MSU LIBRARIES n. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ./ f) .. .’ t“... O ‘ ,"'"‘.’" _.--'.9 use; no, 3‘51) Jag COMPUTER SIMULATION or THE POPULATION DYNAMICS or LAKE WHITEFISH IN NORTHERN LAKE MICHIGAN By Peter Charles Jacobson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Fisheries and Wildlife 1983 (:3) Lb I (\ M \ ABSTRACT COMPUTER SIMULATION OF THE POPULATION DYNAMICS OF LAKE WHITEFISH IN NORTHERN LAKE MICHIGAN BY Peter Charles Jacobson The effects of the present trap and gill net commercial fishery on a population of lake whitefish (Coregonus clupea- formis) in northern Lake Michigan were investigated. A mathematical model was developed to simulate the current fishery and investigate possible management strategies in re- lation to sustainable yield and annual yield variability. The model consisted of an age/size structured, dynamic pool model incorporating a stock-recruitment function subject to environ- mental variability. Trap and gill net selectivity functions specified the fishing mortality rates operating on each cohort. Three scenarios, representing possible directions of the fishery, were investigated. The first scenario involved a multi-gear fishery with gear-specific fishing mortality rates as manageable (controllable) inputs. The second and third scenarios investigated the possibilities of either an exclu- sive trap net fishery or an exclusive gill net fishery with fishing mortality rates, minimum siae limits (trap net simu- lations) and mesh size restrictions (gill net simulations) as controllable inputs. Combinations of the controllable inputs were identified which resulted in a large sustainable yield and a low level of annual variability. ACKNOWLEDGEMENTS The Michigan Sea Grant Program provided the funding for this study. The Michigan Department of Natural Resource and the United States Fish and Wildlife Service provided much use- ful information and I appreciate the helpful suggestions offered by several members of their staff. The cooperation and assistance from the commercial fishermen of northern Lake Michigan was extremely valuable. I am especially grateful to Ross Lang, Bill Carlson, King's Fisheries and the Frazier brothers. - I would like to thank the many graduate students at Michigan State University who helped in the collection of field data. Special thanks to Paul Scheerer for his help- ful suggestions, hard work and friendship. Dr. William Taylor, Dr. Erik Goodman and Dr. Niles Kevern provided helpful assistance and guidance throughout the study. I also extend my deepest appreciation and love to my wife Heidi. ii TABLE LIST OF TABLES. . . LIST OF FIGURES . LIST OF APPENDICES. INTRODUCTION. . . . MODEL DEVELOPMENT . Growth . . . . Mortality. . . Recruitment. . Yield. . . . . OF CONTENTS Computer Implementation and Simulation Organization. . . . PARAMETER ESTIMATION. Field Methods. Parameter Estimates. . . . . . SIMULATION ANALYSIS Scenario #1. . Scenario #2. . Scenario #3. . Other Model Applications . . . Model Limitations. CONCLUSION. . . . . APPENDICES. . . . . LIST OF REFERENCES. Page iv vii 11 11 17 18 20 an an 26 37 37 an 1:6 us 52 56 57 7o Table Table Table Table Table Table LIST OF TABLES Page Sampling dates, locations and numbers of lake whitefish, 1980-82. . . . . . . . . . Mean lengths of lake whitefish in northern Lake Michigan estimated by Scheerer (1982). . Population estimates of lake whitefish by .age class in November, 1980 in northern Lake Michigan (Scheerer, 1982). . . . . . . . Simulated yield values for different time steps. Controllable inputs set at F* = 0.65u, F* = 0.55“, MSL = 432 mm and m = 22 mm with n8 random recruitment factor (9 = 0). . . . . Means, standard deviations and coefficients of variation at five separate simulations made at four different run lengths. Con- trollable inputs set at F* = 0.654, Fa = 0.55A, MSL = 432 mm and mG = 228 mm . . . . . Results of the sensitivity analysis performed on the model parameters. The deviation in simulated yield which resulted from a 110% change in each parameter is listed. . . . . . iv 25 27 29 33 35 53 LIST OF FIGURES Page Figure 1. Map of the study area with sampling ' and tagging locations. . . . . . . . . . . . 