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An investigation of fisheries yield equations with particular reference to annual state models and seasonal… Wild, Alexander 1981

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AN INVESTIGATION OF FISHERIES YIELD EQUATIONS WITH PARTICULAR REFERENCE TO ANNUAL STATE MODELS AND SEASONAL PULSE FISHING by Alexander Wild B .A .Sc . , University of Br i t ish Columbia, 1956 M . S c , University of Bri t ish Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Zoology We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1981 (c) Alexander Wild In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Bri t ish Columbia, I agree that the Library shall make i t 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of Bri t ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 DE-6 BP 75-51 1 E i i ABSTRACT Traditional methods of stock assessment rely on the calculated value of effective effort (f) and the assumed constancy of the catchabil i ty coeff ic ient (q) to provide estimates of abundance and the effects of f ish ing. It is d i f f i c u l t to account for or to quantify a l l the variables that affect f and the r e l i a b i l i t y of this term is often questionable. The uncertainty associated with f and q is reflected in assessments and undermines confidence in management recommendations. Under certain conditions the objective of circumventing these d i f f i -cult ies is realized in the development of the annual state models. At the end of each f ishing year, the f ishing (F) and natural mortality (M) coe f f i -cients and the apparent abundance of each age group (i) are estimated without reference to f , q, the number of recru i ts , or the entire history of a year c lass . The data requirements that make this analysis possible are the catch (C^) and y ie ld in weight (Yw )^ of each age group and estimates of the growth equation parameters. The derived quantity for the mean weight of f ish (Y^ y' C.) is independent of abundance and provides a deterministic solution for the total mortality coef f ic ient , . Consecutive year class estimates of Z | and Z ^ - j , when coupled.with a ratio of catch equations, y ie ld estimates of F^, F . + ^ , and M. The assumptions of this particular model are that growth and mortality are concurrent and continuous during the f ishing period, the stock is closed to immigration and emigration, and M i s constant for a l l age groups. Alternative models are developed that provide a simultaneous solution for the stock using equations for mean i i i length and weight or mean weight alone when .M is constant or a function of age. The equation for is based on a generalized growth model and integrated by an approximate technique. The effect of seasonal pulse f ishing on equilibrium y ie ld is examined for hypothetical species having twenty different patterns of seasonally distributed growth and natural mortality. Each pattern is subjected to ten fishing strategies that vary in seasonal intensity and annual distr ibut ion. The effect on y ie ld of increasing values of M, F and the growth parameter K is also explored. Relative to .continuous f ish ing , the greatest increase in y ie ld is consistently achieved by concentrating the f ishing act iv i ty into a single season. While the magnitude of this increase varies from three to thirty per cent, in any particular situation the optimal; time and the potential y ie ld improvement is a function of K, M, F and the growth-mortality pattern. i v TABLE OF CONTENTS Page TITLE PAGE i ABSTRACT i i TABLE OF CONTENTS iv LIST OF FIGURES vi LIST OF TABLES v i i LIST OF APPENDICES v i i i ACKNOWLEDGEMENTS ix INTRODUCTION 1 FISHERY YIELD EQUATIONS ' 8 1. General Principles 8 2. Fishing and Natural Mortality - Catch in Numbers 15 3. Growth Models 25 3.1 Generalized derivation 25 3.2 Allometric, asymptotic growth parameter estimation 31 4. Yield Models: Combined Growth and Mortality 35 4.1 Generalized y ie ld model 35 4.2 Beverton and Holt constant recruitment, equilibrium y ie ld model 38 4.3 Virtual population and cohort analysis models 43 4.4 An asymptotic, allometric growth y ie ld model 56 4.5 Seasonal pulse f ishing 63 5. Regenerative Yield Models 69 ANNUAL STATE MODELS 74 1. General Remarks 74 2. Assumptions of the Annual State Models and Bases of Analysis . . . 85 V Page 3. Model 1: Methods of Estimating Growth Parameters and Mortality Coefficients 91 3.1 Init ial estimates of L , K and Z 94 00 " 3.2 Init ial estimates of W and b 96 • 00 3.3 Final estimates of parameters, coeff icients and the c r i t i ca l age 97 4. Model 2: Methods of Estimating Growth Parameters and Mortality Coefficients 101 5. The Dependence of the Mortality Coefficient on t 0 and the Effect of Variable Natural Mortality 104 DISCUSSION AND CONCLUSIONS 112 LITERATURE CITED 124 APPENDICES 131. vi LIST OF FIGURES Figure Page 1 A schematic diagram of the simplif ied relationship between recruitment, growth and mortality for a multi age group, closed system fishery 13 2 Seasonally intensive periods of growth (g) and natural mortality (m) for a variety of fishes 65 3 (a) Twenty cases of seasonally distributed growth (g^) and natural mortality (m) patterns. Each year in the l i f e of an age-group (i) is divided into quarters. (b) Ten, seasonally applied f ishing strategies (FS) 66 4 Schematic y ie ld isopleth diagrams for two different multi age group stocks: (a) constant recruitment and (b) density dependent recruitment 7 9 5 Response of 18 conventional and potential f ishery s ta t is t ics in relation to eumetric f ishing curve 82 6 Model 2, residual sums of squares contours for various values of the growth curve intercept, t Q , and the total mortality coef f ic ient , Z 92 Cl (a) Twenty cases of seasonally distributed growth (g-j) and natural mortality (m) patterns, (b) Ten seasonally applied f ishing strategies (FS) 191 vi i LIST OF TABLES Table Page I Age-group catch, mean weight and length for two consecutive years of f i s h i n g . . . . . 93 II Comparison of Model 1 and Model 2 estimates of growth equation parameters and mortality coeff icients with actual values 100 III Expected change in s ta t is t ics W.. and L^ . due to a change in fishing mortality of 0.1 102 IV The effect of an error in t on (a) Z for a single year of f ish ing , and (b) the mortality coefficients for two years of f ishing 106 A l Simulation results and comments. Typical analysis of a Schaefer (1957) general production model. Method 4 140 A l l Results of simulated, experimental f ish ing. Modified Beverton and Holt (1957) year-class model containing a Ricker (1958) recruit function 157/9 Al 11 Total y ie ld per recruit (Y-p/R) associated with age of f i r s t capture (t c ) and fishing mortality (F) . . 166 AIV Predicted total y ie ld and recruitment index 173 CI Equilibrium y i e l d , rank values and y ie ld gain (+) relative to continuous f ishing (FS 10) for growth-mortality Case 1 193 CII Relative equilibrium y i e l d , seasonal pulse f ishing results for F=1.5 196/ 202 v i i i LIST OF APPENDICES APPENDIX Page A Analysis of f isheries models using a gaming technique 131 B Derivation of optimal stock s ize , recruitment and fishing mortality at MSY in a multi age group, single species fishery 178 C Seasonal pulse f ishing results 190 ACKNOWLEDGEMENTS I am indebted to Dr. P. A. Larkin for his infectious enthusiasm, encouragement and support throughout the preparation this dissertat ion. The pleasure of working with Dr. J . Schnute through many mathematically inspiring moments is also gratefully acknowledged. 1 INTRODUCTION The early 1960s witnessed the beginning of a rapid escalation in global catches of f ish and shel l f ish products. The nominal catch grew from 40.2 mil l ion metric tons in 1960 to 70.2 in 1971, and declined s l ight ly to 65.7 tons in 1973. A similar but proportionately greater increase occurred in the Convention area of the International Commission for the Northwest Atlantic Fisheries (ICNAF) off Canada's east coast. There, the nominal catch rose from 2.3 to 4.3 mil l ion tons in the 1960-71 period. In ICNAF's experience the heightened demand for fishery products was met by a substantial increase in f ishing effort being directed against conventional and non-conventional species. The effect of increased harvest rates became quickly apparent. By the mid 1960s certain f ish stocks were being over exploited, and i t was recog-nized that some form of protection was needed short of direct effort control . As a resul t , quotas (total allowable catches) were internationally agreed upon in 1970 for two halibut stocks, and i t has since been estimated (Regier and McCracken, 1975) that, over a l l , the number of stocks requiring quota pro-tection wil l reach 56 by 1975. In fac t , the total proved to be 59 and one additional stock was cited for conservation in 1976 (Akenhead, 1976). The preceding remarks focus attention on one of the chief d i f f i c u l t i e s of f isheries management; i t i s , to prepare stock and stock-complex assess-ments of improved precision at an increasing rate. Solutions to this problem are fundamentally important, not only in terms of conservation, but because of their future implications. Without rel iable assessments the sc ient i f i c c red ib i l i ty of management is placed in doubt and i t becomes increasingly d i f -f i c u l t to support regulatory measures when they are most needed. Without 2 reliable knowledge of the present state of a stock and the fishing mortality inflicted on i t (assessments), yield prediction is also uncertain. From this perspective, assessment is a central issue and the problem of improving its speed and reliability of analysis formed the original basis of this thesis. In the past seventy years a substantial body of theory, computational skil l and practical information has been developed to carry out stock assess-ments. To effect an improvement in the speed and reliability of such anal-yses i t is possible to review this literature and isolate areas in which a gain in information or the development of a new theory might contribute towards these objectives. A problem to recognize in such an approach is that the reviewer is exposed to pre-conceptions that have either a common basis of historical development or are highly specific in their application. It is often beneficial therefore to reappraise a long-standing problem by reducing i t to its simplest elements, construct solutions at this level of under-standing, and introduce realistic complexity in stages. This second approach was used init ial ly on the assessment problem by adapting a gaming technique to fisheries models. The essential features of the process are as follows: An interactive simulation model of a fishery is f irst constructed by an outside party and a second individual, known as the "manager," is then allowed to vary the effort level (boats) in successive years of fishing. Following each application of effort, relevant information from the fishery is printed out and, on the basis of these data, the manager must (1) determine the underlying model system, 3 (2) estimate the coordinates of MSY ( y i e l d , e f f o r t ) , and (3) achieve th is goal in the shortest possible time by a repet i t i ve sequence of y i e l d predict ion and f i s h i n g J The f i r s t type of f ishery examined here by gaming was a discrete-t ime version (Lark in , 1974) of Schaefer 's (1954, 1957) general production model. The object ives were those given above in the process descr ip t ion. The im-portant conclusion reached was that the time needed to achieve MSY was minimized only when data on catch, e f fo r t and recruitment became ava i lab le . This resu l t suggested the importance of es tab l ish ing - - based on er ror - f ree data - - the minimal time needed to ident i fy the unknown parameters of a y i e l d model, estimate MSY, and s t a b i l i z e the f ishery at that point. Such a standard could act as the target in reducing the analysis time of future assessment models. For th is purpose a determin is t ic , multi age-class Beverton and Holt (1957) model of a v i rg in f ishery was developed which included a Ricker (1968) recruitment funct ion. The author was only aware of the model type, and that the instantaneous rates of morta l i ty and the ca tchab i l i t y coe f f i c ien t were constant for a l l age groups. The information obtained from the f ishery was free of er ror . On the basis of these assump-t ions , complete parameter estimates were obtained af ter two years of f i sh ing and the system was s tab i l i zed at the point of MSY within ten addit ional years. In th is case, as in the Schaefer model, data on recruitment were needed before a sa t i s fac to ry solut ion could be reached. The resul ts and management impl icat ions of the two models jus t described are reported elsewhere (Appendix A) . Here, i t - i s more c r i t i c a l to note the ^See Tyler (1974) for a f ishery simulation model involv ing graduated cost penalt ies for time and quantity of information. 4 lessons or principles that emerged from the gaming experience to redirect the emphasis of this report. The remarks that follow are purposely s impl i f ied. Consider an existing fishery in which the management objective is to maxi-mize the average or long-term y i e l d . If this goal is attainable, i t must f i r s t be assumed that a condition of surplus production wil l exist and can be generated by a positive response in recruitment or growth, or a decline in natural mortality. Given that a change in one or a combination of these ele-ments occurs, then four interrelated factors must be known to direct the fishery towards the objective. F i r s t , management must be able to determine the present state of the fishable stock at prescribed intervals of time. The minimal amount of information needed is an estimate of stock abundance or of year-class strengths. The description is substantially improved i f estimates of the rates of growth and natural mortality are also avai lable. Second, a measure of recruitment or forthcoming additions to the fishable stock must be obtained. It is the only measurable safeguard available to management to pre-dict a fa i lure or temporary decline in the f ishery, even i f the cause cannot be immediately ident i f ied. Prior notice allows for the preparation of a response to the situation and not a reaction after the event has disturbed the f ishery. This principle holds equally well in the case of an improvement in the strength of recrui ts . If values can be assigned to both factors, then the instruments to detect changes in the fishery have been established. The principal agent of change, and the third factor, is then readily identif ied as the effort directed against the fishable stock, and a measure of i ts effectiveness over time is the f ishing mortality rate. The reader wil l recognize that the conditions needed to place the fishing process on an experimental basis are almost complete. If management can establish its present position in a fishery (factor 1), and recognize the changes (factors 1, 2) induced by a disturbance of known magnitude (factor 3), then it is possible to move towards the objective in a time-consuming but regulated manner. To complete the experimental analogy, what is missing in the overall procedure is a method of hypothesis testing. The fourth item, yield prediction, fu l f i l ls this function and more. It provides a unifying model or conceptual framework for all three elements. Each change in effort can now be considered a separate experiment to test the predictive value of the model. Through repeated fishing trials and adjustments confidence in pre-diction can be improved by degrees. The benefit of each improvement has an experimental counterpart. Changes of greater magnitude can be imposed on the fishery, thereby reducing the number of time steps needed to reach the final objective. The dual potential for reducing assessment time and increasing predictive accuracy appears to rest in the study of yield models. The essence of such models, however, is fisheries yield equations, and i f improvements are to be expected in assessments and predictions, the assumptions underlying the com-ponents of these equations must f irst be understood. The initial objective of this thesis i s , therefore, to reappraise the basic formulae used in yield equations and contrast their assumptions against some of the known realities of a fishery. Both the practical yield models that use these formulae and the statistics gathered in the course of a modern fishery form a necessary part of the analysis. They are examined specifically for their usefulness in diagnos-ing the present state of a stock. There are two outstanding characteristics of a fishery that negate a simplistic approach. The f irst is that environmental factors predominantly 6 influence the number of recruits and this quantity is effectively a variable. Yield equations which incorporate a deterministic, stock-recruit relation-ship can therefore lead to inaccurate predictions. For this reason, recruit functions are usually excluded from yield equations and estimates are pre-pared separately. The remaining elements of fishable-stock abundance and expressions for growth and both mortalities are then descriptive of only the present state of the stock. A yield in weight can s t i l l be calculated from the contracted equations (or catch in numbers i f the growth term is removed), but full predictability is no longer possible. The second characteristic refers to social, economic, or political fac-tors that can constrain management from changing or even controlling-fishing effort at wi l l . As a result, the experimental use of effort to gain informa-tion about the system may be substantially reduced, and management must rely heavily on the interpretation of past events in the fishery. With both char-acteristics operating simultaneously, i t would therefore be desirable i f a group of statistics could be developed to accomplish the following: (1) Assess the present state of the fishable stock in terms of year-class abundance and individual growth rates by age group, and supply an average value for the fishing mortality rate. (2) Evaluate the fishing mortality independently of the effort used in the fishery. (3) Estimate the fishing, natural mortality, and growth rates independently of-variation in year-class strength. (4) Allow these entire calculations to be performed at the end of each fishing season, without relying on historical data. 7 With this a b i l i t y , estimates of parameters and variables can be updated seasonally. The second objective of this thesis is to demonstrate the feas ib i l i t y of obtaining this complete assessment. Given the assumption of random dist r ibu-tion of effort or f i s h , the data requirements from the fishery are those of total y ie ld in weight and length, and catch in numbers by age group. An independent estimate of the von Bertalanffy (1938) growth equation constant, t 0 , is also required. Appropriate y ie ld equations are developed and the technique applied to a theoretical model of a multi-age class trawl f ishery. Growth in length and weight is also assumed to take place according to the von Bertalanffy equations, but the restr ict ion of a cubic exponent in the weight-length relationship is removed. Practical application of the model is presently limited because y ie ld in length and weight data in a desirable form are not avai lable. 8 FISHERIES YIELD EQUATIONS 1. General Principles Ideally, a biomass model of a fishery should include all time-dependent factors that contribute to and detract from the fishable mass of the target species. To achieve this level of understanding it is necessary to know how the environmental, biological and fishery systems interact to affect:.tne,bio-mass over the course of time. A full appreciation of these processes must also include the self-regulatory response of the population. The value of such a comprehensive model is to minimize the residual error associated with yield prediction and stock assessments, and the research efforts to identify the underlying relationships are therefore closely linked with this goal. The degree to which the ideal model can be approximated depends partly on the quality and quantity of data that is available from the fishery and from inde-pendent research activity. Collectively, this information must be rendered into concise mathematical form for two purposes: 1) to establish the working relationships between the component parts of the model, and 2) to provide the means of estimating the parameters. This last point is critical to the solution of the model and is therefore a vital part of the model building pro-cess. If values cannot-be assigned to the parameters, the model is reduced to a theoretical conceptualization that is neither testable nor appli-cable in a practical sense. The greater source of departure from the ideal case stems from incomplete knowledge of the processes that affect the biomass and hence the yield from a fishery. As a result, simplifying assumptions are introduced into yield models 9 that lead to an ordered decrease in complexity and an increased departure from reality. A general description of these models together with the con-sequences of simplification are given below. The practical requirements of a yield model dictate that each element in its structure be an estimable quantity in either absolute or relative terms. If the time-dependent factors or process are unknown or cannot be quantified then their combined effects must be compressed into variables that can be measured or estimated. Thus, the environmental, biological and fishery systems are known to interact to affect the biomass, and although the details of their inter-relationships may not be clear, their resultant effects are identifiable in terms of six key variables. These quantities are recruitment, growth, mortality due to fishing and all other causes (natural mortality), and fish movement in the form of immigration and emi-gration. Together, these quantities represent the first-order simplification of the idealized model, and each principle component is readily associated with a contribution to or a loss from the existing biomass. Compared to the ideal case the complexity of the model, the number of parameters and the amount of information needed to estimate the parameters are substantially reduced. The apparent gain in simplicity, however, is not achieved without cost. By focussing attention on the end results of the system's interactions, each variable now represents a composite of multiple effects, and the sources of variation within each component can no longer be identified. As a consequence the reliabil ity of the model is affected in two critical areas. First, the derived values of recruitment, growth, mortality and fish movement must each carry a full variance load. Similarly, by reducing the number of parameters 10 i t is less likely that the remainder will each have a constant value. The common effect of these features is to increase the degree of uncertainty attached to each principle component and parameter, and ultimately to the predicted yield i tself . Secondly, since the cause-effect relationships that influence the components and parameters are not determined, it is extremely difficult to interpret correctly the past and present fishing results or make accurate predictions of yield. To offset these combined diff icult ies, to keep pace with events in the fishery and simultaneously offer a sense of direction, management must re-estimate the parameters and monitor the components on a frequent basis. Additional stages of simplification in a yield model may be achieved by reducing the number of key variables through combination or outright elimina-tion. For instance, it is understandably difficult to estimate the change in biomass over time that arises from immigration or emigration. Eliminating these variables from the model does not deny or ignore the net effect of fish movement. It is simply a mathematical convenience that models a closed-population fishery without specifying how the yield is affected by immi-gration and emigration. By not including these variables, however, the unex-plained variance of the predicted yield is increased beyond the first-order simplification, and this condition holds for the removal of any of the six key components. The only exception that might occur is i f the fishery oper-ates over the total surface area of a closed body of water. In this case the immigration and emigration terms can be safely ignored and the resulting model is justifiably descriptive of a closed population. A third variable that may or may not be directly incorporated into a yield model is recruitment. In contrast to the movement terms it is not a 11 question of excluding the recruitment component and allowing the residual error of the yield to increase. To do so would be to ignore a major factor contributing to the fishable stock, and the resultant yield predictions would bear l i t t le relationship to the amounts actually taken by the fishery. It is a problem, rather, of choosing between alternative methods of estimating re-cruitment, three examples of which are: 1) direct sampling of pre-recruit abundance 2) assuming constant recruitment as in the simple Beverton and Holt (1957) model; or 3) by means of analytical models which rely totally or in part on the assumption of a density-dependent relationship between stock-size and the number of young produced. This method in-troduces new parameters into the yield model, however, and the means of estimating these quantities must be considered. The last simplification method that must be mentioned is that of combin-ing two or more of the key components. Recruitment and growth offer an immediate choice since these variables represent the principal means of in-creasing the fishable biomass. Natural mortality may in turn be combined with this new variable to achieve a net measure of stock that is subject to fishing. Each of these combinatory steps, however, reduces the breadth of the yield model and its ability to account for changes in the biomass. The cost of sim-plification in this case, as in variable elimination, can also be equated to a loss in predictive accuracy.(Watt, 1956). It is generally to be expected that as the number-of variables and parameters are reduced the remaining"components will be less able to explain the observed variation in yield. 1 2 Despite the limitations introduced by s impl i f icat ion, the methods just described provide the most rapid means of gaining information on a f ishery. For management purposes, the emphasis on speed is essential i f the time-lag between events in the fishery and suggestions for corrective action is to be minimized. In general, practical y ie ld models ref lect this purpose by using a simplif ied approach, and either by default (due to lack of information) or -design, the f ish movement component is not included. In the forthcoming discussion a similar viewpoint is taken. To begin the analysis, i t is helpful to visualize (Figure 1) the relationships between recruitment (R), growth (G), and mortality due.to f ishing (F), and al l other causes (M). In the diagram these components appear as independent processes that are linked within a regeneration cycle. The f ish movement terms are not included in order to represent a closed system, multi age-class fishery for a single stock. A preliminary condition for the sustained existence of the fishery is the ab i l i ty of the stock to generate a surplus production. It may occur by means of an increase in recruitment or growth or a decline in natural mortality, or a combination of these responses in response to f ishing pressure. Under these conditions the opportunity exists to achieve the greatest y ie ld that is coincident with maximum, surplus production. To assess the progress of the fishery towards this goal, a model is needed to relate the change in fishable stock biomass (B) to the accumulated fishing mortality over a given period of time. Using Figure 1, the necessary model can be constructed from a material balance centred on the fishable stock. If B-i and B ? represent the biomass at the beginning and end of the time 13 YIELD FISHING MORTALITY (F) PRE - RECRUITS t\ FISHABLE STOCK BIOMASS (B) N: -3»—GROWTH (G) / I I N, RECRUITMENT K >• I (R) I GROWTH JUVENILES / ADULTS NON- FISHING MORTALITY (M) R E C R U I T M E N T . Figure 1. A schematic diagram of the simplified relationship between recruitment, growth and mortality for a multi age-group, closed system fishery. Legend: N = number of fish in age group i ; n = last fishable age group; ••• dotted section indicates that a juvenile age group(s) may or may not be included in the fishery. 1 4 interval, the change in biomass is represented by the equation AB = B 2 - B1 = (R+G) - (F+M), (1.01) and since yield is of primary importance in a fishery, the above expression can also be rearranged in terms of the yield equation In either case the reader will recognize Russell's (1931) associative model of a closed system fishery in which the biomass increments and decrements are represented by (R+G) and (F+M), respectively. Since these terms each refer to a change in biomass they are compiexed variables that include a product of numbers and weight integrated over the time interval. For ana-lytical purposes this form of representation is unsuitable, as Russell (1911) himself realized. Separate and independent expressions are needed for re-cruitment, growth and mortality to reflect changes in numbers and weights. The problem of representing these variables in a suitable mathematical form, and their amalgamation into yield equations must now be considered. An essential part of this process is to indicate clearly the assumptions used in developing the formulae, and the remainder of this section is devoted to this purpose. F = (R+G) = AB - M (1.02) 15 2. Fishing and Natural Mortality - Catch in Numbers The simplest representation of mortality rests on the assumption that at a given time, t , the rate of change in f ish numbers, N, is directly proportion-al to the number present. Baranoff (1918) or ig inal ly introduced this idea into fisheries l i terature and i t has been in continuous use since that time. Ex-pressed mathematically, i t follows that where t and N are recognized as the independent and dependent variables, respectively. Since death due to fishing (F) and al l other causes (M) re-sults in a decline in numbers, expression (1.03) can be rewritten as an equal-i ty to represent the rate of change due to each mortality source: In these f i rs t -order equations the proportionality terms F(t) and M(t) are ident i f ied , respectively, as the instantaneous rates of f ishing and natural mortality. That their units involve a rate term can be seen by rearranging the above equations to indicate speci f ic rates of change, i . e . , dN/dt.N, the dimensions of which are "per unit time". The functional notation (t) wil l be used on f i r s t appearance of a time-dependent variable and thereafter i f needed for emphasis. dN(t) dt N(t) (1.03) ( f ) p - - F(t)N, (1.04) (1.05) 16 Similarly, i f fishing and natural mortality act concurrently the instan-taneous rates may be added to obtain the total rate of decline in fish numbers, dN = . ( F + M ) N = _ z ( t ) N j ( 1 > 0 6 ) where Z(t) = the total instantaneous mortality rate. It should be added at this point that more complex expressions of mortal-ity can be derived which include second or higher order terms, such as, = - (aN + b N2 + c N3 + .. .k N n), but there are practical limits to this form of theoretical expansion. The evaluation of the additional parameters (b, c, . ...k) must be considered. If they cannot be determined, or i f the residual error in the final yield model is not reduced, there is no practical justification for the increase in com-plexity. Since one objective of this thesis is to develop a practical assess-ment method, on the basis of general acceptance the treatment of mortality will be limited to the simplest assumption. The relationship between the rate of capture in numbers, F, and the effort expended in obtaining the catch must now be developed. The sources of information used in this task, to which the reader is referred for an enlarged treatment of the subject, are those of Ricker (1940, 1944, 1968, 1977), Widrig (1954a, 1954b), Gulland (1955, 1969), Beverton and Holt (1957), Paloheimo and Dickie (1964) and Cushing (1970). In their final-form the formulae presented here-do not differ from those of the;above authors. The method of -17 presentat ion, however, is somewhat d i f ferent in order to emphasize the assumptions, the meaning of and the units attached to common terms. I n i t i a l l y , i t i s necessary to define two terms: e f fo r t and standardized e f fo r t . Consider a unit of gear (g) that i s in operation for a short time interval At . In a physical sense the gear may be interpreted as the force or agent of f i sh ing morta l i ty and At the duration over which i t acts . The product of these terms const i tutes e f fo r t ( f ) , a measure of morta l i ty work that may be expressed in pract ica l units such as "boat-days". Over the in terval At the number of gear units in operation is a time-dependent v a r i -able and e f fo r t must therefore be expressed in the generalized form Af ( t ) = g( t )At . (1.07) The catching power (catch/hour, say) of two descr ip t ive ly s im i la r vessels need not be the same despite an equal expenditure of e f fo r t . Traw-l e r s , fo r example, may d i f f e r in horsepower, gear capturing e f f i c i ency , net s i z e , operating ve loc i t y , or experience of the crews. I f a s ingle vessel and i t s associated gear i s selected as a standard (g, Af) the catching power of another vessel ( g 1 , A f 1 ) can be expressed in terms of th is standard by means of a power factor (p) , or ra t io of catching powers. I f i t i s assumed that both units operate on the same density of f i s h , then the fol lowing expression i s v a l i d : Af = g(At) = p g ' (At) = p A f To develop a re la t ionship between catch and e f fo r t i t i s usual ly assumed that the number of f i sh captured (Ac) over a short time interval (At) is 18 proportional to the number of encounters between fish and gear. It is immaterial at this point whether the gear is stationary and fish move towards i t , or the gear is drawn through the water. It is easier to visualize the capturing process in terms of mobile gear and therefore a trawler is used here as an example. For a unit operation of a standard vessel i t follows that Ac(t) = q' , : :Af(t) P ( t ) , (1.08) where Ac(t) = the difference in accumulated catch in the interval At, q 1 ' = a proportionality constant, and p(t) = the local density of fish encountered by the net If both sides of equation (1.08) are divided by At, then in the limit as At •->• 0, Ac/At -* dc/dt, and Af/At ->- df/dt, then the rate of capture is expressed as f = q " ^ P , (1.09) and the rate of capture divided by the effort rate yields dc/dt _ dc _ n , , , , ,_ N d f 7 d t ~ d f - q p Equation (1.10) establishes the common relationship that the catch per unit effort is proportional to the local fish density. Despite the fact that the expression is given in instantaneous terms, i t is clear that the 19 derivation rests on the "the catch rate per unit effort rate". The validity of this expression in terms of comparative rates is preserved in subsequent formulae involving stock density and longer periods of time, such as a fish-ing year. The above relationship (equation 1.09) is valid for any single boat operating on the fishing ground. It bears no relationship to the stock den-sity (D) as a whole, and does not indicate how the rate of capture is influ-enced by additional units of gear distributed in time and space. To make this transition and establish continuity between the rate of capture and the rate of decline in fish numbers, one of the following assumptions is necessary: (1) that either the fish or the fishermen are distributed at random over the fishing grounds, or (2) the fish are highly mobile and capable of re-invading locally depleted areas in order to maintain a random distribution (Ricker, 1940, 1944). Two conditions follow from these assumptions. First, the proportionality is established that P(t) = 3 D(t) = 6 H{t) (1.11) A where g = a dimensionless proportionality constant A = the area occupied by the total fishable stock, and N(t) is specifically redefined as the abundance, or total number of fish that may be subjected to fishing. Secondly, the catch rate is directly proportional to the number of units of gear in operation and df(t)/dt is interpreted in a plural sense. 