67 .69 1.0 XJ=0X2=1 .85 .97 .63 1.0 Xi = lX2 = 0 .99 .91 .70 .82 1.0 xrx2=o xrx2=o.5 xrx2=ioo x2=ox2=i x2 = ix2=o n = 25, df=23, r 0.05 = 0.40, r 0.01 = 0.50 English sole X value r value xrx2=o 1.0 XrX2 = 0.5 .95 1.0 xrx2=ioo .79 .88 1.0 x,=ox2=i .95 1.0 .88 1.0 x2 = ix2=o .95 1.0 .88 1.0 1.0 xrx2=o XrX2 = 0.5 xrx2=ioo ox2=i ix2=o n=24, df=22, r 0.05 = 0.40, r 0.01 = 0.51 rock sole X value r value xrx2=o 1.0 XrX2 = 0.5 .96 1.0 xrx2=ioo .56 .69 1.0 x,=ox2=i .97 1.0 .66 1.0 xi = ix2=o .89 .98 .77 .96 1.0 xrx2=o XrX2 = 0.5 xj;x2=ioo X2 = ox 2=i ix2=o n=23, df=21,r 0.05 = 0.41,r 0.01 = 0.53 207 English sole abundance estimates from VPA and CAGEAN were very similar (r=0.66, P<0.05), and catchability estimates obtained from the two techniques were highly correlated (r=0.97, P<0.01). Both catchability and abundance estimates obtained from the two methods were highly correlated for rock sole (r=0.96, P<0.01 for q; r = 0 . 8 9 , P<0.01 for N). Neither method was particularly more sensitive to alternative input parameters. The magnitude of estimates from both methods were sensitive to alternative M values but insensitive to alternative F values (CAGEAN; q values - VPA) for different reasons. In general, I would expect CAGEAN estimates to be least sensitive to alternative F values, particularly when auxiliary effort data are used, because CAGEAN is a non-sequential technique and F values are estimated from fits to the catch-at-age and effort data. Thus, errors in terminal F values do not propagate back through the cohorts in CAGEAN as they do in VPA. S u m m a r y o f R e s u l t s a n d C o n c l u s i o n s The sensitivity analyses have demonstrated that the magnitude of abundance and catchability estimates for all three species depends on the choice of M value (VPA & CAGEAN) and the X values (CAGEAN only), but is insensitive to the choice of F^  values (CAGEAN) or qf values (VPA). However, in most cases alternative input parameters resulted in abundance and catchability estimates that were highly correlated and had very similar time trends. Thus, in the context of my analysis of functional responses which follows in chapter 8, it is unlikely that the use of abundance and catchability estimates obtained from alternative input parameters using either technique would result in 208 different qualitative forms of functional responses. However, because of the affect of alternative input parameters on the magnitude of abundance and catchability estimates obtained from VPA and CAGEAN, the magnitude of particular parameters for a given functional response model (e.g. a and /3, in Fox's (1974) model) will depend on the mortality rates and/or X values used. Therefore, my abilities to address specific quantitative consequences of functional responses are limited. Similar temporal trends in abundance and catchability estimates were obtained from VPA and CAGEAN using the best estimates of input parameters, but the correlations were some what lower than those found for the sensitivity analyses for each method. Another source of potential differences in functional responses derived from abundance estimates using VPA and CAGEAN is that CAGEAN provides estimates of predicted catches. Thus in chapter 8, I will examine the implications of the alternative methods (i.e. VPA vs. CAGEAN) for the assessment of functional responses of trawlers in Hecate Strait. 209 Chapter 8: Functional Responses of Trawlers Introduction In this chapter I examine the functional responses of trawlers to the abundance of three species that are caught in a multispecies fishery located in Hecate Strait. Most of the examinations of functional responses in fisheries reviewed in Chapter 6 involved "single" species fisheries. One purpose of my study was to evaluate possible alternative functional response models and consider their implications in a multispecies situation. Thus, in order to put my research in perspective, I will briefly highlight the major areas of multispecies fisheries research before proceeding with further details on the present study. Many aspects of multispecies fisheries have been examined, and the investigations can be divided into four main categories. First, a few studies have identified multispecies effort patterns by examining temporal and spatial patterns in' species composition of catches (e.g. Tyler et al. 1984; Murawski et al. 1983; Biseau and Gondeaux 1988). The main purposes of these studies are to identify assemblages of species that could potentially be managed as a group (e.g. Tyler et al. 1982), or alternatively to identify subfleets for determining components of bioeconomic models (e.g. Biseau and Gondeaux 1988). The second area of research, concerns the development of techniques for determining the effort for single species that are caught with other species (e.g. Westrheim 1983; Stocker and Fournier 1984) or that compete with other species for space on (or in) fishing gear (e.g. Rothschild 1967). Single species effort is needed primary for calculation of abundance indices (i.e. LPE) that are commonly 210 the only available short term indicators of the response of a population to fishing. Papers concerned with the multispecies assessment of fish populations fall into the third category. The impetus behind the development of these methods is that interactions among species (i.e. predation and competition; e.g. Sissenwine et al. 1982) may alter the growth and natural" mortality rates of fish populations (particularly of younger ages classes) and therefore these interactions should be included in assessment models. Thus researchers have developed multispecies VPA models (e.g. Daan 1987) and multispecies yield per recruit models (e.g. Shepherd 1988). One of the drawbacks of multispecies models is that they require data that are often either unavailable or quite scarce (Ursin 1982). The fourth category includes a variety of papers that investigate alternative harvest or management strategies for multispecies fisheries. This area of research can be divided into a few subcategories. One subcategory includes papers that consider the theoretical effects of species interactions on potential yields (e.g. Horwood 1976; Pope 1976). Another subarea of research investigates the theoretical implications of exploiting stocks or species with different sustainable harvest rates (e.g Paulik et al. 1967; Hilborn 1976). A third subarea includes papers that use one of the multispecies assessment techniques or other techniques (e.g. linear programming) to consider alternative management options or to determine optimal policies for particular multispecies fisheries (e.g. Overholtz 1985; Murawski and Finn 1986; Pikitch 1987). Despite the broad areas covered by multispecies research, there are very few studies that consider the responses of fishermen in a multispecies system, beyond a description of patterns in effort allocation among species (e.g. Tyler et 211 al. 1984). Specifically, one aspect that to my knowledge has not been investigated is the functional response of fishermen in a multispecies setting. My investigation of functional responses has three main parts. Part 1 evaluates three alternative single species models for functional responses and mortality curves (e.g. Figs. 5.1 and 5.4) for each of the three species that are the major components of the multispecies fishery in Hecate Strait. Part 2 examines two multispecies functional response models that take into account the effects of the abundance of alternative prey on the functional response. Finally, in part 3, I present tests of Murdoch's (1969) switching hypothesis to examine a potential mechanism for generating type III functional responses by fishermen. Data and Methods Data Sets In order to examine functional responses in the Hecate Strait trawl fishery, I required estimates of LPE (i.e. NA/T in the functional response equations of Chapter 5) and abundance for each of the three species. I used estimates of abundance obtained from VPA and CAGEAN using the "best" estimates of input parameters as presented in Chapter 7. For VPA, I estimated LPE by L/E where L is the total catch landed (i.e. estimated total numbers landed) and E is the total fishing effort (hours trawled) as described in Chapter 7 (see section "Data Preparation and Background Methods"). Note that hours trawled is analogous to pursuit time in a natural predator (i.e. not T); it does not include time spent searching (while not trawling) for trawl sites, time spent moving between trawl sites or time spent handling catches. Hecate Strait trawlers may search some portion of the time while trawling because most of 212 the trawl sites are well known and some species, particularly flatfish are difficult to detect on the echo sounders. However, I do not expect to find evidence for handling time effects (i.e. saturation in the functional response), since handling time is not included in the effort measure. My main reason for chosing hours trawled allotted by species as the effort measure was that it is the effort measure used by fisheries biologist and managers and I wanted to consider the potential implications of functional responses on assessment and management. In order to maintain consistency between the catches and abundances obtained from CAGEAN, I used the catches predicted by equation 7.7 and the same effort data used for VPA to calculate the LPE for CAGEAN. In my fitting of the mortality curves (e.g. Figs. 5.4, 6.2b; catchability vs. abundance), I used estimates of catchability coefficients obtained from VPA (by eq. 7.14) and CAGEAN (by eq. 7.15). In addition to using estimates obtained from the two catch-at-age methods, I also considered two age ranges for each species and method: (1) all fished ages and (2) fully recruited ages only. I examined functional responses for these two age ranges because the functional responses fit to data for all ages are affected by factors that change the partial recruitment of younger ages (e.g. density dependent growth). For example, if recruits were not completely vulnerable and fluctuations in abundance tended to be dominated by recruitment (as is common among exploited fish populations), then in years of large recruitment, abundance would be large and q would be small and vice versa for years of small recruitment. Thus, catchability would appear depensatory, but not because of the functional response of the predator. In contrast, the functional responses fit to data from the fully recruited age ranges should be determined more by factors underlying the dynamics of the functional response (e.g. 213 aggregation and switching). Thus for each species, I used a total of four data sets (2 age ranges X 2 methods). Alternative Single Species Models I fit the three alternative models below to each of the four data sets for the three species: 1. Y = aX (8.1) 2. Y = aX/(6 + X) (8.2) 3. Y = aX/[X +6exp(cX)] (8.3) where Y is LPE, X is abundance and, a, b and c are parameters to be estimated. Model 1 is a type I response without an asymptote. Model 2, the disc equation (i.e. equation 5.2, where c = 1/t^  and b = 1/at^ ), generates a type II response unless a and b are very large relative to X, in which case the response is linear with slope alb. The third model is Fujii et al.'s (1978) generalized equation that can mimic response types I-IV. Model 3 is a rearrangement of equation 5.8b from Peterman (1980) where Y = NA/T^, and the numerator and the denominator of the right hand side of 5.8b have been multiplied by N/t^ and where c = 1/t^  and b = l/(dt^). Models 1-3 were used to compare alternative functional response within species for each of the data sets. For models 2 and 3, I used the same equations to fit mortality curves except that Y = q (the catchability coefficient) and the right hand side of each of the models was divided by X. In the case of the type I mortality curve (model 1), I fit a simple linear regression with an intercept (i.e. Y = a + 6X). The slope 6 = 0 (i.e. constant catchability) is 214 expected from a type I response. Parameter Estimation For both functional response curves and mortality curves, I fit model 1 by linear regression and models 2 and 3 by nonlinear least squares. Linear regressions were performed using the statistical package Midas (Fox and Guire 1976) and the technique is explained in detail in most statistical texts (e.g. Zar 1974). Nonlinear least squares is much less commonly applied by biologists and therefore, a few comments concerning nonlinear least squares techniques are warranted. Nonlinear least squares methods require starting guesses of each parameter. Given these starting values, the methods use a variety of techniques to determine values that are used in subsequent iterations. The goal of any nonlinear least squares method is to find the parameter combinations that result in the so called global minimum SSE while -avoiding convergence on slightly larger SSE (called local minima). For nonlinear least squares problems, the sums of squares surface can be quite irregular with several local minima. Thus, most nonlinear least squares fitting procedures are prone to false convergence on local minima, particularly if (1) initial guesses of starting parameters are poor, (2) only a narrow range of starting parameters are used in consecutive trial fits and (3) convergence criteria (i.e. the minimum acceptable difference between consecutive sets of parameter estimates) are too large (Vaessen 1984; Bates and Watts 1988). To avoid these potential problems, I chose initial starting values of a equal to the maximum observed LPE for each data set and values of b and/or c 215 that resulted in a slope of the rising portion of the curve being roughly equivalent to the slope of a regression fit to the data (with an intercept). Since I had no independent estimates of parameters, I also compared the fit (i.e. the SSE) of each model fit to each data set with varying combinations of starting parameters. Thus, in addition to the initial set of starting values, I compared fits using double the initial parameter values, half the initial values and combinations with each individual parameter multiplied by 10 and 0.1 while holding the others constant. Thus for model 2, I used seven different starting parameter combinations and for model 3, I used nine different starting parameter combinations and these combinations covered a wide range of starting values for .g each of the models. I used 10 as my convergence criteria in all fits of the models. The parameter estimates that yielded the smallest SSE were considered the "best" fit estimates. All models were fit using a Fortran program that employed nonlinear fitting subroutines described in UBC NLP (Vaessen 1984). I used the routines FLETCH and FNMIN. Both routines employ a quasi-Newton search method, however FNMIN is less prone to false convergence (Vaessen 1984). Comparison of Models Note that each more complex model(s) (i.e. the model with more parameters) reduces to the simpler model(s) for certain parameter conditions. For example, model 2, the disc equation, reduces to model 1 when lib approaches zero (i.e. when b and a are large relative to X). Similarly, model 3 reduces to model 2 when c = 0 (and l/a>0), and model 3 reduces to model 1 when c = 0 and 1/6 = 0 (i.e. as for model 2 above; for other parameter conditions where model 3 reduces to models 2 and 1 see Ch. 5). Thus models 1-3 can be 216 considered nested models; each simpler model is just a special case of the next most complex model. Therefore, to compare these nested models (i.e. to determine the simplest model that adequately fits the data), I used the extra sums of squares test (Bates and Watts 1988). The test statistic is [(SSE^-SSE^. ) /v e ]/[SSE^ (8.4) where SSE^ is the residual sums of squares for the partial model (i.e. the model with fewer parameters), SSE^. is the residual sums of squares for the full model. v g is the degrees of freedom for the sums of squares due to the extra parameter(s), and is equal to the difference in the numbers of parameters of the full (f) and partial (p) models (i.e. P^ .