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Alternative harvest strategies for Pacific herring Hall, Donald Lincoln 1986

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Alternative Harvest Strategies for Pacific Herring by Donald Lincoln Hall B.Sc, Humboldt State University, 1981 y A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in The Faculty of Graduate Studies Department of Zoology We accept this thesis as conforming to the required standard The University of British Columbia March 1986 © Donald Lincoln Hall, 1986 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ^ - c ^ t - o ^ y The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e fipt/c ~2S\ h<rr(o Abstract A simulated Pacific herring population (Clupea harengus pallasi) is used to evaluate alternative management strategies of constant escapement versus constant harvest rate for a roe herring fishery. The biological parameters of the model are derived from data on the Strait of Georgia herring stock. The management strategies are evaluated using three criteria: average catch, catch variance, and risk. The constant escapement, strategy provides highest average catches, but at the expense of increased catch variance. The harvest rate strategy is favored for large reductions in catch variance but only a slight decrease in mean catch relative to the fixed escapement strategy. The analysis is extended to include the effects of persistent recruitment patterns. Stock-recruitment analysis suggests that recruitment deviations are autocorrelated. Correlated deviations may cause bias in regression estimates of stock-recruitment parameters (overestimation of stock pro-ductivity), and increase in variation of spawning stock biomass. The latter effect favors the constant escapement strategy, which fully uses persistent positive recruitment fluctuations. Mean catch is de-pressed for the harvest rate strategy, since the spawning biomass is less often located in the productive region of the stock-recruitment relationship. It is recognized that management agencies never have perfect information. Management ob-servation error is modeled as uncertainty in spawning biomass estimates, based on data for errors in the measurements of spawn length, width and egg layers. Mean catch is reduced for both strategies. Reductions in the constant escapement strategy are more pronounced, and vary with the assumed form of the assessment error probability distribution. ii Table of Contents Abstract ii List of Tables v List of Figures vi Acknowledgements ix 1 Introduction 1 The Fish and Fishery 3 Theoretical Analyses of Harvest Strategies 4 2 The Population Model 10 Virtual Population Analysis . 10 Stock and Recruitment 15 What to Model? 20 Model Mathematics 21 3 Harvest Strategies and Evaluation Criteria 23 The Harvest Strategies 23 Strategy Definitions 26 The Evaluation Criteria 27 Evaluation Criteria Definitions 30 Model Test Standards 30 DFO's CUTOFF Policy 31 4 Results of the Basic Model 32 Basic Model, Independent Deviations 33 Basic Model, Correlated Deviations 36 Basic Model, DFO CUTOFF Policy 39 iii 5 Herring Stock Assessment 43 Escapement Method of Stock Assessment 44 Estimating Spawn Deposition 45 Simulation of Biomass Estimation Errors 50 6 The Population Model with Stock Assessment Error 51 Results of the Extended Model 51 Three General Forms of Stock Assessment Uncertainty 57 7 Harvest Rate or Constant Stock? 60 Information Gain 60 Unresolved Issues 61 Practical Applications 61 Conclusions 62 References 64 Appendix A — Catch, Catch Proportion and Weight Data 71 Appendix B — FORTRAN Source Code . 76 iv List of Tables 2.1 Catch at age, 1951-1985 12 2.2 Number at age from VPA, 1951-1985 14 2.3 Age specific parameters used in the model 16 2.4 Stock-recruitment data, 1951-1982 18 3.1 Four harvest strategy categories 24 3.2 Measures of evaluation criteria 27 A l Total catch by gear type, 1951-1985. 71 A2 Catch proportion at age for the reduction fishery, 1951-1967 72 A3 Catch proportion at age for the food and bait fishery, 1968-1985 72 A4 Catch proportion at age for the seine fishery, 1972-1985 73 A5 Catch proportion at age for the gillnet fishery, 1972-1985 73 A6 Weight at age for the reduction fishery, 1951-1967 74 A7 Weight at age for the food and bait fishery, 1971-1985 74 A8 Weight at age for the seine fishery, 1972-1985 75 A9 Weight at age for the gillnet fishery, 1972-1985. 75 v List of Figures 2.1 Strait of Georgia herring landings, 1951-1985 11 2.2 Stock and recruitment data for the Strait of Georgia herring, 1951-1982 17 2.3 Residuals from the stock-recruitment regression 17 2.4 The bias of autocorrelated deviations on the Ricker stock-recruitment a parameter. . 19 2.5 The bias of autocorrelated deviations on . the Ricker stock-recruitment /? parameter 19 3.1 Graphic representation of harvest strategies 27 4.1 Mean catch versus mean spawning biomass for the basic model (random, uncorrelated deviations) 34 4.2 Catch CV results for the basic model 34 4.3 Risk measurement for the basic model 35 4.4 Mean—variance curves for the basic model 35 4.5 Mean catch versus mean spawning biomass from the model ver-sion including autocorrelated deviations 37 4.6 A time' series of simulated deviations 37 4.7 The effect of correlated deviations on catch over a range of har-vest rates 38 4.8 The effect of correlated deviations on spawning biomass over a range of harvest rates 38 4.9 Catch CV under correlated deviations • 40 vi 4.10 Risk measurements from the model with correlated deviations 40 4.11 The mean—variance curves under the model with correlated deviations 41 4.12 A composite figure of the results from chapter 4 42 5.1 Fishery officer versus SCUBA diver measurement of spawn length in meters 46 5.2 Fishery officer versus diver measurement of spawn width in me-ters 47 5.3 Fishery officer versus diver measurement of egg layers 47 5.4 Corrected fishery officer width measurements of spawn deposi-tion versus SCUBA diver width measurements 48 5.5 Egg density estimates for fishery officers versus divers 48 5.6 Fishery officer versus diver estimates of spawning biomass 49 5.7 The difference in spawning biomass estimates (tons FO — tons diver) versus diver spawning biomass estimate 49 6.1 Mean catch versus mean spawning biomass for the model in-cluding stock assessment uncertainty. 53 6.2 Catch CV under stock assessment uncertainty. 53 6.3 Measures of risk for the model including stock assessment un-certainty. 54 6.4 Mean—variance curves for the strategies under stock assessment uncertainty 54 vii 6.5 The effects of uncertainty on prefishery stock estimates for the harvest rate strategy 55 6.6 The effects of uncertainty on catch for the harvest rate strategy 55 6.7 The effects of uncertainty on prefishery stock size for the con-stant escapement strategy 56 6.8 The effects of uncertainty on catch for the the constant escape-ment strategy 56 6.9 The effects of 3 different forms of uncertainty on the evaluation criteria for both harvest strategies 58 viii A c k n o w ledgements No thanks would be too much for the initiation and early guidance provided by Dr. Ray Hilborn, and the continuing support of Dr. Carl Walters. Dr. N.J. Wilimovsky and Dr. Max Stock er assisted with helpful criticism of the penultimate draft. The herring group of the Canadian Department of Fisheries and Oceans were especially helpful in providing data and answering my frequent questions: special thanks to Vivian Haist, Dr. Doug Hay, Jake Schweigert, Carl Haegele, Dr. Dan Ware, and Lloyd Webb. Support was provided by a University Graduate Fellowship. Fellow students and friends at the Institute of Animal Resource Ecology aided the final drive: Mike Lapointe and Wilf Luedke reviewed the thesis, Chris Foote provided intangible assistance, and Arlene Tompkins helped me through the most difficult of times. ix Introduction In 1982 the herring management section of the Canadian Department of Fisheries and Oceans made a quiet change in management policy. The harvest strategy of trying to maintain constant spawning biomass was replaced by a strategy of trying to remove a constant 20% of the spawning population. The new strategy would .. provide less variable yields, than fixed escapement policies, and permit informative variation in spawning stock" (Stocker, Haist and Fournier 1983). These spec-ulations were based on comments in an analysis for the Commission on Pacific Fisheries Policy (ESSA 1982) that led Pearse (1982, p. 40) to state: Another difficulty in developing long-term plans is the controversy over how to achieve maximum sustainable yields. Some scientists recommend managing the harvest to pro-vide for an equal spawning escapement every year despite fluctuations in the stock; others suggest maintaining the harvest, leaving the escapement to vary. We have little evidence to show the relative efficacy of the different techniques, and so management policy lacks a firm scientific foundation. These questions should be resolved as quickly as possible, and the most effective way of doing so is through careful experimentation. I therefore recommend — 4- The long-term plan for herring management should include provision for experimenting with alternative management strategies to determine their relative ef-fectiveness in maximizing long-term yields. Thus far, deliberate management experimentation is not a policy of the Department of Fisheries and Oceans (DFO). At present, the best available tool for gaining some insights about alternative strategies is simulation modeling. The modeling should serve two purposes: help define the range of experimental choices, and provide interim justification for management strategy until experiments are devised and concluded. The objective of this study is to examine alternative management strategies by the technique of computer simulation for Pacific herring in British Columbia. The objective can be recast in the language of scientific hypothesis testing. The null hypothesis is that we should expect no difference between harvest strategies. A model of the Strait of Georgia herring population and fishery is constructed to test this hypothesis by evaluating differences between strategies with meaningful criteria: catch, catch variance and risk. If differences do exist, a "best" strategy will emerge based on the criteria. The results from the theoretical study should then be tested by large scale management experiments. Many studies have "tested" this hypothesis with a variety of different models and harvest strategies. For two strategies I consider—harvest rate and constant stock—the general conclusions are always the same. The constant stock strategy produces the highest mean yields but at the expense of 1 unwanted catch variance. The harvest rate strategy sacrifices long term yield in favor of substantial reductions in catch variance. A third, or "CUTOFF" strategy, is curently used in British Columbia (Stocker 1985); this strategy mixes the two extremes. Why submit the Strait of Georgia herring stock to the same analysis? The first reason is that it is a different stock and fishery. Management credibility is a significant issue with the fishing industry. Results borrowed from analyses of baleen whales, Pacific salmon or even Bering Sea herring would be suspect. Shepherd and Horwood (1979) and May et al. (1978) found that differences between generalized models produce contrary results. I credit the failure of general analytical models with the return to species and fishery specific simulation models in recent years. The second reason is to extend the basic analysis to include ideas about cyclic abundance patterns and stock assessment uncertainty. Instead of assuming independent, random "environmental effects" every year, the first extension recognizes that there may be persistent periods of good or poor recruitments. The second extension replaces perfect stock size information with a model of the herring assessment methods used in British Columbia. Both extensions represent significant practical problems that have received only minor attention in the theoretical literature. The results of this study favor the harvest rate strategy. The highest average yields are produced by the constant stock strategy, but with an unacceptable level of catch variance. The harvest rate strategy sacrifices a small amount of average yield for significant gains in reducing catch variance. The extended versions of the basic model do not change the qualitative results. Mean catch decreases for the harvest rate strategy with persistent recruitment perturbations; the constant stock strategy is largely unaffected. With the added effect of stock assessment uncertainty, mean catch for the constant stock strategy decreases, but is still higher than the harvest rate strategy over a wide range of escapement levels and harvest rates. In all versions, catch variance for the harvest rate strategy is significantly lower. The harvest rate strategy reduces catch variance by partitioning natural population fluctuations into spawning biomass and catch. The constant stock strategy transmits all population variation directly to the catch. This chapter continues with a brief introduction to Pacific herring and the British Columbia herring fishery, and a review of the harvest strategy literature. Chapter 2 describes the model of the herring population and the techniques that were used to construct it. Chapter 3 defines the harvest strategies and evaluation criteria, returning to the literature for a look at other possibilities. The results of the basic model, the model with correlated recruitment deviations, and a spasvning 2 biomass reserve variation are presented in chapter 4. Field methods of British Columbia herring stock assessment and the technique of assessment error simulation are described in chapter 5, with the results from the extended model presented in chapter 6. Chapter 7 discusses and summarizes the results of the thesis. The Fish and Fishery Pacific herring (Clupea harengus pallasi) range from southern California to Arctic Alaska in the northeast Pacific and from Korea to Kamchatka in the northwest Pacific (Blaxter 1985). Herring are pelagic planktivores. Schooling is standard behavior. Onshore migration occurs in the winter-spring prior to spawning in protected intertidal and subtidal waters (Hay 1985). The demersal eggs adhere to vegetation. Larvae hatch after 6-18 days incubation (Haegele and Schweigert 1985). Juveniles recruit to the spawning population from age 2-5. The maximum age may be over 20 years but ages over 10 years are uncommon in exploited populations. Herring have been exploited off the coast of British Columbia since 1877 (Pearse 1982). A domestic reduction industry developed in the 1930's after the collapse of a pilchard fishery. The 1962-63 season was the peak of the reduction fishery, with 240 thousand metric tons landed. After 1965 the stocks collapsed, and the reduction fishery was closed indefinitely in 1968 (Pearse 1982, Hourston 1980). The fishery for herring sac-roe (ovaries) began after partial stock recovery in the early 1970's. This fishery targets on sexually mature, prespawning herring. The egg skein is processed into Kazunoko, a Japanese delicacy (Trumble and Humphreys 1985). Fishing occurs in late February through April. The objective of maximum prespawning ovary development, combined with too many fishing vessels, creates an intense, often frenzied fishery. Openings are measured in hours and minutes. An entire fishing season may consist of one purse seine set or a few hours of gillnetting. Two other fisheries also exploit herring. A small food and bait fishery has persisted since the 1960's and a new fishery for roe-on-kelp is emerging in British Columbia. The management strategy for herring fisheries in British Columbia has changed with time. Prior to 1967 the fisheries were managed by catch quota and seasonal closures (Stevenson and Outram 1953). The roe herring fisheries were managed with area specific constant escapement strategies until 1981 (Hourston and Schweigert 1981). In 1982 a constant harvest rate of 0.20 became coastwide management policy (Stocker, Haist and Fournier 1983). A modification of the constant harvest rate strategy that 3 provides a minimum escapement level (CUTOFF policy; Stocker 1985) is the current management practice. Theoretical Analyses of Harvest Strategies In the following review I use three guidelines to classify past studies. 1. The type of analysis; analytical or numerical (including simulation). I define analytical studies as those that use mathematical relationships to substantiate a result. Numerical studies use repeated measurements of a process to describe a relationship. For example, Ricker (1958) uses numerical methods to compare the relationship between stock size, environmental variability, and yield for different harvesting strategies. Gatto and Rinaldi (1976) confirm some of Ricker's results in a more general analytical study. 2. The model type. Three general model types have been used. The first is a set of curws that use a stock-recruitment relationship as the basis for population regulation. Second are models that consider the dynamics of the population as a whole unit. The third group are age-structured models. 