20 Substituting equation (1.11) into (1.09) and (1.10), the relationships follow: dc dt 7W= q' df dt df A dt (1.12) and dfTA = q' N , (1.13) where q 1 = q"e, and has the units of area/boats x time, and df/A dt = the fishing intensity-measured as total standard effort/unit area/unit time. If the area occupied by the stock is assumed to be constant, then q = q'/A, and equations (1.12) and (1.13) are reduced to dc dt and (1.14) f = q N , (i.i5) where q, the catchability coefficient, takes on the dimensions (boat x time)~\ Under the assumed condition of redistribution the rate at which fish are being captured is now equal to the negative rate of decline in numbers due to fishing. From equations (1.04), (1.13) and (1.14) i t therefore follows: 21 dN(t) = dc(t) dt dt = F(t) N(t) (1.16) or, after simplification, and depending on the constancy of A, F(t) = q or F(t) = q df(t) dt (1.17) In these last equations the instantaneous fishing mortality rate is seen to be proportional to either the fishing intensity or the fishing effort rate. The respective catchability coefficients, q' and q, are also recognized to be neither equal dimensionally nor in value. In practice, the instantaneous catch per unit effort or its average value over a short time interval is not likely to be directly proportional to N. Local concentrations (or depletions) of fish and effort, and cooperation between fishermen upset the assumptions of random distribution, and may contri-bute to large variations in catch rates. It is usual, therefore, to derive fisheries statistics on a longer time basis over which the fluctuations may be averaged. Mathematically this simply involves integration of the above for-mulae, but the realistic interpretation of F(t) [or M(t)] as a function of time represents a problem. The difficulty is examined here only in terms of the variable F(t), i .e . , M is assumed to be a constant. The number of fish at any time t may be calculated by init ial ly rearrang-ing equation (1.06) as d N = d(ln N) - -Z(t) dt N 22 and integrating the left hand side of this equation between the indicated limits of N N(t) t t / d (In N) = - / F(t)dt - • / M dt , No o o to obtain, after taking exponentials and rearranging, t t t - / F(t)dt - / M dt ' - / F(t)dt - Mt N(t) = No e 0 0 = No e 0 (1.18) where No = the initial abundance at time t = o. The integral of F(t) may be treated separately for the moment, and remains unchanged i f multiplied and divided by J*^ dt, i .e . , t f l F W d t t A ( t ) dt = • fl dt 0 / dt 0 o By definition (Smail, 1949) the integral of the function /gF(t) dt divided by the integral of time over the same limit is equal to the average value of the function, i .e . , F. Similarly, dt = t, and substituting these values into (1.18), N(t) = No e " ( F + M ) t (1.19) The function M(t) can be similarly dealt with to emphasize seasonal mortality, for instance, and to obtain the average values M and Z. Although the seasonal and local variation in f, F and M may be appreciable, equation (1.19) usually appears in the literature as N(t) = No e -(F + M)t = No e -Zt (1.20) indicating that the coefficients F, M and Z are assumed to be constant over the time interval. In the derivations that follow, the latter form of presen-tation is retained although i t is clear from the foregoing that average values are intended. Substituting the value of N(t) from equation (1.20) into (1.16) the instantaneous increment in accumulated catch is expressed as dc(t) = F N(t)dt = F No e " Z t dt , and the total accumulated catch (C) is arrived at by integrating this equation between the limits c (o,C) for t (o,t): d c(t) = F No/J e " Z t dt ( 1 . 2 1 ) and C(t) = F No (1.22) Z where F/Z = the fraction, by numbers, of total deaths (N0-(l-e" ; Z t).) that is attributed to fishing. 24 It is more desirable, however, to express the catch in relation to the average abundance N" over the time interval rather than its initial value, N . To arrive at this point equation (1.21) may be rewritten as indicated below and divided by dt, i .e . , 'o = F N o o^ e " Z t d t '$ d t " f l d t and integrating, CR(t) = £ ( t ) = F No ( 1 ~ e z Z t ) = F N(t) (1.23) where C R = the average catch rate, and N" = the average abundance, both take over the total time t. If the time interval is set equal to one year (t = 1), then equations (1.22) and (1.23) are reduced to the common form C" = CR = F No Z ) = F N (1.24) the catch equation of Baranoff (1918), and on an annual basis the accumulated catch (numbers) and the average catch rate (numbers/year) are numerically equal. Similarly, integrating equation (1.17) over the period of a year leads to the expressions F = and F = qf , (1.25) 25 where F (and incidentally M and Z) is an instantaneous mortality rate, or coeff ic ient , expressed on an annual basis, f/A = the total f ishing intensity, e . g . , total boat-hours/sq. mile/year, and f = the total effort rate; e . g . , total boat-hours/year F inal ly , by substituting equation (1.25) into (1.24) and rearranging, two interpretive formulae are derived: £ / A = q' N or £ = q' | = q D . (1.26) The conclusions follow: i f A is not constant, the total catch per unit f ishing intensity is proportional to the average abundance; but i f A is assumed to be constant, the total catch/total effort (catch per unit effort) is proportional to the average stock density. The principal formulae relating the catch in numbers over time and during the concurrent operation of fishing and natural mortality have been developed. The assumptions that support these relationships have also been specif ied. The primary components of growth and recruitment remain to be discussed. 3. Growth Models 3.1 Generalized derivation With few exceptions growth models that have been adapted to or evolved for use in f isheries have used physiological considerations as a basis of 26 development. Thus, the ra te o f change i n weight has been a t t r i b u t e d to the d i f f e r e n c e i n anabo l i c and k a t a b o l i c ra tes (von B e r t a l a n f f y , 1938) , to the d i f f e r e n c e i n energy content o f r a t i o n s l e s s metabo l i c expend i tu re (Paloheimo and D i c k i e , 1965) , and to an e m p i r i c a l f u n c t i o n of s i z e tha t p a r a l l e l s s t and -a r d i z e d p h y s i o l o g i c a l ra tes (Parker and L a r k i n , 1959). S i m i l a r l y , T a y l o r (1962) proposed a g e n e r a l i z e d growth formula i n which the parameters are i n -tended to have metabo l i c s i g n i f i c a n c e . Appropr ia te m o d i f i c a t i o n o f t h i s l a s t model i nc ludes the " s e l f - i n h i b i t i n g " , o r asympto t i c , growth equat ion o f von B e r t a l a n f f y (1938) , the p a r a b o l i c exp ress ion of Parker and L a r k i n (1965) , and the exponent ia l growth formula o f R i c k e r (1968) as s p e c i a l cases . Because of i t s g e n e r a l i t y , the T a y l o r (1962) model i s used here to i l l u s t r a t e the d e r i -va t i on o f growth equa t i ons . To avo id symbol ic con fus ion the o r i g i n a l no ta t i on o f the author i s not r e t a i ned e n t i r e l y . Under constant environmental c o n d i t i o n s , i f i t i s assumed tha t 1) the ra te o f change i n weight i s equal to the d i f f e r e n c e i n anabo l i c and k a t a b o l i c r a t e s , and 2) these l a t t e r ra tes are p r o p o r t i o n a l , r e s p e c t i v e l y , to the abso rp t i ve su r face area ( s ( t ) ) and weight (w( t ) ) o f the f i s h , the r e l a t i o n s h i p can be w r i t t e n = h s ( t ) - k'w(t) , (1.27) where the constants h = weight syn thes i zed per u n i t t ime per un i t su r face a r e a , and k'= the k a t a b o l i c weight l oss per un i t t ime per un i t we igh t . 27 If it is further assumed that: s(t) = P A t ) , and the weightrlength relationship is adequately described by w(t) = a £ b ( t ) , (1.28) where p, a = proportionality constants £( t ) = the fish length at time t m = an exponent relating length to absorptive surface area, and b = an exponent relating length to the "metabolically effective" weight (Brody, 1945), these formulae can then be substituted into (1.27) and rearranged to obtain dt = E £(m-b+l)_K £ _ ( U 2 g ) where E = hp/ba; and K = k'/b>, a growth coefficient. Equation (1.29) may be integrated between the limits of Jl (& , £ ( t ) ) and time (0, t) to yield: £ ( t ) ( b - m ) = E/K - (E/K - £ (b-m))e-K(b-m)t where iQ = the intercept length at time zero. In the limit as t -> °°, £^ - n i^-> E/K, a constant identified as L«>, the asymptotic length, and i f 28 i =0 when t = t , (the theoretical time when the fish length is zero) the o o • above equation reduces to the generalized form: Three well-known growth formulae may be derived from (1.30) depending on the assumptions concerning K and the exponent (b-m). 1. Asymptotic, isometric growth: b-m = 3-2 =1. In addition to a constant environment, i f i t is assumed that growth is iso-metric and the specific gravity of the fish remains constant, then b = 3 and m = 2. Under these conditions (1.30) is reduced to the familiar von Bertalanffy (1938) equation a(b-m) = Loo(b-m) (1-e -K(b-m)(t-to) (1.30) £( t ) = L~ (1 - e (1.31) and substituting for length in terms of weight from (1.28) the relationship is derived: w(t) = W (1 - e -K(t-t 0 ) } 3 (1.32) where W = a(L°°) , the asymptotic weight. 2. Exponential growth: b = m From equation (1.29), when b = m, the relationship follows that: dt = (E-KU 29 and integrat ing between de f in i te l im i t s for i and t , A ( T ) = IQ e ( E " K ) t = IQ e H t , (1.33) and, subst i tu t ing (1.28) in the above equation, w(t) _ w o ( e H t ) b or w(t) = wQ e G t , (1.34) where H = the instantaneous growth rate .in length, and G = bH; the instantaneous growth rate in weight, both expressed on an annual basis (see F, M and Z, page 25). wQ = the i n i t i a l weight at t = 0 3. Parabol ic Growth: m<b; K = 0, Returning again to equation (1.29), and l e t t i ng K = 0, separating variables and integrat ing between de f in i te l i m i t s , the re la t ionsh ip is obtained: In the notation of the o r ig ina l authors (Parker and Lark in , 1959), where a = (b-m)E and z = b-m, subst i tu t ion y ie lds £ Z ( t ) = at + lZ 0 30 and letting t = tQ when £ q = 0, the equations for length and weight (from 1.28) are derived as *(t) = [a(t-t ) ] 1 / Z and w(t) = [ a z / b a ( t - t Q ) ] b / z The preceding growth models have purposely stressed the physiological interpretation of parameters as intended by Taylor (1962), and particularly by von Bertalanffy (1938) in the case of isometric growth. The derivation of Ricker's (1968) exponential formulae and that of parabolic growth, however, are mathematically inconsistent with the implied theory. In order to derive the former relationship it must be assumed that m = b, a clearly untenable proposition unless the "fish", for example, is tube shaped and its net weight never exceeds that of its own intestine. Similarly, for parabolic growth, katabolic metabolism cannot be eliminated simply by setting K(= k'/b) equal to zero in order to achieve the correct formulae. The intention of the original authors (Parker and Larkin, 1959) to represent growth as a function of size and as the resultant of complex physiological and ecological effects, is clear-ly summarized in their basic equation 4-git> = k w x (t) where k is a proportionality constant "...and x is a fractional exponent less than unity". The relationship is in keeping with observed physiological rates 31 and there is no need to invoke a special biological meaning for the parameters. The physiological interpretation of L°°, K, W°° and b in the von Bertalanffy (1938) formulae have similarly been challenged by Parker and Larkin (1959), Ricker (1960), Paulik and Gales (1964) and Knight (1968). Taylor (1958, 1962) has also indicated that K is not constant, but a function of temperature. While the search for biological meaning in these growth formulae persists, their immediate practical value is recognized as empirically descriptive models of growth. In this sense their f lexibi l i ty, and particularly that of the Ricker (1968) model, offers a wide range of choice to meet the observed growth patterns of individual species. With one possible exception the flexibil ity achieved by a suitable choice of parameters becomes a valuable property that can be readily used in assessments and yield predictions. The exceptional reference is that of isometric growth, but even here a generalized, allometric weight formula can be derived. If regression analysis indicates that the best f i t to length-at-age data is obtained by means of a diminishing growth-rate equation (1.31), and the weight:length relationship is adequately described by w(t) = a £ b ( t ) , the appropriate weight formula (Richards, 1959) is w(t) = Woo (1 - e ' ^ ^ o V , (1.35) where b can now take on values in the reported range of about 1.4 to 4.0 (Taylor, 1962). 3.2 Allometric, asymptotic growth, parameter estimation An extensive amount of work has been carried out on the techniques used to estimate the growth parameters K, t , and L°°. For an introduction to the 32 literature and a critical review of the methods the reader is referred to the computational bulletins of Ricker (1968, 1975), and to Allen (1966) for best least-squares estimates of these parameters when fitting to observed data. Here, only a brief outline of the formula derivations are given to support the fishery statistics developed later in Section III. If i t is assumed in general that mortality between age-groups is not size selective, and in particular that growth in length follows the von Bertalanffy (1938) curve, then under constant environmental conditions the length relation-ship between successive age groups (i) is described by A . = L j l - e - K ( l ' - V ) (1.36) and = L j l - e - ^ - V ) , (1.37) and subtracting (1.36) from (1.37) and simplifying, " *i = Lw e - ^ - V (l-e- K) . But from (1.36) e - 1 ^ 1 ' - ^ = L^-J l . . Substituting this relationship into the above expression, simplifying and rearranging, £ i + 1 = L j l - e " * ) + e _ K A . (1.38) Equation (1.38) is of the linear form y = a + bx and a regression (Ricker, 1973) of successive ordinate values of plotted against those of £^  33 (absissa) yields a line of slope e~ (the k of Ford, 1933) and y-axis - K intercept L^l-e ). Values of K and can therefore be readily calculated. The intercept of the graph with a 45° diagonal drawn from the origin (0,0) can also be used to estimate Lro directly. In its original form equation (1.38) was derived empirically by Ford (1933), later by Walford (1946), and its graph-ical representation is commonly known as the Walford line. Equation (1.36) can also be expanded in the form i • = L - L e • e 0 T 00 CO Rearranging this expression and taking natural logarithms gives In (L, - z.) = (ln L o o + KtQ) - Ki. (1.39) A regression (Ricker, 1973) of successive values of ln (L^-Jo.) on i , i+1, etc. yields a straight line of slope -K and intercept I = lnL ro + Kt , from which t can be calculated as tQ - 1 - 1 n L " (1.40) o K The weight:length relationship w(t) = a I (t) is implicit in the derivation of the weight formula (1.35) given earlier. In order to estimate the intra-age-group values of parameters a and b for integer values of age, the equation b w i = a £ i 34 can be expanded as a linear formula ln(w i) = In a + b ln(&.) Again, a regression of ln(w )^ on ln(a.) is used to determine the slope b , and a from the intercept In a. From a previous estimate of L o t , W^  is esti-mated directly from W m = a ( L J * 5 . An alternative method of estimating b, K and W^  is derived as follows: Given that w. = Wro ( l - e " K ( i p t o ) ) b and w. + 1 = W j l - e ' ^ ^ ' V ) b , subtracting the former expression from the latter, rearranging and simplifying 1 1 1 b b ,.b - K ( i - t ) , , -K* w i + 1 - wi = Wro e v o ; ( l -e ). 1 1 1 But Wb e ~ ^ - t o ^ = Wb - wb , and substituting this expression into the above relationship the following equation can be obtained 1 1 1 \ w b + ] = Wb (l-e"K) + e"K wb (1.41) Equation (1.41) will assume a linear relationship i f the correct value of b 1 1 is chosen in a regression of b on b . wi+l w i To determine this quantity I have written an iterative search program (BSRCH) to minimize the unexplained sums of squares about the line of best f i t . When - K 1 the optimal value of b is found, the slope, e , and intercept, b^ ^ - j _ e - K j , provide estimates of K and W , respectively. 35 4. Yield Models - Combined Growth and Mortality 4.1 Generalized yield model To inject some reality into a discussion on yield models the following observations are necessary. First, there is ample evidence to indicate that for temperate water fishes recruitment is not constant but a variable quantity, and its magnitude can be measured in terms of year-class strength. It is therefore more realistic to develop yield models on the basis of a generalized age group ( i ) , of numerical strength N^(t) in the year of fishing, and to calculate the annual yield as the sum of all contributing ages. Again, it is well established that individuals within a given age group differ in size, and in yield equations using length {i^) or weight (w^ ) these symbols must be interpreted as average quantities. The conversion of any of the previous growth equations to that of a particular age group is readily accomplished by adding the integer value of i to the independent variable t, e .g. , from w(t) = W ( l -e" K ^ t _ t o^) b , o < t < « to w.(t) = w (l-e-K<i+t-to>)b ' ° ^ - t ^ 1 ( K 4 2 ) The growth models given earlier imply a continuity or a continuous growth pattern from one age to the next. As mentioned earlier, to retain this quality the critical assumption must be made that mortality, in whatever form, is not size selective. Finally, there is no mathematical or biological reason why a concept of yield should be limited to that of weight. Fish grow in length as well as weight and it is as easy to conceive of yield in terms of "biolength" (YL and B^ ) as i t is in "biomass" (Yw and B w ) . In fact, as shown above, the _ For a yield model based on length see Jones (MS, 1974) 36 majority of growth models make use of a weight:length relationship and therefore both types of yield models can be derived simultaneously. If all of these points are kept in mind, two basic yield equations can now be developed. In the previous section fishing and natural mortality are represented as independent, concurrent processes. The catch is calculated on this basis and supported by assumptions involving the distribution of fish (or fisher-men) in space and time. If these concepts are retained and it is additionally assumed that: 1) gear selection is "knife-edged" (Beverton and Holt, 1957), and 2) all fishable age groups are subject to common mortality rates of F and M, then the basic equations for rates of yield (y) are: for weight djW*) = ^ ^ _ ^  {] A 3 ) a n d l e n g t h d ( y ^ i ( t ) = F(B^)i (t) = F N.:(t)£.(t) (1.44) where N (^t) = the fishable age group abundance w (^t) = the average, age group weight, and A.(t) = the average, age group length, all determined at time t , (o < t <_ 1). Equations (1.43) and (1.44) illustrate three major points and they are: 1) the growth functions w.(t) and ^ ^(t) are free of any mortality component and are therefore independent processes 2) mortality, from whatever source, is associated with the rate of decline in numbers, e .g . , F N (^t) 3) the rates of change of biomass and biolength, and therefore their yields, are dependent on the concurrent and opposing processes of growth and mortality. To determine the average rates of yield ([Y^ R).j , (Y^ R)..] over a year both sides of equations (1.43) and (1.44) are multiplied by dt, divided by dt, and integrated between the limits of t(0, 1) to obtain , (VPJT = < V i = F < V i = F « T ( ] - 4 5 ) and ( Y L j R ) i = (Y L). = F (B L). = F (NT). (1.46) where (Y..). and (Y.). are, respectively, the annual yields in weight and W 1 L 1 length, and are numerically equal to their average annual rates. (B )^.j and (B )^^  represent the average biomass and biolength, and are respectively equal to the average product of numbers x weight (Nw)^  and length (Nft). over the year. The above method can be illustrated using the integrable functions of weight and length of Ricker (1968). Substituting (No)n.e"Z t for N^t) from equation (1.20) and the exponential formula for weight (1.34) in the general expression (1.43), and transferring dt 38 ' l V l d < V i • F 'I <No>i e - Z t . ( w 0 ) i e G l t d t i (G.-Z)t n , (G.-Z)t,. <Vi -F <v„'i 4 e d t • F<BS>i 'Je where (B )^n- = the initial age group biomass at t=0 . Following integration, <Vi - F'Oi ^ e 1 ! ! ' - 1 ' - ' F ( Vi (1'47) i and similarly for length (V L), - F ( i f ) , ( e ^ _ z -1) . p , where G^  and are, respectively, the age specific instantaneous rates of growth in weight and length, expressed on an annual basis. 4.2 Beverton and Holt, constant recruitment, equilibrium yield model In relation to the biomass yield that can be taken from a fishery Cushing (1972) distinguishes two areas of major concern to management: growth overfishing and recruitment overfishing. The former refers to the potential loss in yield i f fish are caught at too early an age and do not achieve maximal growth rates. Peterson (1894) originally drew atten-tion to the problem with regard- to North Sea plaice, and subsequently Ricker (1945) defined the point of optimal capture as the "critical size". It occurs at the average size at which the net gain in biomass is zero in the absence of 39 ffshing, i .e . , G..-M.. = 0. Only yield models that separate the growth and recruit functions, such as the Ricker (1968) model given above, are therefore capable of providing a solution to the growth overfishing problem. Addition-ally, an outstanding feature of these models is that under equilibrium condi-tions of constant recruitment, the growth and mortality functions can be used exclusively to calculate the anticipated yield for a given age of f irst capture and a range of fishing mortality (F) values. By combining the projected yields for all ages and values of F a yield isopleth (Beverton and Holt, 1957, p. 318) diagram showing contours of yield per recruit can then be constructed to esti-mate the critical age and MSY per recruit, and estimate the eumetric fishing curve. The growth and mortality model that has probably received the greatest attention in fisheries investigations is that popularized by Beverton and Holt (1957). In its original "simple" form it has found wide application in trawl fisheries and for temperate water fishes that exhibit isometric growth accord-ing to the von Bertalanffy (1938) equations. The principal assumptions and conditions under which the simple model is applicable are as follows: 1. It pertains to a multi age-class, single species stock in which growth and mortality (F and M) are continuous, concurrent and independent processes. Additionally, the mortality rates are constant for all groups in the commercial stock. 2. Technically the model applies to a closed system. The equations contain no immigration or emigration terms. 3. Environmental conditions are assumed to be constant, or their average effect over time demonstrates no measurable trend. 40 4. The fish (or fishermen) are randomly distributed at all times. The basic assumptions used to define the mortality functions given earlier therefore apply. 5. The trawl gear operates with "knife-edge" selection. 6. A constant number of fish are recruited on the same date each year regardless of the fishing mortality. 7. Growth in length for the commercial sized stock is asymptotic (von Bertalanffy, 1938), isometric (b = 3), and weight and length are re-3 lated by the expression w(t) = a I (t). To develop the simple model let R equal the number of young fish in a year class that f irst become exposed to the gear at an average age a^, but are not actually captured until some later, average age;; a . Conveniently, since time and age are both measured in the same units (years), these ages can be replaced by their time equivalents, t and t respectively. The number of fish survi-ving to age a c is then represented by R' = R e ^ V V and the number available at any time after capture begins is -Z(t-t ) -M(t -t ) -Z(t-t ) R'e = R e c r e c , t < t < t' (1.48) where t' = an integer; the age of exit from the fishery. Similarly the average weight of a fish at time t is -K(t-t ) , w(t) = Wjl-3 0 ) 3 (1.49) The product of equations (1.48) and (1.49) is equal to the biomass B^(t), 41 and the rate of capture is therefore defined as d(yw)(t)/dt = F Bw(t). For a single year class the total yield (YW) T over its fishable l i fe span, X(= t' - t c ) , is calculated from ( Y \ _ , (VT t' -M(t -t ) -Z(t-t ) -K(t-t ) 3 (VT " / 0 d(yw)(t) = F / Re c r e c Wjl-3 0 ) dt c Following integration and simplifying the resulting expression -M(t -t ) 3 -nK(t -t ) f - r . M ^ i / u ( Y w ) T = F W o o R e c M a n . e 0 0 ( ^ - ( F + M + n K ) ^ ( K 5 Q ) n=o F+M+nK where a n = (1, -3, +3, -1) for n = (0, 1, 2, 3). The biolength yield for the year class is also derived as -M(t -t ) _ z - ^ r ^ J -C7+K)X (Y . ) - r = L R e c r [(1-e Z X) - e c 0 (1-e ( Z + K ) A ) ] : L 1 Z Z+K Under conditions of constant, annual recruitment the numbers of fish in successive age groups are serially related by a common mortality term. The integrated biomass (biolength) of a year class throughout its fishable l i fe span must therefore also be equal to the biomass (biolength) sum of its separ-ate age groups taken over a period of a year. The total yield from a year class must then be equal to both the fishing mortality times the average bio-mass and the annual yield. This conclusion was originally arrived at by Thompson and Bell (1934) in their studies of equilibrium yield in relation to fishing effort. 42 Equation (1.50), its shortened forms (Ricker, 1968; Gulland, 1969) and a simplification prepared by Jones (1957) can all be used to determine yield under constant recruitment. Additionally, i f the basic model is separated into its constituent age groups the expected yields during the transition period from one equilibrium state to another can also be calculated. As men-tioned earlier, even i f R is unknown, the critical size and the maximum sustained yield per recruit can s t i l l be estimated. The model is thus best viewed as having theoretical value, but its practicality is limited by, and the results are dependent upon, isometric growth and constant recruit-ment and mortality. It cannot be used to accurately predict yield i f R should vary from year to year or be subject to upward or downward trends. One method of dealing with these problems is discussed under subsection 4.3. The flexibil ity of the Beverton and Holt (1957) model can be greatly extended i f the total annual yield in weight or length is calculated as the sum of contributions from each age group (i) (Walters, 1969). In the process the identity of the simple model is-essentially .lost. Instead, it becomes-a more generalized age-group model for isometric, asymptotic growth. The . -integrated annual .yield formulae may be readily derived as i ' i 1 3 -nK(i-t ) -(F.+M.+nK) ( YW }T = E ( V i = W o o E V i ( V i E i . O-e 1 1 ) W 1 k W 1 k 1 1 0 V O F.+M.+nK and i ' i ' -(F.+M.) -K(i-t ) -(F.+M.+K) L 1 k L 1 k 1 1 0 1 Fi+Mi F.+M.+K 43 where k is the f irst and i ' the last age group taken by the fishery, $. = the average, selectivity coefficient of the trawl for age group i ; 0<$ .<!), F. and are age-group specific mortality rates, and (NQ).j = the starting population size at the beginning of the year. The application of the yield in weight formula, coupled with a Ricker (1968) recruit function is given in Appendix A, part 2. 4.3 Virtual population and cohort analysis models. The virtual population method developed independently by Fry (1949) had led to the development of one of the most useful analytical tools available in fisheries work. In its original form, Fry (1949) recognized that the sum of the catches taken from a year class throughout its existence in the fishery provided a minimum estimate of the original recruit abundance (in the absence of natural mortality). In addition, i f data on the entire catch history of a year class was available, a maximum estimate of F could be prepared for each fishing year by interpreting the catch per unit effort as an index of avail-ability rather than one of abundance. Subsequently, the method has been adapted and modified by Fry (1957)5, Jones (1961, 1964, MS 1967, MS 1974) and See Ricker (1975, p. 184) for references to earlier Russian papers. Presented at the joint ICES, ICNAF, and FAO fisheries conference in Lisbon, 1957. Following this meeting several authors suggested using the Beverton and Holt (1957) method of plotting Z against effective effort (f) to estimate M and q. The linear relationship is based on Z = qf + M, and the virtual population method is used to estimate Z for a range of f values. Subsequent-ly, Bishop (1959) and Paloheimo (1958, 1961) drew attention to the bias in individual estimates of M and q. Paloheimo prepared an analytical solution which recognized the positive correlation between Z and f. Bishop showed :i that under conditions of either increasing fishing mortality or effort fluc-tuations without trend, regression estimates of M were too low and the slope (q) too high. The reverse condition existed i f effort was decreasing. Ricker (1975) indicates that the regression method is no longer used, but the above criticisms are nevertheless valuable. 44 Gull and (1965) to estimate Z and F, independently of effort, for each year that the year class is fished. Schumacher (1970) has reviewed three varia-tions of the virtual population technique, and the conditions and assumptions common to all three are as follows: 1. The basic catch equation of Baranoff (1918) applies, i .e . , x C n = A x < N o > n < 1 - e " x Z n > x Zn where x = the year class n = a generalized subscript for the year of fishing C x n = the age group catch of year-class x in year n All previous assumptions concerning the distribution of fish (or fishermen), and the continuous, concurrent operation of both mortalities are therefore in effect. 2. The mortality rate M is constant for all age groups and must be estimated independently. 