3 Figure 2. Commercial harvest of lake whitefish in Lake Michigan from 1900 - 1981 (Baldwin et. al. 1979) . . . . . . . . . . . 5 Figures 3a'and 3b. Commercial harvest (Fig. 3a) and effort (Fig. 3b) for lake whitefish in statistical district MM-3 (Smith et. al. 1961) of Lake Michigan, 1948 - 81. Units of effort are thousands of lifts of trap and pound nets and millions of feet of gill net lifted. (Data from the Great Lakes Fishery Lab, United States Fish and Wildlife Service, Ann Arbor, Michigan) . . . . . . . . . . . . 7 Figures 4a and 4b. Gill net selectivity curve (Fig. 4a) from McCombie and Fry (1960) and trap net selectivity curve (Fig. 4b) from Eshenroder et. a1. (1980) in terms of girth to perimeter ratio and actual fish length for whitefish in northern Lake Michigan. Actual fish length scale is for a 114 mm stretched mesh net . . . . . 14 Figure 5. Stock-recruitment curve for lake whitefish in northern Lake Michigan. The shaded area represents 11 standard deviation around the fitted line . . . . . . 32 Figures 6a and 6b. Time trace of the first 100 years of annual yield in a 1000 year simulation (Fig. 6a) and the resulting histogram (Fig. 6b). Controllable inputs were 9; - 0.65u, F’ = 0.554, MSL = 432 mm and mG ; 228 mm. 9 . . . . . . . . . . . . . . . 36 Figures 7a Figure 8. Figures 9a and 7b. Isopleths of simulated mean annual yield (Fig. 7a) and coefficient of variation (Fig. 7b) for lake whitefish in northern Lake Michigan in a multi-gear fishery. Controllable inputs of MSL = 432 mm and m = 228 mm. Shaded area' represents pgssible OSY's. Black dots represent the current state of the. fishery. . . . . . . . . . . . . . . . . Age compositions of lake whitefish in northern Lake Michigan from commercial catch samples in the spring of 1981. . . and 9b. Isopleths of percent trap net yield (Fig. 9a) and spawning biomass (Fig. 9b) of lake whitefish in northern Lake Michigan in a multi-gear fishery with MSL = 432 mm and m = 228 mm. Black dots represent the currgnt state of the fishery (1981) . . . . . . . . . . . . . . . . . Figures 10a and 10b. Isopleths of simulated mean annual yield (Fig. 10a) and coefficient of variation (Fig. 10b) for lake whitefish in a hypothetical exclusive trap net fishery in northern Lake Michigan with MSL = 432 mm. Shaded area represents possible OSY's . . . . . . . . . . . . . Figures 11a and 11b. ISOpleths of simulated mean Figure 12. Figure 13. annual yield (Fig. 11a) and coefficient of variation (Fig. 11b) for lake whitefish in a hypothetical exclusive gill net fishery in northern Lake Michigan, with m = 228 mm. Shaded area represents pgssible OSY's . . . . . . . . . . . . . Simulated mean annual yield at different trap net minimum size limits for lake whitefish in a multi-gear fishery in northern Lake Michigan. Other controllable Page 39 41 43 45 1:7 inputs set at their 1981 values: F* = 0.654, T F* = 0.554 and m = 228 mm.. . . . . . . G G Simulated coefficients of variation in yield of lake whitefish in northern Lake Michigan at different mean annual yields when each gear is fished exclusively. Controllable inputs of MSL = 432 mm and mG : 228 mm. . . . . . . . . . . . . . . vi 49 51 Appendix Appendix Appendix Appendix LIST OF APPENDICES .Page Glossary of variable names used in the text. .‘. . . . . . . . . . . . . . 57 Fortran program of lake whitefish model. . . . . . . . . . . . . . . . . . . 59 Procedures used to calculate spawning biomass and recruitment of lake whitefish in northern Lake Michigan. . . . 65 Estimated spawning stock and recruitment for lake whitefish in northern Lake Michigan. Units are kilograms for spawning biomass and numbers of three- year-olds for recruitment. . . . . . . . . 