3. The geographical area and species of interest. All studies use a random effect for stochastic analysis. The method of noise introduction varies. Most studies use serially independent and identically distributed errors (white noise). The models that consider serially correlated random effects are highlighted. A discussion of the specific harvesting strategies and criteria of evaluation that each reference uses is delayed until chapter 3 (see Tables 3.1 and 3.2). The first harvesting strategy study is by Ricker (1958). It is noteworthy as the only numerical study that does not use a computer. Ricker compares constant effort and constant stock size for 25 generations of a salmon-like population using a set of eight stock-recruitment curves. The results show that a stock maintained at the optimum stock size (fixed escapement) will produce the maximum long-term average yield. This conclusion holds for two levels of environmental variability and all eight curves. Special consideration is given to the dangers of low stock size operating under a depensatory stock-recruitment relationship. Larkin and Ricker (1964) and Tautz, Larkin and Ricker (1969) extend Ricker's original work to include 200 generations (this time on a computer) and different assumptions about the form of the environmental noise. Both additional studies confirm Ricker's original findings. 4 In an infrequently cited study Southwood (1968) evaluates the management practices of the International Pacific Halibut Commission by simulation. Southwood compares three types of stock assessment; empirical, potential yield curve (Schaefer), and yield per recruit analysis. His work should receive credit for first use of many modern techniques. He used an age-structured model, with replicate runs of 25 years each. The model is verified by historical comparison. Southwood's is the only study to use relative cost as a criterion to evaluate management practices. Ricker (1958) identified a crucial problem in the optimum fixed escapement strategy; all the variation generated by random environmental noise is transmitted to the catch; thus the single objec-tive of maximum long-term average catch ignores catch variability. Allen (1973) recognizes that catch stability is also an important objective for the fishing industry. Allen proposes alternatiw strategies for Skeena River sockeye ranging from optimum fixed escapement to fixed exploitation rate; between these extremes he considers modified strategies that temper the drastic on/off fishing policy (no fishing if the stock size is below the optimum) with policies that allow some yield during periods of sub-optimal stock size. The result is reduced average yields of about 15%, but with half the catch variance. Walters (1975) considers this tradeoff between maximum long-term average catch and mini-mum catch variance with the technique of stochastic dynamic programming. He compares a variance minimizing utility function for three levels of average catch against the maximum yield strategy (fixed escapement) for Skeena sockeye. For a small sacrifice in average yield a considerable reduction in catch variance is possible. Walters describes a strategy that is simple to carry out (harvest levels based on increments of stock size) and provides a near optimal catch/variance tradeoff. Harvesting strategy studies began on the east coast with Doubleday (1976) and Sissenwine (1977). In independent analyses they reach similar conclusions. First, controlling stock biomass maximizes yield, but at the expense of high catch variability. Second, a strategy that removes a constant catch sacrifices yield to obtain short term stability (but long term instability due to depensatory effects). Third, a strategy of constant effort (or fishing mortality) will compromise between the high yield of a constant biomass policy and the low yield of a constant catch policy. Doubleday simulates groundfish in the northwest Atlantic using a logistic model. Sissenwine simulates a hypothetical population, also using logistic dynamics. I credit Sissenwine with two important considerations. First, the length of the simulation may influence the results. Second, he considers the effects of autocorrelated error. Most studies model only uncorrelated (white) noise. Tautz et al. (1969) imply a correlated structure to 5 environmental noise by adding uncorrelated deviations to a sine wave function but Sissenwine is the first to specify autocorrelated error as a driving variable in fisheries simulation studies. Within the chronological sequence there are subgroups of researchers and periods of thought that stand out. Such are the British works led by R.M. May and J.R. Beddington in the late 1970!s. That period was dominated by the view that the goal of fisheries management was Maximum Sustained Yield (MSY). The harvesting strategy work done by this group strongly reflects the idea of stable population equilibria central to the MSY objective. Much of their analysis was concerned with the effect of harvesting on the stability and resilence of the population about an equilibrium. Beddington and May (1977) compare constant effort and constant catch strategies using the Schaefer logistic model (Schaefer 1954). They conclude that populations harvested for sustained yield take longer to recover from environmental disturbances than unharvested populations. As harvesting effort increases to the point of MSY, variability in population size and yield also increase. Three studies apply their work to various models of baleen whale population dynamics (Beddington 1978; Beddington and Grenfell 1979; Horwood, Knights and Overy 1979). The most significant contribution from these studies is the search for a satisfactory measure of the reaction of a population to a perturbation (discussed in chapter 3 under evaluation criteria). Shepherd and Horwood (1979) and May et al. (1978) dispute the results of Beddington and May (1977). They show that population stability and resilence may either increase or decrease with increased harvesting. The results depend on the mechanisms regulating the population and the method by which noise is introduced into the model. Their work made it clear that any type of population response can be manufactured by manipulating the form of the population model. Reed (1978) uses a discrete time Markov population model to compare constant escapement and constant catch strategies. His results prove that a policy of taking a catch equal to the deterministic MSY is not sustainable if the population suffers even a small degree of random, environmentally induced fluctuations. Mendelssohn (1980) applies a multiobjective Markov decision model to Bristol Bay sockeye salmon populations. The dual objective is the same as Walters (1975); minimize variance while maximizing mean catch. Mendelssohn wisely stops short of stating which of three harvesting strategies—constant stock, fixed effort, and fixed harvest—is the best. He is willing to conclude that the best fixed harvest policy significantly reduces average yield, and that true optimal policies will rarely be constant. 6 Ludwig (1980) explores simple alternatives to constant strategies, by examining a variety of switching policies that control whether fishing occurs based on stock size. Ludwig compares three production models; Beverton-Holt, logistic and Pella-Tomlinson. At low levels of variation there is little difference between the strategies or models. About this time age-structured analytical models began to appear in the harvesting strategy literature. The deficiencies of models without age structure had long been realized. Models that ignore age structure neglect the importance of age specific mechanisms for regulating populations. But the problems of keeping track of age specific parameters and variables are not easy to solve (Mendelssohn 1978). The first attempts tried to simplify the mathematics. Horwood and Shepherd (1981) and Getz and Swartzman (1981) split the population into two age groups; immature fish and a pool of mature age classes. Both Horwood (1982, 1983) and Getz (1984) realize the inadequacies of their first attempts and develop more sophisticated models the second time around. I consider all six of these studies and Reed (1983) to be on the fringe of harvesting strategy work. They are concerned with developing a stochastic age-structured alternative to the classic deterministic yield-effort models and contribute little new information to harvest strategy research. Horwood (1982) does consider the persistence of recruitment perturbations in North Sea herring. Correlated' deviations can lead to greater than 30% underestimate of yield variance. At this point a shift in the literature occurs away from the analytical work of Horwood, Getz, Reed and associates toward a period of species specific simulations. Swartzman et al. (1983) apply an extended Getz and Swartzman (1981) model to simulations of the Pacific whiting fishery. Swartzman et al. develop a "new policy algorithm" (similar to a harvest rate, in that an annual quota, is adjusted based on stock forecasts) and compare it to constant quota and constant effort strategies. Not surpris-ingly, they find that the constant catch policy can lead to stock crashes in years of poor environmental conditions. The constant effort policy produces slightly higher average yields and slightly lower catch per unit effort than the management algorithm. Trumble (1983) presents a simulation study of Pacific herring in Washington State using a traditional yield per recruit model. He compares three harvest strategies; constant quota, harvest rate, and constant stock. Trumble's study is the first to consider observation error, through random log normal error added to biomass estimates. No specific mention of the effects of this error is made in the results. The constant stock strategy produces higher average yields than the harvest rate strategy, but with 2-5 times the catch variance. The highest average yields are from a combination of 7 strategies, where the harvest rate is 0.40 above a constant stock of 100,000 tons. This strategy suffers from moderately high catch variance and frequent years of zero catch compared to the harvest rate strategy. No mention is made of the quota strategy, presumably because it is always found a poor option. Armstrong (1984) simulates a South African anchovy fishery. He looks at fishery-specific har-vesting strategies; one based on age-specific fishing mortalities, a constant quota, and an annually adjusted quota based on a variation of the MSY theme. It is difficult to compare the strategies over the narrow range of levels he uses. Doubleday (1985) warns of the dangers of a too high harvest rate on northwest Atlantic herring. The simulations suggest that a harvest rate of 0.20 is safe for herring. Ruppert et al. (1985) conduct a simulation study of the Atlantic menhaden fishery. Using a stochastic egg-recruitment model, density-dependent juvenile growth, and age-specific biological parameters they compare constant catch, constant fishing mortality and an egg escapement policy. The egg escapement policy is similar to the fixed escapement strategy that has been used until recently for British Columbia herring. They conclude that the egg escapement policy is best, even though it produces higher catch variance than constant fishing mortality. The analysis includes uncertainty in biological parameters by assuming different levels of natural mortality. Fried and Wespestad (1985) use a simulation model to compare different levels of exploitation rates and inshore/offshore fishery splits for eastern Bering Sea herring. The study advises that extra caution is necessary with mixed stock fisheries. Jacobson and Taylor (1985) compare constant effort and constant quota strategies for a simulated lake whitefish commercial fishery. Their conclusions are consistent with earlier studies. The constant effort policy produces larger sustainable yields, but with higher catch variance. They recognize that their model is only a tool that ignores many management considerations, but nevertheless will aid the choice between the two strategies. Hilborn (1985) considers optimal" harvest policies for a mixed stock fishery. If stocks have uncorrelated natural variation, maximum average catch is produced by a harvest policy very much like a constant harvest rate. When natural variation is correlated between stocks, a fixed escapement strategy maximizes average catch. If the objective is not maximum average catch, but rather the sum of logarithms of catch, the best strategy appears to be quite similar to constant harvest rate. Deriso (1985) presents results that optimize a risk adverse management objective. He defines risk as the "reluctance on the part of managers to freely gamble with current quotas in an attempt 8 to increase the odds for a higher future catch". Deriso defines a logarithmic utility function as a risk adverse objective. He finds that a constant harvest rate strategy maximizes log-catch. He shows that parameter estimation is better for a constant harvest rate strategy than a fixed escapement strategy. There are some general conclusions that can be made from this review. 1. The MSY management objective influenced and hindered harvest strategy research. 2. There has been a complete cycle of research in 25 years. From early species-specific numerical work, to generalized analytical work, and now back to simulation models of specific fisheries. 3. Most studies compare harvest strategies over a narrow range of alternatives. The comparisons are usually the "best" results for each strategy. Fisheries management often operates at subop-timal policy levels for economic, social, political, biological or other reasons. The widest possible range of intra-strategy alternatives should be presented. 9 Chapter 2 The Population Model To evaluate the effects of harvesting strategies on a population it is necessary to begin with a model that encompasses some essential dynamic processes: growth, reproduction and death. To establish a benchmark for comparisons between the model and our perception of actual population dynamics, I reconstruct the history of the Strait of Georgia herring population. Virtual Papulation Analysis (VPA, or cohort analysis; Ricker 1975) is a tool that can be used to rebuild a stock history based on information that is often collected in a managed fishery. The following section describes the VPA 1 conducted for the Strait of Georgia herring. A relationship between parent stock and juvenile production based on the results of the VPA is presented in section two. The third section discusses alternative hypotheses regarding population dynamics. Section four describes the age-structured model that I constructed for the Strait of Georgia herring, based on the results of the VTA and subsequent analyses. Virtual Population Analysis There are two basic information requirements for VPA; catch at age and an estimate of natural mortality. For the analysis to include estimates of younger age classes in recent years, an estimate of fishing mortality in the most recent year is needed. Unfortunately for analysis purposes, fisheries and management practices change over time. Most of the work for this VPA was spent organizing the data into a consistent format. The calculations are trivial. I begin by summarizing the available information, and then describe how I use this information to generate a table of catch at age that can be used for VPA. The data provided by the Department of Fisheries and Oceans, Pacific Biological Station are presented in appendix tables A1-A9. Data are available for total catch and catch proportion at age. The data are from three fisheries for various periods from 1951-1985. The data for 1951-1966 are from the reduction fishery. Data for 1967-1971 are from a small food and bait fishery. Data since 1972 are from seine and gillnet catches in the roe fishery and the persistent food and bait fishery. Though seine and gillnet catch data are from two gear types within one fishery, it will simplify later descriptions if I refer to the data from seine and gillnet as if from separate fisheries. Figure 2.1 presents catch for 1951-1985. The table of catch at age used by VPA (table 2.1) is estimated from the general equation 10 100 BO SS 00 63 70 73 80 83 Year Figure 2.1 Commercial herring landings in the Strait of Georgia, 1951-1985. 3 Ca,t = ^(total catch gear, t) p,,0tt (2.1) where p is the catch proportion determined by catch sampling, t refers to the gear type, and a the age group. In the years of one fishery (1951-1971), the equation reduces to Ca,t — Ctpa,t- The summation is needed only for 1972-1985, when catch at age is summed for the three operating fisheries. The other requirement for VPA is an estimate of natural mortality (m). Any estimate of natural mortality is uncertain because of the problems of the estimation procedures confounding m with mortality due to fishing, the catchability coefficient q, and the proportion offish at age maturing. The standard estimate of m for Strait of Georgia herring is 0.45 (Ware 1985). Walters et al. (ESSA 1982) suggest that standard methods for estimating natural mortality will underestimate m if the proportion of fish at age maturing is not included. The estimates that Walters et al. arriw at are somewhat higher than the standard DFO estimates. For analytical stock assessment procedures that allow m to vary over time, the estimates range from 0.30-0.73 (Stocker, Haist and Fournier 1985). I use the age specific annual survival rates from Walters et al. (table 2.3). 