—P^ ). The degrees of freedom for the full model is v^ . . = N —P^ . . The statistic is compared to F(vg,v^. ,a) and the extra parameter is retained if the calculated mean square ratio is greater than the table value (I used F tables in Zar 1974). I calculated the test statistics for three model comparisons: (1) model 1 vs. model 2, (2) model 1 vs. model 3, and (3) model 3 vs. model 2. The null hypotheses for these comparisons were: 1. H : 1/6 = 0; i.e. c and b >> X. o 2. H : c=0 and 1/6 = 0; i.e. a and b >> X. o 3. H : c = 0. o I restricted my statistical comparisons to the functional response fits only because q estimates used for the mortality curves came directly from abundance estimates (e.g. abundance is in the denominator of eq. 7.14). Thus, an apparent inverse relationships between q and N (i.e. indicating a type II response) may 217 arise simply from random errors in abundance estimates (i.e. the X variate; Shardlow et al. 1985). My main purpose in fitting the q vs. abundance curves was to describe the shape of the relationships for each of the alternative models. Thus, I restricted my comparisons of mortality curves to qualitative comparisons 2 of r and residual mean squared errors (MSE). However, for the linear model I tested the null hypothesis that the slope, 6 = 0 (using a 2-tailed t test), as would be expected from a type I functional response. I calculated the significance of each single species mortality model using equation 8.4 with SSE^ equal to the total sum of squared deviations from the mean q for each data set. Calculation of Power I calculated the statistical power (i.e. the probability of detecting an effect when it exists; Cohen 1977) for all cases where I failed to reject the null hypotheses in the extra sums of squares tests. The purpose of calculating power was that a failure to reject the above null hypotheses means that effect of the additional parameter is not significant and therefore the simpler model is as adequate as the more complex model in explaining the data. However, it is incorrect to assert that the simpler model is adequate (i.e. accept H q ) , if the probability of making a type H error (i.e. p\ the probability of failing to rejecting H p when it is false) is high. Power is simply 1-/3; thus when power is low there is a high probability of accepting H q when it is false. In contrast, a, which is normally set at 0.05, is the probability of rejecting the null hypothesis when it is true. The power of any given test is a function of three factors: (1) the chosen a level, (2) the sample size (N), and (3) the effect size (ES). The effect size is 218 the true magnitude of the effect or "the degree to which the null hypothesis is false" (Cohen 1977). For example, the effect size in the test of the null hypothesis for model 1 vs. model 2 above, would be the degree to which the true lib is different from zero, or graphically the degree of nonlinearity in the functional response. If the data have come from an underlying functional response that is very nonlinear (i.e. saturating), then for a given sample size and a (i.e. 0.05) there will be a higher probability of rejecting H o (i.e. higher power), than if the true underlying response is less nonlinear. Similarly, because the critical values of a test statistic increase with decreasing N and increasing a, a large sample size will have a higher power than a smaller sample size, for a given a and ES, and a smaller a will result in lower power for a given N and ES. Details of the calculations for power can be found in various statistical texts (e.g. Dixon and Massey 1969; Cohen 1977). In the case of F tests, procedures use the noncentral F distribution and therefore require the calculation of the noncentrality parameter (usually called X; not to be confused with the weighting factors for CAGEAN's auxiliary sums of squares, Ch. 7). For the power of the extra sums of squares F test, the noncentrality parameter measures how far the true curve (assumed to be the more complex model) is from the hypothesized curve (i.e the simpler model). In a graphical sense, the noncentrality parameter quantifies the degree to which the curve for the simple model fails to fit the curve the more complex model. The noncentrality parameter X is estimated by the following formula: X = E C Y . - Y )2/v ]/[SSE. / v , ] . (8.5) where Y .^ are the predicted values from the full model using the best fit parameter estimates, 219 are the predicted values for the partial model when it is fit to the predicted values for the complex model and all other terms are as defined in equation 8.4 (R. Lockhart, Dept. of Statistics, Simon Fraser Univ., pers. comm.). Y^.-Y^ is the deviation of the partial model from the full model at each observed value of the X variate (i.e. abundance). To estimate the sums of squares of these deviations, I fit each simpler model to the predicted values of the more complex models for each of the three comparisons above. Thus, for the comparisons of models 2 and 3 vs. model 1, I used simple linear regression through the origin (i.e. I fit model 1 to the predicted value from models 2 and 3). For the comparison of model 3 vs. model 2, I used nonlinear least squares to fit the model 2 to the predicted values of model 3. Given an estimate of X, the test statistic, <f>, used to estimate power is simply v/[X/(ve + 1)], where \ £ is the difference between the number of parameters in the full and partial models as define above. I used the power charts in Pearson and Hartley (1976) to determine the power given 0, the degrees of freedom, v j = v e > v £ = vf ' a n c * a = 0.05. I also estimated the power for cases where I failed to reject the null hypothesis of zero slope in the linear mortality model. In this case, the noncentrality parameter, 5, is calculated by a nearly identical formula used for the t statistic. That is, 6 = ABS(0 - py/SE^, where ABS denotes absolute value, 0 O = O, and SE^ is the standard error of the estimated slope (/3). Given 6, df=N-2, and the chosen significance level a, power can be determined using tables given in Dixon and Massey (1969). I used a Fortran program that calculated power exactly by linear interpolation for 5 values that are between Dixon and Massey's tabulated values (R. Peterman, Simon Fraser Univ., pers. comm.). 220 Examination of Residuals Statistical tests for linear and nonlinear least squares methods assume normality and homoskedasticity of residuals. Violations of either or both of these two assumptions may result in biased parameter estimates and/or biased test statistics, although the F test is reportedly robust to deviations from normality (Zar 1974, Neter and Wasserman 1974). Extreme deviations from normality and homoskedasticity suggest that the fitted model is inappropriate for the data. I tested the residuals from fits of each model for normality using the 2 Anderson-Darling test statistic, A , as modified by Stephens (1974).. Monte Carlo 2 trials have shown that the modified A statistic provides a more powerful test of normality than the Kolmorgorov-Smirnov statistic and others (Stephens 1974). The null hypothesis for the test is that the residuals came from a normal 2 distribution. I used a Fortran program to calculate the modified A values (Paul Higgins, Natural Res. Mgt. Prog., Simon Fraser Univ., pers. comm.) and compared the calculated values to the critical values in the table in Stephens (1974; Case 3) with N—P degrees of freedom. To assess the homoskedasticity of the residuals, I calculated rank correlations (Kendall's T ^ ; Seigel 1956) between the absolute values of the residuals and the values of the X variates (Neter and Wasserman 1974; Peterman 1981). The null hypothesis is that there is no association between the absolute value of residuals and the X variate (i.e. abundance). I calculated values and their significance using Midas (Fox and Guire 1976). 221 Multispecies Models I fit the multispecies disc equation (MSDE; Murdoch 1973) below to data for all three species in an attempt to model the whole system as a unit. Y= aX/[I + bX + cX1 + dX£] (8.6) where Y is LPE (i.e. NA/T^), X is abundance the species being consider in Y, X^ . is the abundance of alternative prey i, and a, b, c and d are parameters to be estimated. The general pattern of landings and effort suggests that usually cod and English sole are caught together in somewhat deeper waters (30-49 vs. 10-19 fathoms) than rock sole (Westrheim and Foucher 1985a). Thus, the most logical structure for a MSDE would consider cod and English sole (i.e. Xs in as the alternatives to rock sole. However, the pattern of landings for the three species varied seasonally and among years (see Section on 'Study area and fishery description' in Ch. 7), and this variability warranted fitting the model to the other species as well. Thus, for cod, the XJS were the abundances of English sole and rock sole, and similarly for English sole, Xs were rock sole abundance and cod abundance. For each species, I used catch and abundance estimates from the same two age ranges and methods used for the analysis of single species responses. Thus I used 4 data sets for each species. I fit equation 8.6 to each of these data sets using nonlinear least squares. In order to obtain some reasonable estimate of initial starting 222 parameters for each data set, I fit the linearized version of equation 8.6 shown below. 1/Y = 6/0 ,+ l/(aX) + cXj/iaX) + dX2/(aX) (8.7) where all variables and parameters are the same as in equation 8.6. I fit equation 8.7 by multiple linear regression and used the resulting regression coefficients (i.e. bla, lla, and cla, and dla) to solve for the initial starting values of a, b, c and d. I took the same precautions described above for the single species models to avoid false convergence on local minima. I also fit the linear model below (by multiple linear regression) for comparison with the MSDE. Y = a + bX + cX2 + dX2 (8.8) Note that when the parameters c, and d of the linear and MSDE equations have opposite signs, the two models predict the same qualitative effects of alternative prey abundances on LPE. I restricted my comparison of the linear 2 model (i.e. eq. 8.8) and the MSDE to r values, because the models were not nested (i.e. each had the same number of parameters), and therefore, the extra sums of squares test was not appropriate (it would have zero degrees of freedom). Similarly, because the data sets for both multispecies models included only the period of overlap in the data sets for each species (i.e. 1961-78 for cod, English sole and rock sole), I was unable to compare the fits of the 2 multispecies and singles species models. However, a sizeable difference in the r s between the best fit single and multispecies model for given data set would at least provide weak evidence as to which approach was best. In addition, I also 223 tested the residuals from fitting each multispecies model for normality by the methods described above. I used equations similar to 8.6 and 8.8 to fit the mortality curves for the multispecies models. For the linear model, I used an equation identical to 8.8, except that Y = q. For the MDSE, I used equation 8.6 with Y = q, and where the right hand side was divided by X as in the single species models. I fit these two models by multiple linear regression and nonlinear least squares, respectively. Again, I restricted my comparison between the multispecies models 2 to r values, and my main purpose was to describe the shape of the functional relationship between q and abundance in a multispecies system. As with the functional response, I limited my comparisons of single and multispecies mortality curves to the values. Tests of the Switching Hypothesis I used the model below to test the switching null hypothesis (cf. with eq. 5.16, Ch. 5): C1IC2 = o(N 2 /N 2 )^ (8.9) where C and N are the landed catches per unit effort and abundances for species 1 and 2, and a and, 0 are parameters to be estimated. A value of /3 > 1.0 indicates that the ratio of C's is increasing faster than linearly as the ratio of abundances increases, and switching is indicated. I estimated a and, 0 using the log transformed version of equation 8.9 shown below. lnfCycy = ln(o) + 01n(N2/N2) (8.10) I fit equation 8.10 by linear regression and tested the null hypothesis of 224 H : 13 < 1.0 using a one-tailed f-test (Zar 1974). o For cases where I failed to reject Ho, I estimated the power using the noncentrality parameter 8 as described above only with /3q = 1.0. In addition to calculating power, I also tested the residuals from fits of equation 8.10 for normality and homoskedasticity by the same tests used for the functional response models (see above). Data for the same age ranges and two methods were used in the tests of the switching hypothesis as for the previous analyses. I expect evidence for switching to be the strongest between cod and rock sole because these two species are generally caught at different depths, and therefore fishermen can easily target on either species while excluding the other. However, I tested the switching hypothesis for all 12 data sets (3 species pairs X 2 age ranges X 2 catch-at-age methods). Results Comparison of Single Species Functional Response Models Pacific cod All three alternative functional response models fit each of the cod data 2 sets reasonably well (all r s ^  0.68; Table 8.1). Best fit parameter values for models 1 (linear) and 2 (disc) resulted in identical curves for all data sets; thus, the best fit disc equation was linear (Fig. 8.1). The best fit parameters for Fujii et al.'s (1978) generalized equation (model 3) were consistent with a type n i (sigmoid) response for all data sets, although the curve for one data set reaches an asymptote at about double the observed maximum in abundance (VPA ages 225 Pacif ic cod CAGEAN VPA Abundance (mill ions) Figure 8.1. Comparison of alternative single species functional response model fits to Pacific cod data sets. Squares are observations and predicted lines were drawn using the best fit parameters (Table 8.1). Numbers beside lines denote the model number; l=linear, 2=disc, 3=generalized equation. Panels A and B use data sets from CAGEAN for ages 4-10 and 2-10 and panels C and D use data sets from VPA for the same age ranges. Note: lines for models 1 and 2 were coincident for all data sets. 226 4-10; Fig. 8.1 C). Extra sums of squares tests were not performed for model. 