3. It is assumed that the catch in any year n represents all of the catches for that age group from year class x. If this assumption is not made, the calculated value of F and the back calculated value of the apparent recruit-x n . r r ment, R' , will be too low. 4. A catch model rather than one of yield is assumed and it is applicable to a closed system. The virtual population, V, for year-class x is defined mathematically as V = I C x x n 45 where the sum is taken over all age groups throughout the fishable l i fe span of the year class. In any given year (n), however, the partial virtual population, V 1 , may be derived from the catch equation, i .e . , A 11 w • C F (N ) ( E N ) V _ x n _ x n xv o n = x o n ,-, r1\ x n = _ j = —j— (1.51) (1-e x n) x n where x^n' = t n e c a t c n i n y e a r n P^u s a ^ future catches to be expected from x i f F and M x n It is a partial sum.  remain constant throught the period, xFn = the exploitation ratio, x (F /Z) n Similarly, xV'n+1 = xCn+l = xFn+l x^Vn+1 = x^E Vn+1 (1.52) fi „ x^n+l\ X^ n+l (1-e ) Equations (1.51) and (1.52) represent the basic formulae used to derive all the methods that follow. Since the discussion will refer to one year class only, the subscript x is removed. Method I Condition: the year class has been fished to extinction and the virtual population, V, is known. -Z •If equation (1.52) is divided by (1.51) and (N Q) n +i is replaced by (NQ)ne n , it follows that V _i_i E -, e Z n = -fl (1.53) n n 46 If it is now assumed that mortality is constant for consecutive years, then as a f irst approximation E +^ = E n and (1.53) reduces to S = V' , = e Z f 1 (1.54) n n+1 V n where = the survival rate in year n. A process of back calculation is initiated by assigning to V .j the sum of catches for the residual age groups that are older than the last age group of x for which catch data are available. Since V' -j, V and M are then known, (S, Z, F) may be calculated directly. It is also possible to derive (S, Z, F) n_2 and preceding years by the appropriate substitution of the virtual population ratios in ( 1 . 5 4 ) . The accuracy of the method is impaired by the assumption that E n = E .j. Method II Conditions: the virtual population, V, is known. An estimate of the exploitation ratio in the last year of fishing must be prepared or chosen. Jones (1964) recognized that the starting population in year n+1 is related to the catch in the previous year in the following way: Given that -1 (N ) = (N ) e n and v o n+1 v on C = E (N ) (1-e n ; ' n nv o'n v then •Z -Z r = (N ) = (N ) e n = e c c i n v oyn+1 v oyn _ z — (1.55) Cn E (N ) (1-e n ) E (1-e n ) nv o nv ' n n 47 Since is a function of F n and M, and M is assumed to be constant, a table of ratios of r n for a given value of M can be prepared for all values of that may be encountered in a fishery. The table can also be extended to in--Z -Z elude values for the terms e , E n and E n O -e n) to avoid unnecessary calculations. From a knowledge of r and Z , F can then be estimated 3 n n n directly. The difficulty at this point is in assigning a value to for the second to last year of fishing. Two relationships provide the necessary insi ght: V ,, = E ,, (N ) , -i and C = V - V' , n+1 n+1v o'n+1 n n n+1 ; and rearranging in the form of the ratio r^  r = (N ) , 1 1 V x l n o'n+1 = -F i n+1 x n c c \ E^ ( ,„.i) (1-56) If i t is once again assumed that V +.| = S^V1^ from equation (1.54), expression (1.56) may be replaced by r n = j Sn E n + l ^ W ( 1 ' 5 7 ) where E +-j represents the last exploitation ratio to be applied to the year class prior to extinction. Its value must be assumed or determined in order to initiate the back-calculation process to evaluate E n , E -|, etc. The relationship between E n and E .j is established by means of equation (1.55) and the expression Cn = W n " W V n + l = <No>n ( E n " En+l e ^ 48 to yield 7 " Z n ' n (N ) , ( N ) e e _ v oyn+l v oyn -Z C -Z E n(l-e n) n (N.) (E -E x l e n) nv v o n n+1 ' Simplifying and rearranging the above relationship i t follows that y 1 - ^ + En+1 e _ Z n = En d ' 5 8 ) -Z where E n(l-e n) = the exploitation rate ("u" of Ricker, 1968) and is the fraction, by number, of fish that will be caught during the year n, and -Z En+-|e = the fraction that is caught later. A point that Schumacher (1970) does not make clear is that i f equation (1.58) forms the basis of the back-calculation technique, the assumption must be reiterated that adjacent exploitation ratios are equal. On this basis E n can be set equal to E -j for the preceding year and the working equation for the model is rewritten from (1.58) as + En+1 ^ - En=l (1.59) To begin the process r n is calculated from (1.57) based on the estimated value of E n + i . The tabular value of r yields the numerical quantities -Z ' -Z E n(l-e ) and e , and substitution into equation (1.59) determines E^ -j. This latter value is again substituted into (1.57) to yield r ^, and the process is repeated until the catch from the f irst age group entering the fishery is encountered. 49 Method III Condition: the year class has not been fished to extinction and therefore the partial virtual populations V1 V ' n , V'n_-|> etc. , are unknown. In this case the catch equation for year n+1, the last year of fishing to date, may be combined with the catch in the previous year to form the ratio and equating this expression to (1.55) ^ j = r " " ~ z • <'.60) The init ial ratio r n is estimated by means of a starting value of E n +^(l-e in equation (1.60). The quantity of E n(l-e n ) , and therefore are obtained directly from tables and (1.60) is used as the basis of back calculation In order to determine the apparent recruitment, R', the following A nomenclature (after Pope, 1972) is adopted for year class x. Let t equal the last age group of a year class for which catch data are available, 50 and let i represent the first age group captured. The catch equation in the f irst year can then be written as C, - ! t ( N 0 ) , O - e " * 1 ) i and expanding and rearranging, R1 = (N ). = C Z i + N..,, x v o'l i -p- i+l i The above expression can be used as a recurrence relationship by substituting for N.j+-|, N-j+2, etc. , as follows: VR' = C. Z i + C . . , Zi+1 + . . . + C. Zt-1 + (N ). (1.61) A 1+1 p t - l r 0 t h i i+l h t - l The value of (N ) t can take on two forms depending on whether the year class has been fished to extinction or not. If fishing is complete, the catch C t refers to age group t and all subsequent residual age groups. In this case (N Q) t = Ct Z t/F t (1.62) Similarly, i f (NQ) t refers to the last age group in an incompletely fished year class (N 0) t - ^ t _ F t ( l -e *) (1.64) and substitution in (1.61) yields X R" = C. F t ( l -e l ) (1.65) To evaluate R', calculations begin with the final age group (t) and an extimate of either F^/Z^ (1.62) or Z^(l-e ) followed by a backward summation Alternate methods of the virtual population technique have also been proposed by Murphy (1965), Pope (1972) and Doubleday (MS 1975).6 The initial method involved an iterative procedure of solving for the. F n values embodied in a sequence of catch-equations ratios between adjacent age groups. Again, the value of M is constant and the initial fishing mortality rate (F..) should be known for ease of computation. The reverse procedure of estimating F^, as in the methods described above, introduces the need for laborious and repeated iterations. In either case the method offers no distinct advantage and has not been accorded wide usage. The "cohort analysis" method of Pope (1972), however, contains several features that are of practical and analytical inter-est. It is an approximate form of Methods II and III given above that: 1. eliminates the use of tables 2. provides estimates of x F n and (N ) n in all fishing years given an estimate of M and the final value F^, and 3. allows an investigation of the errors introduced into x F n , R' and (NQ)n by the arbitrary choice of F^, and sampling errors in catch-at-age data. The approximate formula is derived by rearranging the equation x (NJ = (N ). e v 0 1 -(F.+M) into (N ) e M M N Q ) i - ( N o ) i ( l - e " F i ) , o ;i+l oyi+l See Jones (MS 1974) for a virtual population method based on length. 52 and expressing the last value of (NQ)^ in terms of catch, <Nn>l+l e M = (Nn>i " C i f d-e F i ) ( F i + M ) ] (1.66) oi+l o i i ITjr+M) ^ (1-e 1 ) The terms inside the square brackets remains unchanged i f multiplied and divided as indicated: 7 e . r ( l - e ) . ( I ) -, . e F,/2 L F. -(F.+M) J (F.+M)/2 (1-e 1 1 ) - 1 e to y i e l d a product of three terms , V 2 " V 2 ,F.+M, ( Fi + M) / 2 (e -e ) . ( i ) . e F. (F,+M)/2 -(F +M)/2 F./2 1 (e 1 -e 1 ) e 1 Within the range F. < 1.2 and M <„0.3, the product of the f irst two M/2 terms is approximately equal to one. The last term reduces to e , and therefore (1.66) can be rearranged and replaced by the following expression of Pope (1972): ( V t = ( V i + i e M + c i e M / 2 ( 1- 6 7) For the last year of fishing (1.67) becomes The author is indebted to Dr. J . Schnute for this insight. 53 and substituting the appropriate value for from either equation (1.62) or (1.64) the process of back calculation can begin. In a comparative examina-tion of Fn and (N ) n estimates prepared by virtual population and cohort analysis, Pope (1972) indicates that the differences do not exceed two percent. Analysis of the cohort method (Pope, 1972) indicates that i f there are no errors in the catch data an underestimate of the true value of leads to a positive and negative error, respectively, in the back calculated values of (N ) and Fn . The converse is also true, but it should be added that as the sum of the fishing mortalities (F^ + F ^ + etc.) increases, the errors in (NQ)n and Fn are reduced. Consequently, the estimates of (N ) n and Fn become increasingly more accurate progressing from older to younger age groups. Sampling errors in catch^-at-age data similarly introduce errors in estimates of both age group numbers and their fishing mortality rates. The magnitude of these errors, expressed as a percentage of the variance ratio in catch-at-age data, also declines and approach asymptotic values at younger age groups. The errors are persistent, however, and cannot be eliminated unless the catch data i tself is error free. To appreciate the variance associated with estimates of Fn and (N ) i t is therefore more critical to know the variance in the catch data for each age group than to use the true value of F^  as a starting point. For this reason both Pope (1972) and Doubleday (MS 1975) indicate that the latter quantity is likely to be chosen arbitrarily. In the virtual population and cohort analysis methods the number of parameters (F n , (N ) ) to be estimated is greater than the number of observa-tions (C ) and a unique solution is therefore not possible. To overcome this diff iculty, Doubleday (MS 1975) has applied a least squares method to a matrix 54 of catch-at-age data for a number of consecutive year classes. An adequate description of the method is beyond the scope of this report, but the follow-ing critical features should be noted: 1. For age group i and fishing year n, the fishing mortality rate ^Fn at any point in the matrix may be considered as a product of n-Qfn> where represents a selection coefficient common to each age, and f is an effective effort term that is applicable to all age groups fished in year n. Both and f may then be expressed logarithmically as q^ = In .Q and f^  = ln f^  to yield In .F = .q + f . Similarly, logarithms of the usual catch equation i n n ^ for .C and the ratio of successive catches -jCn/-j + -| C n +-| yield equations that are potentially linear and may be subjected to least squares analysis. Expo-nential terms in the equations of In and ln( i-Cn //+i •C n + 1 ) are expanded by Taylor series approximations to complete the linearization. By these proce-dures, and given sufficient data, i t is possible to obtain more than one observation to support each parameter. In.common with the virtual population and cohort analysis methods, the least squares technique also requires prior estimates of certain parameters. If q^ for the last age group and f^  for the last year of fishing are fixed, the total number of parameters that can be estimated is 2(A+J)-1, where A and J. represent the total number of ages and year classes, respectively. On this basis, as Doubleday (MS 1975) indicates, nine ages and eleven years of data are required in order that each parameter be supported by about two and one-half observations. 2. The principal benefits of the method are that the results lend themselves to statistical analysis. Variance estimates (reliability) can be attached to the parameters, anomalies in the data can be examined, and " . . . the amount of 55 information contained in the catch data about population sizes" (Doubleday, MS 1975) is also indicated. 3. Unfortunately the model may not offer a unique solution. Different starting values of ..q and f^  in the final year may produce different results in terms of ^ and n-(N ) n . Because the catch ratios in successive years are negatively correlated (the denominator -+-|C ^ in one year becomes the numer-ator in the next) the parameter estimates of .jF and .j(N ) tend to move in opposite directions. It is then possible for two or more sets of ^(F, NQ) n values to generate the same catch observation. The quantity q^ + f^  always appears as a sum and an increment in ..q coupled with a decrement in f^  by the same amount does not alter their sum. Finally, for a meaningful analysis of variance, data for well sampled catches over an extensive period of time are required, but there is l i t t le assurance that selection criteria will remain stable during the interval. The assumptions and conditions underlying the virtual population, cohort analysis and a least squares method of dealing with catch-at-age data have been described. Depending on the amount of information available, either technique can be used to prepare estimates of the fishing mortality and starting popula-tion sizes by age group. It is also clear that these parameter estimates are conditional upon the accuracy of F^  used as a starting value in the final year of fishing, the assumption of constant M, and the sampling error in the catches. If these limitations are recognized, the parameter values may be used to develop two secondary estimates concerning recruitment and yield prediction. In the f irst case, i f complete catch data exist for a series of year classes, the apparent recruitment (xR' , _-|R' , etc.) may also be calculated. If sufficient information exists on the adult stock sizes that gave rise to these successive 56 recruitments, then the potential exists to establish the beginnings of a stock-recruit relationship. At the opposite extreme, i f the fishing mortal-ity for each age group at the end of the present year of fishing (n+1) can be estimated, the partial age structure of the starting population in year n+2 can be predicted. Together with an estimate of the apparent recruitment in the same year, the total age structure may then be combined with: ! 1) average net-selection coefficients ($ ), 2) average weight-at-age data, and 3) anticipated fishing mortalities ( - jF^K to predict the annual yield. The procedure is currently used by ICNAF (Hodder, 1975) to prepare estimates of the total allowable catches for single and mixed stocks for single species that are capable of being aged. 4.4 An asymptotic, allometric growth, yield model Solutions to the Beverton and Holt (1957) yield equation when growth is allometric have been proposed by Jones (1957), Paulik and Gales (1964) and Kutty (1968). In more general terms, the yield model can be established by substituting equations (1.20) and (1.35) into (Y w) i = F fl N.(t) w.(t) dt i .e . , 1 _7t -K(i+t-t.) b ( V i = F fo [ ( N o ) i e W l - e 0 )b] dt (1.68) The uti l i ty of this equation can be extended to include the allometric -K(i+t-t ) b conditions of b = 2 or 4, but i f b f integer the expression (1-e ) cannot be expanded prior to integration. Jones (1957) recognized that i f a o For an analytical solution to the enmetric fishing curve for isometric and allometric growth, see Kutty, 1968. 57 suitable substitution is made for this expression, subsequent integration of (1.68) leads to a simplified difference between two incomplete beta functions. On an age group basis his yield formula can be rederived as where X = e u -K(i+l-t ) X ] = e 0 3 = the incomplete beta function, and b can have an integer or non-integer value. To evaluate this equation the beta functions associated with the coordinates (X, 7_, b+1) and (X , 1, b+1) can be obtained from tables prepared K K by Pearson (1948) or the more practical edition by Wilimovsky and Wicklund (1963). The method is simple and direct, but non-tabulated values of b+1, Z/K and X require interpolation, and i f extensive calculations are necessary the method may be time consuming. Alternative solutions to equation (1.68) are possible and they all depend on the interpretation of the number and weight functions in the general equation If w-(t) is treated separately, two solutions that can be integrated directly are: I. Approximate the weight function by means of annual arcs of exponential growth. The risk involved here is that for each age group of interest to the fishery growth will be represented by a convex arc, whereas (Y..). = F (N ). W e Z(i-t ) ( V i = fo N i ( t ) w i ( t ) d t (1.69) 58 the actual curve may be concave. The applicable, integrated formula can be readily derived as (Ricker 1968) (Bw)i = (N 0w o ) i ( e ^ - l ) (1.70) G.-Z II. Approximate the weight function as annual chords of linear growth. The appropriate weight formula is then wn-(t) = ( w Q ) . + g . t , 0<t<l where g. = the linear growth rate for age group i . Substitution into the general biomass equation (1.69) and integrating gives: ( B w ) 1 = ( N Q ) . [(wQ) i (l-e~ Z)- ? ! (1-Z e' Z -e- Z ) ] (1.71) Z z2 The third and fourth methods rely on approximate integration techniques and deal directly with the product (N.(t) w (^t) as a biomass term. Attention is now focussed on the curve generated by plotting the biomass for each age group versus time. The area under the curve for a given age group taken over one year may then be approximated as follows: III. Trapezoidal rule of approximate integration (Smail, 1949). The method is based on representing a definite integral by an area under a curve and approximating this area by a set of inscribed trapezoids, or replacing the arc of the curve by a set of chords. To generalize the method, let: y = the biomass yg and = the initial and final biomass, respectively, for a given age group in the time interval h. 59 x = N.(t) w.(t) a = x evaluated at y Q } and c = x evaluated at y-j; therefore y = f(x) A = the area under the curve. From these re lat ionships the area to be approximated is equal to A = f\ f(x) dx = y dx , a a and i f the in terval h i s set equal to one, the above equation is : approximated by A = h ( y 0 + y i ) . Subst i tu t ing the appropriate biomass values for y^ and y^ , and d iv id ing both sides by f^ dt (=1), i t fol lows that vi- + \ 7 -K( i+ l - t ) . " K ( l - t ) . -If. „ 0 | D n n 7 9 \ (B„ ) . = (N ). W [(1 - e 0 ) b - e ( 1 - 6 > ] ( 1 ' 7 2 ) 2 IV. Simpson's rule of approximate integrat ion (Smai l , 1949).. In th is case the def in i te integral is again interpreted as the area under the biomass curve, but the curve i s approximated by a ser ies of parabol ic arcs . If the time interval of one year is divided into two equal parts (h = 1/2), three biomass values are needed to describe two parabol ic arcs : the i n i t i a l biomass yg , y-j at the midpoint and y^ at the end of the per iod. It can then be shown that the area under the curve i s approximated by A = h (y 0 + 4 y i + y 2 ) = Y ' ( y 0 + 4 y l + y 2 } Again, subst i tu t ing biomass values for y Q , y-j and y^ and d iv id ing both sides by f\ d t (= l ) , the general biomass formula is 60 -K(i-t ) L n r / v \ -K(i + 0.5 - t ) , (B w) i = (N (J)1 Wm (0^5) [ ( 1 - e 0 ) b + 4 e - ° ' 5 ( Z ) ( l - e 0 ) b -Z " K ( i + 1 " t n ) b + e Z ( l - e 0 ) ] (1.73) How well equations (1.70), (1.71), (1.72) and (1.73) approximate the actual value of the average biomass can be determined in the following manner. In order to maximize the yield in weight from a multi age-group fishery one criterion of interest to management is the critical size (Ricker, 1968). On a curve of biomass versus time this size is reached when the derivative dB/dt = 0 at the critical time (age) t ^. In the absence of fishing mortality the biomass at time t (0 _< t _< °°) is evaluated by M . -K(t-t ) , (Bw) (t) = N(t) w(t) = NQ e~ M t Wjl-e 0 ) b (1.74) Setting t = t differentiating this equation with respect to t £ r t and setting the result equal to zero, ^W = 0 = NW [ b K e - M ( e " K ( t c r t " t o ) ) ( l - e " K ( t c r t " t ° ) ) b " 1 d ^ r t - M e - M ( l - e " K ( t c r t " t o ) ) b ] , and solving for t .^ : trrt = t n + 1 l n (bK + 1) (1.75) K M For the west Newfoundland cod stock, Wiles and May (1969) estimate the growth parameters as: K = 0.14, t = -0.2 and L O T = 93 cm. The value of WOT may be calculated as 7263 gm for combined sexes i f b is conveniently set equal to three. Choosing an arbitrary value of M = 0.22, substituting these para-meters into (1.75) and solving: t ^ = 7.4 years. If (N Q)y = 10,000 f ish, 61 the actual average biomass in the critical age interval of seven to eight years is calculated directly from 1 -Zt "K(7+t-t ) 3 (B,,)7 n = (N ) 7 W f e L Z (1-e 0 T dt = 3997.48 Kg W 7-8 o 7 0 0 o Similarly, by substituting these same parameters into equations (1.70), (1.71), (1.72) and (1.73) the comparative biomass estimates are Method Type B~7_8 % difference Kg compared to actual I Exponential 3981.49 -0.400 II Linear 3997.82 -0.008 III Trapezoidal 3982.15 -0.383 IV Simpson's rule 3997.48 negligible For all practical purposes the average biomass estimate prepared by Simpson's rule is identical to the actual value. For this reason it is selec-ted as the method of approximating the average biomass in the case of both allometric and isometric growth. It is used exclusively in the development of the analytical model in Section IV. Despite the apparent complexity of equa-tion (1.73) it is readily dealt with by computer, i f required. The appropriate annual yield formula for weight, by age group, is therefore: w -K(i-t ) . K , -K(i+0.5-t ) . ( V i * F(N0). ^ [ ( 1 - e 0 ) b + 4e"- 5 Z (1-e 0 ) b 6 , -K(i+l-t ) , +e"Z(l-e 0 )b] (1.76) 62 Similarly, i f growth in length occurs according to the von Bertalanffy (1938) equation, the annual yield equation for length, by age group, is (Y. ) = F(NJ. Lro [(l-e' Z) - e " K ( l " t p ) ( l - e - ( Z + K ) ) ] (1.77) L i o n z Z + K An interesting feature of equation (1.76) is that i t contains an estimate of the instantaneous age-specific growth rate, G-.;--If the terms-within the square brackets-are rewritten as / -K(i-t ) . „ ' -K(i+0.5-t ) , 7 -K(i+l-t ) . (1-e 0 ) b [1 + 4e" - 5 Z (1-e 0 ) b + e"Z (1-e 0 )b] , -K ( i - t ) -K(i-t ) 1-e 0 1-e 0 Z G i the terminal expression, excluding the term e , is equal to e . The derivation is arrived at by means of the exponential relationship for integer values of age and the allometric growth formula for weight, i .e . , G. -K(i+l-t ) s wi+l = e 1 = W (1-e 0 )D and w i ~K(i-t ) . 1 W (1-e 0 ) b G. -K(i+l-t) K ( i - t ) K(i-t ) |/ e 1 = r(1-e , , ° )(e 0 ),b = f e , 0 - e ^ y b e 0 - 1 e 0 -1 and taking natural logarithms, K ( i - t ) K G. = b ln re 0 -e'\ . (1.78) J . - u in re -e i 0 -1 63 Since = b H. from equation (1.34), the instantaneous age-specific growth rate in length is therefore M 1 K ( i _ t ° ) " K HL = In re - e i 1 l K(i-t ) J e 0 -1 4.5 r Seasonal pulse fishing Larkin (1977) has described maximum sustained yield (MSY) as an illusionary concept. For a single stock the fishing mortality needed to main-tain MSY may erode the genetic variability and reproductive stability that support the original MSY estimate. Mixed species or stocks of single species are similarly reduced to the MSY level of their most productive elements. In each case the long term value of the sustained yield is less than the hypo-thetical MSY. In the original analysis of a fishery, however, MSY serves as a theoretical limit beyond which biological productivity of a target species is reduced. The interpretation of MSY in this limited sense is common to all the yield models presented so far. In each case the basis of estimating MSY is also restricted to the assumption of continuous fishing, growth and natural mortality. Mathematically it is convenient, but it may not be realistic i f the species undergoes seasonally intensive growth or mortality. In fact, the value of MSY estimated from continuous rates is conditional and may not repre-sent the maximum possible sustained yield. A fishing strategy adapted to the actual seasonal growth and mortality pattern of a species may prove superior. The potential improvement in yield is explored here under conditions of sea-sonal pulse fishing. 64 The seasonally most active period of growth and mortality of several species is illustrated in Figure 2. For temperate water species the most important point is that growth does hot take place continuously. Active periods may span one or two consecutive quarters of a year, they may be bi-modal, or divided seasonally on the basis of maturity or size. These results are both sufficient and encouraging in terms of a preliminary analysis of seasonal pulse fishing (SPF). A greater degree of precision cannot be ex-pected because the original investigations did not focus on the seasonal nature of growth. Information on when fish die is practically non-existent, but there is a suggestion of seasonality for the brown and rainbow troutas well as the walleye and alewife. To overcome the limitations of these data, a simulation model is used to calculate equilibrium yield for 200 differ-ent fishing situations. The details of the model, results and interpretation are reported in Appendix C. A brief description of the model and results are also given here. A hypothetical, multi-age group population is selected as the target species. Growth and mortality are independent processes that can occur in a seasonally condensed or continuous mode. In the seasonally intensive mode annual growth (mortality) takes place in any one of four quarters. Moderate growth (mortality) occurs during any two consecutive quarters, and the exten-sive phase involves three consecutive periods. A total of 240 seasonal and continuous growth-mortality patterns are possible, but only twenty cases are examined by the model (Figure 3a). Ten fishing strategies (FS) are similarly arranged into intensive, moderate, extensive and continuous categories (Figure 3b). SPECIES BROWN TROUT Salmo trutta WHITEFISH Coregonus sp. Coregonus clupeaformis Coregonus clupeaformis HERRING Clupea harengus Clupea harengus Clupea harengus Clupea harengus PLAICE Pleuronectes platessa ARAL SEA BREAM Albramis brama bergi, Grieb ATLANTIC COD Gadus morhua, L. WALLEYE Stizostedion vitreum vitreum LAKE TROUT Cristovomer namaycush RAINBOW TROUT Salmo gairdneri YELLOWFIN TUNA Thunnus albacares ANCHOVETA Cetengraulis mysticetus ALEWIFE Alosa pseudoharengus S e a s o n a l Growth (g) and N a t u r a l M o r t a l i t y (M) P e r i o d s Jan Feb Mar Apr May Jun Jul Aug - - Sep Oct Nov Dec } non- larval g} larval immature _g_ nursery grounds mature large cod _g_ small cod (speculat ion) 9 1 <2 REFERENCE Swif t (1955) Brown (1957) _g_ Nikolsky (1963) Healey (1975) Van Oosten and Hi le (1947) Nikolsky (1963) Das (1972) Bowers (1952) Jensen (1950) Nikolsky (1963) Nikolsky (1963) Kohler (1964) Kelso and Ward (1972) Kennedy (1954) Tody (1964) Hennemuth (1961) Howard and Landa (1958) Brown (1972) Figure 2. Seasonally intensive periods of growth (g) and natural mortality (m) for a variety of fishes. Legend: reduced r a t g ^ g r o w t n o r mortality. .. i > increased 3 cn cn a) CASE 1 10 * * m m * ic rk * * * * * * * m * * * •k rk rk * rk * * ic ic ic rk rk rk rk m m * * * *. * * * * 9, 7 i CASE 11 12 13 14 15 16 17 18 19 20 m g i m m m 9i m * * * * * ic ic rk ic * * * * * * * * * * * * * * rk rk rk rk •k rk rk rk m 9i m 9i m * * * * * * * m * * b) Intensi ve FS 1 * 2 * 3 * 4 * Moderate FS 5 6 7 * * Extensive * * * rk rk rk Continuous FS 10 * * * * Figure 3. a) Twenty cases of seasonally distributed growth (g.) and natural mortality (m) patterns. Each year in the l i fe of an age group (i) is divided into quarters, b) Ten, seasonally applied fi-shing strategies (FS). 67 Initially, the von Bertalanffy (1938) growth constant is set at K = 0.2, M at 0.1 for all age groups, and for a selected growth-mortality pattern, the critical age ( i c r t ) is calculated to the nearest quarter year, using a sea-sonal Ricker (1968) model. The age of f irst capture (AFC) is set equal to the integer value of i .^ . Fishing strategies one to ten are then applied separately, beginning at AFC and F = 0.1 and the seasonal yields summed (YT) from AFC to 100 years, or until the annual growth rate (G) falls below 0.0002. For each fishing strategy (YT)p is calculated for all odd numbered values of F from 0.1 to 1.5 . At F = 1.5, (YT)-| g represents the conditional value of MSY- that is a function of K, M, FS and the growth-mortality pattern. All results calculated to this point represent one data set. Additional sets are derived for K = 0.4 and 0.6, M = 0.2, 0.3 and 0.4, and the remaining case studies. The final results appear in Table C l i t , Appendix C , and for each data set the (YJ)-J g values are expressed as a percentage gain (±) in yield rela-tive to that obtained from continuous fishing, FS 10. The conclusions may be summarized as follows: 1) In general, as the values of K and M increase, the potential equilibrium yield resulting from seasonal pulse fishing in-creases relative to continuous fishing. The degree of benefit, however, depends on the values of K, M, and more cr i t ical ly , on the growth-mortality pattern. If the entire growth period is free of seasonally intensive natural mortality, and annual growth is completed before mortality begins, relative yield due to SPF will be greater. 68 2) Seasonal pulse fishing is least effective when the species' growth and natural mortality are continuous, as in Case 9. In general, as the duration of active growth and mortality is seasonally reduced, the potential for yield improvement increases. 3) As F increases, the ranked order of yields from the 10 fishing strategies does not remain constant. Relative to continuous fishing, the rank of a particular strategy may either increase, decrease, or remain unchanged. In other words, the yield is a complex function of the growth-mortality pattern, the fishing strategy and the value of F. Provided the catchability coef-ficient remains constant, the interaction of these variables can be used to exploit two opportunities. The yield from a seasonally expanded fishing strategy may be maintained or mar-ginally improved by reducing F and increasing the seasonal fishing intensity. Alternatively, only a portion of the avail-able fishing fleet can be used to obtain a regulated yield by an appropriate choice of fishing strategy. 4) Without exception one intensive fishing strategy will provide the greatest yield benefit relative to continuous fishing. A moderate strategy offers a second-best choice and with one ex-ception, extensive fishing is the third alternative. The magni-tude of the benefit increases with increasing values of F. Considering all the growth-mortality.cases, the range in rela-tive yield improvement for the intensive strategy is 3.4 to 30.5 percent at F = 1.5. In an individual case the increase in yield 69 is dependent upon the relationships outlined in item one above. 5) While one intensive strategy leads to the greatest improvement in yield, another choice in the same mode introduces a nearly equivalent loss. The risk of an inappropriate choice can only be eliminated by knowing when fishing should take place. A simple answer cannot be given, but the seasonal relationship between growth and mortality is instructive. With one exception (Case 19), i f the end of seasonal growth coincides with or ex-tends beyond the last quarter of mortality, then the optimal choice of intensive strategy is FS 1. In each case the critical age is equal to an integer. In all other growth-mortality patterns, including Case 19, the critical age contains a frac-tional value, i .e . , 0.25, 0.50 or 0.75, and the situation becomes more complex. The optimal strategy then depends on the relative position, duration and exponential value of g and m and requires a mathematical solution. 5. Regenerative Yield Models The yield models described so far include expressions for the principal components of growth and mortality, and as such they can only explain past but not future events. Unless a recruit function is incorporated in the model, as in Appendix A, part 2, or separately estimated, predicted yields and the direction of the fishery cannot be estimated. Regenerative yield models incorporate recruitment directly. They assume that a stock is capable of generating a surplus production that can be cropped as yield, and that the 70 rate at which biomass increases is a function of both the present biomass and its departure from an environmentally limiting maximum. Graham (1935) and subsequently Schaefer (1954, 1957) formalized the biomass approach based on the Verhulst-Pearl logistic in the following terms: ^ | = g(Bt) - qfBt = rB t ^Bmax " M - qfB t , m^ax where g(B )^ is a density-dependent function of biomass, B^; r is the instantaneous net rate of population increase; B m , v is the limiting biomass; max and the product of the catchability coefficient (q) and effort rate (f) equals the instantaneous fishing mortality rate, F' , in terms of biomass. F' is not to be confused with F, the mortality rate applied to fish numbers. At equilibrium, the rate of stock increase dB/dt = 0 and the rate of natural increase rB.