69 vii INTRODUCTION The lake whitefish (Coregonus clupeaformis) is a his- torically important, commercially utilized species in Lake Michigan. With most of the other valuable species now either extinct or seriously depleted, the whitefish has be- come even more important to the commercial fishing industry. Over two million kilograms have been harvested annually from Lake Michigan since 1971 (Baldwin et. al. 1979). Increased demand for whitefish from a Chippewa and Ottawa Indian ' treaty fishery coupled with a large state-licensed commer- cial fishery has resulted in a significant buildup of fish- ing effort. Fisheries managers are concerned that the pres- ent level of exploitation is excessive (Patriarche 1977). A collapse of the whitefish stocks would be devastating to the Lake Michigan commercial and treaty fishery. The objec- tives of this study were to analyze the vital statistics of the lake whitefish populations in northern Lake Michigan, and to develop and utilize a simulation model to investi— gate the effects of the present fishery on the long term yield and stability of the system and to explore the effec- tiveness of various other management strategies. The lake whitefish is widely distributed over the northern half of North America, including all of the Great Lakes (Scott and Crossman 1973). In Lake Michigan, they 1 2 inhabit the cool water at depths of 10 to 60 meters. White- fish grow as large as 10 kilograms, but average about 1 kil- ogram in the lake Michigan commercial catch. The adult diet consists primarily of benthic insect larvae, small molluscs and amphipods (Koelz 1929). Lake whitefish spawn in Novem- ber in water less than 10 meters over rock and sand (Scott and Crossman 1973). The study area consisted of the northeastern section of Lake Michigan (Figure 1). Several broad lake habitat types are represented within the area, ranging from the sharply, sloping shoreline and deep water (>60 meters) a- round the Leleenau Peninsula (Leland) to the relatively shallow (<40 meters) and productive north shore area. A large island complex is located in the central portion of the study area. Scheerer (1982) identified at least three separate stocks of whitefish within the study area (he defined a stock as a manageable unit of reproductively isolated fish). One inhabits the waters off the western side of the Leleenau Peninsula, another near the Beaver Island complex and one a- long the north shore of the lake. He estimated a fall 1980 biomass of 1.4 million kilograms of whitefish in the north shore population. A trap net fishery operates at depths of less than 30 meters and only during the open water season (approx. April through December). A gill net fishery operates during both the open water and ice-cover seasons and at depths of up to Naublnway ‘ fipoutette A I. ”'3' s Martinique“ 37.. 'i’nx'm 3%..“ c/ 3 ‘ :12: l ;' ' "'7 ‘2' o’.. I l "9'- ’ ' an .. "' ,3! Soul. Choix Pt ‘6 ‘ ’I'i' : \ “’5 i' 5,? I L-.g9rth_ -Sbooe ---_ ’3.“- .------_-_-_-----.----.----. .‘-"" tock Bounds 1:" cf ]/ ry & H09 .‘ \n°-< ooooo LEGEND Sampling Location * Tagging Site Figure 1. locations. Map of the study area with sampling and tagging u 50 meters and deeper. The season is closed during the month of November (spawning season) for both gear types. Most of the gill nets and the trap net pots use a 114 mm stretched mesh net. The north shore population has supported a highly pro- ductive commercial fishery operating out of the ports of Naubinway, Epoufette, Brevort and St. Ignace. The large ex- panse of moderately shallow water (10-40 meters) provides for the high production of benthic invertebrates required by adult whitefish. Several whitefish population studies have been conducted on this north shore stock (Roelofs 1958, Jensen 1976, Patriarche 1977, Scheerer 1982). The exten- sive north shore pOpulation data from these studies, com- bined with unpublished reports from the Michigan Department of Natural Resources, represented the most complete data set for any of the stocks within the study area. The simu- lation model was developed and parameterized specifically for the north shore population. Data were collected from other areas for comparative purposes. The annual commercial harvest of lake whitefish in Lake Michigan has fluctuated dramatically (Figure 2). The recent history (1948—1981) of annual whitefish production in statistical district MM-3 (Smith et. al. 1961 - refer to Figure 1) is displayed in Figure 3a. Pre - 1948 production peaks occurred in the 1880's, the late 1920's and the 1940's. The recent period of high production follows the years of 1957 - 1960 when there was essentially no commer- .Amnma .Ha .um .efiaeammv $3-82 act 5.35:: 9:3 5 £332... 