1 vs. model 2 comparisons because model 2 had slightly larger SSE than model 1 for all data 2 sets (Table 8.1). Model 3 consistently had the smallest MSE and largest r s (Table 8.1). However, the only significant differences among curves were for the data set from CAGEAN ages 2-10 where model 3 had significantly improvement in fit compared to either model 1 or 2 (Table 8.1). Where the null hypothesis for the extra sums of squares test was not rejected, power was low (<0.35; Table 8.1). Thus, for these cases, it would be incorrect to conclude that the simpler model was sufficient because of the large probability (>0.65) of making a type H error. Residuals from fits of models 1 and 2 were generally positive at high abundance and negative at low abundance with the exception of the data set from VPA ages 4-10 (Fig. 8.1). There was no evidence for heteroskedasticity in the residuals from any of the model-data set combinations as all T ^ were not statistically significant (Table 8.1). The null hypothesis of normality was rejected for 3 out of 4 data sets for models 1 and 2, but for only 1 data set fit by model 3 (CAGEAN ages 4-10). Thus, the model 3 fits were most consistent with the assumptions of least square methods. Models fit to data from the two catch-at-age methods had similar curves. However, fits of models 1 and 2 had consistently steeper slopes and fits of model 3 had consistently larger asymptotes for VPA data sets than for CAGEAN data sets (Fig. 8.2). 227 Table 8.1. Summary statistics for single species functional response model comparisons for Pacific cod. For left to right are the data source (age range and melhodX model number (Y=T denotes fitting data with mean Y, 1= linear, 2 = disc, 3=generalizcd equation), the parameter values a, b, 2 and c as appropriate, the residual sums of squares (SSE), the coefficient of determination (r ); the degrees of freedom (df; n = 25), the residual mean square error (MSE), the F statistics for the extra sums of squares test (top 1 vs. 2, middle 1 vs. 3, bottom 2 vs. 3), and corresponding P values (— indicates F statistic and P value not computed because SSE, > SSE , see text). For those cases where the F was not significant at P<0.05, / P the noncentrality parameter, phi, and the power of the F test are also given. Statistics for tests of residuals include for the normality test, Stephens (1974) modified A statistic and corresponding P value, and for the homoskedasticity test, Kendall's tau and corresponding P value. Normality Homoskedasticity Data Parameters Modified source Model a b c SSE 2 i df MSE F P phi Power 2 A P tau b P Ages Y=Y 50829 24 2118 -4-10 1 0.0518 15141 0.70 24 631 — . — — — 0.78 <0.05 -0.06 >0.69 CAG 2 3.40E+6 6.55E+7 15145 0.70 23 658 1.34 >0.25 0.68 <0.15 0.78 <0.05 -0.06 >0.69 3 205 9043 -8.50E-4 13499 0.73 22 614 2.68 >0.10 1.18 <0.35 0.83 <0.025 -0.07 >0.63 Ages Y=T 67075 I 24 2795 4-10 1 0.0544 18201 0.73 24 758 — — — — 0.36 >0.15 -0.05 >0.73 VPA 2 1.37E+7 2.51E+8 18202 0.73 23 791 1.04 >0.25 0.59 <0.15 0.36 >0.15 -0.05 >0.73 3 115807 2.65E+6 -9.40E-5 16632 0.75 22 756 2.08 >0.10 1.02 <0.25 0.51 >0.15 -0.07 >0.63 Ages Y=Y 537040 24 22377 2-10 1 0.0245 172570 0.68 24 7190 — — — — 0.79 <0.05 0.13 >0.63 CAG 2 2.26E+6 9.21E+7 172587 0.68 23 7504 5.76 <0.01 n/a n/a 0.79 <0.05 0.07 >0.63 3 574 174342 -2.40E-4 113241 0.79 22 5147 11.53 <0.005 n/a n/a 0.48 >0.15 -0.03 >0.84 Ages Y=? 871680 24 36320 2-10 1 0.0274 244230 0.72 24 10176 ' — — — — 1.12 <0.01 0.20 >0.17 VPA 2 2.38E+6 8.69E+7 244296 0.72 23 10622 1.12 >0.25 0.63 <0.15 1.04 <0.01 0.20 >0.17 3 933 68691 -9.23E-5 221602 0.75 22 10073 2.25 >0.10 1.10 <0.30 0.65 >0.05 0.07 >0.63 Ages 4-10 Pacific cod Model 1 A 800, .VPA Ages 2-10 Abundance (millions) Figure 8.2. Comparison of alternative single species functional response models from VPA and CAGEAN fit to data sets for Pacific cod. Squares and triangles are observations from CAGEAN and VPA. Predicted lines are drawn using the best fit parameters (Table 8.1). Labels beside lines (VPA and CAG) denote catch-at-age methods. Model numbers are indicated above each pair of horizontal panels. Panels A, B and C use data sets for ages 4-10 and panels D, E and F use data sets for ages 2-10. 229 English sole All of the alternative models fit the data sets for English sole poorly, although fits to the data sets for fully recruited ages (6-10) were somewhat better (0.21<r2<0.37) than fits to data sets for all ages (3-17; 0.02<r2<0.12; Table 8.2). In contrast with the fits to cod data sets, all fits of model 2 were nonlinear and consistent with a type II response (Fig. 8.3). Parameter values for model 3 were consistent with a type III response for 3 out of 4 data sets (exception was CAGEAN ages 3-17; Fig. 8.3 B). However, none of these qualitative differences was statistically significant; the null hypothesis for the extra sums of squares test was not rejected for any of the model comparisons (Table 8.2). While model 1 had the smallest MSE for each data set, power was low (<0.15) for all model comparisons. Therefore, as for cod, it is incorrect to conclude that the simpler models were sufficient because of the high probability (>0.85) of a type II error (Table 8.2). Residuals from all models were not significantly heteroskedastic by the rank correlation test (Table 8.2). However, the distributions of residuals were significantly nonnormal for all model fits (P<0.01; Table 8.2). Data sets from VPA and CAGEAN resulted in similar curves, except that the model 3 fit to data from ages 3-17 was sigmoid for VPA and saturating for CAGEAN (Fig. 8.4 F). .230 Table 8.2. Summary statistics for single species functional response model comparisons for English sole. From left to right are the data source (age range and methodX model number (Y=T denotes fitting data with mean Y, 1= linear, 2=disc, 3=generalized equation), the parameter values a, b, and c as appropriate, the residual sums of squares (SSE), the coefficient of determination (r ), the degrees of freedom (df; n=23), the residual mean square error (MSE), the F statistics for the extra sums of squares test (top 1 vs. 2, middle 1 vs. 3, bottom 2 vs. 3), and corresponding P values (— indicates F statistic and P value not computed because SSE > SSE , see text). For those cases where the F was not significant at P<0.05, f P the noncentrality parameter, phi, and the power of the F test are also given. Statistics for tests of residuals include for the normality test, Stephens' (1974) modified A2 statistic and corresponding P value, and for the homoskedasticity test, Kendall's tau and corresponding P value. Normality Homoskedasticity Data Parameters Modified source Model a b c SSE 2 r df MSE F P phi Power 2 A P tau b P Ages Y=Y 390070 23 16960 -6-10 1 0.1380 308450 0.21 23 13411 0.37 >0.25 0.37 <0.10 1.91 <0.01 0.00 >0.98 CAG 2 947.62 5221.42 303331 0.22 22 13788 0.54 >0.25 0.29 <0.10 1.80 <0.01 -0.11 >0.48 3 279 6508 -1.96E-3 293427 0.25 21 13973 0.71 >0.25 0.44 <0.10 2.10 <0.01 0.03 >0.88 Ages Y=Y 483890 23 21039 6-10 1 0.1370 368810 0.24 23 16035 0.80 >0.25 0.55 <0.15 2.12 <0.01 0.12 >0.42 VPA 2 862 4443 355908 0.26 22 16178 2.21 >0.10 0.41 <0.10 2.03 <0.01 -0.10 >0.51 3 310 3678 -1.17E-3 304713 0.37 21 14510 3.53 >0.05 0.52 <0.15 2.00 <0.01 0.03 >0.86 Ages Y=Y 1398100 23 60787 3-17 1 0.0588 1402400 — 23 60974 Oil >0.25 0.56 <0.15 1.45 <0.01 -o!o6 >0.71 CAG 2 705 5410 1370808 0.02 22 62309 0.24 >0.25 0.32 <0.10 124 <0.01 -0.12 >0.42 3 705 5410 1.06E-10 1370808 0.02 21 65277 — — — — 124 <0.01 -0.12 >0.42 Ages Y=Y 2098200 23 91226 3-17 1 0.0673 1893400 0.10 23 82322 0.22 >0.25 0.29 <0.10 1.82 <0.01 0.14 >0.36 VPA 2 1583 16812 1874338 0.11 22 85197 0.27 >0.25 0.25 <0.10 1.87 <0.01 0.04 >0.79 3 520 39276 -5.95E-4 1845755 0.12 21 87893 0.33 >0.25 0.34 <0.10 1.78 <0.01 0.12 >0.45 CD Q_ 1200-_rz Z5 D 800 CAGEAN ages 6 - 1 0 ages 3 - 1 7 English sole A 600 400 200 1200-800 '2,3 400 VPA ages 6 - 1 0 i 1 3 ages 3 - 1 7 D 2 ' 4 ' 6 Abundance (mill ions) 10 Figure 8.3. Comparison of alternative single species functional response model fits to English sole data sets. Squares are observations and predicted lines were drawn using the best fit parameters (Table 8.2). Numbers beside lines denote the model number; 1 = linear, 2 = disc, 3=generalized equation. Dotted line is for model 1. Panels A and B use data sets from CAGEAN for ages 6-10 and 3-17 and panels C and D use data sets from VPA for the same age ranges. 232 Ages 6-10 English sole Model 1 A 1200 800-400-Model 2 B 1200 800-1 400 Ages 3—17 Model 3 1200 800 Abundance (millions) Figure 8.4. Comparison of alternative single species functional response models from VPA and CAGEAN fit to data sets for English sole. Squares and triangles are observations from CAGEAN and VPA. Predicted lines are drawn using the best fit parameters (Table 8.2). Labels beside lines (VPA and CAG) denote catch-at-age methods. Model numbers are indicated above each pair of horizontal panels. Panels A , B and C use data sets for ages 6-10 and panels D, E and F use data sets for ages 3-17. 233 Rock sole The fits of alternative functional response models to the rock sole data 2 sets had intermediate r values compared to the fits for the cod and English 2 sole data sets. As found for English sole, r values were higher for all models 2 when fit to data for fully recruited ages (6-10; 0.33<r S0.56), than when fit to data for all ages (3-17; 0.21<r2s0.34; Table 8.3). Best fit parameters for models 1 and 2 resulted in nearly identical curves for all data sets. Thus as found for cod, model 2 fits were linear (Fig. 8.5). Model 3 fits to all data sets resulted in sigmoid curves, although the asymptote for one data set (VPA ages 6-10) occurred outside the range of observed abundances (at about 4000; Fig. 8.5 C). Model 3 had the smallest MSE for 3 out of 4 data sets, and three model comparisons found significant differences (model 2 vs. 3 for CAGEAN ages 6-10 and models 1 and 2 vs. 3 for VPA ages 6-10 ;Table 8.3). Power was generally low, although somewhat higher for model 2 vs. 3 comparisons (min. power<0.35; max. power<0.40) than for the two other model comparisons (min. power<0.05; max. power< 0.30; Table 8.3). Thus for all cases where the extra sums of squares test was not rejected, it is incorrect to conclude that the simpler models were sufficient. Data sets for rock sole were the only ones where I found significant heteroskedasticity in residuals by the rank correlation test. Residuals were significantly heteroskedastic for the fits of models 1 and 2 to VPA data sets for ages 6-10, and for all model fits to VPA data sets for ages 4-15 (Table 8.3). 234 Table 8.3. Summary statistics for single . species functional response model comparisons for rock sole. From left to right are the data source (age range and method), model number (Y-Y denotes fitting data with mean Y, 1=linear, 2=disc, 3=generalized equation), the parameter values a, b, and c as appropriate, the residual sums of squares (SSE), the coefficient of determination (r ), the degrees of freedom (df; n=23), the residual mean square error (MSE), the F statistics for the extra sums of squares test (top 1 vs. 2, middle 1 vs. 3, bottom 2 vs. 3), and corresponding P values (— indicates F statistic and P value not computed because SSE > SSE , see text). For those cases where the F was not significant at P<0.05, / P the noncentrality parameter, phi, and the power of the F test are also given. Statistics for tests of residuals include for the normality test, Stephens' 2 (1974) modified A statistic and corresponding P value, and for the homoskedasticity test, Kendall's tau and corresponding P value. Data source Model Parameters a b ' c SSE r2 df MSE F P phi Power Normality Homoskedasticity Modified 2 A P tau, P b Ages Y=Y 1390800 6-10 1 02160 769600 0.45 CAG 2 3.01E+6 1.40E+7 769634 0.45 3 897 44387 -1.65E-3 615120 0.56 Ages Y=Y . 5290500 6-10 1 0.2780 3569200 0.33 VPA 2 1.43E+7 5.13E+7 3569346 0.33 3 2128 267564 -1.82E-3 2596418 0.51 Ages Y=Y 2814000 4-15 1 0.1040 2216700 0.21 CAG 2 6824 60711 2213325 0.21 3 628 367251 -1.46E-3 2031069 0.28 Ages Y=Y 7214500 4-15 1 0.1280 5711700 0.21 VPA 2 5832 39720 5694728 0.21 3 871 2.93E+7 -2.48E-3 4793049 0.34 22 63218 -22 34982 — — — — 2.62 <0.01 -0.10 >0.53 21 36649 2.51 >0.10 0.91 <0.30 2.62 <0.01 -0.10 >0.53 20 30756 5.02 <0.05 n/a n/a 1.72 <0.01 0.26 >0.08 22 240477 22 162236 — — — — ^ 2.59 <0.01 0.32 <0.03 21 169969 3.75 <0.05 n/a n/a 2.59 <0.01 0.32 <0.03 20 129821 7.49 <0.025 n/a n/a 1.77 <0.01 0.26 >0.09 22 127909 22 100759 0.03 >025 0.11 <0.05 120 <0.01 0.21 >0.17 ?1 105396 0.91 >0.25 0.57 <0.15 123 <0.01 0.18 >0.25 20 101553 1.79 >0.10 0.99 <0.35 0.64 >0.05 0.30 <0.05 22 327932 22 259623 0.06 >0.25 0.14 <0.05 2.07 <0.01 0.40 <0.007 21 271178 1.92 >0.10 0.77 <0.15 2.24 <0.01 0.34 <0.03 20 239652 3.76 >0.05 1.33 <0.40 . 1.09 <0.01 0.35 <0.02 VPA C ages 6 - 1 0 0 2 4 6 8 10 0 2 4 6 8 10 Abundance (mill ions) Figure 8.5. Comparison of alternative single species functional response model fits to rock sole data sets. Squares are observations and predicted lines were drawn using the best fit parameters (Table 8.3). Numbers beside lines denote the model number; 1 = linear, 2=disc, 3 = generalized equation. Dotted line is for model 1. Panels A and B use data sets from CAGEAN for ages 6-10 and 4-15 and panels C and D use data sets from VPA for the same age ranges. 2500 2000 CAGEAN ages 6 - 1 0 rock sole A 2500T 2000-236 However, residuals were significantly heteroskedastic for only one model-data set combination from CAGEAN (model 3 fit to ages 4-15; Table 8.3). Residuals from all model fits were highly significantly nonnormal (P<0.01), with the exception of model 3 fit to the CAGEAN data set for ages 4-15 (Table 8.3). Curves for each model fit to data sets from VPA and CAGEAN were similar. However, as found for the cod data sets, curves for models 1 and 2 had steeper slopes and curves for model 3 had larger asymptotes when fit to VPA data sets compared with fits to CAGEAN data sets (Fig. 8.6). Single Species Mortality Curves Pacific cod 2 All three mortality models fit the cod data sets poorly (all r < 0.24; Table 8.4). Only the fits of the linear and Fujii models to the data set from CAGEAN for all ages (2-10) were statistically -significant (Table 8.4). The slope of the fit of the linear model was significantly different from zero for only 1 data set (CAGEAN ages 2-10), where the slope was positive indicating a increase in q with abundance (Table 8.4). Power was low (<0.29) for the other three data sets, indicating it would be incorrect to assert that q is density independent. All fits of model 2 were equivalent to fitting a model of q = constant. Data sets from VPA resulted in larger intercepts (i.e. higher avg. q; Fig. 8.8 B and E; Table 8.4). Model 3 fits to CAGEAN data sets were dome shaped, which is consistent with a sigmoid functional response (Fig. 8.7). Model 3 fits to VPA data sets were linear with zero slope for ages 2-10 and linear with a slightly decreasing slope for ages 4-10 (Figs. 8.7, 8.8). Thus, mortality relationships 237 Table 8.4. Summary statistics for single species mortality model fits to Pacific cod data sets. From left to right are the data source (age range and methodX model number (Y=Y denotes fitting data with mean Y, 1=linear, 2=disc, 3 = generalized equation), the parameter values a, b, and c as appropriate, the P values for the test of H :b=0 for the linear model, the power for the same test, the residual sums of squares (SSEX the o coefficient of determination (r \ the degrees of freedom (df; n=25 for CAGEAN and 24 for VPA), the residual mean square error (MSE), the F value for test of the significance of the fit, and the P value. Note:- — indicates F and P value not computed because SSE of model was greater than or equal to SSE of Y=7. Data Parameters source Model a b c P(b|b=0) Power SSE 2 r df MSE F P Ages Y=7 2.35 24 0.10 4-10 1 0.61 1.44E-4 0.15 0.29 2.14 0.09 23 0.09 2.18 0.15 CAG 2 5.19E+8 6.30E+8 2.35 — 23 0.10 — — 3 4042 9348 -7.20E-4 2.08 0.11 22 0.09 1.39 0.27 Ages Y=Y 4.92 23 0.21 4-10 1 1.09 -7.98E-5 0.55 0.08 4.84 0.02 22 0.22 0.38 0.54 VPA 2 2.31E+8 2.39E+8 4.92 0.00 22 0.22 — — 3 240273 204075 1.31E-4 4.80 0.03 21 0.23 0.55 0.59 Ages Y=Y 0.41 24 0.02 2-10 1 0.20 1.42E-5 0.04 — 0.34 0.17 23 0.01 4.69 0.04 CAG 2 3.19E+7 9.14E+7 0.41 — 23 0.02 — — 3 10286 101582 -1.67E-4 0.31 0.24 22 0.01 6.78 0.01 Ages Y=Y 0.72 23 0.03 2-10 1 0.40 2.13E-7 0.98 0.03 0.72 0.00 22 0.03 0.00 1.00 VPA 2 3.53E+8 8.69E+8 0.72 0.00 22 0.03 0.00 1.00 3 3.61E+6 8.94E+6 -7.01E-7 0.72 0.00 21 0.03 0.00 1.00 ZJ 2500 O Ages 6-10 Ages 4-15 ' 2 ' 4 ' 6 ' 8 ' I'O A VPA • CAGEAN Model 2 a B 250-200-Abundance (millions) Figure 8.6. Comparison of alternative single species functional response models from VPA and CAGEAN fit to data sets for rock sole. Squares and triangles are observations from CAGEAN and VPA. Predicted lines are drawn using the best fit parameters (Table8.3). Labels beside lines (VPA and CAG) denote catch-at-age methods. Model numbers are indicated above each pair of horizontal panels. Panels A , B and C use data sets for ages 6-10 and panels D, E and F use data sets for ages 4-15. 