11 Table 2.1 Catch at age used for Virtual Population Analysis. Numbers are in millions of fish, 1951-1985. Age Year 1, 2 3 4 5 6 7 8 9 10 Total 1951 0 22 268 149 36 7 2 1 0 0 485 1952 68 87 294 138 40 9 3 1 0 0 640 1953 0 10 57 29 4 1 0 0 0 0 103 1954 0 21 367 250 80 23 6 1 0 0 750 1955 0 32 348 282 49 11 3 1 0 0 726 1956 0 47 239 217 173 39 6 2 0 0 724 1957 0 39 411 155 81 47 5 1 0 0 740 1958 0 27 143 56 15 13 7 1 0 0 262 1959 5 110 435 121 23 5 5 2 0 0 706 1960 0 103 364 285 48 16 4 2 0 0 823 1961 0 149 159 137 88 30 5 1 0 0 569 1962 1 121 485 142 46 29 9 3 0 0 834 1963 0 169 458 248 41 11 5 1 0 0 932 1964 0 48 517 280 41 10 4 1 0 0 902 1965 27 120 310 173 27 9 3 1 0 0 670 1966 0 110 168 118 86 15 6 1 0 0 503 1967 50 157 270 83 19 7 1 0 0 0 588 1968 20 17 24 10 3 1 1 1 0 0 77 1969 3 7 3 1 0 0 0 0 0 0 15 1970 0 3 7 1 0 0 0 0 0 0 11 1971 1 4 6 6 1 1 0 0 0 0 18 1972 0 10 41 39 22 4 1 0 0 0 119 1973 0 2 41 30 28 21 5 1 0 0 130 1974 0 1 6 13 13 10 3 1 0 0 47 1975 0 2 12 21 16 6 2 0 0 59 1976 0 4 11 38 31 13 5 1 0 0 103 1977 0 3 53 35 40 13 4 1 0 0 150 1978 0 3 57 75 33 30 10 1 1 0 210 1979 0 2 19 51 56 15 7 0 0 153 1980 0 1 12 8 14 9 2 1 0 0 47 1981 0 3 23 28 17 18 7 1 0 0 97 1982 0 3 25 23 23 9 11 1 0 101 1983 0 2 24 37 30 22 8 7 . 2 0 133 1984 0 6 20 25 19 11 5 1 0 0 89 1985 0 12 17 13 10 7 3 1 0 0 63 12 Virtual population analysis works backward from the oldest age class in the most recent year (age 10, 1985). To estimate incomplete cohorts that are currently being fished and new cohorts that are entering the population, estimates of the current exploitation rate, or catchability at age (qa) and annual effort [Et), are required. I estimate the numbers of fish in age classes 1-9 in 1985 by Na = Ca/Eqa, where = number of fish at age, Ca = catch at age, E = effort, and qa = average catchability at age. Catchability at age is calculated as 9 a Et(Nait-Na+1,t+1-Ca,t) { ' ' Average catchability at age (qa) is calculated as the mean qa since the start of roe fishing in 1972. Estimates of effort are notoriously poor for clupeid purse seine fisheries (MacCall 1976). The shift from the reduction fishery to the roe fishery, increasing gear efficiency, and multiple gear types confound any standard estimate of effort. Instead I use an index of effort (ESSA 1982), E — — l n ( l — harwst rate), where the harvest rate equals catch/(catch + spawning biomass). With the 1985 number at age estimates in place, the cohort analysis procedes from 1985 working backward to 1951 by Na,t = Na+lit+1/Sa + Ca,t (2.3) where Sa is age-specific annual survival. Equation (2.3) is from Ricker (1975, p.198), with 1/S = e m . This approximation of the standard VPA equation is possible when natural mortality can be ignored during the short fishing season (Type 1 population, Ricker 1975, p.10). The abundance of age 11 fish is assumed to equal 0, so the number of age 10 fish is catch plus spawners, estimated by the age 10 catch divided by the harvest rate in year t. The results of the VPA are number of fish at age before harvest (table 2.2). 13 Table 2.2 Millions of fish at age for 1951-1985 from Virtual Population Analysis for the Strait of Georgia herring. Age Year 1 2 3 4 5 6 7 8 9 10 Total 1951 5868 2005 865 344 87 14 4 1 0 0 9187 1952 8945 2934 991 298 97 26 4 1 0 0 13296 1953 5745 4438 1424 349 80 29 8 0 0 0 12073 1954 3020 2872 2214 683 160 38 14 4 0 0 9004 1955 3758 1510 1426 923 216 40 7 4 1 0 7885 1956 4189 1879 739 539 321 84 14 2 1 0 7768 1957 6911 2095 916 250 161 74 22 4 0 0 10433 1958 4296 3456 1028 252 47 40 13 8 1 0 9142 1959 3230 2148 1714 443 98 16 14 3 3 0 7668 1960 5574 1613 1019 640 161 38 5 4 0 1 9055 1961 5375 2787 755 327 177 57 11 1 1 0 9491 1962 5431 2688 1319 298 95 45 13 3 0 0 9891 1963 2628 2715 1284 417 78 25 8 2 0 0 7157 1964 1643 1314 1273 413 85 19 7 1 1 0 4755 1965 1540 821 633 378 67 22 4 1 0 0 3467 1966 648 757 351 161 102 20 6 1 0 0 2046 1967 834 324 324 92 22 8 2 0 0 0 1606 1968 1979 392 83 27 4 1 1 1 0 0 2488 1969 2347 980 188 30 8 0 0 0 0 0 3553 1970 2434 1172 486 92 15 4 0 0 0 0 4203 1971 1787 1217 585 240 46 7 2 0 0 0 3883 1972 2291 893 607 289 117 22 3 1 0 0 4224 1973 3961 1146 441 283 125 47 9 1 0 0 6013 1974 3495 1980 572 200 126 48 13 2 0 0 6436 1975 5714 1747 990 283 93 57 19 5 0 0 8908 1976 3803 2857 873 489 131 39 25 9 2 0 8226 1977 1470 1901 1427 431 226 50 13 10 3 0 5530 1978 1839 735 949 687 198 93 18 4 3 1 4527 1979 1352 920 366 446 306 83 31 4 1 1 3509 1980 1101 676 459 173 198 125 34 12 1 0 2778 1981 729 550 338 224 83 92 58 15 4 0 2092 1982 655 364 274 157 98 33 37 25 5 1 1650 1983 1194 328 180 125 67 38 12 13 7 1 1964 1984 5814 597 163 78 44 18 8 2 2 1 6727 1985 290? 296 71 27 12 3 1 0 1 3318 14 Stock and Recruitment The results from the virtual population analysis are used in this section to define, a relationship between parent stock and juvenile production in the theory of stock and recruitment (Ricker 1954). I assume that density dependent and environmental effects can be represented by the stochastic Ricker model: N1M1 =Eggs t e a-" E«» + " (2.4) wh ere = number of age 1 recruits, Eggs = number of eggs spawned, a = maximum recruitment per Eggst, /? = unharvested equilibrium stock size, vt — error term; mean = 0, variance = tr^. The parameters a, /? and a\ are estimated from the linear regression ln(A r M +i/Eggs t) = a-j9Eggs t + «t (2.5) where a is the intercept, /3 the slope and o~% the variance of the residuals (vt). The estimate of the number of eggs spawned in year t is calculated differently for the reduction and roe fisheries. For 1951-1971 i o Eggs, = 53 (TV.,, - Ca,t) Pa,t Fa 0.5 (2.6) and for 1972-1985 10 Eggs, = (Na,tPa,t ~ Ca,t) Fa 0.5 (2.7) 0=1 where N is number, C is catch, P is the proportion mature, F is fecundity, and 0.5 is the proportion female. The difference between equations (2.6) and (2.7) is the position of the catch term. Equation (2.6) describes the offshore reduction fishery removing fish from the entire population. In equation (2.7) , catch is removed from only that part of the population that has migrated onshore to spavn. 15 Table 2.3 Age specific parameters used in the model. Age Parameter 1 2 3 4 5 6 7 8 9 10 Annua] Survival 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.37 0.3 0.25 Proportion Mature 0.0 0.1 0.4 0.7 0.9 1.0 1.0 1.0 1.0 1.0 Fecundity (1000 eggs/female) 0 13 17 25 30 30 29 31 32 33 Weight (g) 13 58 83 109 128 144 159 169 171 192 There is some evidence for this difference in the catch proportion of age 1 fish in the offshore reduction and food and bait fisheries versus the onshore roe seine fishery (appendix tables A2-A5). Age specific fecundities and maturities are listed in table 2.3. Fecundity estimates are from Hourston, Haist and Humphreys (1981). The maturity schedule is from Walters et al. (ESSA 1982, p. 110); it is similar to recent data presented by Hay (1985) for ages 5+, but assumes less proportion mature for ages 4 and under. The proportion female is assumed to be 0.5 for all ages. The stock and recruitment data are presented in figure 2.2 and table 2.4, and the residuals from the regression equation (2.5) are plotted in figure 2.3. The assumption of random, normal deviates (vt) from the stock-recruitment equation (2.3) appears to be violated. A "periodic" nature to the recruitment process is evident. Walters (1985) outlines two sources of bias that confound standard regression techniques nor-mally used to estimate the Ricker parameters (or and 0) of equation (2.4): 1. Large measurement errors in the independent variables, such as spawning biomass estimates. 2. Correlations between deviations and subsequent levels of the independent variables. It is likely that both biases affect the herring stock-recruitment data. To measure the effect on parameter estimates of a and /? I did Monte Carlo simulations of the deviation feedback bias in the presence of autocorrelated deviations, using known or and /? parameters and an autoregressive function discussed at the end of this chapter. Figures 2.4 and 2.5 show the dual effect of the biases on a and /?. The Ricker parameter or is overestimated by about 10% and the parameter P is overestimated by about 18%. Correcting for these biases will make the stock appear less productive (lower a) and to have a higher unfished equilibrium biomass (higher or//?). The results of the simulation were used to correct the standard regression estimates of a and /?: a is decreased by 10% to 2.35 and (3 is decreased by 18% to 0.00133. Tester (1948) first speculated that catch trends might reflect some cyclic phenomenon. The factors that might cause such a periodic fluctuation have received much _attention. Stocker, Haist 16 10-— 8-10 a (0 U u a GC ai oi < 6-8-- 1 — 10 I— 12 14 Eggs Spawned (xlO 9 ) Figure 2.2 Stock and recruitment data for Strait of Georgia herring, 1951-1982. Number of age 1 recruits ( x l O 8 ) versus number of eggs spawned ( x l O 9 ) . Years of observations are connected to show time trends. Note poor recruitments since 1976. w a> cc -2 ai \ aa 84 V -83 \ 70 ™ SO — I — 55 — I — 60 — I — 63 - n — 70 — I — 75 — I — B0 63 Year Figure 2.3 Residuals from regression of In (recruits/eggs spawned) against eggs spawned plotted over time. Note cyclic tendency of residuals, representing persistent periods of good or poor recruitment. 17 Table 2.4 Stock-recruitment data from VPA, 1951-1982. Year Eggs Spawned (xlO B) Age 1 Recruits (xlO 6 ) Year Eggs Spawned (xlO e ) Age 1 Recruits (xlO 6 ) 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 590 672 1198 1349 1318 858 571 809 996 831 717 690 678 539 414 175 8945 5745 3020 3758 4189 6911 4296 3230 5574 5375 5431 2628 1643 1540 648 834 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 44 61 164 347 559 588 574 707 880 1074 1238 1175 974 824 597 417 1979 2347 2434 1787 2291 3961 3495 5714 3803 1470 1839 1352 1101 729 655 1194 a - 2.61 P = 0.00162 0.79 and Fournier (1985) correlate the fluctuations with environmental effects of sea surface temperature, Fraser River discharge, sea level and hours of sunlight. Ware and McFarlane (1985) describe a negative correlation for adult Pacific hake (Merluccius productus) and sea surface temperature with juwnile herring abundance on the west coast of Vancouver Island. Walters et al. (1986) study a northern stock of British Columbia herring and conclude that cycles might be caused by the predator-prey interactions of Pacific cod (Gadus macrocephalus) and herring. Walters et al. consider the Strait of Georgia herring, but decide that the southern cod populations are too low to drive a similar predator-prey system. If we return our frame of reference to modeling herring population dynamics, the details become less important. The model need not be concerned with the cause of population fluctuations, provided that it can mimic those fluctuations in a statistically realistic manner. Likewise, the choice of the stock-recruitment model is not particularly significant. The suggestion from figure 2.2 is that the best recruitments are from intermediate stock sizes, so I use the Ricker model. The recruitment variability expressed in the residual variance of the regression (equation 2.5) is so high that any model fits poorly. My use of a stock-recruitment function is best described by Walters (1986). . . . [I]t has been suggested that recruitment should be modeled as a fundamentally stochastic and unpredicatable process, with the stock-recruitment function viewed as a collection of probability distributions (one for each level of spawning stock). In this view, the stock-recruitment curve is a description of how average recruitment varies with reproducing stock (and is therefore useful for analysis of long-term average responses to 18 Ricker Alpha Parameter Figure 2.4 The effect of bias in correlations between deviations and between deviations and sub-sequent levels of recruitment on the Ricker stock-recruitment a parameter. The dotted line shows the true a given the model. The probability density function (Chambers et al. 1983) describes the frequency of occurrence (1000 trials). Correlated deviations bias estimates of Q by about +10%. 1.0 0.0 0.001 0.002 0.003 Ricker Beta Parameter 0.004 o.oos Figure 2.5 The Ricker stock-recruitment /J parameter for the same simulation as in figure 2.4. Correlated deviations bias estimates of /? by about +18%. 19 various harvesting policies), without pretense that the curve is at all useful for short-term predictions. What to Model? Just as there are many alternatives for stock-recruitment models, there are an infinite number of hypotheses regarding population biology and population response to environmental change. For ex-ample, density dependent growth, mortality, maturity, fecundity, predation, and environmental effects have all been suggested as mechanisms of population regulation for herring. The measurement of each mechanism is confounded by the others, so that it is virtually impossible to separate the effects. For example, recruitment since the late 1970's has been well below average. It is possible that environ-mental conditions have been poor. But it is just as possible that predation by cod, sablefish, jellyfish (Purcell 1986) or any other predator has adversely effected herring population numbers. Perhaps the high spawning levels of the late 1970's produced offspring with a reduced intrinsic capacity to survive. Another alternative is that the effect of roe fishing near shore (on spawning grounds) is showing up as reduced recruitment, caused by disturbing the spawning behavior of the herring. Of course, the only evidence for any of these mechanisms is the correlation between the cause and effect. As long as the correlation holds (or anomalies can be explained) the mechanism cannot be rejected. Understanding such mechanisms can be useful to management if the mechanism increases the precision and accuracy of stock predictions or leads to alternative harvesting methods. Usually the mechanism itself can not be controlled to elicit the desired response in the herring population. For example, a positive correlation between salinity and recruitment is useful if knowing that lew salinity in year t will adversely effect recruitment in year t + lag. Salinity can not be made higher to produce a good recruitment. It is generally accepted that the one mechanism a fisheries manager can control is catch. That is why this thesis is concerned with different strategies for .controlling catch and the population response to each strategy. To eliminate as much of the confounding effects of different population regulation mechanisms as possible, I chose to model the biology of the herring in its simplest form so that the effects of different harvesting strategies can be clearly shown. The age specific biological parameters (mortality, maturity, fecundity, weight, and percent female) are assumed to be constant over time and population size. The stock-recruitment parameters (a, /?, and v) are determined from historical data and remain constant. Recruitment variation follows two patterns; white noise and structured variation. The variation is not defined to be caused by species interactions, environmental effects, or 20 other causes, though it may reflect any or all possibilities. The parameters and equations that control the model are presented next. Model Mathematics The equations that follow may obscure what is essentially a simple model. I will first describe the action of the model in words. 1. The model always begins with the same size population. 2. Catch is removed from the adult population. 3. The remaining adult population reproduces. 4. Next year's age 1 fish are determined from reproduction. 5. Next year's age 2-10 fish equal this years age 1-9 times survival rates. 6. The model returns to step 2. The following equations elaborate the above outline. For a complete description the source FORTRAN code is listed in appendix B. The parameters of the model are presented in table 2.3. The model begins with a starting stock size at age 1 of 2 billion fish, the approximate mean Ni since 1972 from VPA. Starting numbers for ages 2-10 are calculated by Na = Na-iSa- The adult spawning biomass (in mt) is 10 Bt=Y,X°.tW°,tp«.t (2-8) where Na is total number at age, Wa is weight at age, and Pa is the proportion of fish at age that are mature and spawning. The catch is subtracted from the spawning biomass (the harvest strategies that determine the catch are described fully in the next chapter). The reproductive potential of the remaining spawning biomass is the number of eggs spawned: 10 Eggst=53jVa,tFa0.5 (2.9) The number of age 1 fish next year is determined by the stock-recruitment model A i l t + 1 =Eggs, e 0-" Ess 8 + « (2.4) The age classes 2-10 are then updated, calculating backward from age 10 to age 2 by Ara = Nc-iSa. The updated age classes are fed into equation (2.8) and the process repeats until the number of years that are being simulated are complete. 21 Stochastic variability is introduced in vt by one of two methods. To simulate a random effect where the deviations in year t are independent from deviations in year t— lag (white noise), vt is chosen from a normal random distribution, with mean 0, and standard deviation equal to the standard error of regression residuals (equation 2.5). The second method reflects the structure of the residuals in figure 2.3. The deviation in year t is dependent on the deviations in the previous years, plus an unpredicatable random effect. vt = f»i t>t-i + 0 2 i>t_2 + normal deviate (2-10) The coefficients oi and a% were determined from autoregression of vt on vt-i and vt-2 (equation 2.5, figure 2.3). The order (number of lags) of the regression was determined by a computer program (Walters, pers. comm.) that locates the minimum Akaike criterion (Ulrych and Bishop 1975; Larimore and Mehra 1985); a second order regression was selected by that criterion, with ai = 0.945 and 0 2 = —0.430. Equation (2.10) is used in forward simulations by recursively updating the Vt-\ag deviations, beginning with v-2 = —1.549 and v-\ = —0.880, the residuals from 1981 and 1982. 