33 Lo 33.3; 13358 "N 3&3 Loo> coup Chap 000.. Omar 013. omop ouop Draw 08w b _ _ . b . . o IN (OOl 3‘ 9)!) PIOIA I'WUV 6 cial production. Overexploitation, exotic species intro- duction, and high year class strength variability have been implicated in these fluctuations. (Smith 1968, Wells and McLain 1973). The recent increase in lake whitefish abundance can be attributed to a decrease in sea lamprey predation (Wells and McLain 1973). This lower lamprey predation rate is the result of effective sea lamprey control and the increased abundance of lake trout which is the favored target of the lamprey. The recent trends in fishing effort by the major gear types are displayed in Figure 3b. All have shown increases since the 1960's, except for pound net effort, which has recently dropped to zero in the district. A required con- version from gill nets to trap nets for state-licensed commercial fishermen in the 1970's has resulted in an in- crease in trap net effort and a decrease in gill net effort. Currently, the primary source of gill net effort is repre- sented by the treaty fishery. 14 _ 12‘- , I Annual Yield (K9 x 105) Eflafl ’\ Pound Not ¢.\. ./°\.’. \_,= :fi“»= ’ I r 1950 1960 ' 1970 1980 Your Figures 3a and 3b. Commercial harvest (Fig. 3a) and effort (Fig. 3b) for lake whitefish in statistical district MM-3 (Smith et. al. 1961) of Lake Michigan, 1948-1981. Units of effort are thousands of lifts of trap and pound nets and millions of feet of gill net lifted. (Data from the Great Lakes Fishery Lab, United States Fish and Wildlife Service, Ann Arbor, Michigan). 7 MODEL DEVELOPMENT An adequate mathematical model of the dynamics of this system requires the incorporation of several important bio- logical phenomona, characteristic of lake whitefish popula- tions in general. For example, the existence of highly vari- able year class strength has been observed in many populations of whitefish. Several mechanisms have been suggested which might generate these fluctuations, but there has been no general consensus of which may be most important. Miller (1952) suggested that winds at spawning time are important in determining year class strength for whitefish in several lakes in Alberta, Canada. Christie (1963) suggested that air tem- peratures at the time of spawning and hatching are important factors. Lawler (1965) concluded that prolonged incubation periods are crucial for good year class strength in whitefish in their extreme southern range (Lake Erie). All of these factors could potentially generate variability in recruit- ment of northern Lake Michigan whitefish populations. The interaction of this variability and the level of fishing is of interest to lake whitefish managers and commercial fish- ermen. The influence of environmental variability on the sta- bility of fish yield has been investigated by several authors, using widely different approaches. Variability has been in- corporated into stock-recruitment models (Ricker 1958, Allen 1973, Walters 1975), population growth models (Beddington and May 1977, May et. al. 1978), and age-structured models (Getz 8 9 and Swartzman 1981, Horwood and Sheperd 1981). All of the authors have concluded that there can be an inherent trade— off between the mean and the variance of yield, and that this tradeoff is an important consideration in the management of a fluctuating population. The relative influence of spawning stock density on the recruitment of lake whitefish is also an important question. Lawler (1965) could not explain year class variability on the basis of spawning population size in Lake Erie. However, Christie (1963) suggested that spawning stock density did play an important role in determining recruitment, especially in years with unfavorable environmental conditions. The rel- ative contribution that both density dependent and density independent factors have towards determining recruitment should be an important component of a lake whitefish manage- ment model. The contrasting ways in which the two different gear types affect the size and age structure of the population is another important consideration in developing the model. Trap nets harvest fish over a much broader range of size and ages than do the relatively more selective gill nets. Each gear type produces a different fishing mortality on each cohort. In addition to incorporating