239 2.5-| 2 ^—^1.5H 2 1" X CD CAGEAN ages 4 - 1 0 ages 2 - 1 0 Pacif ic cod A I 1 I 1 1 —1 1 1 2 3 . 4 2.5n 2-1.5-1 •5H 0 .8-1 VPA ages 4 - 1 0 1 1 T 1 1 1 1 1 2 3 4 ages 2 - 1 0 D '1,2,3 10 15 20 25 30 Abundance (mill ions) Figure 8.7. Comparison of alternative single species mortality model fits to Pacific cod data sets. Squares are observations and predicted lines were drawn using the best fit parameters (Table 8.4). Numbers beside lines denote the model number; 1 = linear, 2 = disc, 3 = generalized equation. Panels A and B use data sets from CAGEAN for ages 4-10 and 2-10 and panels C and D use data sets from VPA for the same age ranges. 240 suggested by the best fit parameter estimates differed between VPA and CAGEAN data sets for models 1 and 3, but were virtually identical for model 2 (Fig. 8.8). Mortality relationships were qualitatively consistent with functional responses for each model and data set, except that the mortality curve for model 3 and the VPA data set was linear whereas the functional response was slightly sigmoid, (cf. Figs. 8.2 and 8.8). English sole All three mortality models fit the English sole data sets poorly as none were significantly better that fitting the data with the average q for each data set (all r2s <=0.07, P^0.22; Table 8.5). The null hypothesis of b = 0 for the linear model was not rejected for any of the data sets, although the power was consistently very low (s 0.17; Table 8.5). Thus, again it would be incorrect to conclude that q is density independent. The predicted mortality relationships varied among models within each data set and among data sets for each model, with the exception that model 2 was nearly linear with decreasing slope for 3 out of 4 data sets (Figs. 8.9, 8.10). The model 2 fit for the data set from CAGEAN ages 3-17 was highly nonlinear with decreasing slope as predicted by a type II functional response (Fig. 8.9 B). Model 3 fits were dome shaped for two of the data sets (CAGEAN ages 6-10, Fig 8.9 A; VPA ages 3-17, Fig. 8.9 D), although the dome shape for the CAGEAN data set occurred outside the range of the data (Fig. 8.9 A). The other two fits of model 3 were nearly linear with decreasing slope and very similar to model 2 fit for data from CAGEAN ages 3-17 (Fig. 8.9). The 241 Table 8.5. Summary statistics for single species mortality model fits to English sole data sets. From left to right are the data source (age range and methodX model number (Y=Y denotes fitting data with mean Y, 1=linear, 2=disc, 3=generalized equation), the parameter values a, b, and c as appropriate, the P values for the test of H :b=0 for the linear model, the power for the same test, the residual sums of squares (SSE), the o 2 coefficient of determination (r X the degrees of freedom (df; n-25 for CAGEAN and 24 for VPA), the residual mean square error (MSE), the F value for test of the significance of the fit, and the P value. Note: — indicates F and P value not computed because SSE of model was greater than or equal to SSE of Y=Y. Data Parameters source Model a b c P(b|b=0) Power SSE 2 t df MSE F P Ages Y=7 • - ' 52.72 23 2.29 6-10 1 2.27 -1.51E-4 0.82 0.04 52.59 0.00 22 2.39 0.05 0.83 GAG 2 32807 14570 52.61 0.00 22 2.39 0.05 0.83 3 2128 14000 -0.01 65.16 — 21 3.10 — — Ages Y=Y 65.63 22 2.98 6-10 1 2.66 -354E-4 • 0.57 0.08 64.61 0.02 21 3.08 0.33 0.57 VPA 2 13710 4995 64.71 0.01 21 3.08 0.30 0.59 3 5641 2482 -5.33E-5 64.62 0.02 20 3.23 0.31 0.74 Ages Y=Y 10.88 23 0.47 3-17 1 1.55 -0.00011 0.30 0.17 10.34 0.05 22 0.47 1.15 0.30 CAG 2 4530 -487 10.13 0.07 22 0.46 1.63 0.22 3 4528 -2519 -3.30E-4 10.09 0.07 21 0.48 1.64 0.22 Ages Y=Y 8.15 22 0.37 3-17 1 1.00 -0.00002 0.80 0.04 8.13 0.00 21 0.39 0.06 0.81 VPA 2 53809 55195 8.13 0.00 21 0.39 0.05 0.83 3 5912 85235 -9.79E-4 7.72 0.05 20 0.39 1.11 0.35 X cr 2.5i £Z 2H CD 1 o — 0 D Ages 4-10 Pacific cod Model 1 A * A VPA • CAGEAN Ages 2-10 Model 2 B 8 VPA CAG 3 ' 4 Model 3 C -s .CAG VPA 3 ' 4 Abundance S iTS 15" 2u i5 3D E VPA CAG • >•> * »• i * 1 1*0 1? Z"!! ?5 3b Figure 8.8. Comparison of alternative single species mortality models from VPA and CAGEAN fit to data sets for Pacific cod. Squares and triangles are observations from CAGEAN and VPA. Predicted lines are drawn using the best fit parameters (Table 8.4). Labels beside lines (VPA and CAG) denote catch-at-age method. Model numbers are indicated above each pair of horizontal panels. Panels A , B and C use data sets for ages 4-10 and panels D, E and F use data sets for ages 2-10. 243 CAGEAN ages 6 - 1 0 English sole A VPA ages 6—10 ages 3 - 1 7 Abundance (mil l ions) Figure 8.9. Comparison of alternative single species mortality model fits to English sole data sets. Squares are observations and predicted lines were drawn using the best fit parameters (Table 8.5). Numbers beside lines denote the model number; 1 = linear, 2=disc, 3 = generalized equation. Panels A and B use data sets from C A G E A N for ages 6-10 and 3-17 and panels C and D use data sets from V P A for the same age ranges. 244 predicted mortality curves were similar for data from VPA and CAGEAN for half of the data set-model combinations (models 1,2 ages 6-10, and model 1 ages 3-17; Fig. 8.10 A, B, and D). However, all comparisons should be interpreted 2 cautiously given the similar SSE values and low r s for all three models when fit to each data set. The mortality curves were generally consistent with the predicted functional response for each model and data set with the exception that mortality curve for the VPA (ages 3-17) data set was linear and the functional response was saturating (cf. Figs. 8.4 and 8.10). Rock sole As found for the other two species, the alternative mortality models fit the rock sole data sets poorly; none of the fits were statistically significant (all. r 2 s <0.07, P2:0.25; Table 8.6). For the linear model the null hypothesis of b = 0 was not rejected for any of the data sets and power was low (< 0.17; Table 8.6). Thus it would be incorrect to conclude that q is density independent. Mortality curves predicted by model 2 fits were equivalent to fitting q = constant for all data sets (Fig. 8.11). Curves for model 3 fits were nearly linear with increasing slope for 3 of the data sets and dome shaped for the data from CAGEAN (ages 4-15; Fig 8.11 B). Mortality curves fit to data from VPA and CAGEAN were similar for all model/age range combinations with the exception of model 3 fit to data from ages 4-15, where the best fit to CAGEAN data was dome shaped and the VPA 2 curve was nearly linear (Fig. 8.12 B and D). Given the low r s, it is highly 245 T a b l e 8.6. Summary statistics for single species mortality model fits to rock sole data sets. From left to right are the data source (age range and method), model number (Y=Y denotes fitting data with mean Y, 1=linear, 2=disc, 3=generalized equation), the parameter values a, b, and c as appropriate, the P values for the test of H :b=0 for the linear model, the power for the same test, the residual sums of squares (SSE), the o coefficient of determination {i\ the degrees of freedom (df; n=25 for CAGEAN and 24 for VPA), the residual mean square error (MSE), the F value for test of the significance of the fit, and the P value. Note: — indicates F and P value not computed because SSE of model was greater than or equal to SSE of Y=Y. Data Parameters source Model a b c P(b|b=0) Power SSE 2 t df MSE F P Ages Y=Y 86.38 22 3.93 6-10 1 1.59 7.72E-4 0.31 0.17 82.16 0.05 21 3.91 1.08 0.31 CAG 2 2iOE+7 9.29E+6 86.39 — 21 4.11 — — 3 9.17E+6 5.85E+6 -3.60E-4 81.49 0.06 20 4.07 120 0.32 Ages Y=Y 22326 21 10.63 6-10 1 1.72 9.96E-4 0.35 0.15 213.65 0.04 20 10.68 0.90 0.35 VPA 2 1.90E+8 . 5.76E+7 22326 0.00 20 11.16 0.00 1.00 3 2.18E+7 1.52E+7 -4.85E-4 207.13 0.07 19 10.90 1.48 0.25 Ages Y=Y 20.37 22 0.93 4-15 1 0.96 9.54E-5 Oil 0.09 19.95 0.02 21 0.95 0.44 Oil CAG 2 1.91E+7 1.41E+7 20.37 0.00 21 0.97 0.00 1.00 3 9648 51711 -7.80E-4 19.04 0.07 20 0.95 1.40 0.27 Ages Y=Y 38.04 21 1.81 4-15 1 0.90 1.38E-4 0.41 0.12 36.74 0.03 20 1.84 0.71 0.41 VPA 2 2.11E+8 1.38E+8 38.04 0.00 20 1.90 0.00 1.00 3 3.28E+7 2.98E+7 -7.08E-5 37.01 0.03 19 1.95 0.53 0.60 cu o o D SZ o "5 Ages 6-10 English sole Model 1 Ages 3-17 A V P A • C A G E A N CAG ~VPA D Model 2 1 Model 3 0 ' 2 ' 4 ' 6 " 8 " 1'0 VPA * ' , • . CAG Abundance (millions) Figure 8.10. Comparison of alternative single species mortality models from V P A and C A G E A N fit to data sets for English sole. Squares and triangles are observations from C A G E A N and V P A . Predicted lines are drawn using the best fit parameters (Table 8.5). Labels beside lines (VPA and CAG) denote catch-at-age methods. Model numbers are indicated above each pair of horizontal panels. Panels A , B and C use data sets for ages 6-10 and panels D, E and F use data sets for ages 3-17. 247 CD O O JQ D O 4-| -f— D O CAGEAN rock sole ages 4 - 1 5 B 6-4-ages 4 - 1 5 D Abundance (millions) Figure 8.11. Comparison of alternative single species mortality model fits to rock sole data sets. Squares are observations and predicted lines were drawn using the best fit parameters (Table 8.6). Numbers beside lines denote the model number; 1 = linear, 2 = disc, 3=generalized equation. Panels A and B use data sets from CAGEAN for ages 6-10 and 4-15 and panels C and D use data sets from VPA for the same age ranges. 248 12 8 ' o ° X c r 0 12 CO o CO o o 7t= 0 15 a o a O 12-Ages 6-10 rock sole Model 1 A Ages 4-15 D A VPA B 4 • CAGEAN * * • * • » * * ( CAG * * • »•" * • ' . * • * i • • • * * * • VPA 6 ' 1 ' 2 1 i ~3 Model 2 6 4^  2 ' 4 ' 6 ' 8 ' 1'0 VPA 'CAG 6 ' 2 ' 4 ' 6 ' 8 '. I'O Model 3 C 6-4-F —, y- ^ 1 ' I Abundance (millions) Figure 8.12. Comparison of alternative single species mortality models from VPA and CAGEAN fit to data sets for rock sole. Squares and triangles are observations from CAGEAN and VPA. Predicted lines are drawn using the best fit parameters (Table 8.6). Labels beside lines (VPA and CAG) denote catch-at-age methods. Model numbers are indicated above each pair of horizontal panels. Panels A , B and C use data sets for ages 6-10 and panels D, E and F use data sets for ages 4-15. 249 unlikely that these qualitative differences would be quantitatively significant. Mortality curves were generally qualitatively consistent with the functional response curves for each data set with the exception of the data set for VPA ages 4-15 where the mortality curve was linear and the functional response was sigmoid (cf. Figs. 8.6 and 8.12). Multispecies Functional Response Models Pacific cod Both the multispecies disc equation (MSDE; eq. 8.6, p. 223) and the 2 linear model fit the cod data sets very well (all r s > 0.73, P<0.0003; Table 8.7). For a given data set, fits of the linear and MSDE models had very 2 similar r s and for a given catch-at-age method, fits of both models to data sets 2 for the two age ranges had virtually identical r s (Table 8.7). However, for a given age range, fits were consistently better for VPA data sets than for CAGEAN data sets (Table 8.7). The distributions of residuals were significantly non normal for fits of both models to the CAGEAN data set that used all ages and for the MDSE fit to the VPA data set for all ages (Table 8.7). The general form of the functional response predicted by the MDSE was that cod LPE increased nearly linearly with increases its own abundance (e.g. Fig. 8.13). This pattern was consistent among all data sets. However, the effect of alternative prey abundance on cod LPE varied among date sets for the two age ranges. For data sets from fully recruited ages, cod LPE decreased with increases in alternative prey abundance but for data sets from all ages cod LPE increased (Table 8.7). The b parameter of the linear model was positive and the 250 Table 8.7. Summary statistics for multispecies functional response model fits to Pacific cod data sets. From left to right are the data source (age range and method), model type (Y=Y denotes fitting data with mean Y), the parameter values a, b, c and d, the residual sums of squares (SSE), 2 the coefficient of determination (r ), the degrees of freedom (df; n = 18), the residual mean square error (MSE), the F statistics for the significance of the fits and corresponding P values and for tests of the normality of residuals, Stephens' (1974) modified \ 2 statistic and corresponding P value. Normality Data Parameters modified source model a b c d SSE 2 r • df MSE F P K2 P ages Y=Y 39992 17 2352 4-10 linear 6.34 0.060 -0.022 7.31E-3 9343 0.77 14 667 15.31 <0.001 0.50 >0.15 CAG MSDE 0.069 4.05E-5 2.57E-4 -8.80E-5 10626 0.73 14 759 12.90 0.0003 0.59 >0.10 ages Y=Y 55393 17 3258 4-10 linear -17.05 0.057 -0.025 0.027 8553 0.85 14 611 25.56 <0.001 0.34 >0.15 VPA MSDE 0.048 4.28E-5 2.78E-4 -3.06E-4 9302 0.83 14 644 23.12 <0.001 0.41 >0.15 ages Y=Y 425700 17 25041 2-10 Linear -168.02 0.032 2.77E-3 0.018 96672 0.77 14 6905 15.88 <0.001 1.01 <0.01 CAG MSDE 0.019 -2.18E-6 -1.59E-5 -2.67E-5 115634 0.73 14 8260 1231 0.0003 0.99 <0.01 ages Y=Y 690990 17 40646 2-10 Linear -285.58 0.028 0.068 -0.025 99599 0.86 14 7114 27.71 <0.001 0.35 >0.15 VPA MSDE 0.016 3.14E-6 -1.16E-4 5.65E-5 119547 0.83 14 8539 22.31 <0.001 0.87 <0.025 Figure 8.13. Predicted multispecies functional response surface for Pacific cod using the data set from VPA for ages 4-10. Alternative prey abundance is English sole plus rock sole abundance for fully recruited ages (6-10) estimated by VPA. Abundance is in thousands of fish. Parameter estimates and other statistics are given in Table 8.7. c and d parmeters of the MSDE and linear models were opposite in sign for all data sets (Table 8.7). Thus, the linear and MDSE models predicted similar qualitative forms of multispecies responses. The form of the multispecies models was generally consistent with the form found for the single species models which also showed very little evidence for an asymptote (cf. Figs. 8.1 and 8.13). The fits of the multispecies models 252 had slightly higher r s than the fits of the single species models in some cases (cf. Tables 8.1 and 8.7). However, because of differences in data sets (i.e. the multispecies models were fit to 196 L78 vs. 1961-85 for the single species models), the relative merits of single species vs. multispecies models cannot be determined. English sole Both the MDSE and linear model fit the English sole data sets for fully 2 recruited ages better than data sets for all ages (0.41 S0.86, P<0.051 vs. 0.08sr2<0.17, P^0.45; Table 8.8). The MDSE fit better for 3 out of 4 data sets (Table 8.8). Both models fit the VPA data sets better than the CAGEAN data sets (Table 8.8). In all cases, the distributions of residuals were not significantly different from normal (Table 8.8). The general form of the multispecies responses for English sole was that LPE increased with increased in English sole abundance and decreased slightly as the abundance of alternative prey increased (e.g. Fig. 8.14). The 6 parameter of the linear model was positive and the c and d parameters of the MDSE and linear models were opposite in sign for 3 out of 4 data sets (Table 8.8). Thus, in most cases, the linear and MSDE models predicted similar qualitative, forms 2 for the functional response, r s for both multispecies models were higher than 2 r s for single species models for data sets that used fully recruited ages, but were similar for data sets that used all ages (cf. Table 8.2 and 8.8). 