22 Chapter S Harvest Strategies and Evaluation Criteria The next, step is to simulate population removal by catch. 1 use two fundamentally different methods of population removal, or harvest strategies; constant harvest rate and constant escapement. To evaluate the population response to each strategy I compare the methods by three general criteria; average catch, catch variance and risk. The chapter begins by exploring the harvest strategy literature to determine the basis for this selection and to examine other strategies and their feasibility for British Columbia herring. The harvest strategies are defined. The evaluation criteria are described in the final section by a similar review of the literature. The Harvest Strategies I define a harvest strategy as a method of population removal that controls for population size and/or catch size. This definition distinguishes harvest strategies from harvest tactics. Harvest tactics are methods that management uses to implement a strategy (Hilborn, 1986). Examples of tactics are bag limits, area and time closures, effort restrictions, mesh size limits, and fish size limits. The different strategies that previous workers have studied fall into four categories: 1. a proportion of the stock is removed, 2. stock size is controlled, 3. catch level is controlled, 4. fishing effort is controlled. Table 3.1 uses these four catagories to summarize the literature. The details for each strategy differ within the categories. A Proportion of the Stock is Removed The simplest form of this strategy is a fixed exploitation rate. Ricker (1958) introduced this strategy and it received scant attention in the literature until recent years (Armstrong 1984; Doubleday 1985; Hilborn 1985, 1986; Fried and Wespestad 1985; Ruppert et al. 1985; Deriso 1985). There are two reasons for the lack of interest in exploitation rates. The idea of yield varying in proportion to stock size does not sit well with the ideology of Maximum Sustained Yield (MSY) that dominated in the 1970's. Second, management by harvest rate requires that the stock size be estimated before the yield is removed (unless q and E are constant). Stock assessment procedures during the MSY period concentrated on determining the location of MSY, not predicting the preharvest stock size each year. As the goal of MSY has been forfeited and the reality of stock variability accepted, there has been an 23 Table 3.1 Four categories of harvest strategies. Only the first author is listed. Proportion of Stock Control for Stock Size Control for Catch Control •for Effort Ricker 1958 Ricker 1958 Southwood 1968 Doubleday 1976 Larkin 1964 Larkin 1964 Walters 1975 Gatto 1976 Tautz 1969 Tautz 1969 Doubleday 1976 Beddington 1977 Doubleday 1976 Allen 1973 Lett 1977 Beddington 1978 Sissenwine 1977 Walters 1975 Beddington 1977 Mendelssohn 1980 May 1978 Doubleday 1976 May 1978 Ludwig 1980 Horwood 1982 Gatto 1976 Beddington 1978 Getz 1981 Horwood 1983 Mendelssohn 1980 Beddington 1979 Horwood 1981 Reed 1983 Trumble 1983 Horwood 1979 Swartzman 1983 Trumble 1983 Hilborn 1985 Getz 1980 Jacobson 1985 Armstrong 1984 Ruppert 1985 Mendelssohn 1980 Doubleday 1985 Deriso 1985 Ludwig 1980 Hilborn 1985 Hilborn 1986 Kirkwood 1981 Ruppert 1985 Swartzman 1983 Fried 1985 Trumble 1983 Deriso 1985 Armstrong 1984 Hilborn 1986 Doubleday 1985 Ruppert 1985 Jacobson 1985 Hilborn 1986 increasing reliance on stock forecasting to determine appropriate catch levels. Predictive assessment methods make it possible to carry out an exploitation rate harvest strategy even when q and E are variable; hence the emergence of stock specific exploitation rate strategies in the literature. A related form of the exploitation rate strategy is the control of fishing mortality (F). In the basic catch equation, C=N—^— ( l - e - < F + m>) (3.1) F + m ' ' F/(F + m) controls the proportion of the population removed by fishing mortality. This differs from the harvest rate strategy in that it is not necessary to know the total population size. The control ofF received moderate attention during the MSY period. Doubleday (1976) and Sissenwine (1977) are the first to use it in simulation studies. May et al. (1978) and Horwood (1982,1983) use different levels ofF to describe the results from their analytical \rork. The problem with this strategy, especially regarding herring fishing, is how F relates to fishing effort (E). Effort is the usual management control for F. The assumption is that F = Eq, where q, the catchability coefficient, is constant over time and stock sizes. It has been shown by many authors that this is a dangerous assumption for clupeid fisheries (MacCall 1976; Ulltang 1980; Shelton and Armstrong 1983; Crecco and Savoy 1985). Increasing gear efficiency, non-random search patterns and the characteristic schooling behavior of clupeids results in 24 higher q at lower stock sizes. The alternative is to control F by some means other than effort. This becomes a question of harvest tactics, not strategy. Control for Stock Size Ricker (1958) shows that maintaining an "optimum'' stock level will produce maximum average long term catch. The optimum stock size is maintained by removing all stock above the optimum as catch. If the stock size is below the optimum, no catch is taken. Ricker (1958) recognizes that the problem of controlling stock size is that all variation in population abundance is transmitted directly to the catch. This led to the introduction of strategies that control for stock size but soften the effect of optimum fixed escapement (Allen 1973; Walters 1975; Mendelssohn 1980). Recently Ruppert et al. (1985) define an "egg escapement policy" that is is similar to the management objective for British Columbia roe herring fishery for 1972-1981 and the constant escapement policy defined below. Control for Catch The harvest strategy type that has received the most attention is the control of catch. There are two reasons. First, for most fisheries, catch is easy to measure and provides a clearly defined objective. Second, since much of the harvest strategy literature revolves around the concept of MSY, the objective of removing a sustainable yield has been a popular notion. This is not to sjy that the harvest strategy literature has focussed myopically on MSY. Both analytical and simulation studies criticize MSY policies as biologically dangerous (Doubleday 1976, 1985; Beddington and May 1977; May et al. 1978; Horwood et al. 1979; Kirkwood 1981; Swartzman et al. 1983; Armstrong 1984). True constant catch policies have zero variance, but such policies are generally unrealistic and destabilizing in variable environments (Shepherd and Horwood 1979). Catch stabilizing policies below the MSY level are better at preserving the population, but at the price of large reductions in long term average yield. In the preliminary analysis for my thesis I included a constant catch harvest strategy. (Before 1966 the British Columbia herring fisheries were managed by catch quota limitations and seasonal closures.) My prelimnary findings agreed with results in the literature. To sustain a true constant catch (zero variance) the annual yield had to be maintained at a level that drastically reduced the long term average yield compared to other strategies. At catch levels approaching the average maximum sustainable yield, the strategy lost its appeal of low catch variance. The quota strategy performed so poorly under all criteria of evaluation that I eliminated it from the study. 25 Control for Effort The strategy of effort control is slightly different from the strategy for removing a proportion of the stock (assuming F = Eq). Since it receives separate treatment by many studies I list the references in table 3.1 as a separate category. Constant effort strategies have generally been accepted as intermediate solutions to catch maximizing and variance minimizing strategies (Doubleday 1976). Problems of effort control in a herring-like fishery were mentioned in the proportion of stock removal section. In addition to the problems associated with q, fleet size is critical. In British Columbia the number of licenses issued to fish herring is fixed, but at a fleet size that is considerably above that which can efficiently remove the harvestable surplus. The primary method of catch control is the amount of time given a fishery. This has led to extreme tactical choices, such as 15 minute seine fisheries (J. Broome, DFO, Nanaimo, pers. comm.). The combined characteristics of high catchability and over-capitalization make the strategy of effort control impractical for B.C. herring. Strategy Definitions For the following definitions "the population" refers to that part of the total population that Has matured and is taking part in onshore spawning ground migrations. This is the segment of the total population that is available for capture by the roe herring fishery. The term "constant" means that strategies do not change between years. • Harvest Rate The constant harvest rate strategy removes a fixed proportion of the population, C = Np, where p is the harvest rate. The remaining population (/V(l — p)) spawns. A harvest rate is also referred to as an exploitation rate. • Constant Escapement The constant escapement strategy controls for population size. A constant spawning biomass that is sufficient to sustain the population is determined. If the population is above the level required for the spawning biomass. the surplus is removed as yield. If the population is below the spawning biomass requirement no catch is removed from the population. This strategy is also known as a fixed escapement policy. Figure 3.1 graphically depicts the differences between the strategies for catch and spawning biomass as functions of population size. 26 H a r v e s t Rate Constant Stock Figure 3.1 Graphical depiction of the changes in spawning biomass and catch with increasing stock size under the harvest rate and constant escapement strategies. Table 3.2 Measures of evaluation criteria. Reaction to Catch mean equilibrium variance st. dev. prop, of equilibrium range discounted catch/variance CV T h e Evaluat ion C r i t e r i a To compare different harvest strategies a set of evaluation criteria are needed. Table 3.2 lists the criteria that have been used in previous studies. The specific criteria are grouped into catagories measuring different aspects of the population and management system. The important aspects of each category are discussed with appropriate references to the literature. 27 Stock Perturbation Risk Effort, CPUE mean st. dev. range CV restoring force return time stable/unstable zero catch stock size less-than critical level mean time to-re ach critical level oppurtunity loss log-utility function mean CV Catch Criteria For obvious reasons catch is the most widely used evaluation criterion. The .most common form of the measurement is the mean catch accompanied by some description of dispersion (range, variance, standard deviation, or coefficient of variation). Doubleday (1976) and Sissenwine (1977) compare simulated catches to analytical maximum equilibrium catches. Allen (1973), Ludwig (1980), Mendelssohn (1982) and Ruppert et al. (1985) use catch discounted over time. Mendelssohn (1982) concludes that optimal harvest strategies and the resulting population dynamics of his model are insensitive to changes in the discount factor. Ruppert et al. (1985) state that discounting does not change the qualitative differences between strategies. Several studies consider the tradeoffs between maximum catch and an acceptable level of catch variance (Walters 1975; Beddington and May 1977; Mendelssohn 1980; Horwood and Shepherd 1981; Ruppert et al. 1985). Their conclusions are usually summarized in a figure of mean catch versus mean variance, the curve defining the set of highest achievable mean—variance combinations. Ruppert et al. (1985), Deriso (1985), and Hilborn (1985) use the sum of the natural logarithm of catch as a measure of a risk adverse management objective. Stock Criteria The mean stock size for a given harvest strategy is often used as a measure of population response. The variance, range, standard deviation, or coefficient of variation (CV) have also been used as criteria. Population Reaction to Environmental Perturbations A considerable amount of analytical and simulation work has tried to measure the reaction of a population to changes in its environment, and how the changes are affected by harvesting. The problem has been to find an adequate summary measure of the patterns. Doubledjry (1976) measures what he calls the restoring force, an index of population ability to increase net production when displaced from an equilibrium. The related index is the return time (TT), first introduced by Beddington and Msf (1977) in the harvest strategy literature; it measures the "typical time it takes the system to recover from a small disturbance". They concluded that TT increases with increasing harvesting pressure up to the level of MSY. Shepherd and Horwood (1979) show that the conclusions of Beddington and May (1977) depend on the regulating mechanisms of the population and the method by which the environmental perturbation is introduced into the model. They describe an alternative scenario where population stability (as measured by a decreasing function of TV) increases with harvesting. The 28 measure of return time is ambiguous depending on the population model used. Unfortunately its use continued (May et al. 1978; Beddington 1978; Kirkwood 1981). The last appearances should mark the end; Horwood and Shepherd (1981) and Reed (1983) declare Tr an ambiguous and poor measure of population response. Criteria of Risk The inadequacy of return time as a measure of population response led to a search for other criteria. It was recognized that interest in population fluctuations about an equalibrium might be overly optimistic. In cases of sufficient environmental variability, a population may never approach some stable equilibrium. Instead, a steady decline in population size is often observed in the simulations, especially at high harvesting intensities. Beddington (1978) introduced the mean time (t ) to reach a critical minimum population level. Beddington and Grenfell (1979) declared the mean time criterion to be insufficient after Horwood, Knights and Overy (1979) produced misleading results based on t. In its place Beddington and Grenfell (1979) propose the probability of the stock declining to reach "protected status" over time as the alternative measure. Getz and Swartzman (1981) and Getz (1984) have continued to use this criterion as a measure of risk. It has the advantage of being useful to both management and fishermen because it is a real measure that management uses to control fishing. Ricker (1958), Larkin and Ricker (1964) and Allen (1973) used the number of years with no catch as a measure of risk. This criterion has real meaning to management, fishermen, and the fishing industry. Recently Ruppert et al. (1985) have borrowed the concept of "opportunity loss" from decision theory, a convenient measure of catch with perfect information minus the catch with management uncertainty. Miscellaneous Criteria Evaluation criteria that measure effects on fishing have received little attention. Southwood (1968) compared the mean and variance of catch per unit effort (CPUE) for three assessment procedures in the Pacific halibut fishery. In a simulation study of the Pacific whiting fishery, Swartzman et al. (1983) compare the mean and variance of effort and CPUE for quota and constant effort strategies. Only one study (Southwood 1968) considers the relative cost of the different management schemes as a possible criterion. 29 Evaluation Criteria Definitions The criteria I have chosen for evaluating the two harvest strategies are now defined. Following the pattern of other harvest strategy studies, I use mean catch as a primary criterion. Catch variance is measured for mean/variance tradeoff curves. The coefficient of variation of catch compares catch variance between different levels of the same harvest strategy and between different harvest strategies. Risk is measured by two criteria. The first is the risk of the stock size decreasing below a defined critical level; the usual risk of concern to management. Type 2 risk is that incurred by fishermen; the risk of zero allowable catch and no fishing. The following five specific criteria are used to compare the two strategies. 1. Mean Catch — The catch is summed for all years and runs for a given harvest strategy level, and divided by the product of the number of years and runs. NR NY i C = _C Ci,k I NY • NR (3.2) where C is the catch, NY the number of years and NR the number of runs. 2. Catch variance — E c2 - ((E C?/N) Var = ^ / (3.3) where ^ 2 C is the sum of catch over all years (NY) and runs (NR), as in eq. 3.2, and Ar is the product of NY and NR. 3. Coefficient of Variation of Catch — The standard deviation of the catch is divided by the mean catch, CV = VVar/C. 4. Risk of Subcritical Stock Size — The critical stock size is 20000 mt, approximately 20% of the biomass at the start of each simulation. The number of years in the simulation that the stock size falls below this level are summed. The total is expressed as a proportion by dividing with N. 5. Risk of No Fishing — The number of years that catch is zero are summed and divided by N. Model Test Standards The harvest rate strategy was tested at 17 levels ranging from 0.05 to 0.85 in 0.05 increments. A zero harvest rate was included to measure unfished stock biomass. The constant escapement strategy ranged from 7500 mt to 120000 mt in increments of 7500 mt. Each strategy level was tested over 15 years after a 15 year stablizing period, during which no measurements were made. The 15 year tests were repeated in 100 runs; for each policy level the total number of years was 1500. 