the important biological characteristics into the model, specific manageable inputs must be identified. Closed seasons, gear limitations, limited entry and minimum size limits have commonly been used in lake whitefish management. Some regulations are designed to con- 10 trol fishing mortality, others are designed to protect cer- tain segments of the population from harvest, such as imma- ture fish or spawning fish. Several of these regulations can be generalized by controlling the gear-specific fishing mor- tality rates in the model. With these considerations in mind, a continuous time, age/size-structured, dynamic pool model incorporating a stock- recruitment function subject to random density-independent variation was developed. Trap and gill net selectivity curves specify instantaneous fishing mortality rates for each cohort. The model allows for the manipulation of four controllable inputs: 1. trap net fishing mortality rate 2. gill net fishing mortality rate 3. trap net minimum size limit 4. gill net mesh size There is no minimum size limit for gill nets. The size range of fish harvested by gill nets can be controlled by the mesh size restriction. Appendix 1 is a glossary of variable names used in the text. The specific components of the model are presented in the following sections. 11 GROWTH Individual body growth is modeled using a modified (t0=0) von Bertalanffy growth equation (von Bertalanffy 1938): 11(t)= L, (1-exp(-Kt)) (1) where, li(t)= length of fish in cohort i at time t L, asymptotic length parameter K growth coefficient Weights are calculated from the standard length-weight equation (Ricker 1975): _ b wi(t)- a 11(t) (2) where, wi(t): weight of fish in cohort i at time t a,b = length-weight parameters MORTALITY The decline in numbers of fish in a cohort over time is assumed to be a negative exponential function (Ricker 1975). The total mortality rate is separated into three components: natural, trap net fishing and gill net fishing. The function can be written in the form of a differential equation using instantaneous mortality rates: dN aEi= -(FT’1+ FG,i+ M1) N1 (3) where, N1 = number of fish in cohort 1 FT i= instantaneous trap net mortality rate operating on cohort i 12 FG i= instantaneous gill net fishing mortality ’ rate operating on cohort i M1 = instantaneous natural mortality rate operating on cohort i The specific trap and gill net mortality rates operating on a cohort are direct functions of fish size and gear select- ivity. Fortunately, the selectivity curves for whitefish have been worked out by McCombie and Fry (1960) for gill nets and 'Eshenroder et. al. (1980) for trap nets. McCombie and Fry estimated the relative selectivity of gill nets in terms of the ratio of fish girth to mesh perimeter. First, a length-girth relationship must be established: 31m. A + B 11(t) (u) where, gi(t)= girth of fish in cohort i at time t A,B = length-girth parameters The relative efficiency can then be calculated as a function of cohort girth and gill net mesh size with a log-normal selectivity equation: REG’i(t)= CGEObVZRgi(t)/mcj'1Eexp(-(ln(gi(t)/mG) U u (5) where, REG i(t): relative efficiency of gill nets on ’ cohort i at time t (ranges from 0.00 to 1.00) m = gill net mesh perimeter (equal to 2x stretched mesh measurement) "G = mean of the log-normal selectivity curve 13 db = standard deviation of the log-normal selectivity curve C = constant (required to adjust the range g of relative efficiency to 0.00 to 1.00) The cohort-specific gill net fishing mortality rate is calculated by multiplying the relative efficiency times an apical instantaneous fishing mortality rate: * 6 FG,i(t)= REG’i(t) FG < ) where, FG i(t): specific instantaneous gill net fishing ’ ’ mortality rate operating on cohort i at time t F = apical instantaneous gill net fishing mortality rate (controllable input) Apical instantaneous fishing mortality is defined as the instantaneous fishing mortality rate at the point of maximum efficiency on the gear selectivity curve. Figure 4a illustrates the shape of the gill net selectivity curve, both in terms of a girth to perimeter ratio and and actual fish length. Maximum efficiency occurs at a girth to perimeter ratio of 1.26 and a fish length of 555 millimeters (for a 114 mm stretched mesh gill net). This is equivalent to saying that a gill net is most efficient for whitefish when the girth of the fish is 1.26 times as large as the perimeter of an individual mesh Opening. Fish with girths either larger or smaller than this value will be captured with less efficiency. The Specific instantaneous fishing mortality rate FG i(t) will be exactly 9 ‘l' equal to the apical instantaneous fishing mortality rate FG F‘q 1.04 Gill N61! .1 0.8- . . d '3 01%- m 0 ,, F I 2" " 5 o. t 44 0 d 3 E E? 2:01? 