253 Table 8.8. Summary statistics for multispecies functional response model fits to English sole data sets. From left to right are the data source (age range and method), model type (Y=T denotes fitting data with mean Y), the parameter values a, b, c and d, the residual sums of squares (SSE), the coefficient of determination (r X the degrees of freedom (df; n=18), the residual mean square error (MSEX the F statistics for the significance of the fits and corresponding P values and for tests of the normality of residuals, Stephens' (1974) modified A statistic and corresponding P value. Data source model Parameters b SSE df MSE Normality modified P A 2 P ages 6-10 CAG linear MSDE 72.44 0.19 0.067 5.50E-4 -0.042 4.66E-4 0.040 -5.31E-4 74411 43538 32367 0.41 0.57 17 14 14 4377 3110 2312 3.31 6.06 0.051 0.007 0.61 >0.05 0.39 >0.15 ages 6-10 VPA Y=T linear MSDE 67.20 0.20 0.045 3.93E-4 -0.066 732E-4 0.083 -5.42E-4 138620 32644 19281 0.76 0.86 17 14 14 8154 2332 1377 5.97 28.88 0.008 <0.0001 0.34 0.34 >0.15 >0.15 ages 3-17 CAG Y=7 linear MSDE 320.75 8386. 0.021 17.85 -0.011 6.04 -3.89E-3 -1.44 314730 282690 290714 0.10 0.08 17 14 14 18514 20192 20765 0.53 0.67 0.33 >0.15 0.39 0.76 0.42 >0.15 ages 3-17 VPA Y=Y linear MSDE 380.37 188.13 -0.017 0.71 -0.010 0.10 0.038 -0.42 628290 559770 522798 0.11 0.17 17 14 14 36958 39984 37343 0.57 0.94 0.64 0.45 0.59 >0.10 0.61 >0.05 Figure 8.14. Predicted multispecies functional response surface for English sole using the data set from V P A for ages 6-10. Alternative prey is cod abundance plus rock sole for fully recruited ages (4-10;6-10) estimated by V P A . Abundance is in thousands of fish. Parameter estimates and other statistics are given in Table 8.8. Rock sole Multispecies models fit rock sole data set reasonably well (0.37<r <0.79, P<;0.08; Table 8.9). The M S D E fit better than the linear model for 3 out of 4 data sets and fits of both models to data sets that used fully recruited ages 255 Table 8.9. Summary statistics for multispecies functional response model fits to rock sole data sets. From left to right are the data source (age range and method), model type (Y=Y denotes fitting data with mean Y), the parameter values a, b, c and <t the residual sums of squares (SSE), the 2 coefficient of determination (r \ the degrees of freedom (df; n = 18), the residual mean square error (MSE), the F statistics for the significance of 2 the fits and corresponding P values and for tests of the normality of residuals, Stephens' (1974) modified A statistic and corresponding P value. Data source model Parameters b SSE df MSE Normality modified P A2 P ages 6-10 CAG Y=Y linear MSDE -131.08 0.087 0.39 -3.49E-4 -0.066 9.59E-5 -0.037 -5.32E-5 1269200 492500 381942 0.61 0.70 17 14 14 74659 35179 27282 7.36 0.003 1.70 <0.01 10.84 0.0006 2.50 <0.01 to tn OS ages 6-10 VPA ages 4-15 CAG Y=Y linear MSDE Y=T linear MSDE 90.59 -1384. 528.74 42.11 0.69 3.07 0.14 -0.056 -0.28 -12.36 -0.019 8.31E-3 -0.28 0.47 -0.082 0.10 506880Q 1777800 1073768 2385400 1391000 1386544 0.65 0.79 0.42 0.42 17 298165 14 126986 14 76698 17 140318 14 99357 99039 14 8.64 0.002 0.92 <0.025 17.36 <0.0001 1.14 <0.01 3.34 0.05 1.04 <0.01 3.36 0.05 1.11 <0.01 ages 4-15 VPA Y=Y linear MSDE 489.94 -8.53E-3 0.27 -8.14E-4 -0.020 3.21E-5 -0.16 5.70E-4 5860900 3670600 2546866 0.37 0.57 17 344759 14 262186 14 181919 2.78 0.08 134 <0.01 6.07 0.007 1.64 <0.01 •were better that fits to data sets that included all ages (Table 8.9). Fits to VPA data sets were better than fits to CAGEAN data sets with one execption (the linear model fit to data from all ages; Table 8.9). As found for single species responses, distributions of residuals from fits of both models to rock sole data sets were significantly non normal (P< 0.025; Table 8.9). The general form of multispecies response found for rock sole was that LPE increased with increases in rock sole abundance but decreased with increases in alternative prey abundance (e.g. Fig. 8.15). The qualitative form of the response was consistent for all data sets. The 6 parameter of the linear model was positive for all data sets and the c and d parameters of the MSDE and linear models were opposite in sign for 6 of 8 comparisons (Table 8.9). Thus in mosts cases, the MSDE and linear models predicted similar qualitative forms of 2 response. For a given data set, r values were consistently higher for the multispecies models when compared to single species models (cf. Tables 8.3 and 8.9). Multispecies Mortality Curves Pacific cod Both of the multispecies mortality models generally fit the cod data sets better than the single species models (0.182:r2<0.58, Table 8.10; cf. Table 8.4). 2 " For a given data set, the MSDE and linear models had similar r s. For a given catch-at-age method, fits were better for data sets using all ages than for data sets using fully recruited ages, and for a given age range, fits were better for VPA data sets than for CAGEAN data sets (Table 8.10). 257 Table 8.10. Summary statistics for multispecies mortality model fits to Pacific cod data sets. From left to right are the data source (age range and method), model type (Y= Y denotes fitting data with i mean Y), the parameter values a, b, c and d, the residual sums of squares (SSE), the coefficient of determination (T\ the degrees of freedom (df; n= 18X the residual mean square error (MSE), the F statistics for the significance of the fits and corresponding P values. Data Parameters source model a b c d SSE 2 i df MSE F P Ages Y=Y 1.80 17 0.11 4-10 linear 0.79 1.28E-4 -2.12E-4 1.O0E-4 1.43 0.20 14 0.10 1.20 0.35 CAG MSDE 0.92 -6.49E-5 2.88E-4 -1.25E-4 1.48 0.18 14 0.11 1.01 0.42 Ages Y=Y 3.66- 17 0.22 4-10 linear 0.76 -5.98E-5 -3.47E-4 4.40E-4 2.16 0.41 14 0.15 3.24 0.05 VPA MSDE 0.81 9.16E-6 4.25E-4 -3.61E-4 1.72 033 14 0.12 5.27 0.01 Ages Y=Y 0.30 17 0.02 2-10 linear 0.08 1.01E-5 1.11E-5 2.62E-5 0.21 0.29 14 0.02 1.93 0.17 CAG MSDE 0.23 -7.69E-6 -2.76E-5 -3.00E-5 024 0.22 14 0.02 129 0.32 Ages Y=Y 0.45 17 0.03 2-10 linear ' -4.91E-3 3.61E-7 1.00E-4 -4.10E-5 0.22 0.52 14 0.02 5.02 0.01 VPA MSDE 0.18 1.44E-5 -1.64E-4 6.43E-5 0.19 0.58 14 0.01 6.33 0.006 Figure 8.15. Predicted multispecies functional response surface for rock sole using the data set from V P A for ages 6-10. Alternative prey abundance is cod plus English sole abundance for fully recruited ages (4-10; 6-10) estimated by V P A . Abundance is in thousands of fish. Parameter estimates and other statistics are given in Table 8.9. The multispecies mortality curve for cod was a flat plane over most of the range of cod and alternative prey abundances for 3 out of 4 data sets (e.g. Fig. 8.16). However, for the V P A data set from all ages, catchability (q) increased sharply when very small cod abundances were coupled with very large abundances of alternative prey, and for the C A G E A N data set from fully recruited ages, q increased sharply as cod abundance increased and q decreased 259 Figure 8.16. Predicted multispecies mortality response surface for Pacific cod using the data set from VPA for ages 4-10. Alternative prey abundance is English sole plus rock sole abundance for fully recruited ages (6-10) estimated by VPA. Abundance is in thousands of fish. Parameter estimates and other statistics are given in Table 8.10. as the abundance of alternative prey increased. Note, however that the fit of the MDSE mortality model was not significant for the latter data set (P=0.42; Table 8.10). The b parameter of the linear model was small and the c and d parameters of the MSDE and linear model were opposite in sign for all data sets (Table 8.10). Thus, the MSDE and the linear model predicted similar qualitative forms of response. The form of the multispecies mortality response is 260 also consistent with the nearly linearly increasing multispecies functional response (Fig. 8.13) and with the single species mortality curves (cf. Figs. 8.7 and 8.13). 2 The r s for the multispecies mortality models were higher than those for the single species models for 3 out of 4 data sets (cf. Tables 8.4 and 8.10). 2 Improvement in r values was particularly striking for the VPA data sets 2 2 (singles species models: 0.0<r <0.03 vs. multispecies models: 0.41:Sr <0.58; Tables .8.4 and 8.10). . , English sole Both multispecies mortality models fit the English sole data sets much 2 better than the single species models (0.21 <r <0.94, Table 8.11; cf. Table 8.5). The MSDE fit better than the linear model except for the VPA data set for all 2 ages (where the r s were identical; Table 8.11). Fits of both models to data sets that used fully recruited ages were better than fits to data sets that used all ages (Table 8.11). The MSDE mortality curves for data sets that used fully recruited ages predicted that catchability decreased slightly with increases in English sole and alternative prey abundance (e.g. Fig. 8.17). However for data sets that used all ages, q decreased sharply with increases in English sole abundance for both VPA and CAGEAN data sets but was insensitive to changes in the abundance of alternative prey for the VPA data set and decreased sharply with increases in alternative prey for the CAGEAN data set. The b parameter for the linear model was negative for all data sets and the c and d parameters for the linear and MSDE were opposite in sign for data sets that used fully recruited ages, but had the same sign for data sets that used all ages. Thus, the linear model 261 Table 8.11. Summary statistics for multispecies mortality model fits to English sole data sets. From left to right are the data source (age range and methodX model type (Y =Y denotes fitting data with mean Y), the parameter values a, b, c and d, the residual sums of squares (SSE), the 2 coefficient of determination (r X the degrees of freedom (df; n = 18), the residual mean square error (MSE), the F statistics for the significance of the fits and corresponding P values. Data source model a Parameters b c d SSE r2 df MSE F P Ages Y=7 21.10 17 1.24 6-10 linear 2.47 -5.62E-4 -7.83E-4 8.66E-4 11.30 0.46 14 .81 4.05 0.03 CAG MSDE 2.36 5.24E-4 5.82E-4 -6.32E-4 4.38 0.79 14 .31 17.80 <0.001 Ages Y=T 36.58 17 2.15 6-10 linear 237 -1.22E-3 -1.14E-3 1.68E-3 8.62 0.76 14 .62 6.76 0.005 VPA MSDE 3.15 727E-4 8.80E-4 -7.12E-4 2.27 0.94 14 .16 38.77 <0.001 Ages . Y=T 3.83 17 023 3-17 linear 1.70 -1.10E-4 -3.07E-5 4.74E-6 2.89 024 14 021 131 026 CAG MSDE -0.45 -3.14E-4 -3.36E-5 6.60E-5 1.31 0.66 14 0.09 9.02 0.001 Ages Y=Y 4.66 17 027 3-17 linear 1.48 -1.67E-4 -2.32E-5 1.15E-4 3.67 021 14 0.26 1.26 0.33 VPA MSDE -0.063 -2.32E-4 -1.89E-5 5.98E-5 3.70 021 14 026 1.21 0.34 Figure 8.17. Predicted multispecies mortality response surface for English sole using the data set from V P A for ages 6-10. Alternative prey is cod plus rock sole abundance for fully recruited ages (4-10) estimated by V P A . Abundance is in thousands of fish. Parameter estimates and other statistics aregiven in Table 8.11. and M S D E predicted the same qualitative form of response for data sets that used fully recruited ages, but they predicted opposite effects of alternative prey abundance for data sets that used all ages. In most cases, the form of the M S D E mortality curve was consistent with the single species fits (cf. Figs. 8.17 and 8.9). 263 Rock sole As found for the other 2 species, multispecies mortality models fit the data sets for rock sole better than single species models (Table 8.12; cf. Table 8.6). Fits were better for data sets from fully recruited age ranges than for data sets using all ages for a given catch-at-age method and fits were better for VPA data sets than for CAGEAN data sets (Table 8.12). Over most of the range of observed abundances the predicted mortality curve for rock sole was nearly flat, indicating density independent q (e.g. Fig. 8.18). However, the form of response varied among age ranges near the edge of the range of observed abundances for both rock sole and alternative prey. For data sets that used fully recruited ages, mortality curves predicted that q increased in response to increases in rock sole abundance and either decreased (VPA) or remained the same (CAGEAN) with increases in alternative prey abundance (e.g. Fig. 8.18). For data sets that used all ages, mortality curves predicted that q decreased with the abundance of alternative prey and increased with increases in rock sole abundance. The b parameter of the linear model was positive for all data sets, but the c and d parameters of the MSDE and linear models varied in sign. Thus, the linear model and MSDE predicted similar qualitative effects of rock sole abundance on q, but the predicted effects of alternative abundance on q varied among models and data sets. The multispecies mortality curves are consistent with the functional response curves because q is essentially the slope of LPE vs abundance (cf. Figs. 8.18 and 8.15). Over most of the range of abundances the multispecies and single species mortality curves are consistent as both predict relatively flat or slightly 264 Table 8.12. Summary statistics for multispecies mortality model fits to rock sole data sets. From left to right are the data source (age range and method), model type (Y= Y denotes fitting data with mean Y), the parameter values a, b. c and d. the residual sums of squares (SSE), the coefficient of determination IA the degrees of freedom (df; n= 18), the residual mean square error (MSE), the F statistics for the significance of the fits and corresponding P values. Data Parameters source model a b c d SSE 2 r df MSE F P Ages Y = Y 49.23 17 2.90 6-10 linear 0.42 1.82E-3 -2.77E-4 -1.85E-4 32.08 0.35 14 2.29 2.49 0.10 CAG MSDE 1.15 -3.26E-4 5.16E-5 -5.61E-5 31.98 0.35 14 2.28 2.52 0.10 Ages Y = Y 134.35. 