30 Pseudo-random normal deviates (mean 0, standard deviation 1) were generated from a FOR-TRAN library subroutine (UBC Computing Centre documentation RANDOM, 1981). Mean and standard deviation were adjusted to suit particular distributions by the general formula i3 m, 8d = mean •+ st. dev. • Z>o,i • DFO's C U T O F F Policy At the 1985 Herring Stock Assessment Committee meeting the Herring Section of the Pacific Biological Station introduced a safety measure to protect endangered stocks (CUTOFF policy; Stock er 1985). The study calculates an unfished equilibrium biomass by simulation and graphical procedures based on a Ricker stock-recruitment relationship. The critical biomass is arbitrarily set at 25% of the unfished equilibrium, or 20800 mt for the combined Strait of Georgia and Johnstone Strait herring populations. The policy, which is in effect for the 1986 roe fishery, stipulates that no fishing will take • place if the forecast population is below the critical stock biomass. At forecast stock sizes "marginally above" (not defined) the CUTOFF level, the catch equals the forecast run minus the critical stock biomass. At stock sizes "well above" the critical level a 20% harvest rate is used. The CUTOFF policy provides a documented method for a preseason justification of a fishing closure. The policy insures a stock buffer against over harvesting. It combines the safety of the constant escapement strategy and the catch variance reducing feature of the harvest rate strategy. To simulate the CUTOFF policy over a range of harvest rates and escapement goals I use 20800 mt as the minimum spawning biomass. To insure that with harvesting the population never drops below this level, I define the upper boundary of the stock size region "marginally above" the CUTOFF level as the biomass that satisfies the inequality Stock — hr • Stock < Critical Biomass (3.4) for the harvest rate strategy and Stock — Escapement Goal < Critical Biomass (3.5) for the constant escapement -strategy. Stock sizes above this upper bound are harvested at the going rate. 31 Chapter 4 Results of the Basic Model The results are presented in a series of graphs. To ease comparisons between different versions of the model a consistent format is maintained .There are four figures to describe each version of the model. The first three compare fixed escapement and fixed exploitation rate strategies by the criteria of evaluation; mean catch, catch coefficient of variation (CV), and risk. The fourth figure plots mean/variance diagrams for the strategies. The first three graphs use mean spawning biomass as the coordinate measure for the abscissa. Spawning biomass is averaged over all years and runs of the model (similar to mean catch, eq. 3.2). Annual spawning biomass varies widely for the harvest rate strategy. For example, in figure 4.1 the 90% limits are 36200 mt and 107700 mt for the 0.30 harvest rate. I emphasize this point to clarify the use of the figures. They are not intended as short term management harvest guidelines. It is incorrect to say that if the forecast spawning biomass is 60000 mt the expected catch is 28000 mt. It is correct to say that if the population is managed to maintain an average spawning biomass of 60000 mt then the long, term average catch will be around 28000 mt. The relationship between escapement levels for the constant escapement strategy and the mean spawning biomass (abscissa) is exactly proportional for low escapement goals. As the escapement goal increases above 20000 mt the probability that the spawning biomass is less than the escapement goal increases. The extreme of this effect is shown in figure 4.1 by a crowding of points about a mean spawning biomass of 100000 mt for the constant escapement strategy. The strategies in this group call for escapements of 105000 to 120000 mt, but the population can average no better than 104000 int. Along the top edge of most figures is a scale for the harvest rate (increments are not evenly spaced). This scale corresponds directly to the 17 harvest rate levels. The maximum values for the abscissa and ordinate scales are consistent between graphs. The axes labels for mean stock size and mean catch are in thousands of metric tons. Catch CV and risk are unitless measures. The plus sign (+) is the symbol for the constant escapement strategy (CE). The "o" is the symbol for the harvest rate strategy (HR). Each point is the measured statistic (mean, CV, risk, variance) for 1500 simulated years. Results from three versions of the basic model are presented in this chapter. The first version has recruitment variation controlled by independent random deviations (white noise). In the sec-ond version recruitment deviations are structured with autocorrelation. The third version considers 32 DFO's CUTOFF policy of maintaining a minimum spawning biomass of 20800 mt with autocorrelated deviations. Basic Model, Independent Deviations White noise is the most common form of induced variation in harvest strategy models. Devia-tions from the recruitment function are modeled as independent and identically distributed variables. The results are consistent with previous studies. The constant escapement strategy provides signifi-cantly higher average catches over a wide range of stock sizes, but at the expense of increased variation in catch (figures 4.1—4.2). High average catches for the constant escapement strategy result from this strategy capitalizing fully on large positive recruitment fluctuations that characterize clupeid fisheries. The relationship deteriorates at extreme stock levels. As escapement goals approach and exceed 80000 mt the population is frequently unsuccessful at meeting the demand, and catch is zero for these years, lowering the mean catch and drastically increasing the catch variance for the constant escapement strategy. At low stock sizes mean catch is the same for both strategies. Low stock sizes produce fewer recruits, and any large positive recruitment fluctuations are cropped as catch by high harvest rates and low escapement levels. The difference between strategies is most apparent in the criterion of catch variance (figure 4.2). For the harvest rate strategy, catch CV gradually increases with increasing harvest rates. As harvest rates increase above 0.60, catch more closely mimics the highly variable population fluctua-tions, increasing catch variance. The dampening effect of the spawning biomass absorbing population fluctuations is reduced. For the constant escapement strategy, all variation in population size is trans-mitted to the catch variance. Catch CV increases with escapement level, as the population becomes less likely to meet higher escapement goals. Figure 4.3 displays the criteria of risk. Risk is measured as the proportion of years that the population falls below a critical stock size of 20000 mt. For the constant escapement strategy, this measure is 1.0 for all escapement goals set below the critical level. As harvest rates increase above 0.45 the proportion of years that the spawning stock falls below 20000 mt increases rapidly. The third line on figure 4.3 (dotted) describes the risk of not fishing. Type 2 risk affects only the constant escapement, strategy, since the harvest rate^trategy always removes some fish. As escapement levels become higher, the probability of meeting the goal decreases. Fishing is shut down in years of inadequate escapement. 33 Harvest Rate .4 .3 m u c ID 0) X 40 -30 20 10 40 60 60 Mean Spawning Biomass 120 Figure 4.1 Results from the basic model (random, uncorrected deviations). The constant escape-ment strategy (CE) provides the highest mean catches over a wide range of management options. Catch and spawning biomass are in thousand mt. Error bars are ± one standard error of the mean. Each point is the mean of 100 runs of 15 years (n=1500). > u sz u m u .3 —i— Harvest Rate .4 .3 • CE 0 HR • -•- + . + - + Jf • — • a • • — i — 20 1 1 1 40 60 60 Mean Spawning Biomass - i — 100 120 Figure 4.2 Catch coefficient of variation for both strategies under the basic model. The harvest rate CV is considerably less over the range of realistic management options. Mean spawning biomass is in thousand mt. Error bars are ± one standard error of the coefficient of variation. 34 Harvest Rate . 4 .3 10 c CD >• c o •rl 4-> t_ o a o c a. 120 Mean Spawning Biomass Figure 4.3 Proportion of years that the spawning biomass drops below 20000 mt as a measure of risk to the stock. At low harvest rates and high escapement goals the risk is zero. The dotted line is the proportion of years of no fishing for the constant escapement strategy (CE). Mean spawning biomass is in thousand mt. 0 0.2 0.4 0.B 0.B 1.0 1.2 1.4 Catch Var iance ( x ! 0 B ) Figure 4.4 The mean—variance curves for both strategies under the basic model. Peak catches from the harvest rate strategy are marginally below the constant escapement strategy (CE), but the catch variance reduction is two-threefold. Mean catch is in thousand mt. Points are simulated mean values for 1500 years at each strategy level (HR, 0.05 to 0.85 by 0.05 increments; CE, 7500 mt to 120000 mt by 7500 mt increments). Smoothed curves are passed through the means. 35 It is included on the figure as a reminder that the constant escapement strategy suffers from this additional element of risk. Mean catch is plotted against catch variance in figure 4.4, and a smoothed curve (Tukey 1977) is overlayed to define the mean/variance boundary region. The points are mean catch and catch variance for 1500 simulated years for each strategy level (harvest rate 0.05 to 0.85 by 0.05 increments; constant escapement 7500 mt to 120000 mt by 7500 mt increments). The curves tend to form a closed loop as harvest rates and escapement goals increase. The differences between the two strategies are striking. For a small sacrifice in mean catch, large gains can be made in reduced catch variance for the harvest rate strategy. For example, comparing maximum average yields and associated variances, a 78% reduction in catch variance can be achieved with only an 11% reduction in mean catch. Basic Model, Correlated Deviations The discussion of subsequent figures will concentrate on comparing versions of the model. The basic reasons for differences between strategies have been explained in the previous section. To reiterate the important points: 1. The constant escapement strategy provides higher average yields over a wide range of stock sizes, but at the expense of increased catch variation. 2. The harvest rate strategy reduces catch variance by transmitting inherent population variability to the spawning stock. 3. The constant escapement strategy maintains higher average yields by fully utilizing large positive fluctuations in stock abundance. These qualitative results do not change for the other versions of the basic model, but the quantitative results do change. The discussion will focus on the reasons for differences between model versions. Figure 4.12 combines figures 4.1-4.5, figures 4.9-4.11, and results from the CUTOFF model version onto one page to ease comparisons. Figure 4.5 shows that the difference between the strategies is greater because the mean catch at all stock sizes for the harvest rate strategy are significantly decreased. The explanation depends on understanding the processes of random versus serially correlated deviations and the resulting popula-tion structure. Figure 4.6 shows the two types of deviations for the same sequence of random normal deviates over a 30 year time period. The amplitude is almost the same, but there is "periodicity" in the correlated deviations. The effect is to "spread out" the deviations; there is a persistence to high 36 H a r v e s t Rate .9 .6 .3 .4 .3 .2 .1 0 0 20 40 60 60 100 120 Mean Spawning Biomass Figure 4.5 Results from the model including serially correlated deviations. Mean catch for the har-vest rate strategy (HR) is decreased compared to the basic model (figure 4.1), increasing the difference between strategies. Mean catch for the constant escapement strategy (CE) is slightly reduced. Catch and spawning biomass are in thousand mt. Error bars are ± one standard error of the mean. Autocorrelated Noise White Noise c o *J ID > a a Annual Time Step Figure 4.6 A time series of simulated deviations showing the difference between random (white noise) and second order correlated deviations. The correlated deviations show a rough periodicity, creating a platykurtotic distribution of deviations. 37 70 60 -SO -C 40 H u *i re U 3Q 20 10 -o C o r r . • whie* • o -r .3 .4 Harvest Hate i 1— 9 .6 i i — I — .8 Figure 4.7 The effect of correlated deviations (o) on catch over a range of harvest rates versus the white noise model version (+). The points are medians. Upper lines range from 75th to 95th quantiles; lower lines from 25th to 5th quantiles. The distribution of catches from the model with correlated deviations (right point-line set) are more skewed and more disperse. Catch is in thousand mt. i n re E a -i-i m a c •r-1 c X re a m 200 ISO -100 so -o C o r r . • M l i U ' T — i — .6 — I — » 8 —1— Harvest Rate Figure 4.8 The effect of correlated deviations on spawning biomass. The spawning population shows increased skewness and dispersion with correlated deviations, the same effect as for catch (figure 4.7). 38 and low deviations, with less clumping about the middle. The effect is transmitted to both catdi and population size. Figures 4.7 and 4.8 compare distributions for catch and spawning biomass for white noise and autocorrelated noise over a range of harvest rates. Both mean catch and mean spawning biomass are more dispersed under correlated deviations. Catch and population variance are increased. A side effect of increased dispersion in spawning biomass is reduced average recruitment compared to the white nobe model. Spawning biomass is more often on the tails of the Ricker curve and less frequently in the more productive middle range. This produces a negative feedback on population size and lower average yields. Tautz et al. (1969) identified the detrimental effects of compensatory losses under a fixed percentage management scheme. Recruitment losses due to low spawning biomass are equally serious. The constant escapement strategy avoids this problem by fixing spawning biomass, reaping the benefits of persistent high recruitment and suffering no worse catches than the white noise model during lean years. The pattern for catch variance (compare figure 4.9 with 4.2) is similar. Catch CV for the correlated deviation model is about 30% higher over the range of harvest rates, and 20-30% higher for the constant escapement strategy. For the criterion of risk (compare figure 4.10 with 4.3), the chance of the spawning stock dropping below 20000 mt increases slightly with correlated errors, since the spawning biomass varies more under a structured recruitment pattern. The difference in the mean/variance diagram is more pronounced (compare figure 4.11 with 4.4). Catch variance for both strategies is doubled. For the same example cited in the previous section, an 18% catch reduction now reduces catch variance by 72%. Basic Model, DFO C UTOFF Policy Results from the CUTOFF policy are presented in figure 4.12, column 3. The CUTOFF policy has changed the rules. The spawning stock is not permitted to decrease below 20800 mt in any year. The first thought might be that this change in policy would tend to lower average yield, since in many years catch level will be reduced to maintain the minimum spawning biomass. Catch is reduced in many years, but the benefit of maintaining a minimum spawning reserve of 20800 mt is a long term positive effect on mean catch. Under the CUTOFF policy high harvest rates are not possible, since stock size is governed by the inequality (equation 3.4) to never drop below 20800 mt. Above 0.30 the 39 > CJ sz u o CJ .7 .6 Harvest Rate .4 .3 tn—i 1 1 1 1 1 1 1 1 — 1 1 i — i — i • CE .0 HB 1 1 i i / i * + at T " ~ — • • . 0 20 40 60 80 100 120 Mean Spawning Biomass Figure 4.9 Catch coefficient of variation with correlated deviations. The CV's for both strategies are higher, but maintain their relative positions. Mean spawning biomass is in thousand mt. Error bars are ± one standard error of the coefficient of variation. Harvest Rate • B .6 .s .4 .3 .2 .1 o 0 20 40 60 80 100 120 Mean Spawning Biomass Figure 4.10 Risk measurements under correlated deviations. The increased dispersion of stock sizes shows up as increased risk in the harvest rate strategy (HR) and an increase in the proportion of years of no fishing for the constant escapement strategy (CE) (dotted line). Mean spawning biomass is in thousand mt. 40 Figure 4.11 The mean—variance curves under the model with correlated deviations. Both strate-gies show a significant increase in catch variance. Mean catch is in thousand mt. Points are simulated mean values. Smoothed curves are passed through the means. CUTOFF rule is effective. Actual harvest rates at strategy level objectives of 0.50 and 0.70 are 0.48 and 0.56. Above the CUTOFF level the curves are nearly identical to the correlated noise model. The curves comparing the criterion of catch variance are also similar. The ascent of the harvest rate curve b held in check by the CUTOFF limit. For rbk the curves are shifted slightly to the right as stock size nears the CUTOFF level. The grouping about the CUTOFF limit b clear in the mean/variance figure. Thb area of the graph highlights an interesting comparbon between the strategies. It would seem that the strategies are almost equivalent at mean yields of about 30000 mt and variance about 4.0 x 10 8. However, by referring to the mean catch figure (row 1, column 3) to find the associated stock size (about 22000 mt) and then to the rbk figure (row 3, column 3), thb region b a location to avoid if we are concerned about the stock size falling below the critical limit. Away from thb region the differences between the strategies provide clearer choices. 