(B - B.)/B , is balanced by the surplus production, or yield t max t max rate of qfB^ . A plot of equilibrium yield versus stock size describes a symmetric parabola with MSY situated at B. = Bm /2 . Ricker (1975) indi-cates that at equilibrium surplus production is also a parabolic function of the fishing rate (F1) and effort, and these relationships provide alternative solutions for locating MSY. Schaefer (1954) perceived that under the usual, non-equilibrium conditions of a fishery, MSY could be empirically derived from a time series of yield and effort data provided that F 1 , and hence q, was separately esti-mated. Later, Schaefer (1957) calculated all parameters directly from the same data stream by an iterative procedure. The basic equation, requiring transformation of the data into mean values of yield per unit effort (V) is reproduced in the present notation: 71 T W W " V-'^L + £ ° - 7 9 ) where AV . in year j is estimated by linear interpolation to be J AV. = (V. L l - V. ,)/2 , and e is a random variable. The initial estimates J v J+l J- l 2 of constants r/q and q*B are obtained by dividing the data base in two ^ ^ max 3 parts and solving the simultaneous equations. Substitution of these esti-mates in a summary equation for all the data provides an initial value of 1/q 2 and hence, second and converging estimates of r/q and q*B m a x . Pella and Tomlinson (1969) state that the iterative procedure leads to a non-unique solution and different values of the constants are obtained depending on where the data is partitioned. The reliability of the estimates is improved i f each data sub-set is chosen to include a series of years in which the change in biomass is thought to be large. Recently, Schnute (1977) criticized equation (1.79) on the basis that the linear interpolation method implies that V.+-j can be predicted without the use or knowledge of the mean effort rate in year j+l. He proposed instead the approximation V^  = /V.» Vj+^ to develop the equation for annual events: , „ L ± L - r g 'V*1+l» _ ! L _ ( 7 i * V ' ' V . ' 2 « B m a x 2 where Vj and V ^ are instantaneous values at the beginning of each year and E is the mean effort rate per year. Subsequently, a model is developed that achieves several important goals. Parameter estimates are prepared directly from the data series involving Y, E and V"(=7/E) , the yield in the immediate 72 future is predicted and confidence limits are assigned to both the parameters and predicted yield. The mathematical development of the Schnute model is rigorous and comprehensive and supercedes that of Schaefer in being dynamic and stochastic. The use of the logistic growth function dictates a symmetric, parabolic yield curve. Ricker (1958), Gulland (1961) and Schaefer and Beverton (1963) indicate that for many fish populations the curve may be positively skewed with MSY occurring at less than B m g x /2 . Gulland (1961) proposed that in an age-structured population the present yield from the most productive age groups may be influenced by the mean value of their past, fishing effort experience (T). Ricker (1975) indicates that the relationship between V and 7 is then described by a negative exponential curve. Fox (1970) derived the necessary equations for this situation based on a model in which the growth function g(B~.) = rB , _ 1 (InB , - InB". ) , but MSY always occurs at 0.37 Bm . t max max t max. Pel la and Tomlinson (1969) generalized both the Schaefer and Fox approaches by introducing a four parameter model of the form dB. - ± - HBm - KBt - qfB, where, in the present notation, H = - r , K = -rB . and m = a constant that max determines the negative or positive skewness of the yield curve. Logistic growth is included by the special case of m = 2 . The modified program GENPROD - 2 (Abramson, 1971) is used to search over a parameter space begin-ning with a fixed value of m and estimated values of q, r and (F 1 , V)^y • The goal of the search routine may be to minimize the sums of squares involv-ing actual and predicted yields over a series of years, but the parameters 73 may also be constrained by the experience or knowledge.of the operator. More recently, Mohn (1980), Roff and Fairbairn (1980) and Uhler (1980) have explored the bias and error propagation introduced by catch and effort data when applying different solution methods to logistic-type production models. Hi 1 born (1979) has also compared the merits of different methods of para-meter estimation of the Schaefer model by simulation techniques when the fishery is subject to various regulation schemes. The assumptions of the surplus production models are more restrictive than those in which separate functions are used for growth and age structure. The features of a closed population, and non-selective (size) and randomly distributed fishing are maintained. It is also assumed that each unit of fishing effort instantaneously removes a fixed percentage of the total bio-mass. The effects of fishing on growth rate, recruitment and age structure, and their interrelated time lags, are essentially ignored. The existing bio-mass is the primary determinant of population increase and the response is instantaneous via the parameter r. Relative to dynamic-pool type models the information content is reduced by collapsing rates of birth, natural mortality and growth into a single constant, r . By further assuming that q is constant throughout the data series, the variance of the parameters is increased and the ability of the model to explain past or future events is impaired. Silliman (1971) concludes that in the early stages of a fishery where information on growth, natural mortality and age structure are missing, the simple surplus-production models are useful in preliminary assessment. Ultimately the goal should be to gather data on relevant biological processes and formulate an extended model that is descriptive of the particular species. In what follows attention is therefore limited to a more critical analysis of age-structured models. 74 ANNUAL STATE MODELS 1. General Remarks A number of models have now been discussed and each represents a mathematically expressed hypothesis relating yield and biomass of fish. The models also suggest the kinds of data that need to be collected to solve for the equation parameters. A few of these methods have been described and a more comprehensive account is available in Ricker (1975). Whatever hypo-thesis is used, its validity is tested by evaluating how well predicted values of past and future yields agree with reality. Ultimately, the objective is to establish the relationship between yield and mean stock biomass or the changes in these qualities over time. If growth is not included in the model the analysis can be carried out on the basis of catch and mean abundance. Usually, direct estimates of biomass and abundance are difficult and costly to obtain and the respective indices of V and U are used instead by monitor-ing yield, catch and effort. Because the parameters are calculated from fishery data and the model is used to determine the f i t to the original data, a high degree of correlation is expected i f the model accurately describes the sys-tem. Yet it is precisely here that difficulties arise. In a graph of U versus f for instance, the model assumptions may indicate a linear relation-ship, but due to the scatter of points curvilinearity cannot be denied. The difficulty is aggravated i f the data base does not include a wide range of f values. Historic data may also indicate that several substantially different values of yield or catch have been obtained by the same amount of fishing 75 effort. These examples illustrate a general deficiency of fisheries models in that the unexplained variance cannot be assigned with certainty to changes in either q, H or abundance, or to inaccuracies in yield, catch or effort data. The bulk of recent fisheries literature is devoted to error detection in the above quantities, to more sophisticated methods of analysis or to model alterations that reflect the realities of a fishery more closely. The focus of these efforts usually begins with the basic catch equation and the implied Constance of q and independence of effort units (DeLury, 1947). Paloheimo and Dickie (1964), amongst others, point out that neither of these conditions are satisfied when fish are distributed contagiously and fishermen are able to concentrate their efforts against fish aggregations. The catchability coefficient is then neither constant nor as small as that assumed by a random fishing model, and may be further augmented by cooperation amongst fishermen. Fishing success, or U, at the point of capture is then an index of apparent abundance (Marr, 1951) and may not be related to a change in actual abundance over time. In an extreme case, i f the rate at which fish invade a region of high aggregation is matched by the catch rate, U is maintained at an arti-f ic ia l ly high level and will give l i t t le indication of the decline in abun-dance until the fishery is near collapse. Despite these criticisms, the illusion of constant q is maintained by assuming that the fishery operates throughout the area! range of the stock and that the same value of q is applicable over a number of years. Since catchability is so critical in determining even apparent abundance it is unfortunate that its magnitude is not arrived at independently. In practice, q is estimated indirectly during the process of solving equations and, to a 76 large extent, its reliability is dependent on the accurate measurement of effort. Yet the effort term, or more correctly, the efficiency with which effort is applied, is subject to three levels of variability that are suc-cessively more difficult to quantify. At the primary level standardization is necessary to account for the differences in fishing power of individual vessels. Increased capacity and horsepower reflect the ability to stay at sea longer, to shorten the time between fishing trips and to reduce searching time. The net result is an increase in effective effort that has a direct impact on abundance. At the fishing site, the interaction between gear type and fish density, gear selectivity and saturation, and competition between gears introduce the second source of variation in effort efficiency. The frequency with which these subjects appear in fisheries literature testifies to their importance, but the effects are usually difficult to quantify. The third source of error includes physical and operational components that should properly be grouped under fishing power, but their effects are more subtle than outright changes in capacity or horsepower. Since each fisherman is motivated by economic competition i t is to his advantage to seek out and apply technological changes that lead to reduced gear-handling time, both during and between fishing operations, or extend the precision and range of fish detection. If individual improvements lead.to an increase in fishing success, the result is inevitably detected by other fishermen and the process of upgrading fleet efficiency is set in motion. Superimposed on this system is the intuition, intelligence and experience of individuals that, together with technological change, directs fishing tactics to take advantage of fish distribution. It is extremely difficult to monitor let alone measure the effects of these physical and operational contributions to fishing success, 77 and a time lag always exists between the present and historical performance of effort efficiency. The assumption that the discrepancy can be revealed by analyzing time-series data implies that either q is constant or fishing con-ditions have stabilized. In practice, neither of these assumptions can be verified and the investigator is led back to the original problem of not knowing whether the change in fishing success is due to a change in q, M, effective effort or abundance. The circular impasse between an immeasurable q and an inadequately measured f can be interrupted by recognizing that the real quantity of inter-est to management is the product of these variables: the time average value of the instantaneous fishing mortality. If this statement needs additional support, the practical significance of.being able to estimate F directly can be summarized as follows. During the course of a year, F represents the mortality inflicted on a stock that is the average of all separate fishing incidents involving an interaction between availability (Marr, 1951), catch-abil ity, different gears and their efficiencies. It reflects the inputs of intelligence, intuition, experience and technology without the disbenefit of a time lag, and summarizes the effects of environmental vagaries in q and f in a single term. If F could be calculated directly, there would be no direct need to estimate f and all the ancillary qualities that are known to modify it but cannot be quantified. An annually calculated F would also provide the most recent and reasonable starting value to initiate back-calculations of recruitment via virtual population techniques. However desirable i t may be to. estimate F directly, the fact is that in all the models considered so far, F appears either in the product of two 78 unknowns (FN or F(No).), or in the sum of the mortality terms, Z. In these circumstances the numerical value of F cannot be calculated unless one of two methods is used. Either M is estimated independently and F is obtained by subtraction from Z, or by the reverse procedure, F is derived from a time series of Z and effort data and M is obtained by subtraction. In the anal-ysis to be outlined presently, a third method of estimating M is introduced that relies only on yield (weight and length), catch data for each age group and an independent estimate of t Q . The immediate difficulty then is how to determine F from these data. Initially, a property of a stock or an age group must be identified that responds directly to changes in total mortality. If this is possible, the characteristic must also be measurable from statis-tics collected from the fishery, and a model must be developed that utilizes the statistic to estimate Z and solve for the equation parameters. Each of these problems is considered below in the context of a regulatory purpose of fisheries management. Figure 4(a) represents the familiar isopleth diagram (Beverton and Holt, 1957) for a multi age-group species at equilibrium under conditions of con-stant q, M and recruitment. Each isopleth, or topographical line, indicates a constant value of total yield per recruit, (YW)T/R. The direction of in-w I creasing levels of (Yw)j/R is described by the arc of the eumetric fishing curve. The diagram can be easily constructed from a yield model provided M and a growth function are known. Growth overfishing and underfishing are possible and the conditions are identified, respectively, by the regions below and. above the age of f irst capture that is coincident with MSY. If it is the purpose of management to regulate such a simplified fishery and guide it towards MSY, the problem is in recognizing where the present fishery 79 F — Figure 4. Schematic yield isopleth diagrams for two different multi age-group stocks: (a) constant recruitment, and (b) density dependent recruitment. Legend: AFC = age of f irst capture; F = fishing mortality rate; GU, GO = growth under- and over-fishing; RU, RO = recruitment under- and over-fishing; E = eumetric fishing curve; 1 to 8 = relative magnitude of yield. 80 exists on the diagram. Provided that the fish can be aged, a horizontal co-ordinate can be fixed by selecting the age of f irst capture. Since nei-ther q nor F is known, effort can be temporarily substituted as an index of F, and given sufficient time and a manipulated range of f values, the rela-tionship between the index (Yw)T/R and F will eventually emerge. The process of analysis may be shortened by sequentially altering AFC and f and monitoring the transitional values of V or U. In either case, however, the methods are too time consuming and it is unrealistic to assume that management can vary f at wi11. In Figure 4(b) the situation is made more complicated and precarious by introducing a density dependent, Ricker (1958) recruit function. The iso-pleths below the eumetric fishing curve are collapsed and the possibility of recruitment overfishing is superimposed on that of growth overfishing (Cushing 1972). In this case, it is not sufficient to select an initial value of AFC and f to begin the analysis of the unknown system. There is a risk of col-lapsing the fishery i f AFC is too low and f too high. To avoid the problem M should be estimated init ial ly to determine the critical age of f irst capture. If growth is independent of fish density, the critical age is not affected by disequilibrium or transitional states and provides a minimal, safe basis on which to begin fishing. It is possible eventually to locate MSY by contin-ually incrementing effort provided the monitored but time-lagged response in recruitment is positive. The circumstances shown in Figure 4(a,b) are knowingly unrealistic. The response of the underlying system to changes in the environment and fishing is unknown and the models are over-simplified, mathematical descriptions. 81 Recruitment may be density dependent, but due to unpredictable environmental conditions the number of recruits entering the fishery is variable even under stable fishing conditions. As a result, the concept of a fixed value for MSY is substantially weakened and the best that can be expected is to maximize average yield over a protracted basis of time. Usually, management intro-duces regulation many years after fishing has begun and an init ial analysis is based on data generated by an unregulated fishery. Additional complications arise from restrictive model assumptions related to fishing strategy and the distribution of fish. Under these conditions i t would be beneficial to mini-mize the time between analysis and a regulative response, and to monitor both recruitment and a biological property that reflects the impact of the fishery. Exactly which property has the desirable attributes is unknown, but the search can be narrowed by considering the statistics that can be generated by a fishery. The response of 18 conventional and potential statistics to changes in either AFC or F are shown schematically in Figure 5. The model used to gener-ate the appropriate data is the same as that of Figure 4(b), but the isopleths have been omitted. For simplicity, i f the response of a group of statistics is directionally similar they are combined in a single diagram. The eumetric fishing curve appears as a solid line i f the rate of change of the statistic is altered by crossing over the curve, and as a dotted line i f the response is unidirectional, or independent of a eumetric curve. The direction of an arrow points towards increasing values of the statistic. By holding F or AFC constant the response to changes in either AFC or F are indicated by the ver-tical or horizontal arrows, respectively. The basic statistics of yield (Y), catch (C), and average values of age (A), weight (W) and length (U) are Figure 5. Response of 18 conventional and potential fishery statistics in relation to eumetric fishing curve. Legend: AFC = age of f irst capture; F =• fishing mortality rate; -»- direction_of arrow indicates increasing values; Y = yield; C = catch; R = recruitment; F = mean weight; L = mean length; A = mean age. Subscripts: T = total stocks; W = weight; L = length; i = age group. Dotted curve in region of arrows indicates unidirectional response. See text for comments on diagrams (a) to (f). 00 ro 83 distinguished by subscripts T (total), i (age group), w (weight) and a (length). The symbol R refers to recruitment in numbers of f ish. The criterion for judging the suitability of a statistic as an indicator of fishing pressure is that for known values of AFC and the statist ic, the corresponding value of F (or Z) should be revealed unambiguously. Super-imposed on the equilibrium conditions shown in the diagrams are the effects of variable recruitment, particularly as they may interfere with the determination of F. In panel (a) for example, data on recruitment, the total catch and yield in weight or length are insufficient to disclose the location of F. Since the values of the statistics increase and decrease as they cross over the eumetric curve, there are potentially two solutions to the value of F. Moreover, i f recruitment is variable, then (YT) , (Y T), , R and C T are dependent upon the unknown quantities of either fish numbers or population age structure. In panel (b) the unidirectional response of (YT) /F, (Yr-)./F and CT/F in I W r L I relation to F is ini t ial ly promising as an indicator of fishing pressure, but F itself is unknown and the statistics are further dependent upon either age structure or fish numbers. In both cases (a) and (b), information in addition to AFC and the statistics is needed to reveal F. Similar reasoning applies to panels (c) and (d). The' only statistic to emerge from this analysis that responds directly to F (or Z) is the average weight of a fish in an age group, (panel e). The result is not entirely surprising. It is a direct result of kill ing fish before they have had an opportunity to complete their annual expectation of growth. That the mean weight of the total stock diminishes with increasing mortality was recognized earlier by Ricker (1968), Gulland (1969) and 84 Ssentongo and Larkin (1973), but the same phenomenon also applies to an individual age group. As a result, the restriction of constant recruit-ment that appears in the mean stock size-mortality models of Baranov (1918), Silliman (1945) and Beverton and Holt (1956) is removed because is inde-pendent of fish numbers. The concept follows directly from the catch and yield equations when expressed annually as (C/F). = N". and (Y /F ) . = B., and hence (Y/C) . = 1 1 w I 1 W I (B/N").. = ¥^W../N. = Wij. If abundance increases for a given value of F.. the yield in weight and catch also increases, but the yield divided by the catch, or W ,^ remains unchanged. This statement implies that: (1) the density of fish per unit of fished area must be directly proportional to abundance, and (2) the density relationship is valid whether the fish are distributed at random or contagiously. The degree by which fish density is autoregulated by abundance is presently unknown. To the extent that i t does occur, however, a change in W. will be a useful indicator of a change in F regardless of fish distribution. The relationship between and F.. is maintained even i f the fishermen are. able to direct their efforts against fish concentrations. In this last case the assumption must be made that (C/F). is only indicative of the apparent abundance of an age group. 85 The last statistic appearing in Figure 5(f) is L\, the average length of a fish in age group i that is influenced by F (or Z). For all practical pur-poses the change in U. is negligible for a change in F of 0.1 or less. On f irst consideration the insensitivity of L. may appear to be a disadvantage, but the usefulness of this property will soon be demonstrated. 2. Assumptions of the Annual State Models and Bases of Analysis In this section two models are developed that use data from two immediate past years of fishing to prepare estimates of the growth parameters and the mortality coefficients F and M. The key statistic that permits this rapid assessment is W. , and together with a known age of f irst capture the state, or position, of the fishery in reference to an eumetric curve can be located. The models that utilize the state property of W^. are termed annual-state mod-els. The assumptions of the models are as follows: 1. The fishery consists of a single, closed stock in which the fully fishable, multiple age groups are identified by the subscripts i through i + n. 2. Growth in weight and length are continuous and occur according to the von Bertalanffy-type equations w. = W o o ( l . e - K ( i - t o ) ) b and I. = L ( l - e " K ( i " V ) where b may be a non integer. 3. The mortality rates F and M act continuously throughout the year and 86 are common to all age groups. The models are sensitive to seasonal changes in growth.and mortality. 4. The natural mortality rate is assumed to be constant for any two consecutive years of fishing. 5. Fishing is hot size selective. 6. The recruitment function is unknown and the number of fish entering the fishery in any year is subject to environmental variability. 7. The density of any age group i to i+n is proportional to apparent abundance, (NQ).j , where the subscript "o" indicates the begin-ning of the year. 8. The age of the fish can be determined without d i f f i c u l t y . 9. Each fish that is included in a sampling program to estimate the annual catch must also be measured for length (Model 1) and weight (Models 1 and 2). The methods used must be consistent. 10. To retain a sufficient number of decimal places in the calculated mean weights (W".) and lengths (U^) the catch in each age group should be no less than T000. 11. The parameter t must be independently estimated. Model 1 makes use of all the available data and i t is used here to describe the general analytical procedure. To begin the analysis two con-secutive years of age-group data on catch, yield in weight and length and an annual estimate of t are needed. Thereafter, the data of the third year are o analyzed in conjunction with the second, and so forth. By this procedure the growth equation parameters Z and t are monitored and updated annually. In addition the analysis always includes the immediate past year of fishing. The examination of the data is divided into two parts. In the f i rs t , data for each fishing year, j , are treated separately by means of the anal ytical equations developed earlier in Section II, (4.4). For convenience they are repeated here: b. , W , ~ C , M N W • -K.( i -t ,) 3 -0.5Z. ( V i , j = F j ( N o } i , J ^ C(l-e J ° ' J + 4 e J ( l -b. -K.(i+0.5-t .) J -Z -K (i+l.O-t .) j e J ° ' J + e J ( l -e J ° ' J ) ] (3.01) , "Zj -K.(i-t .) -(Z+K). ( Y L } i i = F i < N o ) i i L ~ i [ 1 T — " e J ° ' J ^ - ) ] ( 3 ' 0 2 ) -L 1 , J J 0 1 , J °°,J £j ^ Z+K) J Division of expressions (3.01) and (3.02) by (3.03) yields the statistics Z. W . -K.( i - t .) b j -0.5Z. -K.(i+0.5-t .) ff 3 - i - ^ [ ( i . e y ° . J ) + 4 e J ( l -e J °»J 1 >J - L. O 1-e J •Z. -K,(i+1.0-t ,) b j +e J ( l -e J ° ' J ) ] (3.04) and L -M i - t „ Z. -(Z+K) 1-e" j IZ^Kj i,j ^ j H - e ^ ^ ( - ^ H ( l ^ J j ] (3.05) J 88 Each age group generates three equations (3.01 - 3.03) with five parameters (W ,^ K, b, t Q)j and two unknowns (F(NQ)1., Z)j . Since the components of the product (F(N-)•),• cannot be estimated separately they are eliminated by restricting the analysis to equations (3.04) and (3.05). For any year j , Z is constant and t • is known, a prior estimate having been made by the method of Beverton (1954), the Walford procedure mentioned earlier or from the known mean size at age of young fish (Fabens, 1965). By either of these methods L - and K. are also estimated simultaneously, but to fully demonstrate the present methodology only the estimate of tQ j will be used. The residual number of unknowns is therefore reduced to five (W , K, b, L and Z) . and the minimum number of equation pairs (W., L^),-. or age groups, needed to arrive at a solution is three. In a multi age-group fishery data are usually available from more than three age groups and the solution must be obtained from an overdetermined system. At least three programs are available at the University of British Columbia Computing Centre Library to deal with such a problem. Specifically, they are: singular value decomposition (SVD), a solution to a non-linear system (NONL), and the International Mathematical and Statistical Library program ZXSSQ. Initial estimates of all unknowns are needed by each program, but SVD and NONL also require the partial derivatives of the independent variables (W. ,1".) with respect to each unknown. Program ZXSSQ, however, uses a modified Levenberg-Marquardt algorithm to find the minimum sums of squares for M functions of N variables and eliminates the need for explicit deriva-tives (Levenberg, 1944; Marquardt, 1963). In the present case ZXSSQ is favored due to cost and programming simplicity to find the least squares solution such that 89 E (W. - W.)2 + -2 (L. - L. ) 2 = minimum. (3.06) The estimates (W., U.). are prepared directly from equations (3.04) and (3.05). Conway (et al . , 1970) indicates that in the general case of a nonlinear system the Marquardt approach is more efficient relative to either the gra-dient methods (Spang, 1962) or Taylor series (Glass, 1967). Whereas the " . . . Gradient Methods are able to converge on the true parameter values even though initial values are far removed, . . . this convergence tends to be slow. On the other hand the Taylor Series Method will converge rapidly providing the vicinity of the correct parameter values has been reached", (Conway, et a l . , 1970). Program ZXSSQ takes advantage of both tactics by init ial ly se-lecting a large value of the internal parameter PARM2 to ensure the steepest descent (Gradient method). As the correct parameter values are approached by iteration PARM-2is reduced to provide rapid convergence by means of Taylor series. One difficulty with this method is that following one appli-cation of program ZXSSQ the final parameter estimates may be determined at a local rather than a global minimum. To overcome the problem ZXSSQ can be applied iteratively such that the parameter estimates from the f irst run become the initial input values of the second run and so forth. The para-meter PARM 2 is thereby re-set to its maximum value at the beginning of each run and a local minimum avoided. The global solution is reached when equation (3.06) is satisfied. The non-linear algorithm is used to estimate (W ,^ K, b, L^, Z) . for each of two fishing years. In the second stage of analysis these results 90 are analyzed by the equation AZ = AF and F. ? / F . , (as on pages 98-99) to solve for F._, and F . = ? and subsequently for M and the age assodiated with the c r i t i ca l s ize. Before proceeding to the section on in i t i a l parameter estimates a comment is necessary on the unusual requirement of an independent estimate of t . In contrast to more familiar fishery models in which M is derived independently or by subtraction from Z, an analysis based on the equations for W- or U. leads to the unpredictable fact that Z and t occur as an in -1 1 o separable sum. Two related examples are chosen to i l lust ra te this point and the f i r s t involves equation (3.05). Since is a mean length, the fraction of a year at which the length of age group i achieves can be equated to time t . Equation (3.05) can then be rewritten as "K(i-t ) , -K(Z+.Kh = L. = L [1-e ° • —=—7 • {L^rnr ' i+t i °° L , -1 Z+K z 1-e - L (1-e •K(i+t -t ) z~V) (3.07) where and e " K t z _ _ J L _ . ( l -e- ( Z + K >) " l -e" Z Z + K •Z, . _ 1 „„r(l-e" ) Z+K i n Q s t z - ^ inl - j - L • (Z+K)) 3 ( 3 ' 0 8 ) The last equation demonstrates that for a known value of K, t is both a function of the unknown Z and a constant for a l l age groups. Program ZXSSQ can be applied to the Lj values derived from a single year's f ishing to 91 establish the difference (t z - t ) in equation (3.07), but further decomposition is impossible unless either t or Z is known. The second example draws attention to a theoretical method that can be used to estimate the correct values of t and Z. Initially the exact values of W.j and L~. for a,number of age groups can be calculated directly from cho-sen values of W , K, b, L , t and Z. Program ZXSSQ can then be used to calculate the residual sums of squares i f W ,^ U. and the remaining parameters are held constant except for t and Z. The results appear in Figure 6. It is clear that the least squares solution lies within the ell iptical trough formed by the residuals contour 0.1, but the surface of the trough is so flat that the correct values of t and Z cannot be estimated. In a more realistic case the errors associated with the yield in weight and length or catch would be substantially greater than in the example and i t is doubtful that a contour value of even 1.0 could be achieved. The extent of the trough also demonstrates the sensitivity and dependence of Z on an accurate-ly determined value of t and suggests that tQ be measured independently. 3. Model 1: Methods of Estimating Growth Parameters and Mortality Coefficients The data presented in Table 1 are used to test the analytical capability of the equations for ¥. and L~. and to develop the techniques of parameter estimation. The model used to generate the data is that of Beverton and Holt (1957). A random multiplier is used to simulate variable recruitment and the resulting catches from the various age groups. The age-group yields in weight and length are not shown but may be calculated from the tabulated values of W. and U. i f required. 