16 8.40 6-10 linear 0.49 3.67E-3 -1.44E-3 -1.21E-3 50.88 0.62 13 3.91 7.11 0.005 VPA MSDE 1.22 -3.71E-4 1.09E-4 2.95E-5 30.29 0.77 13 2.33 14.89 <0.001 Ages Y=Y 16.53 17 0.97 4-15 linear 2.71 2.12E-4 -6.34E-5 -2.62E-4 11.30 0.32 14 0.81 2.16 0.14 CAG MSDE -7.89 9.36E-4 -1.88E-4 -1.66E-3 11.42 0.31 14 0.82 2.09 0.15 Ages Y=Y 29.38 16 1.84 4-15 linear 3.07 5.96E-4 -3.35E-5 -6.66E-4 19.42 0.34 13 1.49 2.22 0.13 VPA MSDE 28314 -10.59 -0.22 13.08 R82 0.39 13 1.37 2.81 0.08 Figure 8.18. Predicted multispecies mortality response surface for rock sole using the data set from VPA for ages 6-10. Alternative prey abundance is cod plus English sole abundance for fully recruited ages (4-10;6-10) estimated by VPA. Abundance is in thousands of fish. Parameter estimates and other statistics are given in Table 8.12. increasing curves (cf. Figs. 8.18, 8.12). The improvement in fit of multispecies models over single species models is particularly striking for the data set shown in figure 8.18 (i.e. ages 6-10 for VPA) where the best fit single species model O had an r of 0.07 compared with 0.77 for the MSDE. However, it should be noted that because estimates of cod abundance were not available, the multispecies data set did not include the period 1956-60 that was a period of 266 rather violent fluctuations in rock sole catchability (Fig. 7.5). Tests of the Switching Hypothesis Switching by fishermen between cod and flatfish is one potential mechanism that could cause type HI responses in the Hecate Strait fishery. 2 However, none of the fits of switching curves were significantly nonlinear (r s ranged from 0.05-0.68; Table 8.13). The fit to one data set was significantly nonlinear at P=0.08 (cod and rock sole, fully recruited ages from CAGEAN; Fig. 8.19; Table 8.13). Power was low for all data sets (<0.54; Table 8.13). I failed to detect any evidence of heteroskedasticity or non normality in the residuals from 10 of the 12 data sets (Table 8.13). The overall results do not provide evidence for switching. However, as indicated by the low power, variability in data and or small effect sizes resulted in weak tests of the switching null hypothesis in most cases. Summary of Results The major results from my investigation of functional responses in Hecate Strait are: 1. A generalized equation capable of mimicking response types I-IV resulted in a type III (sigmoid) functional response for 11 of the 12 data sets. 2. Five comparisons of alternative single species models found statistically significant differences; all five indicated that a sigmoid response was most consistent with the data. 3. In many cases, the best fit disc equation (type n- response) was linear, not saturating, and for cases where a saturating response 267 Table 8.13. Summary of tests of the switching hypothesis. From left to right are the data source (speciesl and age range/species 2 and age range), the catch-at-age method, the estimate of alpha, the degress of freedom (df; n = df+2), the estimate of beta and its standard error, the t value and 2 the probability (P) for test of H : beta l^, the power to the t-test and the coefficient of determination ( r ) . Also given are for tests the normality o of residuals, Stephens' (1974) modified A statistic and corresponding P value ,and for tests of homoskedasticity of residuals, Kendall's tau and b corresponding P value. Normality modified Homoskedasticity Data source method alpha df Beta SE(beta) t P(^ 1.0) Power 2 T A2 P tau b P Cod410/Eng610 CAG 0.76 17 1.11 0.22 0.50 0.31 0.12 039 0.32 >0.15 -0.26 >0.12 Cod210/Eng317 CAG 0.87 17 0.99 0.43 -0.01 0.51 0.05 024 0.43 >0.15 -0.12 >0.49 Cod410/roc610 CAG 0.86 16 1.34 . 023 1.45 0.08 0.41 0.68 0.21 >0.15 -0.06 >0.77 Cod210/roc415 CAG 0.73 16 1.05 0.39 0.12 0.45 0.06 0.31 0.28 >0.15 -0.14 >0.45 Cod410/Eng610 VPA 0.81 17 0.92 0.20 -0.38 0.65 0.10 035 0.31 >0.15 -0.18 >0.30 Cod210/Eng317 VPA 1.29 17 0.43 0.48 -1.19 0.87 0.30 0.05 0.31 >0.15 -0.23 >0.19 Cod410/roc610 VPA 0.97 16 1.30 0.36 0.83 0.21 020 0.44 0.73 <0.05 0.19 >029 Cod210/roc415 VPA 0.59 16 1.18 0.36 030 , 0.31 0.12 0.40 0.47 >0.15 -0.07 >0.71 Eng610/roc610 CAG 14.01 21 037 024 -1.81 0.96 0.54 0.21 0.51 >0.15 0.38 <0.01 Eng317/roc415 CAG 0.42 21 0.72 031 -036 0.71 0.13 0.08 026 >0.15 -0.08 >0.64 Eng610/roc610 VPA 13.60 21 036 0.31 -1.46 0.91 0.39 0.14 0.65 >0.05 020 >0.19 Eng317/roc415 VPA 0.37 21 0.77 0.45 -031 0.69 0.13 0.12 0.27 >0.15 -0.08 >0.64 o O <u o o o o o o o CSI 0 .5 1 1.5 2 2.5 3 (Cod abundance)/ ( rock sole abundance) ( N 1 / N 9 ) Figure 8.19. Predicted switching curves for data set from CAGEAN for cod and rock sole and fully recruited ages (4-10;6-10). Squares are observations. Dotted line is the line of no preference (i.e. where a and 0 = 1.0). Statistics for the fit of the curve are given in Table 8.13. was indicated, it did not have a statistically better fit than either a linear (type I response) or sigmoid model. 4. The single species mortality models fit most data sets poorly, although in most cases for a given data set and model, the qualitative form of the response was consistent with the corresponding functional response. 5. The statistical power of the tests of alternative single species functional response models and of tests of significance of the slope in the linear single species mortality models was low in all cases where I failed to reject the null hypotheses. 269 6. Multispecies functional response and mortality models often resulted in better fits than single species models for all species. 2 In some cases, the increase in r s were quite large. However, formal statistical comparisons of multispecies and single species models were not possible because of differences in the data sets used by the two types of models. 7. Switching was not detected in any of the data sets, although power was low in most cases. Discussion Outstanding Issues Statistical Issues I examined residuals to test for violations of two principal assumptions of my least squares fits (normality and homoskedasticity), however there are other assumptions that were not investigated. Violations of these other assumptions such as (1) measurement errors in abundance estimates (i.e. the X variate; Ludwig and Walters 1981; Walters and Ludwig 1981), (2) autocorrelated residuals (Glasbey 1980), or (3) residuals correlated with subsequent levels of the X variate (i.e. the so called "time series" bias; Walters 1985), or (4) combinations of (1) and (3) above (Caputi 1988) may lead to biased estimates of functional relationships, and/or imprecise parameter estimates. Abundance estimates from VPA and CAGEAN are certainly imprecise and possibly biased under certain conditions, particularly if input parameters are in 270 error (reviewed in Ch. 7). Thus, I restricted my, statistical comparisons to the functional response models, because the mortality curves are particularly sensitive to measurement errors since errors in abundance affect both variates (Shardlow et al. 1985). For simple linear regression, it is possible for a given set of model parameters and MSE to determine how imprecise the abundance estimates would have to be in order to change particular conclusions regarding the functional relationship (e.g. Shardlow et al. 1985). For nonlinear fits, estimating the effect of measurement error requires extensive Monte Carlo trials, preferably with "fake data" where the correct parameters are known (e.g. Walters and Ludwig 1981). However, Walters and Staley (1987) note that using Monte Carlo simulation to derive correction factors for actual data sets does not always completely correct for the bias. I did not perform such simulations, but drawing analogy from measurement error studies of stock-recruitment relationships, I suspect that measurement errors would cause an overestimate of slope parameters (i.e. the rate of effective search) and an underestimate of asymptote parameters (1/t^ ; i.e. overestimate of handling time). I did not examine the residuals from alternative model fits for autocorrelation. For linear regressions, standard corrections are available that generally involve transforming the data (by differencing) to correct for autocorrelation (e.g. Kmenta 1986), and similar procedures are used for nonlinear least squares problems (e.g. Glasbey 1980, Bates and Watts 1988). However, failing to account for autocorrelated deviations does not generally result in biased estimates, but rather incorrect estimates of the precision of parameter estimates (Kmenta 1986). In particular, Glasbey (1980) notes that failing to correct for positively autocorrelated deviations in nonlinear least squares applications may lead to parameter variance estimates that are too small. Thus, correcting for 271 autocorrelation might have changed the P values and power of the f-tests used to test certain parameters (e.g. b in the linear single species mortality model, or j3 in the switching model). However, given that power of these tests was generally very low a P values were generally quite high, precision would have to change substantially in order to affect the results. My tests for homoskedasticity did not identify many cases where the ranks of the absolute value of residuals were correlated with the ranks of the X variates (i.e. abundances). Furthermore, it is unlikely that deviations in LPE would cause deviations in abundance, except perhaps for long lived stocks subject to severe overexploitation (e.g. through recruitment overfishing). Thus, while the tests for homoskedasticity do not provide the exact evidence necessary to discount a potential time series bias, I suspect that the effect of the time series bias was small in my case. Finally, as demonstrated by Caputi (1988), effects of combinations of measurement errors and time series biases may interact. Therefore, he suggests that the two sources of error should be evaluated simultaneously through carefully designed simulations in order to determine the overall effect. In my tests of goodness of fit to alternative functional responses, I assumed that errors were normally distributed. In some cases, particularly for fits to the rock sole data sets, the test for normality showed that there was only a small chance (i.e. P<0.01) that the observed residuals could have been drawn from a normal distribution. Furthermore, other studies (e.g. Bannerot and Austin 1983; Sweirzbinski 1985) suggest that observations of L/E are probably distributed lognormally or as a negative binomial. Thus, assumptions of the least squares techniques may have been met more consistently if I log transformed the 272 data. Indeed, this possibility was confirmed in my tests of residuals from fits of Fox's (1974) loglinear model (i.e. eq. 6.10 fit by eq. 6.12, Ch. 6; results not reported here), where I found no evidence for non normality for the rock sole data sets, and only one case of significant non normality (for English sole) for the remaining eight data sets. However, the Fox model does not permit comparisons of type III vs. either type I or type H responses. Biological Issues My functional response models ignored potential interference effects. There are various mechanisms that could cause interference effects in fisheries (e.g. disinformation). However, I suspect interference effects to be most important over short time scales (i.e. days or possibly weeks) and small spatial scales, not over Hecate Strait as a whole on an annual time scale. Given the large variability about most of the fitted functional relations, it is worth considering some factors that may have contributed to the variability. Some of the variation in annual LPE may simply be due to individual variation in LPE among vessels (e.g. Hilborn and Ledbetter 1985). However, I suspect that the effect of individual variation should to some extent average but over large spatial and temporal scales (e.g. for years and areas the size of Hecate Strait). \ Ironically, the same factors (large spatial and annual temporal scale) that may have reduced the effects of individual variation, may introduce variability from other sources. For example, spatial changes in fleet distribution within Hecate Strait among years would cause variation in LPE and q. Similarly, intra-annual changes in fleet distributions for a given subarea within Hecate 273 Strait could also cause variability in LPE and q. Another potential cause of some of the variation may be nonrepeatability perhaps due to nonstationarity of the parameters . of the functional relationships. Thus, the paramaters of the functional responses may have changed over the time series resulting in different observed LPE or q values for the a given abundance level (Walters 1986, 1987b). Evidence for nonrepeatability is presented in Figure 8.20. Nonrepeatability is most evident in the plots for English sole LPE and q; note in particular, the difference in the response in the early 1960's compared the rest of the time series (Fig. 8.20). The plots for cod show evidence for nonrepeatability, particularly at low abundance levels. The plots for rock sole LPE and q appear to show good repeatability with a few outliers (i.e. '58, '61, 67'). Nonrepeatability may have resulted from a variety of factors including incompletely standardized effort data, long term shifts in the distributions of fish, or changes in fleet composition in Hecate Strait due to changes in alternative fishing opportunities. In the case of the English sole, some of the nonrepeatability is probably due to affects of cod abundance For example, the; decrease in English sole LPE and catchability during the early 1960's occured when English sole abundance was decreasing and cod abundance was increasing (Fig. 8.20; Fig. 7.4, p. 185). Also, fitting a multispecies model that included cod abundance considerably improved the fit of both functional response and mortality curves. Errors in estimated abundances (discussed above) may also be responsible for some of the observed variability about functional relationships (e.g. q vs abundance; Shardlow et al. 1985). 