41 0 4 0 8 0 1 2 0 M e a n S p a w n i n g B i o m a s s 1 . 7 8 . 8 . 2 8 1 \ 0 .1. • . 0 4 0 8 0 1 2 0 M e a n S p a w n i n g B i o m a s s 0 4 0 8 0 1 2 0 M e a n S p a w n i n g B i o m a s s 0 . 4 0 . 8 1 . 2 Variance (xlD ) 0 0 . 4 0 . 8 1 . 2 Variance (xiO9) Variance (xlO ) Figure 4.12 A composite figure of the results from chapter 4. Column 1 panels are from the basic model (random deviations). The figures in column 2 are from the model with correlated deviations. DFO's CUTOFF policy is compared in column 3. The rows correspond to the general figure types: row 1 is mean catch, row 2 is catch CV, row 3 is risk and row 4 is mean—variance. The dashed line is the constant escapement strategy. 42 Chapter 5 Herring Stock Assessment Model results so far have been obtained by ignoring the observation error that plagues analysis in the real system. Harvest rates and escapement goals have been set as though there was perfect information about stock size. The real situation is far from perfect. Fisheries management is full of uncertainty; partial information from collected measurements is used to extrapolate an assessment of the entire population. Each measurement has error. The propagation of error from the individual measurements to the final assessment has received little attention in fisheries research (Wilimovsky 1985). The purpose of this chapter is to analyze field data used for stock assessment and prediction so as to develop a realistic model of stochastic errors in the assessment process for herring. For British Columbia herring, the most important description of the population is the predicted stock size for the next fishing season. There are at least eight measurements that can be used in combination to make this prediction. . 1. total catch 2. age composition of catch 3. spawn length 4. spawn width 5. spawn egg layers (density) 6. weight at age 7. fecundity at age 8. percent female In addition to these annual measurements, there are parameters that are updated periodically, such as annual survival rate and average recruitment. DFO currently uses two methods to forecast stock size. The age-structured model was designed by Fournier and Archibald (1982) and modified by Haist and Stocker (1984). The age-structured model is a complex, predictive catch-at-age analysis, with the unique ability to incorporate errors in aging, though this extension is not utilized in current assessment procedures. The second method of assessment is the escapement method (Schweigert and Stocker 1986). The number of eggs spawned in year tis used to predict the biomass of repeat spawners in year t+1. The biomass of first time spawners (new recruits to the fishery) is estimated as the average biomass of recruits from historical records, plus or minus one standard deviation as environmental conditions and other indicators warrant. In both methods recruitment to the fishery is unknown, but may account for over 60% of the fishable biomass. 43 In the following analysis I consider only the escapement method. The equations of this method are simple enough to summarize in a few lines of FORTRAN code. The age-structured model is too complex for similar treatment. Attempts to conduct a verification of the age-structured model (by entering known data from my model into the age-structured model) met with logistic failure. The escapement assessment method is outlined in the first section. In the second section I describe the measurements of spawn deposition; field measurements of length, width and egg layers form the basis of the escapement assessment method. These are the only measurements for which there is any information on associated error. The data are from a series of dual fishery officer— SCUBA diver surveys that have been conducted since 1975 (Haegele, Humphreys and Hourston 1981). Spawn length, width and egg density are used to estimate the spawning biomass for each measured spawn.. The distributions for diver and fishery officer estimated tonnage are compared. The third section compares simulated error distributions with the observed distributions and describes how the simulated measurement errors are generated in the model. The Escapement Method of Stock Assessment Pre-fishery biomass in year t + 1 is predicted from spawning biomass in year t by where B is stock biomass and S the average survival rate across ages (0.64 in recent years; Schweigert, pers. comm., 1985). Recruitment is estimated from the historical average (1951-1985 or 1972-1985). Adjustments are made to the average recruitment (± one standard deviation) if it is suspected that recruitment will be either above or below average. Examples of altering conditions are: recent El Nino events; high or low recent growth rates; poor spawn deposition in natal years; and high incidence of juvenile age classes in the fall food and bait fishery. The estimate of the spawning biomass (Bt) is the crux of the assessment. The number of spawning adults are back calculated from the number of eggs laid. Assuming a 1:1 sex ratio and 200 eggs/g of female weight (100 eggs/g for both sexes; Hay 1985) the biomass in metric tons is estimated Bt+i = Bt S + recruitment (5.1) by DFO as Bt = Eggs /(100 eggs g"1 )(lxl0-6mt g"1) (5.2) The total spawning biomass is the sum of the spawn estimates for all spawns surveyed. 44 Estimating Spawn Deposition The number of eggs in any one spawn is estimated by the product of the area of spawn coverage and the density of egg deposition. Area (m2) is most often simply the length (m) times the width (m) of the spawn. Unfortunately, the methods of measurement have changed over time, confounding the spawn deposition history of B.C. herring. Mean spawn length has decreased steadily since 1951, but it is not clear if this is a real difference or an artifact of changing measurement practices (Hay and Kronlund 1985). Mean width has increased since the mid 1970's, corresponding with the beginning of diver spawn surveys and the knowledge of extensive subtidal spawn that was missed with surface measurements (Haegele et al. 1981). As fishery officers became aware of the possibility of subtidal spawn they began looking for it (with grappling hooks, towed rakes, and view boxes), significantly increasing width measurements. It is reasonable to assume that current spawn measurements are as accurate as the methods allow. The simplest way to estimate the accuracy of each measurement would be to take repeated measurements on each spawn. No such data are available. DFO adjusts for measurement error by defining functional relationships between observed fishery officer and diver data from the dual surveys. The assumption is that diver measurement error is insignificant compared to fishery officer measurement error. This assumption appears valid, considering sampling techniques of the two groups. It is much easier for a diver to determine where a spawn ends, and to collect a random sample of egg-covered vegetation, than it is for a fishery officer using a grappling hook. The data for the dual fishery officer—diver surveys are presented in figures 5.1-5.3. The fishery officers slightly under-estimate spawn length, under-estimate spawn width, and substantially over-estimate the number of egg layers. DFO recognizes and attempts to adjust for these biases, except in measurements of spawn length (Schweigert and Stocker 1986). For width measurements, they use correction equation (based on data in figure 5.2) ]n(W) = 0.297 ln(Wp0)+ 3.948 (5.3) where the variable W is the adjusted width measurement, Wfco the fishery officer measurement, 0.297 a regression parameter, and 3.948 an area specific parameter for the Strait of Georgia (Schweigert, pers. comm., 1985). The correction for egg layers is more complicated. The measurement is a short cut to estimating egg density. Determining egg density per square meter is a time consuming and costly operation. 45 E x: *J 01 c -I o 10 8 -2 4 6 8 Diver Length (km) Figure 5.1 Fishery officer versus SCUBA diver measurement of spawn length in meters (n=65). Data are from dual surveys, 1977-1985. Solid line represents 1:1 ratio (no error). A regression model has been developed that predicts egg density from diver egg layer observations (Schweigert and Fournier 1982). DFO combines two conversion steps (fishery officer egg layers to diver egg layers to egg density) into a single equation (Schweigert, pers. comm., 1985) Y = 75.86 + 105.32 LYFO + e (5.4) where Y is egg density, and LYro the fishery officer egg layer count. The adjusted widths and densities are displayed in figures 5.4 and 5.5. The obvious bias has been alleviated, but a another bias is now more apparent. The figures suggest that fishery officers overestimate short measurements and underestimate long measurements. The bias is apparent in all three spawn measurements, and is reflected in the plot of fishery officer versus diver spawning biomass estimates (figure 5.6) and isolated in figure 5.7. Figure 5.6 suggests that fishery officer tonnage estimates are practically independent of the stock size that is measured more accurately by diver surveys. 46 600 300 100 200 Diver Width (m) 600 Figure 5.2 Fishery officer versus SCUBA diver measurement of spawn width in meters (n=65). Data are from dual surveys, 1977-1985. Solid line represents 1:1 ratio (no error). 47 0 100 200 300 400 SOO BOO Diver Width (m) Figure 5.4 Corrected fishery officer width measurements of spawn deposition versus SCUBA diver width measurements in meters (n=65). Solid line represents 1:1 ratio (no error). 0 4 B 12 16 Diver Egg Density (xlO3) Figure 5.5 Egg density estimates for fishery officers versus diver estimates (n=65). Fishery officer estimates are from regression equation (5.4). Diver estimates are from egg counts (1975-1983) and an egg prediction model (1984-1985; Schweigert and Founder 1982). Solid line represents 1:1 ratio (no error). 48 a* u c <u c 0) CO 0) ID E O m -2 Diver Spawning Biomass Figure 5.7 The difference in spawning biomass estimates (corrected tons FO — tons diver) plotted against diver spawning biomass estimate. The bias of fishery officers over-estimating tonnage for small spawns and under-estimating tonnage for large spawns is apparent. Biomass is in thousand metric tons. 49 Simulation of Biomass Estimation Error To model the effects of measurement error I used the relationship between fishery officer and diver spawn surveys described in the previous section. The process combines the errors of each spawn measurement—length, width, and egg layers—into three probability distributions that relate to spawning biomass estimates (figures 5.6 and 5.7). 1. The diver spawning biomass estimate. 2. The fishery officer spawning biomass estimate. 3. The percent difference between the fishery officer and diver estimates. These real distributions are simulated with log-normal approximations; these were compared to actual distributions by the Kolmogorov-Smirnov test for goodness of fit. The simulations were considered ad-equate when the null hypothesis of like distributions was accepted in at least 80 out of 100 comparisons (a = 0.20). The error is accumulated from individual spawns. The number of spawns per year varies with stock size, ranging from 0 to 200. In every simulation year the procedure is as follows: 1. A known spawn biomass (diver) is chosen from a truncated log-normal distribution (nor-malized mean = 6.55, standard deviation = 2.10, biomass < 9000 mt). 2. The percent error is selected from a truncated log-normal distribution, (normalized mean = 4.80, standard deviation = 1.20, -98.0% < percent error < 1000%) The percent error is further constrained (percent error • tons spawned < 414000) to model the bias of decreasing percent error with increasing spawn tonnage. 3. The estimate of spawn biomass by fishery officers is calculated as the known biomass ± (percent error • known biomass). This relationship is constrained to mimic observed data (18 mt < FO tonnage < 4500 mt). This procedure continues until the sum of diver spawn estimates (step 1) is about equal to the known spawning biomass (eq. 2.8). The effect of the combined errors and associated bias is twofold. First, a false sense of precision in total spawn assessment is generated. The variance of all surveyed spawns is less for fishery officers than divers, since fishery officers tend to overestimate small spawns and underestimate large spawns. Second, the overall tendency is to underestimate the true total spawning biomass. This critical error is a result of the underestimation of large spawns, which has a greater effect than the overestimation of small spawns. The effect of these biases on the harvest strategies is examined in the next chapter. 50 Chapter 6 The Population Model with Stock Assessment Error Observation error is included in the model by using information from uncertain stock assess-ments to set annual simulated catches. The management rules to use the simulated assessments are the same as in chapter 4. The harvest rate strategy is to remove some fixed proportion of the esti-mated mature population. The constant escapement strategy is to remove estimated surplus above the escapement goal. The difference is that the true population size is unknown. Harvest rates and escapement goals are applied to an estimated population. The only additional constraint is that the catch cannot exceed the true population. It is important to understand the flow of misinformation through the model. Allowable catch is determined from an uncertain stock size, but the catch that is removed is recorded without error. The reproductive population is whatever part of the true population that avoids capture. For example, suppose the prefishery stock estimate is 30000 mt. The forecast catch is 6000 mt with a 0.20 harvest rate, and the remaining 24000 mt are to spawn. However, the true returning stock is 40000 mt. The forecast catch is still removed, leaving 34000 mt to spawn at a true harvest rate of 0.15. Recalling equation 5.1, the forecast population is a function of the previous year's spawning population plus recruitment. The recruitment forecast is the weakest part of DFO's assessment. It would be simple to model recruitment prediction error by comparing published forecasts with results from the cohort analysis, but the added noise \wuld obscure the effects of the spawn deposition measurement errors. Instead, I provide the DFO forecast in the model with the true number of age 2 and 3 recruits. This isolates the effect of spawn measurement errors on the Bt S component of eq. 5.1. The results of the extended model are presented in the same format as results from the basic model. The objective is to determine how uncertainty alters the results of the basic model. Additional figures comparing and explaining the results of the two models are presented. A second section examines three general patterns of stock assessment uncertainty and evaluates the effect of each. Results of the Extended Model The standard results are presented in figures 6.1-6.4; these results are qualitatively consistent with the basic (no uncertainty) model. The constant escapement strategy still maintains the highest mean catches and highest catch variance. However, there are interesting quantitative differences 51 between the two versions of the model. The mean catch curve for constant escapement (figure 6.1) is decreased. The coefficient of variation (figure 6.2) is decreased for the constant escapement strategy, except in the extreme escapement goals. The probability of the spawning biomass dropping below 20000 mt is increased for the strategies (figure 6.3). The harvest rate strategy still provides significantly reduced catch variance for now an even smaller sacrifice in mean catch (figure 6.4). Catch variance for the constant escapement strategy is about halved for escapement levels above 20000 mt, and about doubled for low escapement goals (figure 6.4). The explanations for the differences between the two model versions are presented in figures 6.5-6.8. The distributions for pre-fishery stock size and catch are compared over a range of management options for the basic and extended model versions. Uncertain stock assessments are negatively biased and less variable than the true population (figures 6.5 and 6.7). Catch determinations made from biased assessments reflect this pattern (figures 6.6 and 6.8), except that the relationship deteriorates at high harvest rates and low escapement levels. Without uncertainty, harvest rates and escapement goals are exactly met. With uncertainty, management objectives are not obtained. The actual harvest rates and escapement levels are printed at the top of the uncertain lines in figures 6.5-6.8. As stock size decreases the difference between the management objective and true strategy level increases. The effect of uncertain stock assessments is to lower mean catch and catch variance. The effect is stronger for the constant escapement strategy since this strategy operates best under high population variance. The estimated stock assessments do not pick up the occurrence of big years, lowering the perceived population variance and associated big catches. The harvest rate strategy is less affected since it partitions population variance between the catch and spawning biomass. Another interesting effect of assessment uncertainty is displayed at the low stock size end (high harvest rates) of figures 6.1-6.4. The strategies stop short of driving the population to extremely low levels, somewhat reminiscent of the CUTOFF policy of chapter 4. The stock assessment usually predicts less fish than there actually are. For both strategies, less catch will be taken than if the stock size were known without error. The unaccounted surplus stock spawns. This amounts to an unrealized population buffer, acting like the known buffer of 20800 mt in the CUTOFF policy. 