92 Table I. Age-group catch, mean weight and length for two_consecutive years of f i s h i n g , i = age group; C = catch; W. and L . = mean weight and length, respect ive ly ; j = f i sh ing year. i C i j = l • ~ l (mm) W . T (gm) C. i j=2-L . (mm) W. l (gm) 4 66,040 445.728 804.173 75,936 444.615 798.156 5 71,105 508.994 1,194.647 80,778 508.026 1,187.834 6. 40,023 563.995 1,623.216 86,973 563.154 1,615.950 7 42,202 611.810 2,070.549 48,955 611.079 2,063.120 8 39,023 653.379 2,520.806 51,620 652.744 2,5T;3.443 9 15,309 689.517 2,961.818 47,732 688.965 2,954.692 10 22,342 720.935 3,384.786 18,726 720.454 3,378.015 11 14,724 748.247 3,783.784 27,330 747.830 3,777.444 12 15,525 771.992 4,155.211 18,010 771.629 4,149.345 13 11 ,215 792.634 4,497.267 18,990 792.319 4,491.891 14 9,672 810.580 4,809,504 13,718 810.305 4,804.617 15 6,716 826.181 5,092.443 11,831 825.943 5.088.030 14 E C i i=4 = 347,180 . 15 E C i=5 . = 424,663 l 94 3.1 Initial estimates of L , K and Z A method has been described earlier to estimate the parameters L^and^K by means of a Walford plot based on equation (1.38), i .e . , - U l - e - K ) + e - \ . . A similar relationship can be developed in the general case for successive values of U.+^ ancj I". . As before, let e • K t z _ i _ (1-e-(Z+K>) ' l-e" Z ' ' Z + K and equation (3.05) for can be rewritten as -K(i-t ) -Kt h = Ld-e ° e- Z) a n d -K(i+l-t ) -Kt L i +1 = L-d-e ° ' e Z) -K(i-t ) -Kt -Subtracting L. from L . , substituting L -L . for L e 0 • e z and 1 l + l - J o o - | C O simplifying, L i + 1 = L o o(l-e~K) + e _ K L, (3.09) Consequently, average length values for each age group can be used in a Walford plot to yield initial estimates of L and K that are unaffected by -Kt 0 0 _ _ the mortality term e z . The regression of L.^, . on L. . for each fish-i+l.J i ,J ing year in Table 1 leads to the estimates: 95 L = 930.000 mm. L 0 = 930.001 mm 0 0 , l °°,2 K] = 0.140 Kg = 0.140 Ricker (1975) points out a disadvantage of the above procedure in that each datum appears twice in the calculations except for the first and last. He proposes instead a rearrangement of the basic length equation that leads to the following expression in terms of mean length at age: ln(l - L.) = £nL - K(t - t ) - K, . (3.10) * co ^ ' 0 0 Z O 1 The linearity of this relationship is sensitive to chosen values of Lro and may be used as the criterion for estimating L o t and hence K.. Applying the method to the data in Table 1 does not alter the initial estimates of L °°»J and K., but the intercept, i .e . , ilnL^ - K(t z - t Q ) , for each fishing year yields the value (t - t ), = 0.6617 and (t - t ) 0 - 0.6453 . Z 0 1 Z 0 2 The probable sign and magnitude of t can be estimated from the difference (t 2 - tQ)^ . If Z-j is set equal to zero in equation (3.08), a Taylor series expansion of the term (1-e )/Z-| reduces to one, t = ( l /K)£n(K( l -e* K ) ) and for K = 0.140 the maximum value of t = 0.4942. Since and K in both fishing years are nearly identical it is unlikely that the growth rate has changed and t is therefore assumed to be the same for both years. Substituting t z = 0.4942 in the difference (t - t )^  yields the estimated t = -0.168 for Z, = 0. It follows that t_ 9 = 0.4778 O I Z ) L and solving equation (3.08) by trial and error, Z 9 = 0.197. The magnitude 96 of Z-j and 1^ are incorrect since cannot equal zero, but two conclusions follow from these results that are independent of the actual values of Z-j and 1^ '• 2^ > Z^  and AZ = AF = 0.197 since M is assumed to be equal for two consecutive years. If the actual value of t is found to be -0.200 for o both fishing years, i t follows that t -j. = .4617, t z 2 ~ 0.4453, and by equation (3.08), Z ] = 0.391 and Z 2 =0.591. 3.2 Initial estimates of W and b. A mortality term analagous to t can be incorporated into an expression for the mean, age-group weight, e.g.: " K ( i + t w i - t o ) b W, = V (1-e w ' n 0 ) (3.11) In contrast to t , however, i t can be shown that t, • is not constant for a z w,i given value of Z; i t is slightly larger than t for all values of i but approaches t as i increases. Despite this limitation it is possible to develop an approximate relationship between W. and that leads to an esti-mate of W and b. oo b If weight is related to length by the equation w = a l , the mean weight can be expressed in terms of by the approximation h h -K(i+t -t ) b ( r . ) b « L r o b (1-e z 0 ) (3.12) Dividing (3.11) by (3.12) and rearranging, -K(i+t .-t ) b (1-e w ' n ° 1 " mflkO + b & n C L / 1 e _ K ( i + t _ t } } ] (1-e z 0 97 If it is further assumed that t . = t for all values of i , the equation w, i z ^ above reduces to £n W. S £n(W /L b) + b£n I. (3.13) "j OO CO •] * ' Regression of the tabulated values of Jin W. .. on £n X . . , and substitution of the appropriate value of . leads to the estimates , 1 = 7252.463 gm w » 2 = 7 2 5 2 • 4 1 4 9m b^ = 2.992 b 2 = 2.992 3.3 Final estimates of parameters, coefficients and the critical age The requirement of program ZXSSQ for preliminary estimates of the unknowns is now complete. For the first year of fishing, substitution of the estimates and a fixed value of t = -0.200 in ZXSSQ requires one program run of seven iterations to produce the final results U 1 = 7262.860 gm K, = 0.140 °°,1 3 1 L c o 1 = 929.995 mm Z-, = 0.401 b1 = 3.000 Convergence is similarly achieved in seven iterations for the second year to yiel d W „ = 7262.545 gm K_ = 0.140 °°,2 3 2 L 0 = 929.992 mm Z„ = 0.602 00,2 2 b2 = 3.000 98 /N /\ /\ /\ and it follows that Z 2 - 1 = F 2 - F ] = 0.201 The catch equations for two consecutive years of fishing on the same year class may be described by Ci,l ' Fl(No»i <'-e"V, and -Z ? Ci + i , 2 = ^ V i + i ( ] : e )/z2 • - z l But (N ).,-, = (N ).e and the ratio between successive catches can 0 1+1 0 1 therefore be written as C i , l F l Z2 (1-e Z ] ) Z1 2 — • • _ i '— • p r • F 7 -Z Ci+1,2 F2 Z l ( 1 _ e Z 2 ) Since F. is assumed to apply equally to all fully fishable age groups within one year, and M is constant for both fishing years, it follows that 14 I C, , . , -Z, •Mn 1 » i=4 ' » ] F l Z2 (1-e ] ) Z l 1 5 " FT * Z7 -H ' e (3-14) i = 5 C i + l , 2 2 1 2) The sums of catches for the appropriate age groups appear at the bottom of Table 1 , i . e . , 14 15 J 4 C 1 ,1 = 3 4 7 ' 1 8 0 a n d ^ 5 Ci+1 ,2 = 424,663 99 All other variables in equation (3.14) are known except for F-j/F^ and the solution follows directly that F - | / = 0.499. Since the difference F^"^ is also known, the final mortality estimates are F1 = 0.200 , F 2 = 0.401 and M - 0.201 By applying a catch equation to the data in Table 1, the apparent abundance (N )• for each cohort in each fishing year can be estimated. If a sufficient amount of data exists for a number of years the magnitude of recruitment for each year class entering the fishery can also be calculated. The age of f irst capture that is coincident with the critical size can be estimated by equation (1.75), i .e . , i . + = t + 1 i n i ^ r + 1) . crit o K M ' Since t , b, K and M are equal for both fishing years, substitution leads to the single value of i c r l - t = 7.86 years. The Model 1 analysis of the data for two consecutive years of fishing is now complete. A comparison of the initial and final parameter estimates and the actual values used to generate the data appears in Table 2. Agree-ment between the final and actual figures is expectantly good because the fishery's data are without error and meet all the model assumptions. Under these conditions the foregoing procedure tests the analytic capability of the approximate equation for w\ in combination with data on mean lengths. The estimates can now be used to construct a yield (weight) isopleth diagram having F and AFC as the x and y axes, respectively. During the two years of fishing AFC was held at four years and the successive locations of the Table II. Comparison of Model 1 and Model 2 estimates of growth equation parameters and mortality coefficients with actual values. Wot - maximum weight (gm); - maximum length (mm); b - length weight exponent; K - growth coefficient; F, M, Z - fishing, natural mortality and total mortality coefficients, respectively; j - fishing year one or two; ~ critical age (years); ( ) - assumed Z value. Parameters and coefficients MODEL 1 j Initial Final estimates estimates MODEL 2 Initial Final estimates estimates Actual values Woo 1 7252.463 7262.860 7260.768 7262.946 7263.000 2 7252.414 7262.545 7260.685 7262.693 1 930,000 929.995 930.000 2 930.001 929.992 b 1 2.992 3.000 3.020 3.000 3.000 2. 2.992 3.000 3.030 3.000 K 1 0.140 0.140 0.140 0.140 0.140 2 0.140 0.140 0.140 0.140 Z 1 0.391 0.401 (1.5) 0.405 0.400 2. 0.591 0.602 (1.5) 0.609 0.600 F 1 - 0.200 - 0.201 0.200 2 - 0.401 - 0.405 0.400 M - 0.201 - 0.204 0.200 V r t - 7.86 - 7.79 7.88 -0.200 o o 101 fishery are identified by the coordinates (0.200,4) and (0.401,4). Since "I'c'rit is greater than four it may be argued that the current program of growth overfishing and reduced yield may be improved by simultaneously in-creasing F and AFC in fishing year three. This plan, however, is premature. Despite the reasonable estimates of F^  , F^ and M it is not known i f the increase in fishing mortality {F^) is due to an increase in the catchability coefficient or effective effort. A comparison of the monitored effort in both fishing years can be used to decide the issue and simultaneously esti-mate the catchability coefficients. 4. Model 2: Methods of Estimating Growth Parameters and Mortality Coefficients  In the development of statistics W. and the statement was made that L.j is relatively insensitive to changes in Z. To illustrate this point a comparison of the changes in W. and I. for a change in F of 0.1 at both a low and high value of Z appears in Table 3. The parameters used to generate the data are the actual values listed for the Model 1 analysis. Several interesting features are revealed by the table and the first is that for a fish that has the capacity to grow to 930 mm ( L j the change in U. corre-sponding to AF = 0.1 is less than a millimeter. The response of L. for the older age groups is also less than for the younger fish due to the declin-ing growth rate. In order that L. be a responsive indicator of changes in fishing mortality will therefore require a large sample size together with minimal error in counting, ageing and measuring the catch. The problem can be circumvented by noting that the mean weight is more sensitive to changes in F and particularly in the region of the critical age (7.88 years). 102 Table HI. Expected change in statistics W. and_ due to a change_ in fishing mortality of 0.1. i - age group; W. - mean weight; L. - mean length; F - fishing mortality coefficient; - instantaneous growth rate. W. (gm/fish) 0. 45 0. 35 0.27 0.22 0.18 0.15 0. 12 i 4 5 6 7 8 9 10 F=0 .3 801. 156 1191. 232 1619.574 2066.825 2517.116 2958.246 3381. 392 F=0 .4 798. 156 1187. 834 1615.950 2063.120 2513.443 2954.692 3378. 015 AW.. 3. 000 3. 398 3.624 3.705 3.673 3.554 3. 377 F=l .5 766. 973 1152. 477 1578.201 2024.495 2475.140 2917.601 3342. 761 F=l .6 764. 360 1149. 509 1575.029 2021.247 2471.917 2914.479 3339. 792 AW. i 2. 613 2. 968. 3.172 3.248 3.223 3.122 2. 969 L. (mm/fish) F=0 .3 445.170 508.509 563.573 611. 444 653.061 689.241 720. 694 F=0 .4 444.615 508.026 563.154 611. 079 652.744 688.965 720. 454 ALT. i 0.555 0.483 0.419 0. 365 0.317 0.276 0. 240 F=l .5 438.811 502.981 558.767 607. 266 649.428 686.083 717. 948 F=l .6 438.321 502.555 558.397 606. 944 649.149 685.840 717. 737 A l i 0.490 0.426 0.370 0. 322 0.279 0.243 0. 211 103 The increased response of W. to changes in Z or F forms the basis of the second annual state model. Once again two year's data of age-group yield in weight, catch and an independent estimate of t are needed to begin the analysis. The absence of length information causes the final parameter values and Z to be derived from the approximate equation for W.. As a re-sult the estimates may not be as accurate as in Model 1. A second conse-quence is that K and the initial estimates of and b cannot be derived from the usual weight-length relationship and an alternative method must be used. The program BSRCH described earlier in Section 3.2 makes use of the exact relationship between fish weight at time t+1 and t. Recast in terms of mean weight, the equation ( W i + 1 ) V b ^ W j / b ( l - e - k ) + e - k ( W . ) 1 / b (3.15) is an approximation, but its linearity is sensitive to the value of b and this property may be used to establish the least squares solution I ( ( W i + 1 ) 1 / b - (ftj- + 1) 1 / b) 2 • If program BSRCH is applied to the W. data for j=l , (Table 1), the following results are obtained: n I W Sums of ,1 K, b, squares •1 " "1 "1 4 to 15 7261.803 0.140 3.030 185.9 x IO - 4 5 to 15 7260.768 0.140 3.020 16.7 x 10"4 6 to 15 7260.531 0.140 3.030 13.1 x l O - 4 104 The greatest reduction in the sums of squares is achieved by age groups 5 to 15 and the underlined values of Wm, K and b are accepted as the in i -tial parameter estimates. A similar treatment of the second year's data for age groups 5 to 15 yields the results W 0 = 7260.685 L = 0.140 b_ = 3.030 . A reliable initial estimate of or Z^  cannot be obtained without data on L. but a reasonable starting value (say, Z = 1.5 for j = 1,2) may be assumed. The parameter t is set equal to -0.200 for both fishing years. From this point the analysis for the final parameter values Z^, Z^  and F is identical to Model 1 except that program ZXSSQ utilizes only the data for W.j. The results are compared to the actual parameters and mortality coef-ficients in Table 2. The discrepancy between the Model 2 estimates of W ,^ b, K and i c r l - t and those of Model 1 or the actual values is negligible. The principal effect of using the approximate equation for W. is to introduce an error in Z, F and M in both fishing years. In each case the error is only two percent or less and indicates that the fishery analysis may be carried out on the basis of W. data alone. l 5. The Dependence of the Mortality Coefficients on t and the Effect of Variable Natural Mortality ° It is not essential that the growth parameters be estimated by a Model 1 or 2 analysis. They can be obtained by alternative sampling methods and the values substituted into the equations for W. and L. . Since t is 1 1 o also estimated independently the analyses are reduced to solving for the mortality coefficients, the only remaining unknowns. Departures from the 105 model assumptions and errors in the independent variables (w\ , L. and t ) can affect Z, F and M, and the reliabil ity of the coefficients must be considered. The most important sources of error include measurement, count-ing, age determination and sampling as well as gear selection and age-dependent fishing and natural mortality. A complete sensitivity analysis of the models for each topic is beyond the scope of this thesis. Only two items are selected for detailed treatment and they are the dependence of the mor-tality coefficients on t and the effect of variable natural mortality. The importance of estimating t correctly cannot be overemphasized. If the variables W. and L. contain random errors the effect on Z is linked directly to t in a sum, and because of the disproportionate relationship, a small error in t is compensated by a large change in Z. To illustrate this effect the first stage of a Model 2 analysis of the data (Table 1) for the first year of fishing is re-examined over the range t = -0.200 t 10 percent. The resulting percentage change in Z^  (Table 4a) is calculated on the basis of the original solution, Z^  = 0.405. A t s t h i s low value of Z, the change in both b and K is negligible, that of is < .001 percent and the error in t is transmitted directly to Z-j . In a complete Model 2 analysis, however, in which all mortality coefficients are estimated over two years of fishing, the effect of an error in t is not limited to Z and Z„ . All coefficients are o 1 2 affected, and i f Z^  and Z^  are relatively large some of the parameters may also be influenced. Table 4b demonstrates these results for a minus six percent error in t and in the situation where Z^  and Z^  are equal to 0.4 and 0.6 (Case 1) and 1.6 and 1.8 (Case 2), respectively. Several conclusions follow from the analyses and they are: Table IV. The effect of an error in on a) Z for a single year of fishing, and b) the mortality coefficients for two years of fishing. In both situations the analysis is based on Model 2. In (b) the error in t is minus six percent and the correct values of 1, and Z ? are either low (Case 1) or high °(Case 2). 1 a) t 0 -.220 -.216 -.212 -.208 ' -.204 -.196 -.192, -.188 T.184 -.180 error -10 -8 -6 -4 -2 2 4 6 8 10 Z .645 .597 .549 .501 .453 .357 .309 .261 .213 .165 error 59 47 36 24 12 -11 -23 -35 -47 -59 b) Case 1 Results Fishing year, j 1 2 Parameter and coefficient estimates and errors, W K "1 7262.786 0.140 3.000 0.549 7262.239 0.140 3.000 "2 0.753 '1 0.155 F 2 M 0.394 0.359 0.394 Actual values % error 7263.000 0.140 3.000 0.400 0.600 0.200 <-0.02 0 0 37 26 22 0.400 0.200 -10 97 Case 2 Results 1 7241.493 0.140 2.996 1.755 — 0.583 — 1.172 2 7241.100 0.140 2.996 — 1.946 — 0.774 1.172 Actual values 7263.000 0.140 3.000 1.600 1.800 1.400 1.600 0.200 7o error <-0.31 0 -0.13 9 8 -58 -52 486 107 1) Regardless of the error in t , i f M is constant for two consecutive years the difference between and 1^ (or and F^ ) is estimated correctly even though their invididual values are incorrect. 2) The difference between the actual and estimated value of a total mortality coefficient is inversely related to the sign and magnitude of error in t . The percentage error in Z is also inversely related to the magnitude of Z. 3) Relative to the percentage errors in 1^,1^, F-j and F 2 that of M is larger and increases with increasing values of Z^  and Z^  . 4) For large values of Z^  and Z^  program ZXSSQ attempts to minimize the residual sum of squares by adjusting the parameter estimates. The order of increasing sensitivity of the parameters is K, b and . No satisfactory procedure has been found to cope with an error in t without increasing the data requirement of the models. The ability to correctly estimate the change in F between years (conclusion 1) is only superficially useful. Unless F-j and F 2 , and particularly M, are correct the resulting isopleth diagram and the position of the fishery relative to the eumetric curve will be inaccurate. If an independent estimate of M is available in addition to't , the information can be combined with the value of AZ or AF to solve for the remaining mortality coefficients. For example, in Case 2, Table 4b, F2 = F-j + .191, the ratio (data not shown) 14 15 V C. . / VC „ = 4.662 and i f M = 0.200 the combined catch equation i=4 ' 5 (3i14) can be expressed on the basis of one unknown, i .e . , 108 ^ { ^ + .391} ( i _ g - ( F 1 + .200)) ( F l + .200) 4.662 = F J g l • ( p + 2 Q 0 ) • -(F. + .391K * e (1-e 1 ' The solution for yields the estimates F}-=.1.398, F 2 = 1.589, 1^ = 1.589 and Z 2 = 1.789 which compare favorably with their actual values (see Table 4b), An entirely different problem is encountered i f M is a function of age rather than a constant for all age groups. Since the annual state models are only capable of estimating a single value for the mortality coefficients, it is reasonable to expect the estimates to approximate the mean values Z. . and ft. i f F. . is constant. In the special case of variable M., however, application of the models lead to incorrect estimates of the mortality coef-ficient means and most of the growth parameters,. To demonstrate this limita-tion, consider the situation in which M. assumes the following pattern: Age group 4 5 6 7 8 9 10 1 1 1 2 13 14 15 M. .2 .2 .23 .26 .30 .35 .40 .46 .52 .61 .70 .81 Two consecutive years of fishing at the unknown but constant rates of F. -, 14 15 0.200 and F. 9 = 0.400 yields the catch ratio I C. , / . I C. 9 = 0.978 as well as the individual values (not shown) of for each fishing year. If the methods described earlier are used to prepare the initial parameter estimates, and t is set equal to -0.200, a complete Model 2 analysis pro-vides the following results: 109 Unknown 1 j = 2 Actual Values W a K b 7239,028 0.140 2.999 7239.297 0.140 2.999 7263.000 0.140 3.000 Z. .(i=4 to 15) 1 »J M. .(i=4 to 15) 0.333 0.353 -0.020 0.533 0.553 -0.020 Z ] = 0.620; Z 2 = 0.820 F1 = 0.200; F 2 = 0.400 M. = 0.420 3 Apparently the parameter K, and to a lesser extent b., are insensitive to the condition of variable M. whereas is more responsive. The least desirable situation is also emphasized in that the mortality coefficients contain the largest errors. An obvious remedy in this case is to estimate IAT, b and K. independently, substitute their values into program ZXSSQ and constrain the solution to the only unknown, Z. , in each fishing year. In the process a method of distinguishing between the two situations of variable or constant M. is revealed. If the former condition exists the parameters b and ]flm estimated init ial ly by not constraining ZXSSQ will differ from those obtained independently. An alternative solution that does not require independent parameter estimates is also possible. The method is approximate and makes use of the insensitivity of -K and b and the fact that the difference Z 2 - Z-j (or F 2 -F-|) is estimated correctly whether is variable or not. Equation (3.04) is also collapsed and summed over all fishable age groups such that no 15 _ Z. W 15 -.bl. 15 -Z. 15 -K ( i - t ) b -k(i+.5-tn) b -k(i+l-t ) b where, A. = (1-e 0 ) , B. = (1-e u ) and D. = (1-e 0 ) . Using the fishing data, the original analyses and the given value of t , the following information is either known or may be calculated: j=l and 2 K - 0.140 b = 2.999 15 15 \k. = 4.7854 YB. = 5.1031 4 1 4 1 . 15 ID. = 5.4180 4 1 15 4 1 >1 15 "^1 2 4 1 •>C-2^ + 0.200 36,824.8 = 36,750.1 1.002 Substitution of the appropriate values into equation (3.16) for each fishing year leads to two equations involving the sums of the mean weights. The term W /^6 is cancelled by dividing one equation by the other, and after rearrange-ment and simplification, the relationship Z, -.51, -Z, ,Z, + .20, - — (1.0 + 4.2656 e '+ 1.1322 e ') = 1 .002 >-1-e^l -(Z-, + ,20) 1-e -.5(Z.+.20) -(Z. + .20K (1.0 + 4.2656 e 1 + 1.1322 e 1 ; has only one unknown, Z^  . The solution Z-j = 0.590 leads to 1^ = 0.790, and by means of the combined catch equation, = 0.197, F 2 = 0.397 and M. = 0.393. m The results differ from the actual values displayed in the preceding table because of the approximate expression involving . The mean 1^ or X, can now be substituted into equation (3.16) to solve for W .^ If this last estimate is combined with F-j or a r , d t n e W. values it is also possible to solve for the individual, NT coefficients associated with each age group and the starting values (N ). • . 3 0 1 , ] The analagous situation involving a constant M. but variable F. for each age can be treated similarly to yield the means F^  and F^  as well as the individual, fishing mortality coefficients. In the more general case in which M. and F.. are both variable, the solution cannot proceed beyond esti-mates of Z. , T. and M. . 112 DISCUSSION AND CONCLUSIONS The main topic of this thesis is the development of an assessment method that does not rely on the use of effort data. Before turning to this subject, however, a few words about seasonal pulse fishing are necessary. The process by which seasonally applied effort leads to an increase in yield is not diff icult to understand. In contrast to continuous fishing, which ignores seasonal variation in growth and mortality rates, pulse fishing is adaptive. By applying effort after the biomass has had an opportunity to increase, or to counteract the loss from natural mortality, the yield will be greater than that of continuous fishing. In the model used for analysis, fishing always begins with the age group containing the critical size and the difference in yield between intensive and continuous fishing is maximized. Clark et a l . (1973) arrived at a similar conclusion based on an economic analysis of the Beverton and Holt (1957) constant recruitment model. In succeeding years the yield difference between these two strategies diminishes and eventually is reversed. However, the gain in yield during the f irst few years is partially retained due to the reduced growth rate of a declining number of f ish. Any species that demonstrates rapid growth (small K), or high mortality (M), is therefore more susceptible to yield improvement by seasonal fishing than one having opposite character-ist ics. Regardless of the growth-mortality (G-M) pattern or value of M, an intensive fishing strategy consistently provides the largest yield benefit i f applied at the correct time, or the greatest loss is inappropriately 113 timed. When to fish is as important as knowing the proper strategy and this subject is taken up in the next paragraph. By expanding the fishing season about the most favorable mode, moderate or extensive strategies also provide an improvement in yield but of diminished magnitude. What is less easily understood is how the remaining variables of fishing mortality (F), the G-M pattern and M interact to affect the potential increase in yield and the optimal time to begin fishing. The study shows that the G-M pattern has the greatest influence in one respect. If growth and natural mortality are seasonally distinct and concentrated in time, the maximum potential exists for yield improvement and the best time to fish then depends on the value of F. If F is low, it is necessary to begin fishing intensively before the active growth period has expired. At high levels of F the yield can be increased either by withholding effort until the end of the growth period, or by fishing counteract!'vely to offset the loss due to natural mortality. An analogy exists between these examples and a pisciculture situation given limited or abundant resources, respec-tively, for harvesting the catch. Case 18 illustrates the gist of these arguments and represents the most favorable conditions for a large-valued K, F and M as well as a non-overlapping G-M pattern. In this instance the increase in yield exceeds that of continuous fishing by approximately thirty per cent. As the period of natural mortality coincides to a greater extent with that of growth, the benefits of seasonal fishing must decline regardless of the strategy used. Continuous growth and mortality (Case 9) represents the maximum degree of conflict between these 1 1 4 processes, and even under the most favorable conditions of K, M and fishing strategy the yield improvement is less than five per cent. Because of the complexity of interactions that exist between the variables, the theoretical benefit of seasonal fishing can only be esti-mated by analysis. To do so requires detailed knowledge of the G-M pattern, M and the anticipated value of F; yet it is here that the f irst practical difficulties arise. If temperate species undergo a period of accelerated growth during the year, the magnitude and duration of the increase, as well as the general growth pattern, can usually be found by sampling the catch. By comparison, i t is substantially more diff icult to obtain a reliable estimate of M and particularly its seasonal variation. The more realistic situation of age-group variation in mortality adds further uncertainty to the choice of optimal fishing strategy. It has also been pointed out that the decision of when to fish is dictated by the magnitude of F. Since total allowable catches, quotas and regulated open seasons are subject to change, they can affect the intensity, duration and time of fishing. The structure for applying seasonal fishing may . already exist in the form of regulations, but the influence of these addi-tional variables should be considered in any future appraisal of its benefits. A substantial part of this thesis is concerned with the assumptions and derivation of three basic types of fisheries models. Specifically, these are: the regenerative, or stock production models, the dynamic pool model, and the virtual population and cohort analysis techniques. The purpose in 115 analyzing them in such detail is to discover which factors contribute to or limit their assessment capability and to use this information in developing the annual state models. The following paragraphs summarize how each model type differs, by choice or capacity, in its approach to the assessment prob-lem. The util ity of the regenerative models depends on the relationship established between catch and apparent abundance or yield and apparent bio-mass when a method of aging the stock is unknown. Of the two components of an assessment, age-group abundance and the effects of fishing, only the lat-ter has the potential of being estimated. Although it is usually not stated explicitly, an assumption of the models.is that stock structure-be reasonably stable. Changes in catch or yield are therefore the result of fishing activity and the displacement of the abundance or biomass from their respec-tive environmental carrying capacities. If there are environmentally induced changes in recruitment, however, the abundance and biomass are al- . tered independently of the effects of fishing or the natural growth rate of the stock. If the catchability (q) is also variable because of behavioral changes, weather or accessibility, the consequence of both activities is to introduce uncertainty (or variance) in the parameter and q estimates. More-over, since q is not determined each year, but only through analysis of a long time-series of data, the observed changes in fishing success (U or V) cannot be attributed with confidence to fluctuations in either the stock or q. Collectively, these problems are further aggravated by the difficulty in obtaining a reliable yearly estimate of fishing effort. Unlike catch or yield, the effort term cannot be measured directly; i t must be assembled from and weighed according to a variety of influences that exist in a fishery. 116 Since the distribution of fish and fishermen is usually non-random, what is required is an estimate of the effective fishing effort (f). It is the quantity of effort that would have been expended to obtain the catch or yield i f fishing had been conducted at random and continuously during the year. Provided f can be calculated U and V will then serve as indices of apparent abundance and biomass, respectively. The derivation of f, however, involves more than just the standardization of vessels and gear. The co-operation of the fishermen is essential in revealing where and when they fished, the quantities and species of fish caught, and the time spent search-ing, fishing and handling gear. For competitive reasons fishermen may be reluctant to divulge all or part of this information, despite management's assurances of confidentiality, and on occasion the data may be intentionally misleading. Taken together, these factors emphasize the difficulty in ob-taining a reliable estimate of f and the corollary product F. In contrast to the problems associated with recruitment, q and f, the regenerative models offer a distinct advantage in not requiring an estimate of M. If i t is possible to determine the age of a fish the potential exists for carrying out a complete assessment. Any doubt concerning the appropriate-ness of the logistic equation to relate yield to biomass, as in the above models, is also removed. Growth equations in length or weight supply the multiplicand in calculating the appropriate biomass product and the remaining unknown is reduced to fish numbers. All of these benefits are reflected in the simple and more complex forms of the dynamic pool model, but it is equally clear that the difficulties of an assessment are compounded. In addition to the fishing mortality coefficient the growth parameters, apparent abundance and two new variables, the total and natural mortality coefficients 117 must be estimated for the stock or each age group. Of these quantities the growth parameters are obtained most easily, and since the abundance estimates depend on the coefficients, the confidence in an assessment rests on the accuracy of the mortality rates. The methods used to calculate these rates, however, rely extensively on the historical and current values of f, the assumed constancy of q over variable periods of time and, less frequently, on a reasonably stable recruitment and application of fishing effort. Similar or identical assumptions and dependencies occur in connection with the regenerative models and their affect on the reliability of an assessment is not repeated here. The virtual population and cohort analysis techniques reconstruct the experience of a year class during its fishable lifespan by interlocking a series of annual catch equations. An estimate of the apparent recruitment is obtained by back-calculation once the year class has left , or nearly left, the fishable stock. Furthermore, data on growth in weight are not required. These simplifications, together with information on age, create the potential for a complete assessment. Calculations can begin with any age group in a year class, the requirement being a known or assumed constant value of M and a starting value of F. Both Jones (1961) and Pope (1972) indicate that the accuracy of back-calculated estimates of (NQ).j and F for the earliest ages improve with the number of age groups included in the analysis, but the improvement is relative to the chosen value of M. Although this feature is important for historical reasons, management is more concerned with the accuracy of a stock assessment for the immediate past year of f ish-ing. In this respect the virtual population and cohort analysis techniques offer no advantage over the dynamic pool model, and they are equally dependent 118 on f and q to prepare the initial estimates of the mortality coefficients. The position taken in the above summaries is that variable recruitment, the reliabil ity of f and the assumed constancy of q restrict the confidence and timeliness of stock assessment procedures. Under certain conditions the annual state models circumvent these difficulties and permit a complete assessment of the stock or an individual age group at the end of each fishing year. The potential implications for management are that the dependence on fishing effort and historical data is reduced, while growth parameters and the elements of an assessment are updated in time to apply regulations to the fishery. The capacity of the state models to provide this rapid assessment is attributed to three modifications of the dynamic pool model and the use of the age-group statistic for mean weight, W".. Initially, it is a reasonable assumption that past events in the l i fe of a year class are summarized in the abundance and growth characteristics of each age group currently being exploited. By reducing the yield model for the stock to that of an age group, the analytical complications of variable recruitment and the histori-cal effects of fishing are avoided. The second modification, that of sub-stituting the generalized Richards (1959) growth equation -k(t-t ) b (wt = Wjl-e u )u) for the von Bertalanffy (1938) model, removes the cubic restriction on the parameter b. The purpose of this change is to allow a more accurate description of a species' growth. The final adjust-ment involves integration of the yield model by an approximate method with-out a serious loss in accuracy. These alterations, together with informa-tion on yield in weight (Y^ .) and catch (C^ .) lead directly to an expression 119 for W. that is independent of abundance. More importantly, the equation for W.j provides a deterministic solution for since the only other un-knowns, the growth parameters, can be readily obtained by sampling. If these methods are applied to the same year class for two consecutive years, the difference between 1-,, and Z. • estimates the change in annual fishing mortality rates assuming that M^  = M.