274 Catch per effort (LPE; n/hr) Catchability coefficient ( q x l O ) Pacific cod ages 4-10 VPA 60-40-200-2 ' i 1 4 English sole ages 6-10 CAGEAN 8 o 1200-1 800 400 rock sole ages 6-10 CAGEAN 58 61 8H 67 6 4-2-0 Abundance (millions) Figure 8.20. Times series plots of LPE and Catchability (q) vs. abundance for selected data sets for each species. Numbers above points denote years, and points for each year are connected sequentially in chronological order. "Loops" of years for LPE and q indicate nonrepeatability in the functional and mortality responses respectively. Data sets used are indicated above each plot. 275 B o t h catch-at-age models t h a t I used to r e c o n s t r u c t a bundances a s s u m e d th a t n a t u r a l m o r t a l i t y (M) was constant. However, i f M changed over time, or p a r t i c u l a r l y i f M was density dependent, a p p a r e n t r e l a t i o n s h i p s between L P E and abundance a n d q and abundance could be a r t i f a c t s . E v i d e n c e for den s i t y dependent M has been found for P a c i f i c cod su g g e s t i n g t h a t M in c r e a s e s w i t h d e c r e a s i n g abundance ( F o u r n i e r 1983). I f M is depensatory, m y e s t i m a t e s of abundance would be un d e r e s t i m a t e s for low ab u n d a n c e s a n d o v e r e s t i m a t e s for hi g h abundances. T h u s , h a d I found evidence for t y p e II responses i n cod, i t could h a v e been a r e s u l t of i n c o r r e c t l y a s s u m i n g a con s t a n t M. However, a l l type II fits were l i n e a r for cod, a n d the o n l y s i g n i f i c a n t difference found suggests t h a t of the three a l t e r n a t i v e s the t y p e III response is most evident. If the re l a t i o n s h i p between M a n d d e n s i t y was dome shaped, the sigmoid response m i g h t be a n a r t i f a c t of u s i n g a con s t a n t M. However, there is no evidence for a dome shaped M p a t t e r n i n P a c i f i c cod. F u r t h e r m o r e , there is no evidence t h a t M is den s i t y dependent for eit h e r of the f l a t f i s h species. O n e of the most consistent p a t t e r n s i n the resu l t s f r o m fitting the fu n c t i o n a l response models was the lack of consistent s t a t i s t i c a l evidence for a n as y m p t o t e i n either the single or mult i s p e c i e s fits. I n order to u n d e r s t a n d the potent i a l reasons for this f i n d i n g , i t is n e c e s s a r y to r e c a l l the potential m e c h a n i s m s t h a t lead to s a t u r a t i o n i n the f u n c t i o n a l response. I n the case of single species models, the a s y m p t o t e a r i s e s p r i m a r i l y because of the effects of h a n d l i n g time on s e a r c h i n g time. H o w e v e r , the effort m e a s u r e I used, hours trawled, does not include h a n d l i n g time, a n d i t m a y even be r o u g h l y proportional to s e a r c h i n g time. T h u s , I m a y h a v e foun d greater evidence for a n as y m p t o t e i n the f u n c t i o n a l response h a d I used a diff e r e n t 276 effort measure, such as days fished. In the case of multispecies models, a saturating response could arise through handling time affects as well, but also through the effect on search time of switching to alternative prey. For example, if (1) total foraging time was composed of time spent foraging for two species (si and s2; i.e. T = T j+T^) and (2) LPE was measured as L/T and (3) fishermen spent more time searching for species 1 as its abundance increased, then the time available for searching for species 2 would decline. Thus a saturating response could arise for species 2, if species 1 was more abundant than species 2, and the two species had similar temporal trends in abundance (e.g. cod and English sole). However, I measured LPE as L/T^. , thus reducing my chances of seeing a asymptote in the multispecies responses. My main reason for chosing hours trawled allotted by species as the effort measure was that it is the effort measure used by fisheries biologist and managers and I wanted to consider the potential implications of functional responses on assessment and management. Some of these implications are presented in the next section. General Implications Several studies have reviewed the various consequences of type II functional responses for fisheries (e.g. Ulltang 1980; Peterman and Steer 1981; Crecco and Savoy 1985; reviewed in Ch. 6). I did not find any quantitative evidence for type H responses in the LPE data for the reasons mentioned above. Thus my discussion focuses on two areas; (1) potential implications of type III responses, and (2) the implications of multispecies responses particularly for less 277 abundant species in a multispecies fishery Implications of Type III Functional Responses Most investigations of functional responses in fisheries (reviewed in Ch. 6) have focused on "single" species fisheries, and therefore most studies have not tested for, or considered the possibility of sigmoid functional responses. Of the studies reviewed in Chapter 6, only Peterman (1980) fit an equation that could potentially result in a type III response, but* he found parameter conditions consistent with type II response. I found statistical evidence for a type III response for 3 of 12 data sets, however a type III functional responses can be expected for multispecies fisheries particularly if fishermen switch their effort among species. I failed to detect switching in the trawl fishery. However, my tests of the switching hypothesis had low power and given the various mechanisms for switching in fisheries reviewed in Chapter 6, I suspect that multispecies fishermen are switching predators. As noted by Murdoch and Oaten (1975), switching commonly but not always leads to a sigmoid functional response. Sigmoid functional responses have implications for several aspects of fisheries including: (1) interpretation of abundance indices based on LPE and the potential management response based on such indices, (2) equilibrium yield vs abundance relationships used to determine maximum sustainable catches, and (3) simulation models used to investigate alternative management strategies. If a manager assumed a linear functional response when the correct response was sigmoid, his LPE indices would overestimate abundance at low stock sizes and underestimate abundance at high stock sizes (e.g. Fig. 8.1 B; 278 data set from CAGEAN ages 2-10). Thus, the manager might tend to be overly optimistic when abundance was low and overly conservative when abundance was high. Fortunately, the mortality curve would compensate for potential over- or under- harvest, particularly for management tactics base on regulating effort, because the manager would tend to overestimate catchability at low stock sizes and underestimate it at high stock sizes (e.g. Fig. 8.7 B; data set from CAGEAN ages 2-10). Fisheries yield models that assume logistic growth and a type I mortality curve for fishermen usually predict a dome shaped relationship between yield and population size or yield and effort (Ricker 1975). In the case of type II responses, the dome may be positively skewed or even bend back toward the origin (Fox 1974; Condrey 1984). The domed shape arises because population models that assume logistic growth and density independent mortality (i.e. a type I mortality curve) have two equilibria: zero, and the natural equilibrium corresponding to resource limitation as some life stage (e.g. P^, Fig. 8.2 IA; Peterman et al. 1979). However, for type ni mortality curves two intermediate potential equilibria may be added (Peterman 1977, Peterman et al. 1979), (e.g. Fig. 8.21B). The points Pj and P^ are points of attraction (or equilibria) since population sizes above these levels will tend to decrease (since N^^/N^ < 1.0) and populations below these levels will tend to increase. The point P^ is unstable for the opposite reasons. The area between points P^ and P^ is known as a predator pit, because if the population decreases to a level between points P^ and P^, it will tend to remain there until more favorable than average survival or reproductive conditions allow it to escape. Note also that the size of the pit (i.e. the distance between 279 0 1 1 1 1 1 1 1 2 3 4 5 6 N t (Population in mill ions) B N t (Population in mill ions) Figure 8.21. Schematic plots population growth rate (N /N ) vs. population size for two hypothetical fish populations. Population in A has one stable equilibrium point (P^) cause by limitations at high abundances (space, food). Population in B has in addition to P , a lower stable equilibrium point P_ and Jl o an intermediate unstable equilibrium point, Pg. The area beneath the 1:1 line between points Pg and Pg is called the predator pit and it is caused by depensatory mortality at intermediate N. 280 and P )^ can be affected by harvesting because N ^ ^ / N is decreased due to the fraction removed by harvesting (Peterman et al. 1979). Thus, the potential for the population to become trapped at low levels increases as harvest rate increases. The potential for a predator pit in fish populations caused by other fish predators was suggested by Peterman (1977), but if fishermen have sigmoid functional responses, they may cause predator pits themselves or act as another component in affecting existing pits caused by other predators. Type III functional responses also have implications for simulation models of fish populations used to explore alternative harvest strategies. If the true response is type III rather than a type I response (i.e. Baranov Catch equation) there will be a greater likelihood of a stock collapse to some low level at high harvest rates because of the predator pit discussed above. Thus, simulations would underestimate the probability of stock collapse or the probability of no fishing (i.e. proportion of simulations the stock falls below some arbitrary cutoff) if they ignore the response pattern. One of the barriers to including type III responses in fisheries models is the lack of an appropriate equation. The various equations I presented for a type III response in Chapter 6 were all instantaneous equations, thus they ignore the effects of prey depletion. The derivation of exploitation models for type III responses is difficult and generally the resulting equations are quite complicated (e.g. see exploitation model for the generalized equation; eq. 36, Fujii et al. 1978). Furthermore, the resulting equations commonly involve NA (i.e. Catch) on both sides and thus must be. solved iteratively. And finally, even if these problems could be overcome, parameter estimates would have to be determined (i.e. a, t^ , c) using the exploitation model because using estimates from fitting the instantaneous form would tend to overestimate the catch. 281 However, in cases where both exploitation and instantaneous models have been fit to the same data sets, they each describe the data equally well (e.g. Rogers 1972). Thus, for the purposes of simply investigating the implications of type III responses on hypothetical populations, instantaneous equations are probably adequate (e.g. Peterman 1977). Implications of Multispecies Functional Responses A common problem in multispecies fisheries assessment is how to determine effective fishing effort for individual species when they are caught together (e.g. Westrheim 1983). The goal is to find an effort measure that results in a LPE index that is approximately proportional to abundance (or is some function of abundance). The problem is particularly acute for species that are not the major target, but generally are bycatch or are sought after when the major target is in short supply (e.g. Westrheim et al. 1988). The poor fits of single species functional response models for the flatfish and'the considerable improvement of in fit of the MSDE (especially for rock sole) clearly indicate that even LPE indices based on allocated effort may depend to a large degree on the effects of alternative species. Thus, an alternative approach to dividing the effort among species would be to model LPE using the total effort (all species) and a MSDE that explicitly takes account of the effects of abundance of alternative species. It is possible to solve the MSDE for each species for abundance and use the set of three equations to predict abundance given LPE and parameter values (c, fe, c, and d) from historical data (Carl Walters, Univ. of British Columbia, Vancouver, BC, pers. comm.). 282 The potential for predator pits resulting from type III mortality curves has implication for multispecies situations as well (e.g. Peterman 1977). For example, if both the major and minor species have predator pits and the major species has a higher optimum harvest rate, harvesting at a rate appropriate for the major species may force the minor species into its predator pit. The situation could go undetected, particularly if the catch of the minor species is underestimated because of discarding. Thus the detection of type III response may be particularly critical in multispecies situations. 283 Perspectives for Future Research This section briefly considers directions for future research in the study of numerical and functional responses in fisheries. I applied one approach to examining movement patterns, hypothesis testing, and discussed three others, models of fisherman choice, search theory applications, and simulation modelling. Which of these techniques or combinations of techniques will prove valuable in the future depends on the objectives for studying movement patterns. If the objective is to improve the management of fisheries, then further hypothesis testing followed by more detailed simulation modelling may prove most fruitful in the short run because these two approaches are most easily applied to existing data. However, theoretical aspects of movement dynamics would be best advanced by pursuing the application of choice models and search theory, and these approaches may also have long term benefits to management. One of the problems inhibiting further progress by any of the approaches is the lack of data gathered specifically for the purpose of studying movement patterns. Thus one key to future advances will be to gather the appropriate data to test the hypotheses, to describe the relative benefits of alternative choices, to describe the underlying spatial distributions of fish, and to fit the simulation models. Specifically, data on fishing costs itemized within trips and for smaller areas (e.g. costs for fuel steaming between grounds, searching and trawling on grounds) and daily area-specific L P E data would be particularly valuable for examinations of within trips movement patterns. Given the influx of electronic equipment on modern fishing vessels, one possible way to collect this data would be to use "on board" computers. 284 Understanding movement patterns requires an interdisciplinary approach, but unfortunately various disciplines have generally followed separate paths; mathematicians have focused on search theory applications, economists have pursued choice models, and biologists have tested hypotheses and built simulation models. One of the challenges ahead will be to get these disciplines to work together, perhaps by focusing attention on a simulation model of a particularly data rich fishery. It is likely that fisheries scientist and managers will continue to rely on landings per unit of fishing effort as an index of the abundance of fish (at least for short term predictions). Thus, the importance of understanding functional responses in the fishery is unlikely to diminish. The fundamental issues are the choice of effort measure for use in the desired functional response model and how to include the affects of alternative prey in multispecies fisheries. Most functional response models assume random search by the predator and predict that catch per unit of time spent searching is proportional to abundance. Thus if managers want to continue to used linear functional response models, catch per unit of search time would be the ideal index. Unfortunately, fishermen probably dp not search randomly, at least not on the large spatial scales commonly used for management, and even if they did, obtaining accurate estimates of search time could be difficult. Yet if my analysis of the British Columbia trawl fishery is at all representative, assuming landings per hours trawling (the common index used in trawl fisheries) is proportional to abundance can be misleading. While fisheries scientists have certainly recognized the potential pitfalls of this assumption, there is still further research needed. I believe that detailed time budget analyses of 285 the fishing process is the next essential step toward improving the accuracy of abundance indices. Time budgets studies could easily provide estimates of handling time and, if combined with spatial mapping of the fish abundance, might even provide estamates of the rate of effective search (o; proportion of area search by one unit of fishing effort). This two parameters could be used to correct L P E indices that use aggregate effort measures such as days fished for saturation effects. Repeated measure of the rate of effective search at different abundance levels be used to develop sigmoid response models. Application of functional response models that assume non random search (e.g. Fujii et al. 1978; Mace 1983) or two stage models (i.e. non random search for patches, but random exploitation within a patch; Griffiths and Holling 1969) would also improve future studies of functional responses in fisheries. The application of multispecies functional responses may be a valid alternative to methods that attempt to allot effort to individual species in multispecies fisheries. Specifically, the multispecies disc equation can be rearranged to solve for N interms of L P E for each species and the resulting set of equations (1 for each species) can be used to correct L P E indices of a given species for the effects of the abundance of alternative species. But further research is needed, particularly in developing models to predict discarding of bycatch species. 