52 xz u u c ID 0) X 40 30 20 10 Harvest Rate .8 .7 .6 .3 .4 .3 .2 .1 - i— i i i i 1 1 — i i—i 1 1 — i — i 1 r •t-• CE 0 HR +>4 4 4 / 20 — I — 40 60 —r 120 Mean Spawning Biomass Figure 6.1 Results from the model including stock assessment uncertainty. The constant escape-ment strategy (CE) provides the highest mean catches over the entire range of management options. Catch and spawning biomass are in thousand mt. Error bars are ± one standard error of the mean. Each point is the mean of 100 runs of 15 years (n=1500). Harvest Rate .B .7 .6 .8 .4 .3 .2 .1 3 • 2 -> u c u 4 J ro U 0 --I 1 1 I 1 — I — I I 1— I I • CE 0 HR / l/l Jt 20 - 1 — 80 — I — 100 —r 120 40 60 Mean Spawning Biomass Figure 6.2 Catch coefficient of variation for both strategies under the extended model. The harvest rate CV is considerably less over the range of realistic stock sizes. The pattern is similar to results from the model without uncertainty (figures 4.2 and 4.9), though the constant escapement CV is decreased due to negatively biased and less disperse stock assessments. Mean spawning biomass is in thousand mt. Error bars are ± one standard error of the coefficient of variation. 53 H a r v e s t Rate .s .4 .a (0 1 0 > c • o a o c a. .73 .23 0 -120 40 60 80 Mean Spawning Biomass Figure 6.3 Proportion of years that the spawning biomass drops below 20000 mt as a measure of risk to the stock. The effect of the assessment bias shows up clearly with the harvest rate (HR) curve right shifted as compared to the versions without uncertainty (figures 4.3 and 4.10). The dotted line is the proportion of years of no fishing for the constant escapement strategy (CE). Mean spawning biomass is in thousand mt. JC u •U m (j c n w x 40 30 20 10 1.4 C a t c h V a r i a n c e ( x l O 9 ) Figure 6.4 The mean—variance curves for both strategies under uncertainty. Peak catches from the harvest rate strategy are marginally below the constant escapement strategy (CE), but the catch variance reduction is two-threefold. Mean catch is in thousand mt. Points are actual data. Curves are fit to smoothed data. 54 m m re B O m >» c 01 i n u c 0. 230 200 150 100 DO -• A c t u a l o F o r e c a s t .OB . 1 8 . 8 4 .38 . 8 8 . 8 8 . 8 8 - 1 — .6 .8 Harvest Rate Objective Figure 6.5 The effects of uncertainty on prefishery stock size for the harvest rate strategy. For harvest rates under 0.40, the forecast stock sizes are lower and less variable than the perfect information case. The points are medians. Upper lines range from 75th to 95th quantiles; lower lines from 25th to 5th quantiles. For each harvest rate objective, the right point-line set is from the basic model without assessment uncertainty. The left point-line set is from the model including uncertainty. Actual harvest rates from the version with uncertainty are listed. Biomass is in thousand mt. u re C J 70 -60 so 40 -30 -20 -10 + A c t u a l o F o r e c a s t . 8 4 . 1 8 . 8 8 . 3 8 . 4 8 . 8 8 . 8 8 I I -r— .1 — i 1 1 1 1 — .2 .3 .4 .9 .6 Harvest Rate Objective —i— .7 Figure 6.6 The effects of uncertainty on catch for the harvest rate strategy. Errors in stock assess-ment show up as a skewed and less dispersed catch distribution. (See figure 6.5 for an explanation of points and lines.) Catch is in thousand mt. 55 w ro E o 1-1 CD >. r_ to sz in 0) c 0_ 200 ISO -100 BO + A c t u a l o F o r e c a s t 8 4 48 1 1 7 8 • 7 0 7 0 t IS —I— 30 —I— 43 — i — 60 - i — 73 —i— 90 Escapement Level Objective Figure 6.7 The effects of uncertainty on prefishery stock size for the constant escapement strategy. At higher escapement levels, stock forecasts are biased and less dispersed. Actual escapements from the model with stock assessment error are printed at the top of each line. (See figure 6.5 for an explanation of points and lines.) Biomass and escapement are in thousand mt. c u 4 J ID C J 120 100 80 • B0 40 -20 -0 -+ A c t u a l o F o r e c a s t 31 48 8 4 7 8 07 87 I— 19 —I— 30 49 —I— 60 —I— 73 —I— SO Escapement Level Objective F i g u r e 6.8 The effect of uncertainty on catch for the constant escapement strategy. The error in preseason biomass estimates is reflected in the catch. (See figure 6.5 for an explanation of points and lines.) Catch and escapement are in thousand mt. 56 Three General Forms of Stock Assessment Uncertainty The biases that result from measurements of egg deposition are somewhat surprising. The typical situation expected when errors are accumulated for a series of measurements would be for the variance of the final measurement to increase. The fishery officer—diver surveys are perhaps a peculiar case resulting from the biases of overestimating measurements for small spawns and underestimating measurements for large spawns. Three other cases are considered. Case 1 Bias is zero; standard deviation is about 1.5 times true standard deviation. Case 2 Bias is negative 10000 mt; standard deviation is identical. Case 3 Bias is positive 10000 mt; standard deviation is identical. The results are presented in the composite figure C.9. The results are consistent with previous versions of the model, except for case 1. For case 1, the constant escapement strategy is adversely affected by the increased variance in stock forecasts. The mean catch curves are almost identical between strategies. The harvest rate strategy maintains its usual advantage in catch variance, and appears slightly better than usual in risk comparisons. For cases 2 and 3, the curves shift as the bias changes, but their relative positions remain the same. For any given harvest rate or escapement level the shifted curves suggest much different mean catches. For example, a 0.40 harvest rate strategy results in average catches 23000 mt with positive assessment bias, but 29000 mt with negative bias. With negative bias, the forecast stock size is always less than the true population. Case 2 is the same negative bias as described for the spawn measurement errors, except with identical variance. The variance of the fishery officer spawning biomass estimates is less than the diver estimates, usually containing stock assessments within actual stock size limits. With negative bias and identical variance, assessments are not contained and will frequently grossly underestimate stock size. The opposite problem of habitually overestimating stock abundance (case 3) drives the population below 20000 mt frequently for both strategies. The mean catch curve is shifted to favor lower harvest rates and higher escapements levels. The dangers of overestimating stock size are serious for escapement levels below 40000 mt and harvest rates above 0.40. Mean catches decrease rapidly as overestimated catches remove most or all of the true population. Above these levels enough spawning takes place to sustain the population. The results for underestimating stock size are somewhat surprising, highlighting the value of computer modeling to understand such situations. By intuition I expected that the constant escape-ment strategy would suffer considerable losses in mean catch. Given the rule that if the prefishery 57 Case 1 Case 2 Case 3 Harvest Rate Harvest Rate Harvest Rate . 7 . 8 . 3 . 1 . 7 . 8 . 3 . 7 . 3 . 1 0 4 0 8 0 1 8 0 0 4 0 8 0 1 2 0 0 4 0 8 0 1 2 0 0 4 0 8 0 1 2 0 0 4 0 8 0 1 2 0 0 4 0 8 0 1 2 0 0 4 0 8 0 1 2 0 0 4 0 8 0 1 2 0 0 4 0 8 0 1 2 0 M e a n S p a w n i n g B i o m a s s M e a n S p a w n i n g B i o m a s s M e a n S p a w n i n g B i o m a s s 0 0 . 8 1 . 0 1 . 8 2 . 0 0 0 . 8 1 . 0 1 . 8 2 . 0 0 0 . 8 1 . 0 1 . 8 2 . 0 Variance (xiO9) Variance (xlO9) Variance (xlO9) Figure 6.9 A composite figure showing the effects of other forms of uncertainty on the evaluation criteria for different strategies. Case 1 is no assessment bias but a st. dev. of « 1.5 times the true stock size. Case 2 and case 3 have identical variance, but are negatively and positively biased by 10000 mt, respectively. Dashed line is the constant escapement strategy. 58 stock (real or estimated) is below the escapement goal the catch is zero, the number of years that this occurs should increase considerably with negatively biased or more variable stock assessments. The harvest rate strategy should not be as strongly affected, since it always removes some catch. The problem with this intuitive argument is that it ignores the feedback of the harvesting strategies on the true population. There is a definite positive feedback when stock forecasts are underestimated in the constant escapement strategy. When catch is zero, both the forecast stock size plus the difference between the forecast and the true population are committed to the spawning stock. The result is a much quicker recovery. Also, there is always an unrealized spawning buffer that helps to maintain the stock. The harvest rate strategy is always chipping away at the population, missing out on the rehabilitative effects of occasional fishery closures. 59 Chapter 7 Harvest Rate or Constant Stock? The question that I have avoided until now is which strategy is best? The answer depends on the objectives of the management agency, fishermen and the processing industry. If the sole objective was to maintain the highest long-term mean catches, the constant escapement strategy is the obvious choice. The decision becomes more complicated with multiple objectives; in this case other disciplines are equipped with methods to aid the decision process (Raiffa 1982). Each objective is weighted relative to its importance, and the choices are ranked by a scoring system. My biases should be apparent by now. I weight catch as the most important criterion, catch variance a close second and risk a more distant third. With this weighting scheme, the key question is: Does the advantage in mean catch for the constant escapement strategy outweigh the advantage of lower catch variance for the harvest rate strategy? My answer is definitely no. Is there a best level for each strategy? Mean catch peaks at a harvest rate of 0.35-0.45 and a constant escapement level of 40000-60000 mt. Catch CV does not change much over this range. Catch CV is lower at other harvest rates and escapement levels, but mean catch at these levels is unacceptably low for both strategies. The risk of no fishing under the constant escapement strategy is low and the risk of sub-critical stock size is zero practically for the range 40000-60000 mt. The risk of sub-critical stock size increases from 0.04 to 0.29 as the harvest rate increases from 0.35 to 0.45 (correlated deviation model). Harvest rates under 0.30 reduce this risk, but with a reduction in mean catch of at least 12%. Information Gain There is no doubt that the harvest rate strategy provides more information about the stock-recruitment parameters of the stock, since it allows more contrasting variation in stock size. By definition, the constant escapement strategy provides no contrast in population size. Both strategies could be modified to "actively" probe (Walters and Hilborn 1976, 1978) over a wide range of stock sizes. Either strategy could accomplish the objective of contrast in stock size by using high or low harvest rates or escapement goals. One danger is that once strategies become variable the tendency is to take big catches in big stock size years and vice versa, recreating the problem of no contrast in spawning biomass. Unless deliberate large scale experiments can be undertaken that eliminate this tendency the preferred option for information gain is to remain passively adaptive with a constant harvest rate strategy. 60 Unresolved Issues The most serious problem with a simulation study of this kind is the assumption of constancy in various model parameters. The actual population dynamics are doubtless more complex than can be fully represented by aiitoregressive deviations from stock-recruitment model. The hope is that the model is sufficiently robust so that unmodeled mechanisms or misspecified parameters would not change the results. The problem of incorrect parameters can be partially approached by comparing results over a wide range of values. Unmodeled mechanisms could be included in different model versions, but soon the choices of parameters and model versions would become overwhelming; the most robust results would then be ignored as explanations are sought for results from increasingly complex, speculative models. No formal verification or validation procedures have been attempted in this study, other than to compare the distributions from a wide variety of indicators to observed values. A serious discrepancy exists between the long-term model results and short-term management objectives. The typical management cycle has a period of one year. In dire circumstances, the outlook might be extended to 2-3 years (the number of years from spawning to recruitment in the herring fishery) to justify a stock rebuilding program. The current population level in the Strait of Georgia appears to be near 1966 "crash" levels. The immediate issue is not harvest rate versus constant escapement, but how to manage so as to move out of this crisis situation. Short-term effects are just as easy to model, but much more dangerous to speculate on than long-term averages. The model becomes predictive rather than explanatory. I am especially concerned about missing population regulating mechanisms at low stock sizes. It is possible that at low population levels the proportion of fish spawning at younger age classes increases, fish grow faster and natural mortality decreases. Misspecifing any one of these mechanisms could change the results. Practical Applications The original objective of this study was to provide justification for the switch from a constant escapement strategy to a harvest rate strategy that occurred in 1982. As is often the case in fisheries, the switch was made without any form of documentation (Wilimovsky 1985). A feeble declaration of policy change was made in several manuscripts and technical reports about the time of the policy change (e.g., Stocker, Haist and Fournier 1983). Though somewhat after the fact, this study provides documented evidence that supports the harvest rate strategy. 61 The results from the analysis of the dual fishery officer—diver spawn surveys should be added to the list of remote measurement versus underwater measurement biases. Recognizing and correcting the bias of overestimating small spawn measurements and underestimating large spawn measurements will improve assessments. The brief examination of other typ es of assessment bias and error confirms that the harvest rate strategy is the best option. The likelihood of serial correlation of recruitments should be explored in depth. The emphasis should change from attempting to determine the causes of cycles to the best possible management given unpredicatable events. There is some hope of improving recruitment predictions by analysis of recruitment deviations using autoregressive models that should be at least as accurate as current methods and far more repeatable. The results from this simulation study should be used to help design coas<wide management experiments. Harvest rates from 0.10-0.60 and a wide range of escapement levels should be explored. The experimental design should include two scales: large areas (i.e., west coast Vancouver Island, Strait of Georgia, Queen Charlottes) and replicate experiments at a level similar to Hourston and Hamer's (1979) management sections. The current low stock sizes and closed fisheries in southern British Columbia provide a uncommon unique opportunity to begin experimental management strategies when fishing resumes. Conclusions 1. The constant escapement strategy produces consistently higher average yields over a wide range of population sizes maintained by either constant escapement levels or constant harvest rates. The range of population sizes includes all realistic management options. 2. Over the same range of population sizes, catch variance is significantly lower for the harvest rate strategy. 3. The threat of the spawning population decreasing below a predefined critical level is greatest for high harvest rates. At harvest rates below 0.30 the probability is always less than 0.05, even with considerable stock assessment uncertainty. 