^ . The ratio j/F^+i » obtained from consecutive catch equations, can then be used to solve for the individual values of F.. and F ^ , and simultaneously, for M.. ^ and the apparent abundances. The above method is one of several model variations that can be used to carry out an assessment. In Section III alternative solutions are presented that deal with either the total stock (Models 1 and 2) or a single age group when M is constant or a function of age. Although these alternative methods are analytically interesting, in some cases their practical application is limited. If a species grows rapidly in length a model based on the mean length of fish (L^) may be adequate. For slower growing species, the length of fish sampled from the annual catch probably cannot be measured with suf-ficient accuracy to estimate Z^  from Lj . The quantity is more sensitive in this respect and is the logical indicator to use. If-L/ is not included in the calculations (Model 2) it is neither practical nor advisable to solve simultaneously for the growth parameters and the total mortality coefficient. The surface over which program ZXSSQ must search for a solution is flatter than that of Model 1 and there is greater chance of being trapped in a local minimum. To avoid this problem the independently determined values of Wro, K, b and t can be substituted directly into the Model 2 equations. The 120 final transition from a stock model to that of an age group is readily accomplished and is described in the preceding paragraph. In all of the above models the key variable that determines the reliabil ity of an assessment is the total mortality coefficient. The accuracy of Z.., however, depends on the uncertainty associated with all of the parameters (W ,^ K, b, , t ) and variables (w\ , U ) . In this sense the limited exploration of errors in t that affect Z^  (p. 105) is insufficient. A complete sensitivity analysis is required to indicate whether the uncertainties in the parameters and variables are acceptable in determining Z^, and how their uncertainties contribute relatively to that of Z.j (Majkowski, 1981, MS). In practice, to improve the accuracy of i t may be necessary to abandon the traditional use of the von Bertalanffy (1938) and Richards (1959) functions as composite descriptions of growth for all age groups, particularly i f growth has age-group varia-bil ity between years, or is a function of sex or stage of maturity. In these cases the sampling program that runs concurrently with fishing operations to establish W. or can be expanded to obtain the necessary information. An assumption that is common to all of the above models is that the stock is closed to immigration and emigration. Although neither condition can be prevented, their occurrence can affect the results of an assessment. For example, i f an immigrant source of fish enters the fishing grounds, the art i f ic ial ly high values of and will reduce the estimate of 1. and increase that of abundance. Throughout the development of the state models i t is also assumed that growth and mortality act concurrently and contin-uously during the fishing year. If there are seasonal changes in either 121 growth or the rate of fishing, W. in particular will be affected and lead to an erroneous estimate of Z . . In this case the models can be partitioned into shorter periods of time in which the original assumption is maintained. The equation for is then modified to: Z. W -0.5vZ. -vZ. W. = —j- • — v(A + 4Be 1 + De ') 1-e 1 6 where v = the fraction of a year; i = the age at the beginning of the interval v , and A, B and D are equal, respectively, to (1 - e " ^ " V ) b , ( 1 _ e - k ( i + - 5 v - t o ) ) b , a n d ( 1 _ e -k( i + v-t 0 ) jb > I n t h e i r p r e s e n t f o r m t h e models apply only to the fully fishable age groups and are limited by the assumption that within-age size-selective mortality does not occur. If the smaller members of an age group are not harvested, to\ will increase and lead to an underestimate of I. and an overestimate of (NQ)^. The opposite situation is expected to occur i f the larger fish in an age group are able to avoid capture. The last important assumption to consider is that M. • = M-j+j j+l o r ' ™ e f f e c t ' that M is constant for all age groups. The state models for two consecutive age groups, as well as the cohort analysis technique (Pope 1972), are particularly sensitive to this assumption. If i t is invalid, the ratio F.. j/F-j+j determined by the catch equations will s t i l l be correct, but the difference between the total mortality coefficients will not estimate the change in fishing mortality alone. In this case it is not possible to solve for the individual values of F.. .. and F^  . directly and two optional methods of analysis must be considered. In the f irst proce-dure i t is immaterial whether M is constant or a function of age, but it must be assumed the fishing mortality rate of all age groups in a given year is 122 constant. If in addition F. f F . + , , then it is possible to solve for the mean values Z. , Z . + 1 and M. . + , , and eventually for F., F. , and the individual values of M. . If both fishing and natural mortality are age dependent, the second procedure is restricted to a stock analysis in consecu-tive years and cannot proceed beyond the mean values of the mortality coefficients. With the exception of seasonal pulse fishing the preceding remarks are summarized in the following conclusions: 1. The potential of the annual state models to provide a complete assessment of a stock is demonstrated. 2. The fishing and natural mortality coefficients and the apparent abundance of each age group is estimated without reference to effective effort, the catchability coefficient, the number of recruits or the entire history of the year class. 3. The data requirements of the models are an age group sample of: 1) the annual catch and yield in length and weight, or 2) the annual catch and yield in weight, and 3) an independent estimate of the growth curve intercept. This information is used to cal-culate the mean length (L^) and weight (W\) of each age group, to develop suitable expressions for each of these quantities and to maintain an accurate description of growth. 4. Equations for yield in weight and mean weight incorporate a generalized growth model by means of an approximate integration technique. Potentially, the same method can be applied to other 123 empirical growth equations to improve the accuracy of the state models and extend their range of application. The assessment capability of the models is also flexible. Depending on the assumed constancy of the natural or fishing mortality coeffi-cients, the analysis can be carried out on a stock or age group basis by combined (T. and W^ ) or individual (W..) equations. The major assumptions of the models are that: (1) the stock is closed to immigration and emigration; (2) growth and mortality are concurrent and continuous during the period of analysis; (3) fishing is not size selective within age groups; and (4) the natural mortality rate of the fishable stock is constant. 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Bul l . , 90(2):141-147. 130 Watt, K.E.F. 1956. The choice and solution of mathematical models for predicting and maximizing the yield of a fishery. J . Fish. Res. Board Can., 13:613-645. Widrig, T.M. 1954a. Method of estimating fish populations with application to Pacific sardine. U. S. Fish and Wildlife Ser. Bull. 94, 56:141-166. . 1954b. Definitions and derivations of various common measures of mortality rates relevant to population dynamics of fishes. Copeia, 1:29-32. Wiles, M. and A.W. May. 1969. Biology and fishery of the west Newfoundland cod stock. Fish. Res. Board Can. Studies, Part 1, No. 1318:487-525. Wilimovsky, N.J. and E.C. Wicklund. 1963. Tables of the incomplete beta function. Inst, of Fish. , Univ. of British Columbia. 291p. APPENDIX A Analysis of Fisheries Models Using a Gaming Technique The results, conclusions and management implications arrived at by gaming with two fisheries models are reported in this section. In each case the general objective was to identify the minimal amount of infor-mation needed from the fishery to solve a given problem in the least amount of time. A gaming technique is readily adapted to such questions and in addition it is a valuable, educational tool. In the models that follow the reader will note that the investigator is constantly weigh-ing the information content of the fishery data against the desired ob-jective. Because of the minimal time limit, information that is mis-leading or ambiguous is quickly identified and rejected. By a process of elimination the "manager" is then led to ask the following question: What piece or pieces of information are at present not being collected from the fishery that would help solve the problem more directly? If such statistics can be identified, a second question must immediately be answered: How can the information be collected from the fishery, or in the present case, is the simulation model programmed to reveal this information? If not, the model can be altered to disclose increas-ing amounts of information in keeping with a greater, understanding of the problem. The equivalent situation in a real fishery is recognizable in that data collection should be oriented towards both economy and high information content. 132 The method of presentation and specific objective is different for each of the models. Details of the Schaefer (.1954, 1957) simulation are summarized while those for the modified Beverton-Holt (J957) model are reported in ful1. 1. The Schaefer general production model. Objective: to determine the catch and effort co-ordinates of.maximum sustained yield (MSY) using the least number of separate applications of fishing effort (f = boats). The only information available to the investigator is that the maximum number of boats cannot exceed 1000, a convenience reflecting the scaling of the computer programme. The operator is not aware of the Schaefer model parameters or those of the stochastic processes. The simulation is based on a discrete time version (Larkin, per-sonal communication) of Schaefer's production model. Variability in recruitment (R) and catchability (:q) is achieved by multiplying each quantity by a normally distributed, random number with mean equal to one and a standard deviation of 0.5 and 0.25, respectively. Given the basic Schaefer model, in time step j , d N j = r N. - r N . 1 - 9 8 , CA 1) W J K J where, r = 1; net rate of population increase in numbers K = 1000; carrying capacity dN./dt = rate of change of stock size N.; equal to absolute recruit-3 3 ment AR. A flow chart for a single time step may be diagrammed as follows; Start AR = l.N. -0.001 N . 1 ' 9 8 J J R = AR.v N.1 = N. + R J J CB = f . q . n F = CB/960, C = F. N . ' -»• Print out catch (c) Res = N.' - c Set ff.+i = Res and, Return to Start where, v = recruitment random number multiplier N-' = fishable stock J n = catchability coefficient (q) random multiplier f = number of boats applied by investigator; o<CB<960 F = effective fishing mortality Res = residual reproductive stock Solution Methods and Discussion 1. It is tempting at f i rst to begin a cautious analysis of the problem by the successive application of low effort levels. A knowledge of the resulting catch and effort, and hence the catch per unit effort CU), may be interpreted as an index of average, fishable stock abundance, A record of catch or effort versus U should then reveal the proximity of MSY. Unfortunately, the above method is extremely wasteful of time, and information on catch or U can be both misleading and ambiguous. For a given percentage change in effort the resulting change in catch may 134 be relatively greater of smaller, and there may be no apparent relation-ship between effort and stock-size index. The investigator is therefore left in doubt as to the cause or causes that induces the change in catch. It may be due to the manipulation of effort, but equally a change in availability or vulnerability can produce similar results. In addition to the loss in time the manager cannot formulate a rational basis of future effort changes based on past events in the fishery. 2. A bolder approach would be to increase effort by a factor of two or three in consecutive years. Hopefully, the variation induced by envir-onmental effects will be overshadowed by the comparatively larger changes in effort, and a positive sense of direction may be revealed. The decision to follow this course in a fishery is potentially dan-gerous to the stock and may also be wasteful of effort. If the init ial stock size is unknown, and particularly i f the productive capacity is both low and unknown, a rapid rise in effort may drive the population to extinction. In addition to this risk the information on catch and U from the exploratory fishing period can be misleading. Repeated trials indicate that i f effort is increased in three successive time steps i t is possible for the catch to show a consecutive decline while the U simultaneously increases. The random number multipliers, v and n, can produce such an effect and the investigator is denied a logical inter-pretation of the data. The rationale for subsequently changing effort is missing and therefore the procedure cannot lead to the objective in minimal time. 135 3. If both small and large changes in effort can produce equivocal results a more informative approach might be to maintain the same effort level for two or more time steps. The variability within the system would then be partially revealed and similar information could be gathered at different effort levels. The above method is the least productive of information over time. By maintaining a constant effort an associated distribution of catch and U may be generated provided the system is at equilibrium. The random multipliers, however, introduce stochasticity and equilibrium cannot be achieved. There is a greater liklihood that the f irst choice of effort will introduce oscillations in the system that can be vaguely monitored by a catch (or U) record. In time, the variability may become damped, but the manager will only achieve information relative to a single effort value. The balance of the system will remain unknown. The cause of variation in U or catch will be similarly confused and unassignable to either a change in fishing efficiency, availability or recruitment. Of the three methods just described the cautious manipulation of effort (methods 1 and 3) appears to involve low risk, and given suffir cient data, the co-ordinatres of MSY could be estimated. The cost of using either approach, however, can be equated to a loss in analysis time and a low rate of return of information for the effort involved. The second procedure, involving large changes in effort, increases the breadth of information taken from the fishery and the perturbations are confined to a short time interval. The exploratory potential of the 1 3 6 method i s appealing, but the benefit i s associated with a high r i s k . Future catches may be jeopardized and be accompanied by severe social and economic dislocation i f the experiment should f a i l . Nor can a large amount of e f f o r t be committed to or removed from the fishery at w i l l . To reduce the risk there i s a need to i d e n t i f y a pa r t i c u l a r action with i t s consequences and the time lag between these events should be minimized to allow for remedial action i f necessary. A measure of recruitment can f u l f i l l both requirements. In addition i t i s the only s t a t i s t i c that can act as a warning device with the least ambiguity, a property that i s not shared by either catch or U data alone. A fourth solution method i s therefore possible. 4. The simulation model can be modified to print out an estimate of recruitment following step two in the flow chart. Since the informa-tion appears prior to fis h i n g i t i s assumed that the recruits form part of the fishable stock and the following analytical model i s use-ful as a f i r s t approximation: Let, = number of recruits entering fishery prior to f i r s t application of e f f o r t N Q = residual stock before fis h i n g begins .N-j = f i r s t fishable stock size C-j = catch taken from N-j Res-j = residual stock after removal of C-j g = symbol to indicate a function = underlined symbol known from fishery data 137 Then, given a starting value of RQ, the analytical sequence is estab-1ished: Start N, = N + R 1 o —o Assume a value for N (= N ) o o Apply effort R e s l = N l " h = \ + ^ ~ C l } R} = q(ReS l) = g (N ] - C 1 ) Return to start and substitute: R-, for R ; and N for N — l — o o o The algorithm above is used recursively to establish a relationship between R_^  and (NQ + R^  - C^). Notice that in the last expression the initial and correct value of NQ is never determined. It is an internally generated number to which an arbitrary, large value is f irst assigned by the investigator and up-dated following each return to start. Using the init ial value of NQ i t is then possible to estimate Res-j and estab-lish a functional relationship between this quantity and the number of recruits (R-j) derived from it . Operationally, the successive values of R become the focus of attention for the region of MSY is associat-ed with the highest average value of R. In this connection the manager will discover that the original objective, to find the co-ordinates of MSY, is inconsistent with its determination, " . . . in the least number of separate applications of fishing effort". A low number of iterations is partly fortuitous because a new value of NQ is used at the beginning of each trial run and cannot be duplicated. Similarly, since a variance component is attached to both R and effort, only the approximate region of MSY, followed by a repetition of this procedure. A typical Simula-tion, with appropriate comments is recorded in Table Al , The solution method used in the example depends crit ical ly on the variance (stochasticity) assigned to recruitment and catchability. If the stochasticity is zero a production curve can be estimated using a minimum of three levels of effort. Alternatively, i f the variance of R and q is high relative to the change in catch with effort, a pre-dictive relationship cannot be established over a short time period. In a real fishery, historical data may offer some guidance, but the length of time over which the data must be collected depends on the stochasticity inherent in the system and the accuracy with which effort can be measured. An analytical solution to the present problem is possible using equation (Al) below. The investigator would, of course, be unaware of the parameter values and the details are offered here for the sake of completeness. The rate of change of stock size is related to the stock by the expression: dN _ N - 0.001 N 1 ' 9 8 (Al) dt At the point of equilibrium (e), where the catch is maximized, the rate of change of d Ng/dt is zero, and the value of Ng is obtained by taking the second derivative of equation (Al): d 2 Ne = 0 = 1 - (0.001) (1.98) N 0 , 9 8 2 dt^ 139 and solving, Ng = 573 fish The maximum catch associated with Ne is derived by substitution in equation (Al): C = ^ e = 1(573) - 0.001 (-573)1'98 = 284 fish e dt The number of boats, f g , needed to obtain the equilibrium catch is ob-tained from the relationship: or C = f_e__ . N e 960 e f e = C e . 960/Ne, and substituting the values of Cg and Ng f = 284 (960)/573 = 476 .boats TABLE AI. Simulation results and comments. Typical analysis of a Schaefer (1957) general production model. Method 4. C / B E s t i m a t e : E s t i m a t e : Time Recruits Boats Catch or Fishable Residual Step R 15 C U Stock, FS Stock, Res Comments 1 35 3 270 8 199 11 317 14 322 16 226 50 44 0.88 2035 2 143 2143 150 221 1.47 300 362 1.21 300 83 0.28 450 276 0.61 500 274 0.55 2192 1545 1577 1597 1528 1991 1922 1830 4 228 2058 300 253 0.84 1805 5 274 2079 450 693 1.54 1386 6 481 1867 450 423 0.94 1444 7 219 1663 450 317 0.70 1346 450 334 0.74 1211 9 82 1303 1220 10 128 1348 300 88 0.29 • 1260 300 190 0.63 1387 12 299 1686 1 4 1 0 13 219 1629 450 354 0.79 1275 1323 15 229 1552 500 250 0.50 1302 Estimate s t a r t i n g stock (N„) = 2000; FS = 2000 + R = 2035 Enter with low e f f o r t to determine system response. Res = FS-C = 1991. R2 = g ( R e s i ) , but funct ion unknown. FS = 1191 + 143 = 2143. Both FS and R Increased. Increase B. Both FS and R cont inuing to increase . Increase B to reduce FS. Note: U seems to be h ighly v a r i a b l e . FS, R and Res d e c l i n i n g . Hold e f f o r t at 300. R improving. FS no worse than in time step 1. Increase e f f o r t . Unexpected high c a t c h . Recrui t response may be excep t iona l . Repeat e f f o r t . Res 6 = R e s 5 . Near MSY? R 6 must have been unusually h igh . Repeat e f f o r t to reduce Res. Res appears to be s t a b i l i z i n g , but r e c r u i t response begins to d e c l i n e . FS f a l l i n g and unse t t l ed . Repeat e f f o r t to see i f recruitment continues to d e c l i n e . FS , Res and R show d e c l i n e . May have exceeded MSY. Upper bound MSY is uncertain but lower leve l of 300 boats appears to generate a favorable recrui tment . Reduce e f f o r t to 300, a l low stock to recover , repeat e f f o r t increase to locate upper l i m i t MSY. U lowest on record , but hasn't been s a t i s f a c t o r y index in past . Both R and FS showing nominal increase . Repeat e f f o r t . Strong r e c r u i t response. FS beginning to increase . Repeat e f f o r t to see i f r e s u l t due to chance or populat ion a c t u a l l y bu i ld ing up. Res beginning to respond p o s i t i v e l y . Recrui t response ho ld ing , FS recovered to time . t ep 7 l e v e l . Begin plan to increase e f f o r t to overshoot MSY. Repeat e f f o r t to determine i f dec l ine in R cont inues. Small change in R and FS, but Res beginning to dec l ine sharp ly . If MSY in excess of 450 boats must determine now while Res s t i l l has the capaci ty to respond. Increase e f f o r t to 500 boats. Repeat e f f o r t to determine r e c r u i t response. REMARKS: Repeated e f f o r t of 500 boats f o r two add i t iona l time steps caused a d e c l i n e in R. Conclude that MSY in region of 400 to 500 boats. At time step 19, return to B=400, al low stock to recover and repeat incremental process . 141 2. A modified Beverton and Holt (1957) model containing a Ricker (.1968) recruit function A critical requirement in the application of the Schaefer (.1957) model is that the fish species have an early age of reproduction and entry to the fishable stock. Under these conditions i t is usually as-sumed that the rate of change of stock size at any time is a function of the stock abundance or biomass at the same point in time. Conse-quently no time lag exists between a change in stock and the net response in recruitment. However, for a large number of commercial species these assumptions are inappropriate, and a model is called for that can simul-taneously accomodate time lags and be descriptive of a multi age class fishery. The "dynamic pool" model of Beverton and Holt (1957) is a suitable choice, but its simplified form (ibid, p. -36) is restricted to the analysis of an equilibrium condition with constant recruitment. A considerable improvement in realism is possible i f the usual yield and catch equations are made to apply to individual age groups. The total yield from the fishery over (say) one year will then equal the sum of yields from each cohort in the fishable stock over the same time inter-val. The modification also allows for variation in year-class strength and the inclusion of a deterministic recruit function. In this final form the modified age-group model is suitable for analysis by a gaming procedure. In the pages that follow several empirical relationships and formu-lae are used without the support of mathematical derivation. At the 142 time of their development the author was in search of rapid and approxi-mate methods of analysis in which these objectives figured more prom-inently than mathematical rigour. Subsequently, the necessary proofs and derivations were carried out in fu l l . They appear in the main body of the report and appropriate references are given in this appendix. The gaming rules are modified in the present case to achieve part-icular objectives. The investigator is forewarned that the following assumptions apply: 1. The fisheries model is deterministic and contains a Ricker 0968). type of recruit function. 2. Information from the fishery is perfect or error free. 3. The distribution of the target species, cod, is random. Together with assumption (2), all fishery and research samples are therefore fully representative of the stock age distribution. 4. Fishing and natural mortaltiy occur simultaneously and act continu-ously throughout a fishing year. 5. The trawl gear operates with "knife-edge" selection, and the average environmental effects over time is zero. 6. The natural mortality rate, M, is constant for all age groups. 7. The investigator is free to change the age of f irst capture [t ). and effort (f = boats) each year. All fishable age groups are equally affected by the effort, and therefore by the fishing mortal-ity rate, F. The catchability coefficient (q) is constant. 143 8. In any year that fishing takes place a nominal number of research vessels ( f = 10 boats) continuously sample all age groups less than t . c 9. Growth is density independent, constant for each age group, and fol-lows the von Bertralanfly (1938) equations for length and weight. 10. In the year before fishing begins the single stock is at equilibrium and in a virginal state. 11. Effort is restricted to the range: o<tVI000 boats. Given these assumptions the objectives are expressed as follows: 1. Determine the minimum number of years needed to estimate the param-eter values in the yield and recruit equations. Specify the in-formation needed to do this and its source, i .e . , fisheries or research data. 2. Estimate the yield, effort and t c co-ordinates of MSY. 3. Equilibrate the fishery at MSY by means of a repetitive process of yield prediction and experimental fishing. 4. To avoid social and economic distress the boats assigned to the fishery by the investigator are "permanently" committed. This is an unusual condition, but i t imposes an additional constraint on the investigator. The interactive, simulation model is only responsive to the number of boats (f) and the value of t £ specified by the manager. Internally, the program converts f into a fishing mortality rate that is used in computing the annual yield and catch from age groups equal to and older 144 than t c- Exit from the fishery occurs at age t' = 13. At the end of each fishing year reproduction is calculated on the basis of the residual sum of adult fish four^ years old and older. Fecundity at age is constant for all adults and i t is further assumed that larval mortality occurs in one instant in time. The recruit function cal-culates the.residual progeny which enter the fishery in the next year as age-group '0' . The exponent in the length-weight relationship is three. The von Bertalanfly constants are taken from Halliday (1972) and apply to the Scotian shelf cod stock complex. Finally, the sequence of events for each fishing year (j) and the internal formulae are out-1ined below: Sequence Model input and calculations - (External operator: pre-select M and t m 1'(Start) Specify f. and t J c 2 F. = q f. and F' = q f , where f = 10 Z. = F. + M and Z' = F' + M J J 3 -nK( i - t ) -(.Z, +nK) 3 Y. . = F (N ). . W» z %_e °_ (1-e J ) 1 , J n=o Z. + nK (A2) 4 C i , j " F J ( N o » i . J ( l -e"^) /Z. (A3) t ' - l t ' - l - z . 5 si= I G N t } i , j e J The age of f irst maturity (t ) is a pre-selected variable; range: l<t <10. m 145 Sequence Model input and calculations a[l-S./S ] 6 R j + 1 = c S . e J e CA4) 7 Print out requested information 8 Advance all age groups by one year, -Z.(orZ') 1 - e - < V i + i , J+i - ( V i , j e J 9 Set R j + 1 equal to age group '0 ' , or (.N0)0> j + 1 10 Return to start for the next fishing year. The notation is standard except for the following: i = age group subscript j = fishing year subscript; j=o for virginal state S. = sum of reproducing stock at the end of the year 3 (N ). . ; (N ). . = number of fish in age group i at the beginning o 1 ,J i ,j and at the end, respectively, of the year = (NQ) 0 j+1 = P r o 9 e n y calculated from S^  which will enter popula-tion as age-group 'o' in the following year. S g = equilibrium adult stock size, 1106 units, in the absence of fishing. Based on equilibrium, age-group 'o' starting value of 1191 units (Re)- (The starting population is calculated internally from -M the recursive relationship, N +^^  = e , where M = 0.28; i .e . , (NQ)1 . = (NQ)o . e"M = 1191 e"" 2 8 = 900 units, etc. Since reproduction takes place at the end of the year, at equili-brium (j = o) the number of adults is , S = P CN.). e~^ = e i=4 z 1 1106 units) c = R e /S e = 1191/1106 = 1.0769 Additional constants: W~ = 11,410 gm K = 0.14 t = 0.07 yr. a = 1.895 M = 0.28; a pre-selected variable; range: 0,2 <^  M<£ 0.3 Method of Analysis: Year One The first decision the investigator must make is to assign values to f and t . A low, initial effort is desirable because the reproduc-tive capacity of the stock is unknown, and there is a social and econ-omic obligation not to retract any boat committed to the fishery. The choice of t is also cr i t ica l . The first few years of fishing must be viewed as an exploratory period during which important information on the system is obtained. A high valued choice of t is cautious and least productive of information. Alternatively, a low value of t can produce four, desirable results: 1. provide data on a broad range of age groups, 2. permit research to establish the age-of f i rst maturity Ct 1, and estimate the age-group instantaneous growth rates, G., from fishing and research samples, 3. initiate the "fishing-up" process for the surplus stock, and 4. produce a strong recruit response in the second year of fishing provided t < tm. 147 There are at least two risks that must also be considered and both argue in favor of a low, init ial effort. The value of t is unknown 3 m and therefore the time lag between recruitment in the second year and the point at which these fish enter the reproducing stock is also unknown. In the interval their numbers will decline due to natural mortality and possibly fishing i f they are removed as juveniles. The second point is also related to the time lag, but concerns the decline in age-groups 1, 2, 3, . . . t immediately following the first year of fishing. The quant-ity of fish in these age groups cannot exceed their in i t ia l , equilibrium numbers. They too will decline due to M and possibly F, and therefore the fishable stock will be substantially reduced for an unknown number of years. Recovery may begin when the f irst year recruits become mature, but the degree of recovery will depend on the variable choices assigned to t in the intervening years. The manager must anticipate and guard against these risks to both conserve the fish stock and protect the investment of fishing boats. In consideration of the above arguments, the levels of f and t for the first year are set to: f 1 = 50 boats t £ = 2 years The only information needed from the fishable stock is yield in weight and catch in numbers by age group. Research data on age groups less than t are also available and are reported as indices of abund-ance (oy = C . / f ) and biomass (V.' = y./f* }. The yield and catch data, however, are used in several repetitive calculations and the model is programmed to carry out these operations directly. The techniques used 148 in preparing the derived statistics are given below, (i) W. The statistic W. is obtained by dividing the yield (Y.) by the catch (C^). It represents the average weight, by number, of fish of age group i taken during the fishing year. For further details on the derivation and meaning of w\ , the reader is referred to Section IV, p. 37. The values of W. for the fishable age groups can be used in an iter-ative, regression technique to provide an estimate of the length-weight exponent, b. The details appear in Section II (3.2) p34. In the pres-ent analysis, the above method was unknown and the f irst estimate of b was taken from the literature on cod (Kohler, et al_., 1970) and set equal to three. K, t , W and L ' o 0 0 °° The length-weight relationship w(t) = a £ b ( t ) can be rearranged into an approximate expression involving : 1 (W\)b ~= QL (A5) where Q is a proportionality constant and L. is a statistic analogous to ¥ . , but refers to the average length, by number, of fish caught in age-group i. From the known values of M^, W^, . . . !