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Age Year 2 3 4 5 6 7 8 9 10 1961 58 369 160 31 5 3 3 0 0 1962 165 223 132 65 26 8 2 2 0 1963 342 543 260 77 17 12 5 1 0 1964 1358 668 279 108 21 4 1 1 0 1965 314 2162 912 197 51 4 4 0 0 1966 384 1554 1043 305 76 10 1 1 0 1967 889 591 339 148 58 18 5 1 0 1968 113 1322 334 150 72 20 6 3 0 1969 314 185 375 96 27 " 13 2 1 1 1970 84 313 86 16 11 4 1 0 1 1971 379 138 94 44 6 . 4 3 0 0 1972 908 288 155 49 7 7 1 4 0 1973 695 811 - 277 66 37 6 2 2 1 1974 1158 465 372 122 32 5 3 2 0 1975 591 926 364 187 22 37 14 7 0 1976 498 1125 308 110 43 20 2 3 0 1977 935 543 272 50 32 10 0 0 0 1978 79 449 213 62. 9 7 2 0 0 1979 1349 345 452 106 33 6 3 0 1 1980 676 934 384 % 34 12 2 1 0 1981 407 438 436 154 22 14 0 0 0 1982 222 382 172 54 15 3 1 1 0 1983 225 432 258 88 21 6 1 0 0 1984 348 247 174 54 19 1 2 0 0 1985 75 237 73 24 13 2 1 0 0 Table A2. Estimated numbers landed-at-age (XlO) of female English sole from northern Hecate Strait (Area 5D) for 1956-79. Age Year 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1956 16 211 600 310 156 85 34 12 12 10 6 0 0 0 0 1957 12 169 140 243 123 56 24 22 12 0 0 0 0 0 0 1958 6 113 252 212 217 100 52 35 11 .3 0 3 0 0 0 1959 8 243 436 335 206 130 68 31 21 6 0 2 0 0 0 1960 37 355 488 363 231 257 82 31 16 10 2 4 0 0 0 1961 2 267 645 307 137 68 40 19 12 4 4 0 0 0 0 1962 5 129 292 152 89 28 11 11 4 0 0 0 0 0 0 1963 8 92 175 204 69 29 8 6 0 2 0 0 0 0 0 1964 16 211 270 127 63 28 5 5 " 1 1 0 0 0 0 0 1965 1 93 233 177 97 45 20 11 2 2 0 1 0 0 0 1966 2 78 190 171 89 38 15 7 0 1 1 0 0 0 0 1967 2 54 388 302 118 54 23 8 5 1 0 0 0 0 0 1968 9 26 161 454 214 90 33 2 2 0 0 0 0 0 0 1969 28 70 107 194 358 ! 132 43 15 0 2 1 0 0 0 0 1970 47 215 145 116 184 213 71 25 4 0 2 0 0 0 0 1971 31 250 136 49 50 55 66 25 8 3 0 0 0 0 0 1972 15 90 155 121 78 43 20 8 3 2 1 1 0 0 0 1973 145 183 221 101 44 34 32 16 10 0 0 0 0 0 0 1974 60 172 153 107 76 33 22 10 3 12 3 10 9 0 1 1975 35 273 381 281 199 101 60 15 3 7 . 4 2 2 0 0 1976 13 132 382 310 205 103 59 12 3 6 4 4 4 1 0 1977 92 238 325 232 147 66 37 13 2 4 2 5 2 1 6 1978 27 79 121 73 52 43 30 8 1 0 2 0 0 0 0 1979 105 200 219 135 98 71 47 14 2 3 2 2 2 0 0 Table A3. Estimated numbers landed-at-age (XlO) of female rock sole from northern Hecate Strait (Area 5D) for 1956-78. Age Year 4 5 6 7 8 9 10 11 12 13 14 15 1956 58 78 164 148 89 49 16 5 4 1 1 0 1957 47 43 87 88 67 36 21 9 2 0 0 0 1958 45 742 148 69 34 40 10 9 2 2 0 0 1959 2 133 101 15 3 2 0 0 0 0 0 0 1960 9 332 508 92 23 8 3 2 3 1 0 0 1961 0 45 498 361 66 5 5 0 0 0 2 0 1962 0 6 91 299 124 23 10 1 0 0 0 0 1963 29 62 98 228 205 78 24 4 5 0 0 0 1964 105 116 60 89 106 44 11 - 6 1 0 0 0 1965 35 102 101 65 33 17 7 4 2 1 0 0 1966 75 729 661 303 197 121 75 46 18 2 2 0 1967 39 414 354 354 116 53 19 12 2 0 0 0 1968 205 392 671 216 55 24 12 4 0 0 0 0 1969 440 362 245 138 ' 29 11 5 0 0 0 0 0 1970 9 116 247 262 79 82 24 21 2 4 0 0 1971 21 228 365 181 154 74 18 3 0 0 0 0 1972 39 114 114 72 37 19 8 4 2 1 0 0 1973 48 30 20 26 22 13 7 7 4 0 0 0 1974 153 182 14 3 0 6 6 3 0 0 0 0 1975 23 72 222 227 168 96 41 13 5 3 8 3 1976 94 153 266 75 28 13 0 0 0 0 0 0 1977 83 166 . 112 80 19 8 4 3 2 0 0 0 1978 124 109 81 72 52 34 32 14 11 10 3 1 Table A4. Fishing effort (hours trawled) for Pacific cod (areas 5C+5D), English sole (area 5D), and rock sole (area 5D). Source: Tyler et al. (1985). Year Pacific cod Species English sole rock sole 1956 1300 1538 1957 — 1653 2631 1958 — 2332 742 1959 — 2468 1099 1960 — 1505 989 1961 2312 1676 402 1962 2195 1232 754 1963 3075 3101 1287 1964 3176 2308 1102 1965 5608 3491 1255 1966 6685 5141 3244 1967 4597 2077 918 1968 14190 3144 3337 1969 9049 2571 2122 1970 5925 2402 1672 1971 6142 2708 2466 1972 3910 2211 2268 1973 3502 2265 2568 1974 4477 2318 2212 1975 8373 3415 5864 1976 10582 6757 5609 1977 9841 • 7594 8085 1978 6146 4714 3453 1979 9863 8651 — 1980 12304 — — 1981 15598 — — 1982 4933 — — 1983 6038 — — 1984 5329 — — 1985 7161 — — 310 Table A5, Age specific net fecundities. eQ (eggs), used in CAGEAN for Pacific cod. English sole and rock sole. Age Pacific cod'' Species 2 English sole rock sole^  2 120,588 3 783,921 45,000 — 4 1,510,883 250.000 125,000 5 2,192.303 595.000 245.000 6 2,849.446 1,080,000 585,000 7 3,447,840 1.500.000 850.000 8 3,957.208 1.700.000 950,000 9 — 1.850.000 1.100.000 10 — 1.950.000 1,200.000 11 2.050.000 1,300,000 1 Source: Ketchen (1961.1964). Thomson (1962) 2 Source: J. Fargo, Pacific Biological Station, Nanaimo, B.C. Table A6. "Best" estimates of starting parameter values used in VPA and CAGEAN for Pacific cod, English sole, and rock sole. Species Parameters MB q^ TXlO ) Et(hours trawled) ^B=<i^t *7 Pacific cod .68 5.79 7161 .41 .5 .5 English sole .22 4.09 8651 .35 .5 .5 rock sole .25 5.17 3453 .18 .5 .5 Table A7. Starting input parameters generated by CAGEAN's routine COHORT given best estimates of starting parameter values (see Table A6) for Pacific cod, English sole, and rock sole. ?a,t denotes population numbers (XlO ) of age a fish in year t. Fr denotes fishing mortality in year t . Sa denotes the selectivity of age a fish, q is the catchability coefficient, and a and b are spawner-recruit parameters. Species Pacific cod English sole rock sole Parameters number name value name value name value ; P8-10,61 11 Pll-17,56 105 Pll-15.56 75 2 P7.61 35 P10.56 69 P10.56 123 3 P6.61 121 P9.56 167 P9.56 233 4 P5.61 211 P8.56 399 P8.56 186 5 P4.61 734 P7.56 689 P7.56 454 6 P3.61 1736 P6.56 1135 P6.56 439 7 P2.61 5487 P5.56 2005 P5.56 426 8 P2.62 8964 ' P4.56 2149 P4.56 936 9 P2.63 12297 P3.56 2406 P4.57 2234 10 P2.64 16579 P3.57 2341 P4.58 3042 11 P2.65 8911 P3.58 2312 P4.59 5649 12 P2.66 6039 P3.59 2479 P4.60 2118 13 P2.67 7983 P3.60 1828 P4.61 1240 14 P2.68 2800 P3.61 1320 P4.62 1276 15 P2.69 7109 P3.62 1751 P4.63 2618 16 P2.70 4729 P3.63 2244 P4.64 2804 17 P2.71 5199 P3.64 2705 P4.65 3060 18 P2.72 8618 P3.65 3976 P4.66 3321 19 P2.73 6690 P3.66 2450 P4.67 2404 20 P2.74 7098 P3.67 1353 P4.68 3070 21 P2.75 7035 P3.68 1460 P4.69 1861 22 P2.76 5762 P3.69 1871 P4.70 1202 23 P2.77 6528 P3.70 1810 P4.71 2357 24 P2.78 5380 P3.71 2091 P4.72 1757 25 P2.79 7699 P3.72 2287 P4.73 1784 26 P2.80 4351 P3.73 2930 P4.74 1933 27 P2.81 4062 P3.74 2549 P4.75 1500 28 P2.82 2977 P3.75 1829 P4.76 1261 29 P2.83 1982 P3.76 1534 P4.77 1489 30 P2.84 2923 P3.77 1846 P4.78 2404 31 P2.85 756 P3.78 1754 F56 .44 32 F61 .24 P3.79 1394 F57 .55 33 F62 .38 F56 .28 F58 .68 34 F63 .52 F57 .22 F59 .09 35 F64 .34 F58 .31 F60 .28 36 F65 .44 F59 .43 F61 .34 37 F66 .57 F60 .92 F62 .22 312 38 F67 .46 F61 .89 F63 .36 39 F68 .69 F62 .58 F64 .18 40 F69 .64 F63 .46 F65 .09 41 F70 .24 F64 .31 F66 .75 42 F71 .21 F65 .60 F67 .67 43 F72 .22 F66 .40 F68 .53 44 F73 .29 F67 .51 F69 .18 45 F74 .36 F68 .38 F70 .49 46 F75 .60 F69 .45 F71 .60 47 F76 .80 F70 .53 F72 .25 48 F77 .52 F71 .37 F73 .13 49 F78 .33 F72 .19 F74 .03 50 F79 .56 F73 .17 F75 .78 51 F80 .71 F74 .16 F76 .14 52 F81 .80 F75 .42 F77 .10 53 F82 .45 F76 .57 F78 .18 54 F83 .71 F77 .49 S4 .35 55 F84 .65 F78 .22 S5 .80 56 F85 .43 ' . F79 .35 Sll-15 1.04 57 S2 .28 S3 .05 q 1.45E-04 58 S3 .74 S4 .31 a 9.12E-04 59 q 7.11E-05 S5 .73 b 1.93E-22 60 a 9.12E-04 Sll-17 1.02 61 b 1.93E-22 q 1.32E-04 62 a 9.12E-04 63 b 1.93E-22 313 Table A8. Numbers-at-age (XlO) of Pacific cod in Hecate Strait (Areas 5C+5D) for 1961-85 estimated from VPA using best estimates of input parameters; M =0.68 and q =5.79X10 S. Age Year 2 3 4 5 6 7 8 9 10 1961 6518 1561 588 187 59 17 9 0 0 1962 9441 3243 535 188 73 26 6 3 0 1963 12357 4642 1480 180 51 20 8 2 0 1964 16671 5988 1965 568 39 14 0 0 0 1965 9662 7460 2556 797 213 ,6 6 0 0 1966 5747 4650 2294 676 268 72 0 0 0 1967 7769 2629 1296 465 139 84. 29 0 0 1968 3352 3301 922 422 135 32 30 12 0 1969 4837 1610 781 240 112 21 3 11 3 1970 4291 2219 684 147 57 38 2 0 5 1971 5474 2104 903 285 63 21 17 0 0 1972 8705 2496 965 390 114 28 8 6 0 1973 6397 3759 1060 380 162 52 9 3 0 1974 6824 2744 1340 346 147 57 22 3 0 1975 7219 2644 1064 423 93 52 26 10 0 1976 5682 3229 710 290 89 32 3 3 0 1977 6508 2519 870 153 73 17 0 0 0 1978 5356 2636 898 254 43 16 3 0 0 1979 7615 2644 1020 307 86 16 4 0 0 1980 4423 2912 1094 215 84 22 3 0 0 1981 3994 1764 838 293 44 19 3 0 0 1982 2840 1732 591 137 47 8 1 1 0 1983 1772 1277 612 182 33 13 2 0 0 1984 2362 738 353 136 33 3 3 0 0 1985 759 951 206 63 33 4 1 0 0 Table A9. Numbers-at-age (XlO) of female English sole in northern Hecate Strait (Area 5D) for 1956-79 estimated from VPA using best estimates of input parameters; M =0.22 and q =4.09X10 ^ . Age Year 3 4 5 6 7 5 9 10 11 12 13 14 15 16 17 1956 2339 2095 1963 955 452 267 . 103 45 16 16 7 0 0 0 0 1957 2217 1863 1492 1042 491 225 138 53 26 3 4 0 0 0 0 1958 2246 1768 1344 1072 620 285 131 90 23 10 2 3 0 0 0 1959 2356 1797 1318 855 672 305 140 60 41 9 5 2 0 0 0 1960 2011 1883 1226 670 389 356 130 52 20 14 2 4 0 0 0 1961 1245 1580 1195 552 219 109 62 32 14 3 3 0 0 0 0 1962 1806 998 1031 391 173 56 28 14 - 9 0 0 0 0 0 0 1963 2781 1444 686 568 179 60 20 13 2 3 0 0 0 0 0 1964 2714 2225 1077 395 275 83 22 9 5 2 1 0 0 0 0 1965 4009 2163 1598 624 204 165 42 13 3 3 0 1 0 0 0 1966 1771 3216 1652 1075 343 79 92 16 1 1 1 0 0 0 0 1967 1084 1420 2512 1157 711 , 197 29 60 7 1 0 0 0 0 0 1968 1087 868 1091 1670 660 466 110 4 41 1 0 0 0 0 0 1969 1453 864 672 732 937 340 293 59 1 31 1 0 0 0 0 1970 1763 1140 631 444 415 435 156 197 35 1 23 0 0 0 0 1971 1837 1373 723 377 254 171 161 63 136 24 1 17 0 0 0 1972 2092 1446 879 459 259 159 89 71 28 102 17 1 13 0 0 1973 2453 1665 1081 568 261 138 89 53 49 20 80 13 0 10 0 1974 1979 1839 1173 671 365 170 81 43 29 30 16 64 11 0 8 1975 1214 1535 1322 805 443 226 107 45 26 . 21 14 10 43 0 0 1976 883 944 989 723 397 180 92 34 23 18 11 7 6 32 0 1977 702 697 640 455 306 139 54 23 17 15 9 5 3 1 24 1978 703 481 349 227 161 116 53 11 7 12 8 5 0 0 0 1979 3292 540 316 173 118 83 55 16 2 4 9 5 4 0 0 Table A10. Numbers-at-age (xio3) of female rock sole in northern Hecate Strait (Area 5D) for 1956-78 estimated from VPA using best estimates of input parameters; =0.25 and q^ =5.17X10~5. Age Year 4 5 6 7 8 9 W // 12 13 14 15 1956 649 434 384 340 177 111 35 13 5 1 1 0 1957 2004 455 270 156 136 61 44 13 5 0 0 0 1958 2941 1520 316 134 46 48 17 16 3 3 0 0 1959 3255 2250 541 118 45 7 3 4 5 0 0 0 1960 1603 2533 1636 333 78 32 3 2 3 4 0 0 1961 1130 1240 1681 831 178 40 . 17 0 0 0 2 0 1962 1230 880 926 874 334 81 27 9. 0 0 0 0 1963 1704 958 679 641 421 152 43 12 6 0 0 0' 1964 2862 1301 691 443 301 150 51 13 6 1 0 0 1965 2924 2136 912 486 267 142 78 30 5 4 0 0 1966 2879 2246 1574 621 322 179 96 54 20 3 3 0 1967 2500 2178 1114 651 221 82 36 12 3 0 0 0 1968 2141 1912 1333 558 201 72 18 12 0 0 0 0 1969 1997 1487 1146 457 247 108 36 3 5 0 0 0 1970 1115 1170 842 677 236 166 75 23 3 4 0 0 1971 918 860 809 440 299 115 58 37 1 1 0 0 1972 979 697 470 315 186 100 26 30 26 1 0 0 1973 1111 728 443 267 182 113 61 13 20 19 0 0 1974 1144 823 540 327 185 122 77 41 4 12 15 0 1975 668 756 482 408 252 144 90 54 30 3 9 11 1976 571 500 526 182 122 52 30 34 31 18 0 0 1977 460 363 255 179 77 70 30 23 27 24 14 0 1978 1559 286 138 101 70 44 47 20 16 19 19 11 Table All. Numbers-at-age (XlO) of Pacific cod in Hecate Strait (Areas 5C+5D) for 1961-85 estimated from CAGEAN using best estimates of input parameters; M =0.68 and F =0.41. Age Year 2 3 4 5 6 7 8-10 1961 4670 1530 557 225 60 26 21 1962 7865 2272 674 227 92 24 19 1963 10045 3748 933 247 83 34 16 1964 15055 4686 1431 305 81 27 16 1965 8139 7177 1926 525 112 30 16 1966 6236 3758 2644 596 162 35 14 1967 5712 2811 1275 720 162 44 13 1968 2602 2587 969 356 201 45 16 1969 4259 1133 780 220 81 46 14 1970 4143 1928 390 217 61 22 17 1971 5942 2008 839 156 87 24 16 1972 9176 2877 871 334 62 35 16 1973 7924 4389 1196 325 125 23 19 1974 6557 3716 1706 402 109 42 14 1975 7236 3063 1426 562 132 36 18 1976 6069 3154 926 325 128 30 12 1977 6068 2648 958 212 75 29 10 1978 5293 2734 898 260 58 20 11 1979 6463 2529 1133 333 97 21 11 1980 5161 2902 846 302 89 26 9 1981 4915 2265 898 200 71 21 8 1982 4031 2101 640 184 41 15 6 1983 3415 1866 781 201 58 13 6 1984 6198 1541 636 214 55 16 5 1985 36% 2912 603 216 73 19 7 Table A12. Numbers-at-age (XlO3) of female English sole in northern Hecate Strait (Area 5D) for 1956-79 estimated from CAGEAN using best estimates of input parameters; M =022 and F =0.35. B B Year 3 4 5 6 Age 7 8 9 10 U-17 1956 2397 1994 1512 804 480 271 154 39 288 1957 2176 1906 1450 937 424 253 143 81 231 1958 2035 1736 1428 971 561 254 152 86 223 1959 1745 1620 1273 906 532 307 139 83 211 1960 1568 1383 1128 705 398 234 135 61 185 1961 888 1230 857 459 188 106 62 36 133 1962 1551 700 808 406 157 64 36 21 101 1963 2060 1231 499 475 197 76 31 18 82 1964 2461 1639 899 313 255 106 41 17 70 1965 3484 1960 1206 574 173 141 59 23 61 1966 1483 2769 1410 726 289 87 71 30 56 1967 1016 1184 2092 < 966 451 180 54 44 61 1968 1212 810 880 1372 559 261 104 31 71 1969 1453 967 608 592 827 337 157 63 72 1970 1959 1156 707 382 319 446 182 85 87 1971 2115 1554 820 410 181 151 211 86 102 1972 2463 1683 1133 511 218 96 80 112 121 1973 3507 1965 1263 762 308 132 58 49 159 1974 2635 2790 1431 786 405 164 70 31 142 1975 1494 2097 2043 904 427 220 89 38 121 1976 1113 1182 1441 1091 375 177 91 37 100 1977 1683 879 795 728 413 142 67 35 82 1978 1927 1321 553 336 207 117 40 19 62 1979 3760 1531 951 334 . 170 104 59 20 55 Table A13. Numbers-at-age (XlO) of female rock sole in northern Hecate Strait (Area 5D) for 1956-78 estimated from CAGEAN using best estimates of input parameters; M =0.25 and F =0.18. Age Year 4 5 6 7 8 9 10 11-15 1956 746 462 364 364 190 132 45 81 1957 1783 551 283 157 157 82 57 71 1958 2632 1327 348 132 74 74 39 , 74 1959 1998 1942 803 147 56 31 31 63 1960 1040 1545 1460 574 105 40 22 70 1961 851 791 1077 870 342 63 24 63 1962 1364 650 564 677 547 215 39 61 1963 1349 1046 471 367 441 356 140 71 1964 2773 1018 704 257 201 241 195 126 1965 3062 2112 715 426 156 122 146 208 1966 2388 2354 1550 482 288 105 82 253 1967 1983 1692 1182 417 130 77 28 161 1968 2084 1453 995 466 164 51 30 112 1969 1696 1541 891 434 203 72 22 85 1970 999 1272 1007 454 221 104 36 69 1971 478 742 794 459 207 101 47 62 1972 759 350 436 312 181 81 40 57 1973 1487 569 229 222 159 92 41 59 1974 1584 , 1134 403 142 138 99 57 68 1975 671 1218 833 272 96 93 67 89 1976 899 484 663 272 89 31 30 74 1977 980 657 281 252 104 34 12 57 1978 1325 731 418 134 121 49 16 44 

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