3".- 2 . ': DJ E M g t I U I I r I 2 V g F'T 3 1.0'1 s 3 - g o F Trap Note a d 0 3; ‘03- FVZ‘ - 03— 055‘ 0 ‘L 0" 1 I I I I I I I 0.8 1.0 1.2 1.4 1.0 Glrth : Perimeter Ratlo I I V 400 500 600 700 Length [mm] Figures 4a and 4b. Gill net selectivity curve (Fig. 4a) from McCombie and Fry (1960) and trap net selectivity curve (Fig. 4b) from Eshenroder et. al. (1980) in terms of girth to perimeter ratio and actual fish length for whitefish in northern Lake Michigan. Actual fish length scale is for a 114 mm stretched mesh net. 14 \ 15 when the cohort has fish with girths 1.26 times the perimeter of the mesh opening. At all other girths, FG,i(t) will be less than F; . Cohort-specific trap net mortality is calculated in the same manner. Eshenroder et. al. (1980) also used the girth to perimeter ratio concept to determine trap net selectivity. Although not presenting a specific functional relationship in their paper, one of the authors (Doug Jester, personal communication) suggested fitting the following equation to their data: RET,i(t): [(gi(t)/mG)*fC1]/[(C1**C2) + (gi(t)/mT)**C1J (7) where, ' RE .(t)= relative efficiency of trap nets on T’1 cohort i at time t (ranges from 0.00 to 1.00) mT = trap net mesh perimeter (equal to 2x stretched mesh measurement) C1,C2 = selectivity curve parameters The specific instantaneous trap net mortality can then be calculated: * FT,i(t)= RET,i(t) FT (8) = 0 if li(t)-< MSL where, FT i(t): specific instantaneous trap net fishing ’ mortality rate operating on cohort i at time t i F = apical instantaneous trap net fishing mortality rate MSL = trap net minimum size limit 16 The trap net selectivity curve is illustrated in Figure 4b. Maximum efficiency occurs at a girth:perimeter ratio greater than 1.10 and at fish lengths greater than 500 mm, for a 114 mm stretched mesh trap net. The specific instant- aneous fishing mortality rate, FT,i(t)’ equals the apical instantaneous fishing mortality rate, F; , at girth:perimeter ratios greater than 1.10 and fish lengths greater than 500 mm. Maximum efficiency for trap nets occurs at a smaller girth:perimeter ratio than for gill nets (1.10 vs. 1.26). Eshenroder et. al. (1980) suggest that this disparity is probably related to different capture mechanisms. Gill nets must retain against more of an escapement struggle than fish impounded in a trap net. This behavioral difference was substantiated by field observations within the study area. While a small percentage of whitefish were gilled during a trap net lift, the vast majority were passively retained within the lifting pot. In contrast, whitefish struggled greatly to escape during a gill net lift. This active motion allows whitefish with a girth:perimeter ratio of slightly greater than 1 to pass through the gill net's mesh. The selectivity curves in Figure 4 illustrate that the relative efficiency at a girth:perimeter ratio of 1.00 is only 15% for gill nets, but near 50% for trap nets. Interestingly, trap nets are also significantly efficient for fish small enough to easily pass through the mesh (girth:perimeter «<1.00). Again, this is due to the relatively passive nature of whitefish in trap nets. 17 Actual numbers of fish in a cohort are calculated by integrating Equation 3: t Ni(t)= Ni(tR) - jf'(FT,i + FG,i + Mi) Ni dt (9) ‘2R where, Ni(t)= number of fish in cohort i at time t tR : time at recruitment RECRUITMENT The recruitment submodel was developed to include both density-dependent and density-independent factors. The Ricker stock-recruitment function presented by Walters (1975) was used: R: as exp(-BS) exp(v) (10) where, R: number of recruits S: spawning biomass 0.3: fitting parameters v: independent random variable, normally