^ and b, the corres-ponding products QL^J QL^• - • QL-j2 are determined and used in a Walford (Walford, 1946; Ford, 1933) line regression of QU.+.| on QL.. The 149 slope provides an estimate of the von Bertalanfly growth constant K, A and the intercept produces an estimate of QLOT in equation (A5). A similar Walford regression of ln (:QLs> - QL^) on age (i + 0.5) yields an estimate of t . It is necessary to increment each age by the con-stant 0.5 to represent half a year, the approximate time at which the weight and length of a f ish reach their average values of W. and * (see Section IV, p. , for an improved estimation technique), h. The instantaneous growth rates are calculated directly from the equation: - - ~ • ^ " V -K G i = b l n (^ s (A6). K( i - t ) . e v o - l The derivation of this equation is reported in Section II (4-3), p The results of the above calculations, together with the research and fishery values of IK and V^, are reported under the heading FISHING YEAR in Table A II at the end of this appendix. ( i i ) Estimation of the natural mortality rate, M. The assumption of an equilibrium condition for the virginal stock permits an estimate of M to be obtained in the f i r s t year of f ishing. Let R equal the number of progeny entering age-group ' o' in year j=o, the year prior to f ishing. The number of f ish in age-groups l 2 ' and '3' at the start of the fishing year (j = 1} wi l l then be: 150 CN )? = R e " 2 M , and (A7) (N )_ = R e " 2 M e" M = R e ' 3 M , (A8) 0 o since the number of fish in adjacent age groups are related by the ex--M pression (N Q ) i + - | = (N )^  e at equilibrium. Equations (A7) and (A8) can be substituted in the catch equation (A3) to y ie ld indicies of abundance: U2 = q (N Q ) 2 = q R e " 2 M (1-e Z l ) / Z 1 (A9) and U 3 q R e "3 M (1-e ^ / Z - , , (A10) where, Zj- is the total mortality rate in year j = 1. Dividing equation (A9) by (A10), simplifying and taking natural logar-i thms: In U 2 - In U 3 = M , or more generally, for the fishable stock where i takes on values (:2, 12) In U. - In U . + 1 = M (All) Equation (All) indicates that i f successive values of In (ordinate) 0 are plotted against age (i + 0.05; abscissa) in a catch curve, the resulting slope wil l equal -M. Again, the ages must be incremented by 0.5 because each U\ term is an index of average stock size over the year. The method is approximate and therefore the derived value of M is only an estimate. A straight-line relationship is obtained from the data in Table A II: ln U. ln (1132) = 7.0317 6.7511 6.4708 6.1924 5.9108 5.6312 Age, i +0.5 2.5 3.5 4.5 6.5 7.5 8.5 In U. 5.3519 5.0689 4.7875 4.5109 4.2341 Age, i + 0.5 8.5 9.5 10.5 11.5 12.5 Slope 0.2800 = - , and M_ 28 ( i i i ) Estimate of the optimal age of f irst capture, t ^ In the absence of fishing the critical size (Ricker, 1968) is ob-tained at the point where the difference between the instantaneous growth rate (.G^ ) and M is zero. The yield will be less than optimal i f t is less than or more than the critical size. A search of the G. values in Table A II, for j = 1, indicates the following: 0.280 = + 0.091 0.280 = + 0.010 0.280 = - 0.049 A 5 = G5 - M = 0.371 -A6 = 0.290 -A 7 = 0.231 -152 Therefore, the f irst estimate of the optimal age of capture is t Q p t = 6 years. Year Two An estimate of q is needed to convert effort into fishing mortal-ity rate and allow the development of a predictive equation for yield. If the f irst year levels of . t and f are maintained for a second . c year it is possible to achieve both goals. The cost of obtaining this information can be equated to a temporary, future loss in catch from all age groups greater than 1 1 ' . It is more important, however, to be able to recognize the present position of the fishery and predict a future direction; and i f need be, the depleted age groups can be al-lowed to recover by setting t c = t in year three. The ability to advance the age of f irst capture reduces the experimental risk and, with this assurance, the second year values are set to: = 50 boats, t = 2 years. The fishing results appear in Table A II, (i) Estimation of the catchability coefficient, q. The following relationship is valid i f the fishing effort is held constant for two, consecutive years: l n i f ^ " J \ J+1 (A12) Ui+1, J+1 Equation (.A12) is analogous to (7\11) in that the logarithm of the ratio of two indices is equal to a mortality rate. A catch curve based on expression (A12) would be incorrect, however, because the indices refer to different sampling years. Calculation of the geometric mean (G.M.) from individual ratios is indicated (Ricker, 1968), and the details appear below. Let the notation i , j/i+1, j+l be abbreviated to i , i+1. Age-group In i_ In Z, ? ratio U i + 1 1 , d - K i 2.3 M = 0.3802 -0.9671 3.4 0.3795 -0.9689 4.5 0.3795 -0^9689 5.6 0.3812 -0.9644 6.7 0.3774 -0,9824 7.8 0.3789 -0.9705 8.9 0.3820 -0.9623 9.10 0.3776 -0.9739 10.11 0.3808 -0,9655 11.12 0.3837 -0.9579 154 Number of samples of Z-j 2 , n = 10 1 11,12 G.M. = antilog - z In Z, , = 0.3798 n 2 j 3 1,2 and Z1 2 = 0.38 An estimate of q and F is obtained directly: Z 1 , 2 = F 1 , 2 + A = q f l , 2 + A = q (50) + 0.28 , and solving, q = 0.002 and F.j 2 = 0.10 , for f = 50 boats (ii) A predictive equation for yield. The average weight of a fish at time t can be approximated by the exponential growth formula (Ricker, 1968): G.t w i (t) = (w ) i e 1 , for o<t<l , (A13) and the decline in age-group numbers by, -(F+M)t -Z.t N i (t) = (N 0) i e J = CNQ)i e J , (A14) where, (w )^  = the in i t i a l , average weight at the beginning of the time interval (t = o). Over a year of continuous fishing, the age-group yield is obtained by integrating the general expression for biomass: 155 Y i , j = Fj <l B i , j C t ) d t = F. f] N. Ct) w. Ct) dt , . . t(o,l) (A15) Substituting equations (A13) and (A14), into (A15), integrating and rearranging: , (G. - Z.)t Y i , j = Fj <No>1 <"o>1 7 o e 1 ' d t (G. - Z.) A. = F. ( B J . . (e 1 J - 1) = F, ( B J . , (e 1 - 1) (A16) bi Zj A i or, to \ Y- . .-. A. V. . A. / A 1 7 x (BJ-i -i = 1 'J i = i ,J i > (Al7) ' J F. / Ai n q A. J Ce 1 - 1) (e 1 - 1) where, (B ). . = the initial biomass at t = o, and o i ,j V. • = an index of biomass (Y. . / f . ) . The biomass at the end of year j , (B ). . , is derived from equation + ' >J ( A 1 7 ) : ( ; B + »i,r ( B o ' i , J e A i = ^ i — — r — < A 1 8> But (B + ) i j = (B 0 ) i + i j + 1 :> and the yield in year j+1 can be predicted by integrating the following equation: -i ( G i + 1 - z. At Y i + l j j + 1 = F . + 1 ( B o ) i + l j j + 1 / J e 1 + 1 ^ dt Substituting equation (Al8) for CB0)^+i J + 1 in the above expression and integrating: Y i + 1 i + 1 = F \l A i (e 1 + 1 - 1). (A19) 1 + 1 , 0 1 J l q ^sr " A T — — The total y i e l d , (-YTJ , from al l fishable age groups must include the biomass of pre-recruits in year j that wil l enter the fishery in year j + l . The generalized, predictive y ie ld formula i s , therefore: (vT)j+1 - F J + 1 V^ i ( 6 t c - u - M ) ( e G t c " Z j + 1 - n ( 1 - e c ) c 11 + i+l=t +1 Y i + 1 ' j + 1 (A 2 0^ c Equation (A20) can be suitably modified to allow for a change in t at the discretion of the investigator, i . e , t . , f t . . The entire C , J T I C , J program is contained in the subroutine PREDY. To predict the total y ie ld in year j+ l , the necessary values of (G. and V . ) . are taken from Table A II and combined with the previous estimates of q and M. At the end of two years of f ishing i t is necessary to assume that the analysis of fishing and research samples have led to an accurate measure of t , the time of spawning and the correct values of K, W ,^ b, t and G.. In Table A II the reader wil l note that estimates of these o i parameters are prepared by the simulation program in year three and thereafter. The recorded values are now known to be inaccurate, but they are included in the table to i l lustrate the departure from the . . real values as effort is increased. In a l l future calculations and predictive subroutines the following quantities wil l apply: TABLE A l l . Resu l ts of s imu la ted , experimental f i s h i n g . Modi f ied Beverton and Hol t (1957) y e a r - c l a s s model- con ta in ing a Ricker (1968) r e c r u i t f unc t i on . Legend: i = age-group s u b s c r i p t ; j = f i s h i n g y e a r . s u b s c r i p t ; V. , = y i e l d per un i t e f f o r t (gm/ f ) ; U i . = catch per un i t e f f o r t ; W. = average weight by numbers (gm); ' J G, = instantaneous growth r a t e ; f = e f f o r t ( b o a t s ) ; 1 t = age f i r s t cap tu re ; K, W^, t = est imated von Be r ta l an f f y (1938) growth equat ion constants ; R = '0' age cgroup progeny. Note: Values of W . . G . , K, W and t only change wi th a change in e f f o r t ( f ) . 1 1 oo 0 j = 1 R 0 1 2 3 . 4 AGE 5 GROUP 6 7 8 9 10 11 12 v i , j b.i w i G i 2.058 1 n o 555 1 1 306 132 271 052 534 .855 624 .687 719 .646 1113 .492 833 .489 1704 .371 873 .369 2365 .290 854 .279 3062 .231 795 .211 3771 .188 712 .159 4469 .154 619 .120 5142 .128 526 .091 5780 .107 438 .069 6375 f K W CO v 50 2 0.138 11,512 0.061 j = 2 V i , 3 U i , j 2.230 1 108 524 1 300 109 483 .774 651 .585 753 .442 790 .334 773 .253 719 .191 644 .144 560 .109 476 .082 397 .062 f fcc 50 2 j = 3 V i , 3 ! i l .J •W i G i 2.375 1 117 652 1 1 308 129 273 042 494 .788 627 .683 613 .550 1116 .490 578 .343 1687 .371 608 .259 . 2346 .289 596 .196 3043 .231 555 .148 3751 .188 498 .112 4450 .155 433 .085 5124 .129 368 .064 5762 .108 307 .048 6359 f K W 00 fco 219 5 0.137 11,574 0.039 j = .4 V i . j U i . j 2.634 1 124 .760 1 334 .224 525 .837 651 .584 567 .336 392 .167 384 .126 358 .096 321 .072 279 .055 237 .041 198 .031 f 219 5 j = 5 V i , j 2.656 1 138 .951 1 355 .303 568 .907 692 .620 602 .357 384 .164 248 .082 231 .062 207 .047 180 .035 153 .027 127 .020 f 219 5 TABLE Al l (contirfued) R AGE GROUP 0 1 2 3 4 5 6 7 8 9 10 11 12 j = 6 f 219 V i . j - 139 394 606 750 639 408 243 149 133 116 99 82 5 U i , j 2 648 1 .968 1.445 .966 .672 .379 .174 .080 .040 .030 .023 .017 .013 K .0.137 * i - • - 273 627 1116 1687 2346 3043 3751 4450 5124 5762 6359 W CO 11,574 G i - - •1.042 .683 .490 .371 .289 .231 .188 .155 .129 .108 -*<> 0.039 j = 7 V i . j _ 139 398 672 799 693 433 258 146 86 75 63 •53 f 219 U i . j 2. 652 1 .962 1.458 1.071 .715 .411 .185 .085 .039 .019 .015 .011 .008 tc 5 3 = 8 f 285 V i , j - 139 396 677 886 694 442 258 146 79 45 38 32 t c 5 U i , j 2. 657 1 .964 1.453 1.080 .793 .413 .189 .085 .039 .018 .009 .007 .005 K 0.137 W i - - 273 627 1116 1680 2339 3035 3743 4442 5117 5756 6353 W» 11,594 G i - - 1.040 .683 .490 .371 .290 .232 .188 .155 .129 .108 - 0.034 j = 9 f 302 V i , j - 139 397 675 893 758 406 241 134 73 38 21 18 t c 5 U i , j 2. 657 1 .968 1.455 1.077 .800 .452 .174 .080 .036 .016 .008 .004 .003 K 0.137 W i - - 273 627 1116 1678 2337 3033 3741 4441 5115 5754 6352 11,598 G i - - 1.039 .682 .490 .371 .290 .232 .188 .155 .129 .108 - 0.033 j = 10 V i . j -139 398 676 890 764 436 218 123 65 34 17 9 f 302 U i , j 2. 657 1 969 1.458 1.078 .798 .455 .187 .072 .033 .015 .007 .003 .002 5 j = 11 f 304 V i , j -139 398 677 892 760 438 233 110 60 31 16 8 t c 5 U i , j 2. 657 1 969 1.458 ' 1.080 .799 .453 .188 .077 .030 .014 .006 .003 .001 K 0.137 W i - - 273 627 1116 1678 2336 3033 3741 4440 5115 5754 6351 W» 11,599 G i - - 1.039 .682 .490 .371 .290 .232 .188 .155 .129 .108 - *0 0.033 TABLE Al l (continued) R AGE GROUP 0 1 2 3 . 4 5 6 7 8 9 10 11 12 j = 12 V i , j - 139 398 678 893 761 435 234 118 54 28 14 7 f 304 U i . j 2.657 1 .969 1 .458 1 .080 .800 .454 .187 .077 .032 .012 .006 .003 .001 5 j = 13 V i . j - 139. 398 678 893 763 436 232 118 57 25 13 6 f 304 U i , j 2.657 1 .969 1 .458 1 .080 .800 .455 .187 .077 .032 .013 .005 .002 .001 5 j = 14 V i , j - 139 398 678 893 763 437 233 118 58 27 11 6 f 304 U i , j 2.657 1 969 1 458 1 080 .800 .455 .187 .077 .032 .013 .005 .002 . .001 5 160 K = 0.14 W*, = 11,410 gm t Q = 0.07 years b = 3.0, and from equati on : (A 6), G. , G. i i i i 1 .1.9873 7 0.2301 2 1.0543 8 0.1867 3 0.6877 9 0.1533 4 0.4917 10 0.1272 5 0.3704 11 0.1064 6 0.2886 12 0.0.895. ( i i i ) Estimation of the recruit function parameters There are two methods of estimating the parameters Rg and S g in the Ricker (1968) recruit function, R a ^ ~ V l 7 V (A21)  5e The f i r s t procedure makes use of q to express S g in absolute terms, and the second leads to a calculation in which the parameter S g and the variables S. and R are given as functions of q. I n i t i a l l y , both tech-niques are used interchangeably, but in developing the final form of the prediction equation the lat ter method is preferred. The fishery information, measured as abundance ind ic ies , can then be used direct ly . A size index of the reproducing population at equilibrium (S ) may be calculated from the knowledge that t = 4, and that spawning takes place at the end of the year. Expressed mathematically, i t follows 161 that: or, replacing the age-group numbers by their respective indices after the f i r s t year of f ish ing , Z 1 2 S p = 1 ( E U . . + U n .) (A22) - Z 5 i o , i (1-e h q The only unknown in equation (A22) is the index y An estimate of this quantity can be prepared from the serial relationship that exists between adjacent age groups, namely: U i , l = r C A 2 3 ) U i + 1 , 1 The geometric mean value of r is calculated as follows: 1 1 U G.M.(r) = a n t i l o g l s ln , i , l x n 1-5 < U . + 1 , 1 = antilog 1 In ,0.489 0.369 0.09K = , 7 l0.369 0.279 0.069 ; ' and, from equation (A23) , U i o i = U12, 1 = 0.069 = 0.052 I J > 1 - 1.323 Substituting known values in equation (A22), where Z^ = 0.38, and (I U l f l + U 1 3 f l l = 1.839, 162 S = 0-38 (1.839) = 2.210/q , o r , S e = 2.210/.002 = 1105 f i sh S i m i l a r l y , Rg can be expressed as a function of .q: Re = VLL Z' = 2.058 (0.30) = 2.382 q _ 7 - . Q n _ *--30\ P ( 1 - e " 2 ' ) 4 ( 1 - e - ) and R /S =1.078 e e An estimate of parameter a (equation A21) can be prepared from the general predict ion re la t ionsh ip : a( l - Sj/2.210) R,, , = Uo,j+1 V = 1.078 S, e q , J 1 n -7' J q (1 - e Z ) where, Z ' / ( l - e " Z ' ) = 0.30/(1 - e " - 3 0 ) = 1.157 , a ( l - Sj/2.210) and, U o, j+l = 0,931 S, e q (A24) q J The var iables S. can occur in two forms depending on the re la t ionship between t and t . For t < t^ : c m c — f m 1 2 1 2 II Z S. = E (,N,). . = E u i , j j , (A25) J t 1 , J t q Z. m m e j _ 1 and, for t > t c m - 1 (, ^ —fr- + } *Li ^ — ) (A26) 163 Using the data from Table A II for the f irst year of fishing, and equation (-A25) , S, = (0.646 + 0.489 + . . . + 0.069) (0.38) 1 = 2.000 , . Ce"38 - 1) q q and Uo,2= 2.230/q q Substituting these values into equation (7\24), rearranging and solving, a = 1.898 The general, predictive equation for the index of recruitment is there-fore given by: 1.898(1 - S ./2.21) U o J + ] = 0.931 q S. e q , (A27) where, S. can take either of the two forms indicated by equations (J\25\ 3 and (A26). The variable q in the above expression cancels..when S. is given as a sum of indicies divided by q. (iv) A refined estimate of the age of f irst capture From the estimates of G., -M and q i t is possible to construct a constant-recruit model to determine the optimal age of f irst capture. The procedure is contained in the subroutine AFC0PT, which will be de-scribed shortly. For a given starting value of pre-recruits entering age-group '1' , the model generates data on the total yield per unit effort (Vj) and total yield per pre-recruit (-Y/R)j for t in the inter-val (2, 12), and for F values in the range (-0.10, 1,50), From the total amount of printed information i t is easy to select the best age [t .) that will maximize VT or (Y/R),, A change in the number of pre opt i i recruits only introduces proportional changes in Vj or CY/Rl T; the optimal age of capture remains unaffected and for this reason the cal-culations are only carried out once. The precaution is taken to re-place the natural mortality rate MI by V (.= 0.30) for all age groups less than t . During the gaming process these same age groups are continuously sampled by research vessels and effectively produce an added mortality rate of 0.02. In Table A II the index of pre-recruits in age-group 'V during 2 the second year of fishing (U )^ ' i s 1,555. In the absence of fishing mortality the starting biomass for i in the interval {2, 12). is read i ly calculated from the relationships: Nn = U,/q B l = N l W l e - 1 G n-Z and , (B Q) 2 = (B Q) 1 e = B -tG,-Z') (1 - e <Bo}3 = <Bo>2 6 <Bo>12 = ( V n 6 - Z 'The subscript j is temporarily removed for clarity. 165 For a given value of t and F, the appropriate starting biomass, (B ). , is selected from the above listing and the following yield 0 zc calculations are performed: Sequence Comments and Calculations  1. Start Select t = 2, F = 0.1, and (B ) 2 from l is t 2. Calculate R = uyq = 1.555/.002 = 900 units 3. Set AZ2 = G 2 - (F + M) 4. Calculate individual yields for fishable age-groups, t (2, 12), using (B ) 2 as the equilibrium starting value: Y 2 = F (B Q) 2 (e ^-1)/AZ2 AZ9 AZ, Y 3 = F (BQ) 2 e *(e J - D/AZ^ : = : i i X AZi A z  Y12 = f{Bo]2 q 2 ( e 1 2 ' 1 ) / A Z 1 2 12 5. Calculate total yield (YT). F and YT/R from Y T = I Y. 1 V t 1 6. Print results from sequence (5.). c 7. Return to Start, increment F by 0.1 and repeat. When F = 1.5, return to Start, reset F = 0.1, increment t by one and sel-ect appropriate starting value of (B ) + . 8. Stop when calculations for t = 12 are completed. A synthesis of the Y^/R output from the program AFC0PT appears in Table A III. Only the data for age groups in the region of t ^ is given. Age group '6* appears to offer the greatest potential YT/R , but its TABLE AIII. Total y i e l d per re c r u i t (Yj/R) associated with age of f i r s t capture ( t c ) and fi s h i n g mortality rate (F). A Ricker (1968) constant recruitment model. Maxima are underlined. Total Yield Per Recruit (gm/R) Age of F i r s t Capture ( t c ) F 4 5 6 7 0.1 263 240 210 175 0.2 380 361 326 281 0.3 431 424 393 346 0.4 453 456 431 387 0.5 462 474 455 414 0.6 463 484 470 432 0.7 462 489 480 444 0.8 460 492 486 453 0.9 456 493 491 460 1.0 453 493 494 465 1.1 450 493 496 469 1.2 447 493 498 472 1.3 444 493 499 474 1.4 441 492 500 477 1.5 438 491 501 478 167 maximum value, as well as MSY, will not occur until F approaches F°°, This feature is a characteristic of constant recruitment models having a fixed natural mortality rate. To avoid this difficulty and introduce a practical compromise fishing should begin one year earlier, and ac-cordingly t j. is set at age five. The maximum Y^/R in this age group demonstrates a broad plateau at 493 gm/R. It is sufficiently close to the relative value of MSY (.501), and the mortality rate is also held in the realistic range of F = 0.9 to 1.3. (v) Estimation of optimal stock size and effort at maximum recruitment In Appendix B, a method is developed to calculate the stock size, recruitment and fishing mortality to yield MSY for the particular fish-ery described here. The exact results indicate that at MSY the ex-pected yield is 508 Kg, recruitment is 1532 f ish, and the optimal effort is 371 boats. Because of the high value of a in the recruit function the yield at MSY is only about one percent higher than that obtained at maximum recruitment, i .e . , R m = v = 1538 and Y T = 503 Kg., but the max i effort needed at R is 22 percent lower (303 boats). In this case max it is not economically justified to stabilize the fishery at MSY. For this reason R is chosen as the target value of recruitment in the max current problem. If the age of f irst capture is theoretically set equal to the age of f irst maturity (4) then equation (A25) is descriptive of the repro-ducing stock size S.. Let U . . . be the index of the average, mature j mat,j stock size. Then i t follows that the predictive recruitment equation 168 (A27) can be rewritten as: 1-898 (1 - ^mat.j ^ Z j ) , ^ 2 Q ) U o , j + l = ° ' 9 3 1 Umat,j - ^ l - e - ~" C e ^ - 1 1 e j - 1 Maximum recruitment occurs where the slope of the stock-recruit curve is zero. Therefore, differentiating equation (A28) with respect to (-IL. . 2 mat,j Z./e J - 1), setting the result equal to zero and simplifying: Z j 2 ' 2 1 ° (e^ - 1) C 2 ' 2 1 0 " ' J Z j . 3 e " 1 In the above equation the term in the square brackets must equal zero, and solving for li . .: 3 mat,j U m a t > j = 1-164 e ^ J . WZ91 3 i. Substituting this value into equation (A28), the (Z./e J - 1) terms can-eel and i t follows that: U O J + 1 - 2 . 6 6 1 , and, the maximum number of recruits is then equal to: R = U o,j+l Z' = 2,661 (0.3) = 1539. fish q ( 1 _ e-VY .002 0 > 3 ^ ^ If the fishery is removing the surplus stock at a rate sufficient to produce R m a x in successive years, then the reproducing stock is at equil-ibrium and at its optimal value (S^).- The derivative (dRm a x /d SQp t) 169 is therefore equal to zero. Substituting the values for R Q /S g , and S g (1105 fish) in equation (A21), i t follows after differentiating with re-spect to S Q p t that; S „. = Se = 1105 = 583 fish o p t T 17898 With the theoretical age of f irst capture s t i l l set at four the terminal stock size S ^ must be equal to the sum of the residual, fishable stock, i .e . , But at maximum recruitment , < N + ) 4 = ( N 0 > 4 6 ^ = Rmax e ^ = 4 6 4 ^ -I. -21. (N + ) 5 = (N + ) 4 e J = 464 e J ! = : -91. (N + ) 1 2 = 464 e J , and summing to obtain S opt ' -Z, -2Z, -91. = AF.A (a opt S nnt = 464 (e J + e J +... + e j ) = 583 The value of Zj that satisfies the above equation could be determined by Newton's method of solving transcendental equations (Smail, 1949). How-ever, in the present case, trial and error methods are equally quick, . and the solution is Z. = 0.583 (:f = 152 boats). Substituting this value of Z. in equation (A29), the catch per unit effort index of the mature J population that will produce R m a x in the following year is obtained directly: IJ . . = 1.164 ( e ' 5 8 3 - 1) = 1.580 m a t " 3 0.583 Year Three The sum of the catch per unit effort figures for the mature stock (U j.) at the end of fishing year one is 2.434 and in year two it is 2.203. (Table A IV). Both figures exceed the optimal value of 1.164 needed to produce R . The age structure of the population in year r max -two, as revealed by the IK figures, indicates both the source of the high U . value and the potential means of correcting the situation. 3 mat In its present state the population is unbalanced. There are too many adult fish aged five and over, and not enough in the age-group range of two to four due to the previous fishing strategy. In addition, the increased index of recruitment in year two (UQ ^ = 2.230). will not be-gin to benefit the reproducing population until fishing year six. The effort in the fishery must be curtailed until this event occurs. The effort to be used in year three is based on the level needed to ensure maximum recruitment in year four, provided all mature age groups are theoretically fished. If this same effort is actually ap-plied to age groups five and over in the simulation model, three pur-poses are accomplished. First, age group '4' is allowed an additional year of recovery; second, the risk of having to remove excess boats from the fishery at some future time is reduced; and third, the over abundant fish in age groups '5+' are removed. 171 The number of fish entering the adult population at the beginning of year three is: 12 11 1 1 li 7 E ( N j . , = z (N ). 9 = E u i ,2 2^ 1198 fish 4 0 1 , J 3 ^ 3 q Z 9 (e 2 - 1) For maximum recruitment the desired number of fish at the end of year three should not exceed S „. therefore it follows that: opt 12 12 -Z„ -Z , E (NJ . - = E (NJ . e J = 1198 e J = 583. 4 ' 4 u i 5 o and solving: and Z 3 = - ln (583/1198) = 0.72 f 3 = (0.72 - 0.28)/.002 = 219 boats. Substituting the values of V. 9 (Table A II) for i in the interval (1, 12) in subroutine PREDY, the predicted yield is 865 kg for t = 5. From this point on the value of t will not change and the predictive recruitment index in year j+1 can be rewritten and abbreviated as fol-lows : From equation (A26), S • 1 2 Z I i = 1 ( u 4 j , z ' + E u i j J ) q q t *>3 7 r 7 7 5 1 ' J Z \ e 1 e J - 1 12 ? z = 1 ,0.8575 U, . + E u. . y l j , q 1 4 ' J 5 ' J Z. ; e J - 1 The numerical value of q is not needed in this equation. It cancels when S./q is substituted into £A27) to calculate U . , , . j o,j+l Similarly, the predictive formulae for the total mortality and effort in year j+l to produce R ^ in j+2 may be abbreviated to: 11 7 1. = - In r(408 - 317.62 IL .)/(428.74 U, . + _ J _ S U. , Zj )-, e J - 1 and, f \ + 1 = ( Z j + 1 - 0.28)/0.002 The actual and predicted total yields and indices of recruitment for fishing years three to fourteen are summarized in Table A IV. The formulae given above are used in each case and the detailed calculations are not reported. Years Four to Seven The effort required in years four to seven to produce R m a v in the following year are: f 4 = 194 boats f 5 = 140 -Each value is less than the committed level of 219 boats, and the adult population from age group '5' onward is purposely overfished during the interval. At the same time the recruitment index Ui . = 4 to 7) obtains o,j a maximum value of 2.657; (the difference is negligible and is due to rounding error in the computer). The reason why the expected recruit-ment is not lower can be attributed to one major cause: at the maximum effort difference of 219 less 140 boats the reproducing stock is driven 173 TABLE A IV. Predicted to ta l y i e l d and recruitment index. Legend: Y^ = to ta l y i e l d (Kg); = ca t ch /e f f o r t , recrui tment; U . = c a t c h / e f f o r t , mature stock. Age u F i r s t T o t a l / . mat Capture Mor ta l i t y (optimum t c Zj 1,580 ) 1 361 - 2.058 - 50 2 0.100 2. .434 2 328 - 2.230 - 50 2 0.100 2. .203 3 864 865 2.375 2. .361 219 5 0.718 1, .805 4 600 605 2.634 2, .637 219 5 0.718 1. .509 5 468 471 2.656 2, .660 219 5 0.718 1. .413 6 410 412 2.648 2. .653 219 5 0.718 1, .428 7 397 397 2.652 2, .656 219 5 0.718 1, .489 8 496 499 2.657 2. .660 285 - 5 0.850 1. ,559 9 511 514 2.657 2. .661 302 5 0.884 1. .572 10 504 506 2.657 2. .661 302 5 0.884 1. .571 11 505 508 2.657 2, .661 304 5 0.888 1. .570 12 503 506 2.657 2. .661 304 5 0.888 1 .571 13 503 505 2.657 2 .661 -304 5 0.888 1 .572 14 503 505 2.657 2 .661 304 5 0.888 1 .572 Fishing Values Values Ef fo r t Year Actual Predicted Actual Predicted f y y U U (Boats) J 'T T U R U R 174 to the left of optimum on the stock-recruit curve, but i t is s t i l l within the region of S t -Years Eight to Twelve The strategy of advancing the age of f irst capture to five initiates, in year four, a continuing recovery of the first reproducing age group. By fishing year eight the increase in the terminal number of four-year olds is sufficient to allow the second, major increase in effort to 285 boats. In subsequent years the effort is incremented to insure maximum recruitment and the final effort of 304 boats is introduced in year eleven. A second estimate of the maximum effort is calculable at the end of fishing-year nine for at that time maximum recruitment has stab-il ized all age groups from '0' to '4' . From Table A II, the equilibrium catch per unit effort of four year olds is 0.800 (U^ g). The terminal number of fish at this age is therefore: (N+), Q = 0.800 V = 0.800 (0.3) = 343 fish q J- 1 i -002 0.3 , ^ e - 1 e - 1 At equilibrium, the optimal reproducing stock is: 12 z 5 S o p t = 3 4 3 + l ( N + } i , 9 = 5 8 3 -Z . -2Z , -8Z . 343 + 343(,e 0 p t + e o p t + . . . + e o p t ) = 583, and solving by trial and error, 175 1. = 0.888, and f . = 304 boats, opt opt The data in Table A II indicates that for practical purposes the introduction of the above effort stabilizes the fishery at the economic level of MSY CMSY } by the end of year twelve. Conclusions: The objectives and assumptions made available to the investigator are detailed at the beginning of this appendix, part 2, Of these assump-tions the one concerning perfect information from the fishery is prob-ably the least tenable, but i t is included for a specific reason: under ideal conditions the minimum, standard time needed to analyze the modal can then be determined. The information is useful as a reference point in determining the conditions that prevent an equivalent analysis in more realistic models. At the completion of this exercise the following conclusions may be drawn: (1) Given the initial assumptions, estimates of all the unknown para-meters can be prepared within two fishing years. Armed with this in-formation the necessary steps to reach MSYg in a controlled manner can then be undertaken. In this sense, what appeared ini t ia l ly to be a problem in fishery population dynamics, is recast as an issue involv-ing parameter estimation. To prepare the estimates two sources of in-formation are necessary: the yield in weight and catch from both the fishable and pre-recruit age groups. Whereas the former data is help-ful in assessing the present state of the stock, yield prediction and 176 parameter estimates for the recruit function cannot be prepared unless pre-recruit information is available. Ultimately, the parameters of the von Bertalanfly (.1938) growth equation must be obtained from research sample analysis. (.2) If the age of f irst capture is set equal to six, the critical size, the fishery must remove all fish of this age and over to maintain maximum recruitment. Provided there is no pre-recruit, sampling mortality the real value of MSY will equal the biomass of fish reaching this age, or 584 kg. With sampling mortality of pre-recruits the potential MSY is reduced to 518 kg. Both conditions require the application of an in«v finite fishing mortality rate. For practical reasons the age of f irst capture is therefore reduced to five, and under this condition, the maximum sustainable yield (MSY ) is limited to 503 kg. The fishery . can be stabilized at this production level in twelve years with a max-imum committment of 304 boats (-F = 0.89). (3) The parameters of the growth equation and instantaneous growth rates can be estimated in each fishing year. The necessary information is limited to age-group values of W. and a separate estimate of the length-weight exponent. At the stage of development shown in this ex-ercise, however, the accuracy of the estimation technique is restricted to low fishing mortality rates. The statistic -W-. also shows strong potential as a 'state' indicator. The values of W^  associated with each age group differ substantially, but they are also individually response to changes in the total mortality rate. Potentially, this rate could be estimated from W. statistics i f the analysis is limited to the fully fishable age groups. In addition, 177 since represents an average weight by numbers, i t s value i s independ-ent, of year-class strength and could therefore be used as a diagnotic aid under non-equilibrium conditions. (4) The analytical method provides for multiple estimates of q to be obtained during the development of the fishery. A l l that i s required i s that the fis h i n g e f f o r t be held constant for two consecutive years. By this procedure the parameter may be updated at the discretion of the i n -vestigator. In the present analysis the objective of s t a b i l i z i n g the fishery in the shortest time only allowed a single estimate of q to be prepared. 178 APPENDIX B Derivation of optimal stock size, recruitment and fishing mortality at MSY in a multi age-group, single species fishery. To calculate the absolute value of MSY in a multi age-group, single species fishery, two questions must f irst be answered. What is the best, average size at which to capture the f ish, and how many of them should be taken in order to generate the maximum yield on a sustained basis? The critical size (Ricker, 1968) is the solution to the f irst question; the average size at which the gain and loss in biomass due to growth and mortality, respectively, are in balance. A solution to the second problem also involves a balance, but the conflict is between the number of above critically-sized fish that are harvested and those that must be left behind to maintain reproduction. At MSY, this balance is achieved at the optimal, reproductive stock size (-S^) that will support the highest, equilibrium yield. The fishing mortality rate at MSY is in turn a function of S j., the stock-regeneration function and the time during the year when reproduction takes place. Given the fol-lowing assumptions about the biology of the species and the conduct of the fishery, the objective is to determine S optimal recruitment (RQp t) and fishing mortality rate .