4. The risk of no fishing due to insufficient stock size is always lower, and usually zero, for the harvest rate strategy. The risk of no fishing increases as escapement goals increase for the constant escapement strategy. Strategy modifications that create a reserve spawning biomass, « 62 along with stock assessment uncertainty, further increase the probability of fishery closures under the constant escapement strategy. Conclusions 1—4 are a function of how each strategy accommodates the natural population fluctuations characteristic of herring. The constant escapement strategy transmits all variation to the catch. This increases mean yield and catch variance as the constant stock strategy fully uses population surplus above the escapement requirement. The harvest rate strategy partitions the population variance between catch and spawning biomass, slightly reducing and greatly stabilizing yields. 5. Serial correlation between deviations that determine recruitment success lowers average yields and increases catch variance for both strategies. 6. Correlation between recruitment deviations and subsequent stock size causes bias in estimates of stock-recruitment parameters. The stock appears more productive than it really is. 7. Stock assessment uncertainty based on spawn measurement errors reduces mean catch for both strategies. The constant escapement strategy is hardest hit. Stock assessments that are less variable than the true population underestimate years of high stock size that the constant escapement strategy thrives on. 8. Stock assessments that are more variable than the true population size (case 1) are the worst situation for the constant escapement strategy. It is the only version for which the harvest rate strategy ties the constant escapement strategy for the criterion of mean catch. Unbiased stock assessments with more variability than the true population may be the most typical situation for most fisheries. 9. Stock assessments that are biased downward (case 2) create an unrealized spawning biomass buffer that helps to maintain the stock for both strategies at low population levels. 10. 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Sci. 42(Suppl. l):258-262. 70 Appendix A — Catch, Catch Proportion and Weight at Age Data Table A l Total catch by gear for the Strait of Georgia herring fisheries, 1951-1985. Catch in metric tons and millions of fish. Catch in metric tons Food Year Reduction & Bait Seine Gillnet Total Reduction & Bait Seine Gillnet Total 1951 47411 47411 485.4 485.4 1952 53458 53458 639.5 639.5 1953 8425 8425 103.2 103.2 1954 71516 71516 750.4 750.4 1955 70513 70513 725.6 725.6 1956 73632 73632 737.8 737.8 1957 72513 72513 740.7 740.7 1958 23468 23468 261.6 261.6 1959 55639 55639 705.6 705.6 1960 76570 76570 822.9 822.9 1961 51818 51818 608.4 608.4 19B2 75600 75600 855.0 855.0 1963 79188 79188 963.0 963.0 1964 89611 89611 907.5 907.5 1965 65153 65153 672.6 672.6 1966 53604 53604 502.6 502.6 1967 45852 45852 586.1 586.1 1968 5275 5275 76.5 76.5 1969 829 829 14.9 14.9 1970 916 916 10.9 10.9 1971 1788 1788 18.1 18.1 1972 2900 10031 154 13085 24.1 93.3 1.1 118.5 1973 4116 9559 3290 16965 30.4 77.6 22.1 130.1 1974 514 1600 4196 6310 6.3 13.1 27.4 46.8 1975 405 1195 5991 7591 7.1 12.7 40.0 59.8 1976 5126 1352 7373 13851 40.6 13.3 49.4 103.3 1977 5700 4825 7797 18322 51.1 46.9 52.1 150.1 1978 13245 4228 7316 24789 120 41.2 49.1 210.3 1979 13720 6762 20482 108.8 44.2 153.0 1980 2541 169 3177 5887 23.6 1.7 21.6 46.9 1981 5072 2081 5065 12218 43.6 20.3 33.3 97.2 1982 3981 3312 5583 12876 31.9 32.5 36.7 101.1 1983 903 7780 8613 17296 6.4 69.8 56.4 132.6 1984 822 4140 6039 11001 6.6 40.3 42.2 89.1 1985 738 2762 3475 6975 7.4 31.5 23.6 62.5 Catch in millions of fish Food Total catch in numbers is calculated from metric tons of fish by ,10 Pa,t Va,t where Wt = total catch in mt in year t, pa = proportion of catch at age a, and ti>0 = weight at age. Weight at age data are presented in tables A6-A9. 71 Table A2 Catch proportion at age for the herring reduction fishery, 1951-1967. Age Year 1 2 3 4 5 6 7 8 9 1951 .045 .553 .308 .073 .015 .004 .001 1952 .107 .136 .460 .216 .063 .014 .004 .001 1953 .003 .098 .558 .283 .043 .013 .002 .001 1954 .028 .490 .333 .107 .031 .009 .002 1955 .044 .479 .388 .068 .015 .005 .001 1956 .064 .331 .300 .239 .054 .009 .002 .001 1957 .053 .556 .210 .109 .064 .007 .001 1958 .102 .546 .214 .058 .048 .028 .004 .001 1959 .007 .156 .616 .171 .032 .008 .007 .002 .001 1960 .125 .443 .346 .058 .019 .005 .003 .001 1961 .262 .280 .241 .154 .053 .010 .001 1962 .001 .145 .581 .170 .055 .035 .011 .003 1963 .181 .492 .266 .044 .012 .005 .001 1964 .054 .574 .310 .046 .011 .004 .001 1965 .040 .179 .463 .258 .041 .013 .004 .002 1966 .219 .333 .235 .170 .030 .012 .002 1967 .085 .267 .460 .141 .032 .013 .002 .001 Table A3 Catch proportion at age for the herring food and bait fishery, 1968-1985. Age Year 1 2 3 4 5 6 7 8 9 10 1968 .258 .222 .308 .137 .044 .016 .007 .009 1969 .231 .474 .232 .051 .010 .002 1970 .004 .245 .603 .094 .035 .016 .004 1971 .048 .218 .320 .311 .067 .028 .008 .001 .001 1972 .103 .318 .346 .182 .035 .013 .001 1973 .001 .010 .279 .310 .254 .125 .017 .003 1974 .174 .733 .081 .012 1975 .011 .221 .604 .123 .021 .014 .005 .002 1976 .001 .063 .204 .413 .215 .063 .028 .009 .004 .001 1977 .019 .503 .233 .182 .048 .010 .003 .001 .001 1978 .023 .355 .399 .107 .077 .030 .005 .003 .002 1979 .017 .176 .370 .287 .087 .043 .014 .004 .001 1980 .006 .030 .433 .231 .181 .083 .022 .012 .003 .001 1981 .036 .335 .351 .154 .084 .031 .006 .002 1982 .033 .372 .321 .165 .058 .040 .010 .001 .001 1983 .031 .193 .304 .219 .123 .045 .053 .028 .005 1984 .148 .318 .231 .171 .081 .034 .010 .004 .003 1985 .280 .408 .191 .076 .031 .010 .004 .001 72 Table A 4 C&tch proportion at age for the roe herring seine fishery, 1972-1985. Age Year 1 2 3 4 5 6 7 8 9 10 1972 .083 .355 .327 .187 .037 .010 .001 1973 .025 .383 .189 .153 .185 .052 .010 .003 1974 .025 .263 .313 .283 .093 .023 1975 .018 .047 .445 .231 .159 .068 .024 .006 .002 1976 .083 .192 .327 .189 .132 .052 .019 .007 .001 1977 .006 .051 .547 .194 .139 .036 .014 .008 .003 .001 1978 .009 .340 .423 .127 .077 .019 .004 .001 .001 1979 .015 .205 .328 .281 .091 .050 .020 .006 .004 1980 .025 .498 .159 .177 .099 .026 .012 .003 .001 1981 .073 .370 .304 .112 .095 .037 .006 .003 1982 .006 .073 .340 .234 .221 .045 .051 .025 .005 .001 1983 .029 .319 .282 .176 .116 .032 .031 .013 .003 1984 .001 .116 .373 .273 .122 .067 .032 .011 .005 .001 1985 .299 .422 .164 .069 .029 .011 .005 .001 Table A5 Catch proportion at age for the roe herring gillnet fishery, 1951-1985. Age Year 1 2 3 4 5 6 7 8 9 10 1972 .044 .115 .465 .290 .069 .015 .002 .001 1973 .148 .273 .398 .151 .023 .007 1974 .035 .333 .332 .214 .071 .015 1975 .050 .431 .359 .122 .033 .006 1976 .006 .338 .400 .177 .064 .015 .002 1977 .033 .274 .460 .172 .047 .011 .003 1978 .007 .204 .301 .358 .113 .014 .004 1979 .012 .232 .547 .135 .058 .015 .002 1980 .021 .092 .448 .332 .084 .020 .003 1981 .001 .022 .182 .230 .374 .161 .028 .003 .001 1982 .044 .154 .285 .148 .208 .142 .017 .004 1983 .005 .278 .295 .236 .093 .074 .017 .003 1984 .071 .302 .311 .192 .093 .017 .007 .009 1985 .001 .031 .261 .325 .236 .092 .037 .008 .008 73 Table A6 Weight (g) at age for the herring reduction fishery, 1951-1967. Age Year 1 2 3 4 5 6 7 8 .9 10 1951 15 42 88 111 134 156 166 183 187 1952 11 41 89 113 136 155 168 160 178 1953 13 33 76 97 132 166 189 162 298 1954 27 37 77 105 137 164 180 195 185 1955 7 49 89 104 129 153 170 186 1956 46 84 107 122 143 161 178 176 201 1957 10 34 82 114 140 153 171 193 194 195 1958 42 75 109 143 157 165 180 191 1959 22 48 80 94 116 127 117 127 111 111 1960 44 90 110 113 114 108 130 87 110 1961 56 79 110 123 142 145 164 94 1962 16 47 88 102 137 150 184 186 216 1963 44 84 105 123 148 170 179 1964 21 54 92 113 131 148 156 164 1965 14 55 102 120 139 146 149 130 91 303 1966 48 100 130 147 161 194 141 Table A 7 Weight (g) at age for the herring food and bait fishery, 1971-1985. Age Year 1 2 3 4 5 6 7 8 9 10 1971 16 52 99 126 147 166 172 209 182 1972 8 65 98 132 152 173 176 207 180 1973 9 57 101 129 160 175 193 198 201 1974 64 80 114 199 1975 9 35 56 91 106 82 125 184 1976 8 50 90 129 150 176 194 213 228 231 1977 63 92 119 140 161 177 193 195 282 1978 7 47 86 115 135 157 171 165 197 198 1979 57 89 119 140 159 173 194 206 231 1980 34 48 84 112 136 157 169 184 200 228 1981 63 92 115 141 164 181 201 199 209 1982 71 102 125 148 172 188 196 161 200 1983 55 101 133 153 167 178 195 205 207 1984 66 103 135 159 171 176 183 185 214 1985 69 94 125 141 156 169 175 181 74 Table A8 Weight (g) at age for the roe herring seine fishery, 1972-1985. Age Year 1 2 3 4 5 6 7 8 9 10 1972 53 82 123 139 157 173 190 190 1973 57 91 119 144 164 179 205 195 1974 79 95 124 136 149 161 1975 8 40 79 105 116 142 155 167 176 1976 49 72 103 117 124 146 164 184 197 1977 10 56 86 116 138 162 198 203 205 222 1978 13 55 83 104 122 143 143 153 162 192 1979 58 78 115 139 164 176 198 197 216 1980 53 75 102 128 154 165 177 177 206 1981 59 82 103 129 149 165 180 180 193 1982 9 55 87 105 114 135 147 157 167 189 1983 58 85 111 129 137 151 163 176 194 1984 17 59 85 110 133 153 155 163 174 195 1985 60 83 111 132 156 170 186 197 232 Table A9 Weight (g) at age for the roe herring gillnet fishery, 1951-1985. Age Year 1 2 3 4 5 6 7 8 9 10 1972 60 130 143 154 170 177 189 208 1973 119 138 159 163 191 194 1974 127 148 152 160 167 186 1975 133 141 155 173 182 202 1976 104 139 150 157 172 181 175 1977 124 143 151 156 165 167 176 1978 93 129 146 157 165 186 197 1979 132 145 152 163 172 184 182 1980 102 130 143 155 165 173 171 1981 62 110 130 147 159 170 177 181 179 1982 124 133 144 156 166 169 174 177 1983 124 135 149 160 173 177 194 217 1984 116 130 146 155 163 181 164 182 1985 77 117 136 144 159 162 161 173 178 75 Appendix B — F O R T R A N Source Code C C AGE STRUCTURED HERRING MODEL C FOR HARVEST STRATEGY COMPARISONS C C C INITIALIZE VARIABLES AND POPULATION PARAMETERS C LOGICAL YESAVE REAL SURV(IO),NUMB(10),FEC(10).MAT(10),PR0PFM(10).WEIGHT(10) REAL HS.SPNAGE(IO).PROPSP(IO),NAGE1.WCTAGE(IO).HSPAWN(IO) REAL LEVFAC,CS0RT(15OO),PS0RT(15OO) INTEGER HSTRAT,HSLOOP PARAMETER(RICKRA=2.35,RICKRB=0.00133,ENVAR=0.79,RESERR=0.55) PARAMETER(RSAVEl=-0.880.RSAVE2=-1.549,ASUB1=0.945.ASUB2=-0.430) C DATA SURV/.5,.5,.5,.5,.5,.5..49,.37..3..25/ DATA FEC/O.0,1.3.1.7,2.5,3.0,3.0,2.9.3.1.3.2,3.3/ DATA MAT/0.0.0.1,0.4,0.7,0.9,1.0.1.0.1.0,1.0,1.0/ DATA PROPFM/0.0,0.5,0.5.0.5,0.5,0.5.0.5.0.5,0.5,0.5/ DATA WEIGHT/13.,58.,83.,109.,128.,144.,159.,169.,171.,192./ DATA CS0RT/15OO*O/,PS0RT/1500*O/ CRITBM=20000. C C OPEN FILES FOR I/O C OPEN(2) OPEN(3,FILE='GRAND') OPEN(7,FILE='DFOERR') OPEN(13,FILE='CSORTF') OPEN(14,FILE='PSORTF') C C INPUT MODEL CONTROLS C READ(2,250)HSTRAT,NYEARS,NRNOUT,NRUNS,ISTART.IEND. +INCRMT.LEVFAC.IDEVAT,INCLER,CUTOFF,RSEED CLOSE(2) 250 F0RMAT(3I3,I4,3I7.F5.2,2I3.F8.O,F4.1) C CHECK IF OKAY PRINT* PRINT*,' STRATEGY ',HSTRAT PRINT*,' YEARS SIMULATED ',NYEARS PRINT*,' RUNOUT ',NRNOUT PRINT*,* NUMBER OF RUNS ',NRUNS PRINT*,' STARTING LEVEL ',ISTART PRINT*,' ENDING LEVEL MEND 76 PRINT*,' INCREMENT ',INCRMT PRINT*,' LEVEL FACTOR ' .LEVFAC PRINT*,' AUTOCORRELATION ',IDEVAT PRINT*,1 ERROR MODEL '.INCLER PRINT*,' CUTOFF LEVEL ",CUTOFF PRINT*,' RANDOM SEED ',RSEED PRINT* C C START MODEL C DUMMY=RANDN(RSEED) NTOTAL=NYEARS*NRUNS RTOTAL=REAL(NTOTAL) NYEARS=NYEARS+NRNOUT C C************** HS LOOP ************************************************ C DO 5000 HSLOOP=ISTART,IEND,INCRMT HS=REAL(HSLOOP)*LEVFAC C SOME INITIALIZATION LC=0 AVEYLD=0.0 SUMYLD=0.0 RISK1=0.0 RISK2=0.0 SUMP0P=0.0 YLDSS=0.0 AVEDC=0.0 SUMDC=0.0 DRISK1=0.0 DRISK2=0.0 SUMDP=0.0 DFOSS=0.0 WRITE(6,*)HS C C*«*********************** RUM LOOP ************************************ C DO 4000 LOOPRN=i,NRUNS NUMB(1)=2000.0 DO 300 1=2,10 NUMB(I)=NUMB(1-1)*SURV(I) 300 CONTINUE RESID1=RSAVE1 RESID2=RSAVE2 C C*************************************** TIME LOOP ********************* C DO 3500 L00PTI=1,NYEARS C IF(LOOPTI.LE.NRNOUT) THEN YESAVE=.FALSE. ELSE 77 YESAVE=.TRUE. ENDIF C C START CALCULATIONS C C COMPUTE TOTAL SPAWNING BIOMASS C SPBI0M=O.O DO 400 1=1,10 SPNAGE(I)=NUMB(I)*WEIGHT(I)*MAT(I) SPBIOM=SPBIOM+SPHAGE(I) 400 CONTINUE IF(SPBIOM.LE.O.O) THEN SPBIOM=0.0 GOTO 57B ENDIF C PROPORTION SPAWNING AT AGE DO 500 1=1,10 PROPSP(I)=SPNAGE(I)/SPBIOM 500 CONTINUE C C ADD UP KNOWN AGE 2, 3 RECRUITS — SUM TO DSPBIO C 575 RECRTS=SPNAGE(2)+SPNAGE(3) IF(LOOPTI.EQ.1) SPNEXT=SPBIOM-RECRTS DSPBIO=SPNEXT+RECRTS C C CATCH CALCULATIONS C GOTO(660,600),HSTRAT C FIXED ESCAPEMENT 600 IF(SPBIOM.LE.HS.OR.SPBIOM.LE.CUTOFF) THEN TCATCH=0.0 WCATCH=0.0 ELSEIF ((SPBIOM-(SPBIOM-HS)).LE.CUTOFF) THEN TCATCH=SPBIOM-CUTOFF ELSE TCATCH=SPBIOM-HS ENDIF IF(DSPBIO.LE.HS.OR.DSPBIO.LE.CUTOFF) THEN DCATCH=0.0 ELSEIF((DSPBIO-(DSPBIO-HS)).LE.CUTOFF) THEN DCATCH=DSPBIO-CUTOFF ELSE DCATCH=DSPBIO-HS ENDIF GOTO 670 C HARVEST RATE 660 IF(SPBIOM.LE.CUTOFF) THEN TCATCH=0.0 ELSEIF ((SPBIOM-(HS*SPBIOM)).LE.CUTOFF) THEN TCATCH=SPBIOM-CUTOFF 78 ELSE TCATCH=SPBIOM*HS ENDIF IF(DSPBIO.LE.CUTOFF) THEN DCATCH=0.0 ELSEIF ((DSPBIO-(HS*DSPBIO)).LE.CUTOFF) THEN DCATCH=DSPBIO-CUTOFF ELSE DCATCH=DSPBIO*HS ENDIF C C APPORTION CATCH TO AGE CLASSES C 670 IF(INCLER.EQ.l) THEN TCATCH=DCATCH IF(TCATCH.GT.SPBIOM) THEN TCATCH=SPBIOM DCATCH=SPBIOM ENDIF ENDIF WCATCH=0.0 DO 680 1=1,10 WCTAGE(I)=TCATCH*PR0PSP(I) WCATCH=WCATCH+WCTAGE(I) 680 CONTINUE C C STORE WCATCH AND SPBIOM IN ARRAYS CSORT AND PSORT C FOR MEDIAN AMD QUAHTILE CALCULATIONS C IF(YESAVE) THEN LC=LC+1 CSORT(LC)=WCATCH PSORT(LC)=SPBIOM ENDIF C C REMOVE CATCH FROM PREFISHERY MATURE POPULATION C SPBI0M=SPBIOM-WCATCH DSPBIO=DSPBIO-DCATCH DO 700 1=1,10 SPNAGE(I)=SPNAGE(I)-WCTAGE(I) NSPAWN(I)=SPNAGE(I)/WEIGHT(I) NUMB(I)=NUMB(I)-(WCTAGE(I)/WEIGHT(I)) 700 CONTINUE C C REPRODUCTION C EGGS=0.0 DO 800 1=1,10 EGGS=EGGS+(NSPAWN(I)*FEC(I)*PROPFM(I)X 800 CONTINUE C CHOOSE RANDOM ENVIRONMENTAL EFFECT 79 IF (IDEVAT.EQ.O) THEN C WHITE NOISE 830 DEVIAT=ENVAR*FRANDN(0.0) IF(DEVIAT.GT.1.6.0R.DEVIAT.LT.-l.6)GOTO 830 ELSE C AUTOCORRELATED NOISE 840 DEVIAT=0.0 DEVIAT=(ASUB1*RESID1)+(ASUB2*RESID2) DEVIAT=DEVIAT+(RESERR*FRANDN(0.0)) IF(DEVIAT.GT.1.6.OR.DEVIAT.LT.-l.6)GOTO 840 ENDIF C RICKER FUNCTION NAGE1=EGGS*EXP(RICKRA-(RICKRB*EGGS)+DEVIAT) RESID2=RESID1 RESID1=DEVIAT C SKIP IF NOT INCLUDING ERROR MODEL IF(INCLER.EQ.O) GOTO 2999 C C+++++++++++++++ OBSERVATION MODEL +++++++++++++++++++++++++++ C C LOOP UNTIL TONS SPAWNED "= SPBIOM C TSPAWN=0.0 DSPAWN=0.0 IF(SPBIOM.LE.0.0) GOTO 1200 DO 1000 1=1,500 1010 TONS=EXP(2.1*FRANDN(0.0)+6.55) IF(TONS.GT.9000.) COTO 1010 C CHOOSE PERCENT DIFFERENCE 1020 TPD=EXP(1.2*FRANDN(0.0)+4.8)-100.0 IF(TPD.GT.1000.0.OR.TPD.LE.-98.0) GOTO 1020 IF((TPD*T0NS).GT.414000.) GOTO 1020 C CALCULATE F.O. TON ESTIMATE TFO=TONS+(TPD*TONS/100.) IF(TFO.LT.18.0.OR.TFO.GT.4500.) GOTO 1010 C TSPAWH=TSPAWN+TONS DSPAWN=DSPAWN+TFO C CHECK SPAWN ACCUMULATION IF((TSPAWN/SPBIOM).GT.0.99) GOTO 1200 1000 CONTINUE C DFO FORECAST OF RETURNING ADULTS 1200 SPNEXT=DSPAWN*.64 C C ACCUMULATE RESULTS SO FAR C C KNOWN STATS C 2999 IF(YESAVE) THEN SUMYLD=SUMYLD+WCATCH YLDSS=YLDSS+(WCATCH*WCATCH) IF(SPBIOM.LT.CRITBM.OR.(SPBIOM-l.).LE.CUTOFF) RISK1=RISK1+1.0 80 IF(TCATCH.EQ.O.O) RISK2=RISK2+1.0 SUMPOP=SUMPOP+SPBIOM C C DFO STATS C SUMDC=SUMDC+DCATCH SUMDP=SUMDP+DSPBIO DFOSS=DFOSS+(DCATCH*DCATCH) IFCDSPBIO.LT.CRITBM.OR.(DSPBIO-1).LE.CUTOFF)DRISK1=DRISK1+1.0 IF(DCATCH.EQ.0.0)DRISK2=DRISK2+1.0 ENDIF C C UPDATE AGE CLASSES C 3000 DO 3300 1=10,2,-1 NUMB(I)=NUMB(I-1)*SURV(I) 3300 CONTINUE NUMB(1)=NAGE1 C C**************************** END OF TIME LOOP *********************** C 3500 CONTINUE C C*************** END OF RUN LOOP ************************************* C 4000 CONTINUE C C STILL WITHIN HS LOOP C VARNCE=(YLDSS-((SUMYLD* SUMYLD)/RTOTAL))/(RTOTAL-1) IF(VARNCE.LE.O.O) THEN STDEV=0.0 SEMEAN=0.0 ELSE STDEV=SQRT(VARNCE) SEMEAH=STDEV/SQRT(RTOTAL) ENDIF AVEYLD=SUMYLD/RTOTAL IF(AVEYLD.GT.O.O) THEN COVAR=STDEV/AVEYLD SECV=(COVAR/SQRT(2.0*RTOTAL))*SQRT(1+(2*(COVAR*COVAR))) ELSE COVAR=0.0 SECV=0.0 ENDIF AVEP0P=SUMP0P/RT0TAL RISK1=RISK1/RT0TAL RISK2=RISK2/RT0TAL C C DFO ESTIMATES C IF(INCLER.EQ.O) GOTO 4050 81 DFOVAR= (DFOSS- ((SUMDOSUMDC) /RTOTAL)) / (RTOTAL-1) IF(DFOVAR.LE.O.O) THEM DFOSTD=0.0 DFOSEM=0.0 ELSE DFOSTD=SQRT(DFOVAR) DFOSEM=DFOSTD/SQRT(RTOTAL) ENDIF AVEDC=SUMDC/RT0TAL IF(AVEDC.GT.O.O) THEN DFOCV=DFOSTD/AVEDC DFOSCV=(DFOCV/SQRT(2.O*RT0TAL))*SQRT(l+(2*(DF0CV*DF0CV))) ELSE DF0CV=0.0 DF0SCV=0.0 ENDIF AVEDP=SUMDP/RT0TAL DRISK1=DRISK1/RT0TAL DRISK2=DRISK2/RT0TAL C C CALL SYSTEM SUBROUTINE SSORT TO SORT ARRAYS C 4050 CALL SS0RT(CS0RT,NT0TAL,3) CALL SS0RT(PS0RT,NT0TAL,3) C C CALL SUBROUTINE QUANTS TO CALCULATE QUANTILES C CALL QUANTS(CSORT,NTOTAL,CMIN,CQ5,CQ25,CQ50,CQ75.CQ95,CMAX) CALL QUANTS(PSORT,NTOTAL,PMIN,PQ5.PQ25,PQ50,PQ75,PQ95,PMAX) C C WRITE CATCH, POPULATION QUANTILES TO FILES CSORTF, PSORTF C WRITE(13,4075)HS,CMIN,CQ5,CQ25,CQ50,CQ75,CQ95,CMAX WRITE(14,4075)HS,PMIN,PQ5,PQ25,PQ50,PQ75,PQ95,PMAX 4075 F0RMAT(1X,F10.2,7F8.0) C C WRITE RESULTS TO FILES GRAND, DFOERR C WRITE(3,4190)HS,AVEYLD,VARNCE,SEMEAN,COVAR,RISK1,RISK2,AVEPOP, +SECV IF(INCLER.EQ.O) GOTO 5000 WRITE.(7,4190)HS,AVEDC,DFOVAR,DFOSEM,DFOCV,DRISK1,DRISK2,AVEDP, +DFOSCV 4190 FORMAT(1X,F10.2,F8.0,F12.0,F9.2,3F6.2,F8.0,F8.3) C C************ CONTINUE FOR HS LOOP *********************************** C 5000 CONTINUE C C CLOSE FILES C CLOSE(3) 82 CLOSE(7) CLOSE(13) CLOSE(14) STOP END C c SUBROUTINE QUANTS (ARRAY,N,QMIN,QB,Q2B,Q50,Q7B,Q9B,QMAX) DIMENSION ARRAY(N) LOGICAL EVEN QMIN=ARRAY(1) QMAX=ARRAY(N) EVEN=M0D(N,2).EQ.O IF(EVEN) THEN QBO=(ARRAY(N/2)+(ARRAY((N/2)+1)))/2 ELSE QE0=ARRAY((N+l)/2) ENDIF QB=ARRAY(NINT((REAL(N)+1)/20)) Q25=ARRAY(HINT((REAL(N)+l)/4)) Q7B=ARRAY(NINT((3*(REAL(H)+1))/4)) Q95=ARRAY(HINT((19*(REAL(N)+1))/20)) RETURN END 83 

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