distributed with a mean of 0.0 and a standard deviation of OR This approach does not require the identification of the specific abiotic mechanism(s) which affect recruitment. Instead, an empirical measure of variability O , is calculated from existing recruitment data. The log-normal distribution of the error term exp(v), is common to a wide variety of fish popul- tions (Allen 1973, Walters 1975 , Peterman1981 ) and is chars acterized by a large number of average and slightly below average recruitments, with an occasional production of a very 18 large year class. The multiplicative nature of exp(v) produces higher levels of variability at greater spawning stock densities. YIELD The rate of yield accumulation is assumed to be a direct function of the instantaneous fishing mortality rate and the number of fish present (Beverton and Holt 1957) and can be represented by the following differential equations: dY T,i = FT,i Ni w1 (11) dt d! G,i - FG,i Ni wi (12) dt. where, dYT i , : rate of trap net yield accumulation for dt cohort i dYG i z : rate of gill net yield accumulation for dt cohort i The actual yield is calculated by integrating the differential yield equations: t YT,i(t)= [FT’i Ni wi dt (13) 15R t YG’i(t).-. f FG,i Ni wi dt (14) t"R where, YT 1(t): trap net yield from cohort i at time t 7 YO i(t): gill net yield from cohort i at time t 19 Yield for the entire population is calculated by summing the contributions from each cohort: IMAX YPOP”): iz-O (YT,i(t) + YG,i(t)) (15) where, YPOP(t): total population yield at time t I : maximum age attained MAX 20 COMPUTER IMPLEMENTATION AND SIMULATION ORGANIZATION The continuous time differential equations are solved by numerical integration. A simple predictor-corrector technique is used consisting of Euler's formula and the trap- azoidal rule (Manetsch and Park 1981). The Euler formula makes a rough prediction of the value of the integral at relatively small steps of time (At) and then corrects the answer by the trapezoidal rule: Predictor Step - Euler Formula Ne(t +At)= N(t) + [f(N(t))] At (16) Corrector Step - Trapezoidal Rule N(t +At): N(t) + At/2[f(Ne(t +At)) + f(N(t))] (17) where, At : time step Ne(t 4» At): rough prediction N (t + At): correction f(N(t)) : differential equation to be solved dY ) (e.g. dN. dY 1 ’ T,i ’ Gzi dt dt dt '9 21 The accuracy of this numerical integration depends on the size of ‘At: the smaller the time step, the more accurate the solution for the range of .At's of interest. A tradeoff exists between the size of .At and computer time, with the smaller time steps being more expensive. An appropriate At is chosen which results in a predetermined level of accuracy (see Parameter Estimation section). The simulation.model is organized to calculate yield at annual intervals. At the end of each simulated year, yields are calculated (Equation 15), spawning biomass is determined (Equation 18), the resulting recruitment is computed (Equation 10) and the cohort integer age is updated. IMAX S: 1:20 Ni(t) wi(t) iff 11“)le (18) where, S: spawning biomass 1M: length at maturity This process runs the entire length of the simulation, at which time the mean and coefficient of variation of yield are calculated: nMAX Y: Z Yn / n n:1 MAX (19) where, Y: mean annual yield 22 Yn: yield in year n “MAX: simulation run length (number of years) and, “MAX 2 _ CV: Z (Yn —'Y) / (n-1) / Y (20) n:1 where, CV: coefficient of variation of yield The first twenty simulated years are not included in the calculation offthese descriptive statistics in an attempt to "wash out" any effect the initial conditions may have on the long term values. For example, consider a simulation which produces a long term yield much smaller than the yield possible in the short run. The inclusion of the early annual yields would aberrantly influence the calculation of mean annual yield. Although the short term behavior of the model may be of considerable interest, the primary objective of this study was to investigate the long term sustainable yield and stability. The short term model behavior could provide estimates of the yield possible in the near future, but would not provide any insight into the sustainablity of the yield. The stochastic recruitment factor,.v, is randomly gener- ated from a normal distribution with a mean of zero and a standard deviation of Oh. An inverse transformation tech- nique (Manetsch and Park 1981) is used to generate these random numbers. 