^)., and the value of MSY. 1. Natural mortality (M) is constant for all age groups and F affects all ages equally. The fish Cor fishermen) are distributed at random at all times, and growth and mortality are continuous and con-current. Growth and natural mortality are density independent. 2. Reproduction takes place at the end of the year (as for the 179 cod stock in Appendix A(2) and according to the Ricker 0 9 6 8 ) . re la-tionship a(i -sys } R j + 1 = c S j . e (,B1) and R a(l - S. /S ) d j+1 ,= c e J e 0 - a S . / S J (,B2) d S. J e where Rj+-| = the number of fish entering age-group '0' at the beginning of year j+1, c = a constant, R e / S E J where Rg and S g are, respectively, the equilibrium progeny and stock size in the absence of f i sh -ing S. = the reproducing stock size at the end of year j . Al l adults are assumed to be equally fecund. 3. The c r i t i ca l s i ze , or age in this case, has been determined, '•< but f ishing actually begins at some ear l ier age (-.t ) to maintain a rea l is -t i c effort rate. The age of exit from the fishery is t ' , and the units of age and time are the same, i . e . , years. Selection of the trawl gear is "knife-edged". The fishery is a closed system and the average envir-onmental effect over time is constant. Under these conditions the annual, equilibrium y ie ld (Y ) may be w represented by the general equation Y = F A' Hit) w It) dt. (,B3) c -Z(t - t ) But N (t) = R' e c , t 1 t < t" , or 180 -Mt -Z(t - t ) N(t) = R e e where R' = the number of recruits entering the fishery at time t c , and it is also equal to the equilibrium progeny, R, times the accumulated mortality rate up to time t c-Substituting these relationships into (C3): -Mt , -Z(t - t ) Yw = F e C R / J e c w (t) dt or -Mt Yw = e c R Q (B4) where Q = F times the integral. At MSY, the rate of change of yield with respect to fishing mortality, d Y / d F , must equal zero, and differentiating (B4), w ^w = e c , R d Q + Q d R . d S x = 0 (-B5) dF K dF dS dF ' -Mt Replacing R by (Bl), dR/dS by (-B2) and canceling the term e c , equation (B5) reduces to S dQ_ + Q (1 - a S dS = 0 dF SQ dF e and dividing by SQ, dQ_ + J - a \ dS = 0 (B6) Q dF lS S e j dF The above equation can be used to test whether the choice of S and F are such that the yield will be equal to MSY. If not, (B6) will provide a residual (y) that will differ from zero. 181 The differential dS/dF can be expressed in three different ways depending on the relationship between the age of f irst capture it \ and the age of f irst maturity C.t ). Each case is developed separately, as fol1ows: t„ > tm  c m The reproducing stock size at the end of the year is equal to the terminal number, (N+).> of adult fish in each age group i . Expressed mathematically it follows that: S = CZ (N ) . * I (N ). - M ( T ™ + " -M -a -"tVtj. - i) S = R e (1 + e M +e m + . . . + e c m -Mt 7 ? 7 -Z ( t ' - t ) + R e c {e~L + e d L + . . . + e c ) (B7) -M The f irst summation of terms involving e may be collapsed into a simpl-er form as follows: Let x = e and . T ] = 1 + x + x^  + . . . + x m then and (t -t ) - xT, = 1 - x c m (t -t ) -M(t -t ) T ] = 1 - x c m = 1 - e c m 1 - x ] _ E-M Similarly, the second summation may be abbreviated to 182 7 -Z(.f - t ). -ZCt'-t ) T 2 = e"Z Q - e ^L) = 1 - e c 1 - e"Z e Z - 1 If and T 2 are substituted into (B7), and R is removed as a common term, then (B7) gives -M(t x l ) -Mt S = R (T] e m + l + T 2 e c) (B8) or, replacing R by equation (Bl), a(l - S /SJ -MUm + 1.) -Mt S = c S e (T-, e m + 1 + T ? e c ) . (B9) Cancelling S, taking natural logarithms, rearranging and solving for S, S = je In c + S + In e m 1 + T 2 e c) a a Only the last term in the above equation is a function of F. Therefore, differentiating with respect to F, the f irst and second terms equal zero, and with the appropriate substitutions for T-j and T 2 , i t can be shown that H q s -"IVV'I d_S = _e re^  x dF a L ( e Z _ 1 } 2 -ZCt'-t ) 7 7 -z(.t'-t ) (t '-t c) e c ( e l - 1 ) - e Z (1 - e c ) "M^ c - t J -M(-t - t -1 ) - Z C t ' - t J ] ( B 9 - 1 } ( 1 - e c m ) + e c m ( 1 - e c ) 1 - e"M e Z - 1 133 2- K = *m c m In this case, i t can be similarly shown that t ' - l -Mt -Z(t ' - t ) . S = z (N+). = R e m (-1 - e m ) m e - 1 and following the above procedures, dS = ^ ( f - tm J j 3 e Z(t ' - t m ) _ , 1 - e ( B g > 2) 3. t <t c m t'-l " ^ V ^ " ^ V ^ + ] ) • Z ( t ' - t m ) S = z (NJ , = R e m c e m c ( 1 - e m ) \ 1 - e " Z and f = ! f ( ^ T 7 T - - e - z 1 7 T - ( t » - t ^ 1 ) ) <B,3, Notice that equations (B9.1, .2 and .3) all include the term S g / a , and therefore when substituted into (B6), the latter formula with the residual y may be rewritten as Q f + ( | f - i ) i j s - y , (BIO) where dS'/dF = only the bracketed portions of equations (B9.1, .2 and .3)'. 184 The expression for Q as well as the d i f f e r e n t i a l dQ/dF may be prob-lematical. Before d i f f e r e n t i a t i o n can take place, Q must f i r s t be integrated, and the ease with which this can be done depends on the weight function w(t). A general solution to the problem i s therefore not presented here, but the method i s i l l u s t r a t e d below using as an example the cubic, von Bertalanfly (1938) growth equation. Example In appendix A (2), the exact values of certain variables and param-eters for a hypothetical cod population are as follows: t = 4 a = 1.895 m t = 5 S = 1106 V = 13 ' c = R e/S e =1191/1106 = 1.0769 t Q = 0.07 S e/a = 583 K = 0.14 M = 0.28 b = 3.0 V = M + 0.02 = 0.03 = 11.41 Kg ' q = 0.002 where Z' = the natural mortality rate plus a small component due to research sampling of age groups less than t c - The value of Z' must therefore replace M in a l l formulae developed prior to the example. The objective i s to determine S ^ , RopJ. and the three v a r i -ables that govern the position of MSY. To begin the analysis a starting value of R must f i r s t be determined. A reasonable choice i s Rm,„, the maximum recruitment at which the slope max of the stock-regeneration function (B2) i s zero. At R the appropriate max stock s i z e , and the f i r s t estimate of S, i s 185 S, = S /a = 583. 1 e and substituting this value of S in equation (Bl) R, = R =1538 1 max At equilibrium, since t £ > t , the sum of the reproducing age groups may be determined by equation (68).. The component terms in the formula are T, =1 - e " Z ' ( t c " t m ) - l - e " 0 - 3 ^ i i -0 .3 1 - e 1 - e y 2 = e" Z (1 - e " Z ( t ' " t c ) ) = e" Z (1 - e ' 8 2 ) = 1 - e" 9 Z 1 - e"Z 1 - e" Z 1 - e" Z and since t = t +1, c m -z'(yi) -z-tc 5Z, e = e = e It follows that equation (B8) may be reduced to S = R e" 5 Z" (1 - e" 9 Z) , (Bll) 1 - e"Z -5Z' and substituting the values of , R, and e , and rearranging 1 - e~ 9 Z = 1.701 1 - e " Z Solving this equation by trial and error..for Z, 186 Z ] = 0.886 and therefore^ = 0.886 - 0,28 = 0.606. Substituting equation (Bl) for in dBll), taking natural logarithms and solving for S-j h - ^  1 n c + S e " % e" 5 Z' + !& [InO - e"9Z) - ln(l - e"Z)] a a a and d S, . S • 0 , d F l a { e 9 Z - 1 e Z - 1 Therefore, replacing Z by 0.886 in the above expression d S l = ^e ( - 0.6985) or d Sl.= -0.6985 d F ] a d F ] Since the value of Z is reasonably large and t' - t = 8, the product of these two terms, when used as a negative exponent of e , should yield a small number. On this basis, the shortened form of the integral Q can be used, i .e . , t . -Z(t - t ) -K(t - t ) . Q = F WOT fx e c (1 - e 0 T dt L c or 1 - K ( V t o ) " ^ W 3 K ^ V t o ) Q = F IT -  3 e + 3 e -  e \ K L Z + K Z + ZK Z + 3K ' Similarly, -K(t -t ) - Z k ( t - t ) dQ = M - 3:e c 0 (M + K) + 3 e c 0 (M + 2K) d F Z 2 tZ + K)2 ( Z + 2K)2 -3K(t„-tJ (M +3K (Z f e!<)2 (B13) - e c 0 (M +3K) 187 Substituting the values of M, and in equations (B12) and (B13), it follows that Q = 0.1290 and dQ = 0.0154 d F l All the values needed to determine the f irst residual in equation (BIO) have now been calculated, i .e . , (BIO) Y, = dQ + ,h - K 1 Q dF1 K a > dF1 and substituting, Y, = 0.0154 + (583 - 1) (-0.6985) = 0.1194 1 0.1290 583 Equation (BIO) can also be used to solve for the value of S that will in fact make the residual zero. On this basis, and using the prev-iously derived values, 0 = 0.0154 + (583 - 1) (-0.6985) 0.1290 S and solving, the second estimate of S is S 2 = 498. The entire procedure can now be repeated by using S 2 to prepare a second estimate of R2 from (Bl), solving for Z 2 and by trial and error, and calculating a second residual y 2- If i t is not zero, a third estimate, S^* is calculated and the procedure is repeated. The results of five iterations are given below. 188 Estimate No. S R 1 583 1538 2 498 1520 3 556 1536 4 518 1528 5 544 1534 6 534 1532 F Z Y .606 .886 +0.1194 .863 1.143 -0.0569 .678 .958 +0.0483 .793 1.073 -0.0276 .713 .993 +0.0209 .742 1.022 +0.0014 The absolute values of the residuals from estimates 4 and 5 are almost equal and are sufficiently small to allow an estimate of S ^ to be prepared graphically. By plotting S on the abscissa, y on the ordinate, and using the co-ordinates (S^, Y )^ and (S^, Yg), a line join-ing these points intersects the S-axis at 534 (-= S Q p t ) . The remaining values of (R, F and Z) .^ are then readily determined. The yield may now be calculated using the exact expansion of the Beverton and Holt (1957) model at equilibrium. To complete the com-parison the yield at maximum surplus production (msp) is also included. R S . F Z f (boats) Yield (.Kg) Ropt = 1532 534 0. 742 1. 022 371 508.2 (MSY) R = max 1538 583 0. 606 0. 886 304 502.7 R . = 1456 408 1. 300 1. 580 650 485.0 msp For the particular case defined in Appendix A(2), the effort (boats) needed at R o p t is 22 percent higher than at R m a x, where as the gain in yield is only about one percent. Economically, the • yield obtained at R is therefore a more justifiable goal. 190 APPENDIX C Seasonal Pulse Fishing Results The effect of seasonally applied fishing mortality on the value of equilibrium yield is examined by means of a hypothetical, constant recruitment model. Annual, instantaneous growth rates (G^ for successive age groups (i) are calculated from equation (1.78) for-t = 0, b = 3.0, i .e . , G = 3 *n <e - e ) ( e l K - l ) using K values of 0.2, 0.4 and 0.6. Annual, instantaneous natural mortality rates of 0.1, 0.2, 0.3 and 0.4 are applied separately for each K value and are constant for each age group. Seasonal growth (g )^ and mortality (m) rates are arrived at by appropriate division for each growth-mortality combination . reported in Figure Cl(a). The processes are independent of each other, and their annual rates may be condensed into a single quarter, or occur consecu-tively for half, three-quarters, or a full year. The last situation represents continuous growth and natural mortality. For a given case the growth-mortality pattern is constant throughout the l i fe span of the cohort. Initially, the critical age is determined to the nearest quarter year by means of the recursive equation B. • = B. . . e ( g 1 - m ) J where j = the quarter index and B. , . , and B. . are the initial and final biomass respectively. Calculations begin by arbitrarily setting the f irst year-old stock (NQ)-j at 1000 fish and = 100. Thus, the biomass of one-year olds at the beginning of the f irst quarter is calculated from 191 ( a ) M S E CASE m * * * 2 9 i * * 12 „ . m * * * * m1 * * * * 1 3 9i * * * m * * m * * * 5 Qi * * * m * * * * m * * * * 1 6 9i * * * * 7 g- * * * * m * * 1 7 91 * * * * ID * * * 8 q,- * * * * m * 9 g. * * * * m * * * * 9-j * m * 10 g. * * 20 g i m * * m (b) INTENSIVE MODERATE EXTENSIVE FS 1 * FS 5 * * 8 * * * 2 * 5 * * g * * * 3 * 7 * * 4 * CONTINUOUS FS 10 * * * * Figure Cl . (a) Twenty cases of seasonally distributed growth (g-j) and natural mortality (m) patterns. Each year in the l i fe of an age group (i) is divided into quarters, (b) Ten seasonally applied fishing strategies (FS). 192 B l , 0 = ( N o } l " J 1 " 6 " * ) 3 = 1 x 1 0 5 d-e"K)3 . Once the critical age is determined, the age of f i rst capture (AFC) is set equal to the integer value of the critical age without regard to the fraction. The f i rst seasonal fishing strategy (FS 1) shown in Figure Cl(b) is then applied to AFC and successive ages for either 100 years or until the annual growth rate (G )^ falls below 0.0002. Quarterly yields are summed to give the equilibrium yield (Y ) by means of the equation i = 100 or G.<0.0002 . (g r f -m) . (e 1 J - 1.0), Y-r = E E f ,B. i=AFC j=l T : ' jD i , j - l (g-f-mjj where f. = the seasonal value of the annual instantaneous fishing mortality rate, F, that is constant for all age groups. For each fishing strategy Y^  is calculated separately for odd numbered values of F from 0.1 to 0.5 inclusive. Intensive fishing describes the situation in which the annual rate is condensed into a single quarter, moderate and extensive i f F is distributed over two or three consecutive quarters, respectively. Strategy FS 10 represents continuous fishing. The amount of data generated by this program is voluminous, but they need not be reported in full to grasp the essentials of seasonal pulse fishing. In Table CI several general principles are illustrated which apply to all growth-mortality combinations. The data refer to the relative equi-librium yield, yield rank and percentage gain (+) in yield relative to continuous fishing. Strategy FS 10 is chosen as the basis of comparison Table CI Relative equilibrium yield, rank values and yield gain (+) relative to continuous fishing (FS 10) for growth-- mortality Case 1. K =0.4; M = 0.3; F = 0.1 to 1.5; cri t ical age = 3.5 y r , ; age of f i rs t capture = 3.0 yr. Range = percentage sum of absolute maximum gain and loss relative to FS 10. CASE 1 G | * | * | | | M | | 1*1*1 FISHING STRATEGY (FS) .1 Relative Equilibrium Yield (Yj) .3 .5 .7 .9 1.1 1.3 1.5 .1 .3 Yield F .5 .7 Rank .9 1.1 1.3 1.5 Percentage Gain .1 .3 .5 .7 (+) Relative to FS 10 .9 1.1 1.3 1.5 1 9.53 16.76 19.12 20.01 20.35 20.46 20.47 20.42 8 9 9 10 10 10 10 10 -1.9 -4.8 -6.7 -8.2 -9.2 -9.9 -10.5 -10.9 2 10.44 18.85 21.91 23.25 23.90 24.22 24.37 24.44 1 1 3 3 3 3 3 5 7.5 7.1 6.9 6.7 6.6 6.6 6.5 6.6 3 10.18 18.70 22.05 23.69 24.60 25.16 25.54 25.81 3 3 1 1 1 1 1 1 4.8 6.3 7.5 8.7 9.8 10.7 11.7 12.6 4 8.76 16.09 18.98 20.39 21.17 21 .66 21.98 22.22 10 10 10 9 9 9 9 9 -9.8 -8.6 -7.4 -6.4 -5.5 -4.7 -3.9 -3.1 5 9.97 17.75 20.40 21.44 21.86 22.00 22.00 21.94 5 6 8 8 8 8 8 9 2.6 0.8 -0.5 -1.6 -2.5 -3.2 -3.8 -4.3 6 10.31 18.77 21.97 23.45 24.20 24.62 24.85 24.98 2 2 2 2 2 2 2 2 6.1 6.7 7.2 7.6 8.0 8.3 8.6 9.0 7 9.48 17.46 20.64 22.22 23.14 23.72 24.14 24.44 9 8 6 5 5 5 5 4 -2.4 -0.8 0.7 2.0 3.2 4.4 5.5 6.6 8 10.03 18.04 20.88 22.08 22.60 22.82 22.88 22.86 4 4 5 6 6 6 6 7 3.2 2.5 1.9 1.3 0.8 0.4 0.0 -0.3 9 9.80 17.95 21.11 22.63 23.46 23.95 24.27 24.48 6 5 4 4 4 4 4 3 0.8 2.0 3.0 3.8 4.6 5.4 6.1 6.8 10 9.71 17.60 20.50 21.79 22.41 22.72 22.87 22.93 7 7 7 7 7 7 7 6 RANGE 17.3 15.7 14.9 16.9 19.0 20.6 22.2 23.5 194 since i t is the usual method of reporting yield. The principles that emerge are as follows: 1. As F increases from 0.1 to 1.5 the yield for each fishing strategy approaches an asymptotic value. The rate of increase is not the same for each strategy, however, and leads to a changing pattern of relative rank yields. 2. To be able to benefit from seasonal pulse fishing i t must be applied on a species basis for which , M, K and the growth-mortality pattern are known. From item 1 above the annual fishing mortality rate and the degree to which i t can be seasonally concentrated must also be estimated. Once these factors are known, a suitable strategy can be selected to improve or maximize equilibrium yield. 3. Alternatively, only a portion of the available fishing fleet can be used to obtain a regulated yield by an appropriate choice of fishing ° strategy. 4. The opportunity to conserve fishing effort (at constant catchability) and obtain an annual yield equivalent to that of a greater effort can be demonstrated. For instance, i f current practice is to fish inten-sively (FS 4) at F = 1.5, the same yield can be obtained with less than half the effort (F = 0.7) by means of the extensive strategy FS 7. Similarly, a change from continuous fishing to a moderate strategy (FS 6), coupled with a reduction in F from 1.5 to 0.7, can lead to a small (2.3 per cent ) but positive increase in yield. The ten fishing strategies are divided into four modes: intensive, moderate, extensive and continuous. Within these categories the strategy 195 options are reduced, respectively, from four (FS 1-4), to three (FS 5-7), to two (FS 8-9) and one for continuous fishing. From the rank values in Table CI it is clear that for each fishing mortality rate there is a best and worst choice of fishing strategy within the f i rst three modes as well as on an overall basis. As F increases, the rank of a particular strategy can adopt only one of three courses: i t can improve, deteriorate, or remain unchanged. At low values of F the rate of change of improvement or deteriora-tion is rapid but tends to stabilize as F assumes higher values. At F = 1.5 the difference between strategies leading to yield improvement and loss is therefore magnified, and the absolute value of the largest difference, or range, also takes on an extreme value. If data on only the best fishing strategies to be used is extracted from the percentage gain column of F = 1.5, then Table CI can be condensed into a single line. This feature forms the basis of Table CII, but additional information on all K and M values and the 20 growth-mortality cases can now be included. The usefulness of Table CII can be judged in relation to two important management questions: 1. Are there general guidelines that can be used to indicate whether seasonal pulse fishing will benefit the yield from a fishery? 2. If the gain is sufficiently attractive, how and when should fishing take place? To answer the f irst question, the following generalizations emerge from a study of Table CII. They are intended to act as guidelines only; a few exceptions exist. Ultimately, there is no substitute for a detailed analysis of a target species. 196 TABLE CII Relat ive equi l ibr ium y i e l d , seasonal pulse f i sh ing resu l ts fo r F = 1.5. The as te r i sks (*) ind icate the quarter during which a f rac t ion of the annual instantaneous growth (G) and natural mor ta l i ty (M) rates apply. Strategy refers to the f i s h i n g s t ra teg ies of Figure CI(b) and the resu l ts are reported as +_ gain in y i e l d re l a t i ve to continuous f i sh ing (FS 10). % Range = sum of f i r s t capture (years) AFC = age of f i r s t capture (years) AFC + f r ac . = the c r i t i c a l age Strategies reported under In tens ive, Moderate or Extensive re fer to the best choice in each f i sh ing mode. CASE 1: G ! M ! * j * I i i * i * i 0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 Worst Strategy % Gain - 2.9 - 6.2 - 7.8 -10.8 - 4.4 - 6.5 -10.9 -20.2 - 3.6 - 6.8 -15.0 -14.1 INTENSIVE Strategy % Gain 3 3 3 3 3 3 3 3 3 3 3 3 3.4 7.2 9.5 13.2 4.8 7.5 12.6 23.0 4.0 7.8 16.7 16.5 MODERATE Strategy % Gain 6 6 6 6 6 6 6 2.6 5.5 7.5 10.3 3.3 5.6 9.0 14.6 2.9 5.7 10.8 11.7 EXTENSIVE % A F C Strategy % Gain Range Frac.=.50 9 9 9 9 9 9 9 9 1.8 3.9 5.0 6.9 2.7 4.0 6.8 12.4 6.3 13.4 17.3 24.0 9.2 14.0 23.5 43.2 7. 14. 29. 30.6 9 6 5 4 5 4 3 2 4 3 2 2 CTi CASE 2: G ! M ! * j * I i * i * i * i Frac.=.50 0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.1 0.2. 0.3 0.4 0.1 0.2 0.3 0.4 • 1.8 • 4.1 • 5.3 • 7.1 - 3.4 - 4.4 - 7.9 -16.3 - 2.5 - 4.8 -12.0 -10.1 2 2 2 2 3 3 3 3 3 3 3 3 I. 9 3.9 5.9 7.5 3.1 4.0 7.3 15.7 2.3 4.4 I I . 4 9.4 6 6 6 6 6 6 6 6 6 6 6 6 1.8 3.9 5.5 7.0 2.5 4.0 6.6 11.4 2.1 4.1 8.4 8.5 9 9 9 9 9 9 9 9 9 9 9 9 1.1 2.5 3.2 4.1 2.0 2.6 4.7 9.6 1.5 2.8 7.1 6.0 3.7 8.0 11.2 14.6 6.5 8.4 15.2 32.0 4.8 9.2 23.4 19.5 6 5 4 5 4 3 2 4 3 2 2 197 CASE 3: G ! M ! 0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 Worst Strategy % Gain - 1.4 - 3.3 - 3.5 - 5.1 - 3.0 - 3.6 - 6.7 -14.9 - 2.1 - 4.0 -11.0 - 8.6 INTENSIVE Strategy % Gain MODERATE Strategy % Gain EXTENSIVE % A F C Strategy % Gain Range Frac.=.50 3 1 7 6 1.3 9 0.9 3 1 9 3 3 8 6 2.9 9 2.1 7 1 6 3 4 3 6 3.6 9 2.3 7 8 5 3 6 3 6 5.0 9 3.3 11 4 4 3 3 1 7 2.1 9 1.8 6 1 5 3 4 1 6 3.0 9 2.2 7 7 4 3 7 4 6 5.0 9 4.1 14 1 3 3 15 9 7 11.7 9 8.9 30 8 2 3 2 3 6 1.6 9 1.3 4 4 . 4 3 4 4 6 3.1 9 2.4 8 4 3 3 11 .5 7 8.4 9 6.6 22 .5 2 3 9 .6 6 6.5 9 5.2 18 .2 2 CASE 4: G ! * ! M ! ! I * i * 0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 • 2.2 - 4.7 • 5.8 - 8.1 - 1.8 - 5.0 - 8.5 - 7.3 - 2.8 - 5.3 - 4.6 -10.9 Frac.=.50 3 - 2 6 6 1.7 9 1 3 4 8 9 3 5 6 6 3.6 9 2 9 10 3 6 3 7 4 6 5.0 9 3 7 13 2 5 3 10 2 6 6.7 9 5 1 18 3 4 3 2 3 6 1.6 9 1 1 4 1 6 3 5 8 7 3.7 9 3 0 10 8 4 3 9 8 7 5.7 9 5 1 18 3 3 3 9 4 6 .6.5 9 4 6 16 7 3 3 3 1 7 2.1 9 1 7 5 9 4 3 6 1 7 4.0 9 3 2 11 4 3 3 5 6 6 4.4 9 2 .6 10 2 3 3 12 8 7 8.7 9 6 .6 23 .7 2 CASE 5: G ! * ! M ! ! i j i * i Frac.=.75 0.2 0.1 1 - 1.1 2=3 0 9 6 0.9 9 0 6 2 0 9 0.2 1 - 2.7 3 2 2 6 2.0 9 1 5 4 9 6 0.3 1 - 2.7 2 2 8 6 2.6 9 1 5 5 5 5 0.4 1 - 4.0 2 3 7 6 3.6 9 2 3 7 7 4 0.4 0.1 1 - 2.4 3 2 1 7 1.9 9 .1 4 4 5 5 0.2 1 -'2.9 3 2 4 6 2.0 9 1 6 5 3 4 0.3 1 - 5.4 3 4 6 7 3.7 9 3 0 10 0 3 0.4 1 -12.3 3 10 7 7 10.7 9 6 8 23 0 2 0.6 0.1 1 - 1.7 3 1 4 7 1.1 9 1 0 3 1 4 0.2 1 - 3.2 3 2 .7 6 2.1 9 1 8 5 9 3 0.3 1 - 9.0 3 7 8 . 7 7.7 9 5 0 16 8 2 0.4 1 - 6.9 3 5 .8 7 • 4.5 9 3 8 12 7 2 1 9 8 Table CII (continued) CASE 6: G ! M ! i * i 1*1*1 0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 Worst Strategy % Gain 0.7 1.9 1.5 • 2.4 • 2.0 - 2.1 - 4.3 -10.9 - 1.3 - 2.4 - 7.9 - 5.4 INTENSIVE Strategy % Gain 3 3 3 3 4 3 3 4 3 3 4 3 0.9 2.3 2.3 3.4 2.4 2.4 4.7 14.4 1.4 2.7 10.2 5.9 MODERATE Strategy t Gain 7 7 7 7 7 7 7 7 7 7 7 7 0.8 2.0 1.7 2.7 2.2 2.3 4.6 12.2 1.4 2.6 8.8 5.9 EXTENSIVE Strategy % Gain 9 9 9 9 9 9 9 9 9 9 9 9 Range A F C Frac.=.75 0.4 1.6 9 1.1 4.2 6 1.0 3.8 5 1.5 5.8 4 1.2 4.4 5 1.2 4.5 4 2.5 9.0 3 6.2 25.3 2 0.8 2.7 4 1.4 5.1 3 4.5 18.1 2 3.1 11.3 2 CASE 7: G ! M ! i * [ i * i 0.2 0.1 0.2 0.3 0.4 1 1 1 1 - 1.4 - 3.2 - 3.8 - 5.3 3 3 3 3 1.7 3.6 4.7 6.5 6 6 6 6 1.3 2.7 3.9 5.2 0.4 0.1 0.2 0.3 0.4 1 1 1 4 - 1.1 - 3.4 - 2.1 - 4.7 3 3 2 3 1.4 3.7 3.4 5.9 6 6 6 6 1.2 2.7 3.4 5.0 0.6 0.1 0.2 0.3 0.4 7 1 1 1 - 0.4 - 3.6 - 2.3 - 7.5 2 3 3 3 1.2 3.9 3.5 8.2 6 6 6 6 1.1 2.8 3.5 5.6 9 9 9 9 9 9 9 9 9 9 9 9 0.9 2.0 2.5 3.4 0.7 2.1 1.5 3.0 0.4 2.2 1.6 4.6 3.1 6.8 8.5 11.8 2.5 7.1 5.5 10.6 1.6 7.5 5.8 15.7 Frac.=.50 9 6 5 4 4 3 5 3 3 2 CO CASE 8: G ! * ! * • * • * • M ! 1 * ! * ! * ! 0.2 0.1 0.2 0.3 0.4 7 4 4 4 - 0.9 - 3.3 - 3.2 - 3.9 2 2 2 2 0.9 1.8 2.6 3.4 0.4 0.1 0.2 0.3 0.4 4 4 4 4 - 1.4 - 4.4 - 6.1 - 5.0 2 2 2 2 0.9 1.9 2.8 3.5 0.6 0.1 0.2 0.3 0.4 7 4 4 4 - 1.5 - 5.4 - 5.6 -10.7 2 2 2 2 1.0 1.9 2.7 3.8 Frac.=.25 5 5 6 6 0.5 0.9 1.5 2.0 CO OO CO CO 0.3 0.6 0.5 0.6 1.8 5.1 5.8 7.3 10 7 5 4 6 5 5 6 0.5 1.3 1.7 1.8 8 8 8 8 0.2 0.8 1.1 0.8 2.3 6.3 8.9 8.5 6 5 4 3 6 5 5 5 0.8 1.7 1.5 3.4 8 8 8 8 0.5 1.0 1.0 2.0 2.5 7.1 8.3 14.5 5 4 3 3 199 Table CII (continued) CASE 9: G ! * ! * ! * i * i 0.2 0.4 0.6 Worst Strategy 0.1 4 - 0.9 0.2 4 - 1.7 0.3 4 - 3.9 0.4 4 - 5.2 0.1 4 - 0.5 0.2 4 - 2.5 0.3 4 - 3.6 0.4 4 - 1.7 0.1 4 - 1.7 0.2 4 - 3.7 0.3 4 - 3.2 0.4 4 - 7.6 INTENSIVE % Gain Strategy % Gain 0.6 1.1 2.5 3.4 0.3 2.1 2.3 1.1 1.1 2.3 2.0 4.8 MODERATE Strategy % Gain 5 5 5 5 5 5 5 5 5 5 5 5 0.4 0.7 1.7 2.3 0.2 1.5 1.5 0.7 0.7 1.6 1.3 3.2 EXTENSIVE Strategy % Gain 8 8 8 Range A F C Frac.=.0 0.2 1.5 10 0.4 2.8 7 0.8 6.4 6 1.1 8.6 5 0.1 0.8 6 0.8 4.6 5 0.8 5.9 4 0.4 2.8 3 0.4 2.8 5 0.8 6.0 4 0.7 5.2 3 1.6 12.4 3 CASE 10: 0.2 0.4 0.6 G i * ! * ! M * ! * ! ! 0.1 3 - 3.4 0.2 3 - 6.7 0.3 3 -10.1 0.4 3 -13.4 0.1 3 - 3.5 0.2 3 - 6.8 0.3 3 -10.5 0.4 3 -13.7 0.1 3 - 3.6 0.2 3 - 6.9 0.3 3 -10.6 0.4 3 -13.7 Frac.=.0 3.2 6.4 10.1 13.6 3.5 6.7 11.2 14.5 3.9 7.0 11.5 14.3 5 5 5 5 5 5 5 5 5 5 5 5 1.3 2.5 4.2 5.7 8 8 8 8 8 8 8 8 8 8 0.2 6.6 10 0.3 13.1 7 0.7 20.2 6 1.0 27.0 5 0.4 7.0 7 0.5 13.5 5 1.5 21.7 5 1.6 28.2 4 0.6 7.5 6 0.7 13.9 4 1.6 22.1 4 1.5 28.0 3 CASE 11: 0.2 0.4 0.6 : G . i * ! * ! M * ! * * ! ! 0.1 3 - 2.7 0.2 3 - 5.3 0.3 3 - 8.0 0.4 3 -10.6 0.1 3 - 2.8 0.2 3 - 5.4 0.3 . 3 - 8.4 0.4 3 -11.0 0.1 4 - 3.3 0.2 3 - 5.5 0.3 4 - 7.5 0.4 3 -10.9 Frac.= .0 2.5 5.0 8.0 10.7 2.8 5.3 9.0 11.5 3.1 5.6 10.7 11.4 5 5 5 5 5 5 5 5 5 5 5 5 1.2 2.4 4.0 5.4 1.5 2.7 5.0 6.2 1.9 3.0 6.7 6.1 8 8 8 8 8 8 0.3 5.2 10 0.5 10.3 7 1.1 16.0 6 1.6 21.3 5 0.5 5.6 7 0.8 10.7 5 1.8 17.4 5 2.1 22.5 4 0.7 6.4 6 1.0 11.1 4 3.4 18.2 4 2.0 22.3 3 200 Table CII (continued) CASE 12: G ! M ! I * i * Worst K M Strategy % Gain 0.2 0.1 3 - 1.7 0.2 3 - 3.4 0.3 3 - 5.3 0.4 3 - 7.0 0.4 0.1 3 - 1.9 0.2 3 - 3.6 0.3 7 - 5.9 0.4 3 - 7.4 0.6 0.1 4 - 2.8 0.2 3 - 3.7 0.3 4 - 7.3 0.4 3 - 7.4 CASE 13: G ! ! * ! * ! * ! M l * ! * i I INTENSIVE Strategy % Gain MODERATE Strategy % Gain EXTENSIVE % A F C Strategy % Gain Range Frac.=.0 0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 3 3 3 3 3 3 3 3 3 3 3 3 2.1 4.1 6.8 8.9 - 2. - 4. - 6. -10 - 2.5 - 5.0 - 8.8 -10.1 1.8 5 0.8 8 0.2 3.5 10 3.5 5 1.5 8 0.3 6.9 7 5.7 5 2.7 8 0.8 11.0 6 7.6 5 3.7 8 1.1 14.6 5 2.0 5 1.1 8 0.4 3.9 7 3.8 5 1.9 8 0.6 7.4 5 6.7 5 3.8 8 1.5 12.6 5 8.4 5 4.5 8 1.6 15.8 4 2.4 5 1.4 8 0.6 5.2 6 4.1 5 2.1 8 0.9 7.8 4 7.0 5 4.0 8 1.7 14.3 4 8.3 5 4.4 8 1.5 15.7 3 2.4 4.7 7.9 10.6 2.9 5.3 7.7 12.1 2.8 5.8 10.2 11.9 5 5 5 5 5 5 5 5 5 5 5 5 0.8 1.6 3.1 4.1 4.9 5.1 Frac.=.0 8 0.1 4.5 10 8 0.1 8.8 7 8 0.5 14.7 6 8 0.8 19.5 5 8 0.3 5.4 7 8 0.4 9.9 5 8 0.5 14.3 4 8 1.4 22.4 4 8 0.3 5.3 5 8 0.6 10.8 4 8 1.5 19.0 4 8 1.3 22.0 3 ro o o CASE 14: G ! ! * ! * ! * ! M ! * ! * ! * ! ! 0.2 0.1 3 - 1.3 0.2 3 - 2.6 0.3 3 - 4.6 0.4 3 - 6.0 0.4 0.1 4 - 1.9 0.2 3 - 3.1 0.3 • 3 - 4.5 0.4 4 - 8.2 0.6 0.1 7 _ 1.7 0.2 4 - 3.8 0.3 7 - 7.4 0.4 4 - 7.7 1.7 5 0.8 3.3 5 1.5 5.7 5 2.8 7.7 5' 3.8 2.2 5 1.2 3.9 5 1.9 5.5 5 2.7 9.2 5 5.0 2.1 5 1.1 4.4 5 2.3 8.0 5 4.6 8.9 5 4.8 Frac.=.0 8 0.2 3.0 10 8 0.4 5.9 7 8 0.9 10.3 6 8 1.3 13.7 5 8 0.4 4.1 7 8 0.7 7.0 5 8 0.9 10.0 4 8 1.9 17.4 4 8 0.4 3.8 5 8 0.9 8.2 4 8 1.9 15.4 4 8 1.8 16.7 3 201 Table CII (continued) CASE 15: G ! ! * ! * ! * ! M i * j * j * j * i Worst INTENSIVE MODERATE EXTENSIVE % ' A F C K M Strategy % Gain Strategy % Gain Strategy I Gain Strategy % Gain Range Frac.=.0 0.2 0.1 2 - 0.6 1 0.9 5 0.3 8 0.1 1.5 10 0.2 2 - 1.2 1 1.8 5 0.6 8 0.2 3.0 7 0.3 4 - 2.2 1 3.4 5 1.5 8 0.6 5.6 6 0.4 4 - 2.9 1 4.6 5 2.1 8 0.8 7.5 5 0.4 0.1 4 - 1.5 1 1.4 5 0.8 8 0.3 2.9 7 0.2 4 - 1.7 1 2.4 5 T. l 8 0.4 4.1 5 0.3 4 - 1.9 1 3.2 5 1.4 8 0.5 5.1 4 0.4 4 - 6.3 1 6.1 5 3.2 8 1.4 12.4 4 0.6 0.1 4 - 1.2 1 1.3 5 0.7 8 0.3 2.5 5 0.2 4 - 2.8 1 2.8 5 1.5 8 0.6 5.6 4 0.3 4 - 7.2 1 5.7 5 3.3 8 1.6 12.9 4 0.4 4 - 5.8 1 5.8 5 3.0 8 1.3 11.6 3 CASE 16: G ! * ! * ! * ! * ! M ! * ! * ! ! ! Frac.=.0 0.2 0.1 3 - 2.1 1 2.1 5 0.9 8 0.2 4 2 10 0.2 3 - 4.0 1 4.0 5 1.7 8 0.3 8 0 7 0.3 3 - 6.7 1 6.9 5 3.2 8 0.8 13 6 6 0.4 3 - 8.9 1 9.4 5 4.3 8 1.1 18 3 5 0.4 0.1 3 - 2.5 1 2.7 5 1.3 8 0.4 5 2 7 0.2 3 - 4.6 1 4.7 5 2.2 8 0.5 9 3 5 0.3 3 - 6.6 1 6.7 5 3.0 8 0.7 13 3 4 0.4 3 -10.3 1 11.1 5 5.4 8 1.6 21 4 4 0.6 0.1 3 - 2.5 1 2.5 5 1.2 8 0.3 5 0 5 0.2 3 - 5.0 1 5.3 5 2.6 8 0.7 10 3 4 0.3 3 - 8.7 1 9.7 5 4.9 8 1.7 18 4 . 4 0.4 3 -10.1 1 10.8 5 5.2 8 1.5 20 9 3 CASE 17: G ! * ! * ! * ! * ! M ! * ! * ! * ! ! Frac.=.0 0.2 0.1 4 1.4 1 1.3 5 0.8 8 0.3 2.7 10 0.2 4 - 2.6 • 1 2.6 5 1.6 8 0.6 5.2 7 0.3 4 - 5.4 1 4.8 5 3.0 8 1.2 10.2 6 0.4 4 - 7.1 1 6.4 5 4.0 8 1.6 13.5 5 0.4 0.1 4 2.4 1 2.0 5 1.2 8 0.5 4.4 7 0.2 - 4 - 3.8 1 3.3 5 2.0 8 0.8 7.1 5 0.3 4 - 5.1 1 4.6 5 2.8 8 1.1 9.7 4 0.4 4 - 9.9 1 8.1 5 5.1 8 2.2 18.0 4 0.6 0.1 < 4 _ 2.2 1 1.8 5 1.1 8 0.5 4.0 5 0.2 4 - 4.7 1 3.8 5 2.4 8 1.0 8.5 4 0.3 4 - 9.6 1 7.4 5 4.7 8 2.0 17.0 4 0.4 4 - 9.5 1 7.8 5 4.9 8 2.0 17.3 3 202 Table CII (continued) CASE 18: G ! M ! Worst INTENSIVE MODERATE EXTENSIVE A F K M Strategy % Gain Strategy I Gain Strategy % Gain Strategy % Gain Range Frac. 0.2 0.1 1=4 - 2.3 3 4.0 6 2.5 9 1.7 6.3 9 0.2 1 - 5.2 3 8.8 6 5.3 9 3.8 14.0 6 0.3 - 7.9 3 11.3 6 7.4 9 4.7 19.2 5 0.4 1 - 8.9 3 15.9 6 10.1 9 6.7 24.8 4 0.4 0.1 1 - 4.0 3 6.1 7 3.6 9 2.7 10.1 5 0.2 1 - 5.5 3 9.1 6 5.3 9 3.9 14.6 4 0.3 1 - 9.7 3 15.8 6 8.6 9 6.8 25.5 3 0.4 1 -19.4 3 30.5 7 19.1 9 13.0 49.9 2 0.6 0.1 1 - 3.1 3 4.9 6 2.8 9 2.2 8.0 4 0.2 1 - 5.9 3 9.6 6 5.5 9 4.1 15.5 3 0.3 1 -14.2 3 22.0 6 10.3 9 9.5 36.2 2 0.4 1 -12.5 3 20.8 6 11.4 9 8.8 33.3 2 c .50 CASE 19: G ! M ! Frac.=.75 0.2 0.1 3 - 3.8 1 2.3 5 0.9 10 0.0 6 1 9 0.2 3 - 7.3 1 4.3 5 1.5 10 0.0 11 6 6 0.3 3 -11.3 1 7.2 5 3.0 10 0.0 18 5 5 0.4 3 -14.7 1 9.2 5 3.8 10 0.0 23 9 4 0.4 0.1 3 - 3.2 4 5.3 5 0.7 10 0.0 8 5 5 0.2 3 - 7.3 4 6.0 5 1.4 10 0.0 13 3 4 0.3 3 -10.2 4 11.7 5 1.2 10 0.0 21 9 3 0.4 3 -11.2 4 29.4 7 • 4.1 10 0.0 40 6 2 0.6 0.1 3 - 3.6 4 3.6 5 0.5 10 0.0 7 2 4 0.2 3 - 7.1 4 6.7 5 1.2 10 0.0 13 8 3 0.3 3 - 8.7 4 20.8 7 2.5 10 0.0 29 5 2 0.4 3 -13.6 4 14.8 5 1.8 10 0.0 28 4 2 ro O CASE 20: G ! M ! Frac.=.0 0.2 0.1 4 - 1.3 1 0.7 5 0.6 8 0.3 2.0 10 0.2 4 - 1.3 . 1 1.3 5 1.0 8 0.5 2.6 7 0.3 4 - 5.6 1 2.9 5 2.3 8 1.1 8.5 6 0.4 4 - 7.4 1 4.0 5 3.1 8 1.5 11.4 5 0.4 0.1 4 - 0.7 1 0.4 5 0.3 8 0.2 1.1 6 0.2 - 4 - 4.0 1 2.1 5 1.6 8 0.8 6.1 5 0.3 4 - 5.2 1 2.7 5 2.1 8 1.0 7.9 4 0.4 4 - 2.4 1 1.3 5 1.0 8 0.5 3.7 3 0.6 0.1 4 - 2.5 1 1.3 5 1.0 8 0.5 3.8 5 0.2 4 - 5.3 1 2.7 5 2.2 8 1.0 8.0 4 0.3 4 - 4.6 1 2.4 5 1.9 8 0.9 7.0 3 0.4 4 -10.7 1 5.6 5 4.4 8 2.1 16.3 3 203 1. As m increases for a given K value, or as K increases for a given value of M, the benefits from seasonal pulse f ishing increase relat ive to the y ie ld from continuous f ishing. In turn, the percentage y ie ld increase depends on the degree of overlap between the seasonal growth and mortality patterns. If the growth period is entirely free of any natural mortality component, the expected y ie ld improvement is enhanced. 2. The c r i t i c a l age depends on the values of K and M and the growth-mortality pattern. In the model used here, i f the c r i t i ca l age includes an annual f ract ion, the age of f i r s t capture (AFC) is reduced to the integer value. For instance, i f the c r i t i c a l age is calculated as 9.75, 9.50, 9.25 or 9.00, then AFC is set equal to 9.00. Apparently the further AFC departs from the c r i t i c a l age, the y ie ld improvement due to seasonal f ishing is increased. In practice, commercial f isheries usually exploit age groups younger than the c r i t i c a l age, and therefore seasonal pulse f ishing may offer a dist inct advantage over the continuous mode. The question of how to f ish on a seasonal basis can be answered without ambiguity. Seasonally intensive f ishing offers a superior y ie ld compared to a continuous strategy, and a moderate program provides a second best choice. With only one exception (Case 19), the extensive category also contains a small but positive y ie ld benefit. These comments must be tempered by the guidelines given ear l ie r . For example, in Case 1, where growth and mortality are seasonally separate, K=0.2 and M=0.4, an intensive strategy (FS 1) can generate a 13.2 per cent increase in y ie ld over continuous f ishing. In Case 9, however, where growth and mortality are continuous for the same K and M values, the y ie ld gain from FS 1 is only 3.2 per cent. The decision to implement an intensive program will therefore depend on the species charac-204 teristics and whether concentrating the fishing activity is considered worthwhile. While one intensive strategy leads to the maximum, relative increase in yield one of the three remaining strategies in that mode usually represents the opposite extreme. The magnitude of the potential gain and loss are nearly equal. The same situation applies in the remaining modes and there-fore it is as important to know not only how to fish but when. The answer : is not clear cut, but the relationship between the growth and mortality patterns is instructive. If the critical age is equal to an integer value, the best strategies in the intensive, moderate and extensive modes are consistently FS 1, FS 5 and FS 8, respectively. In intensive fishing the worst strategy is also removed from the best choice by two or three quarters. The difficulty in assigning the best time to fish arises when the critical age contains a fraction. If the growth period is free of natural mortality, the optimum intensive strategy usually falls in the season immediately after growth is completed. Where growth and mortality overlap, the decision is influenced by the seasonal intensity of growth and mortality. For example, in Case 2 one third of the annual mortality rate coincides with the terminal but moderate growth period. For values of K = .2 and M = 0.1 to 0.4, intensive fishing in the second quarter is advantageous compared with either FS 1 or FS 3. As K increases, however, intensive fishing can be delayed until FS 3 to achieve the most favorable yield. In Case 4, growth is distributed over three seasons, mortality confined to two, and the relative intensity of these patterns is the reverse of Case 2. Here, because of the exponentially diminished value of g^, intensive fishing must begin in the third quarter (FS 3) to offset the increased loss from natural mortality. In summary, the question of when to fish cannot be answered on the basis intuition or the appearance of the growth-mortality pattern. The inter-action between K, g^  and m is too complex, and the decision requires a mathematical solution. 

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