23 The coefficient of variation, (CV), was selected as the most appropriate statistic to describe the yield variability. Although standard deviation was considered, it does not take into account the relative magnitude of annual yield. For example, a high standard deviation at a low mean annual yield would have a more "noticeable" effect and impact on the comm- ercial fishery than would a high standard deviation at a high mean annual yield. The coefficient of variation, however, does relate the magnitude ofvariability to the mean. The model was implemented in Fortran 77 on a Cyber 750, Control Data Corporation computer. The program prints annual statistics and an end-of-simulation summary (Appendix 2). PARAMETER ESTIMATION The data used to parameterize the model came from sev- eral sources. Field data collected from October 1980 to June 1982 (Table 1) were used for many of the estimates. Paul Scheerer, an MSU graduate student also involved in this study, analyzed much of the data and estimated many of the population statistics. Other sources of information included data from the fisheries literature, the Michigan Department of Natural Resources and the United States Fish and Wildlife Service. Data from the north shore population were used to parameterize the model. Data from other areas of the lake were collected for comparative purposes. FIELD METHODS ‘ During the 1980 and 1981 closed season (November 1 - November 30), 3239 whitefish were tagged and released. The majority of the fish were tagged in the Naubinway-Epoufette area (2207), and the Leland-Empire area (532), with fewer numbers tagged near Beaver Island (19) and Grand Traverse Bay (163) (Table 1). Several commercial fishermen provided their time, nets, boats and expertise during the tagging operations. The fish were captured in 114 millimeter stretched mesh commercial trap nets. Each fish was measured 'for length, tagged with Floy FT-1 dart tags in 1980, and Floy FD-68C anchor tags in 1981, directly below the dorsal fin, then released. A $1.00 reward was offered for each tag returned and considerable contact with fishermen was made throughout the study to encourage the return of tags. 24 25 Table 1. Sampling dates, locations, and numbers of lake whitefish, 1980-1982. Port Date Sample Type N North Shore Area Naubinway 10/29/80 SLW 513 Naubinway ll/04/80 Tagged 1683 Epoufette 06/29/81 SLW 107 Epoufette 08/24/81 SLW 264 Epoufette 08/25/81 SLWF 36 Naubinway lO/l7/81 SLW 170 Epoufette 10/24/81 SLW 161 Naubinway 11/03/81 Tagged 1024 Epoufette 05/17/82 SLW 263 Epoufette 05/17/82 CL 86 Epoufette 05/18/82 SLW 63 Leland Area Leland 10/29/80 SLW 62 Leland 10/30/80 SLW 52 Leland 11/07/80 Tagged 415 Leland 06/15/81 SLW 81 Leland 08/27/81 SLW 94 Leland 10/21/81 SLW 134 Leland 10/22/81 SLW llO Leland 11/02/81 Tagged 117 Leland 05/20/82 SLW lll Leland 05/24/82 SLWG l4 Leland 05/24/82 SLW 107 Leland 05/24/82 SLWG ll Leland 05/24/82 SLW 16 Beaver Island Area Beaver Island ll/05/80 Tagged 19 Beaver Island 06/16/81 SLW 219 Charlevoix 05/18/82 SLW 169 ' 05/19/82 Grand Traverse Bay Area Northport 06/14/81 Tagged 163 Northport 06/14/81 SLW 140 scale sample, L = length, W = weight, G = girth, (D II II fin rays 26 The commercial trap net catch was sampled for length, weight, and age composition from October 1980 through June 1982 (Table 1). Total length was measured to the nearest 5 millimeters and weight to the nearest 10 grams. A scale sample was removed from the area midway between the dorsal fin and the lateral line. Scale aging techniques were used to determine the age of each fish. Scales were magnified 22x on a Bell and Howell AER-1020 microfiche reader. Crossing over on the posterio-lateral radius and disruptions of circuli were the primary criteria used for distinguishing annuli (Van Oosten 1923). Scales proved to be reliable estimators of (age (Scheerer 1982). A ninety-four percent agreement was found between ages determined from fin ray sections and ages determined from scales (n:36). There was an 82% agreement between 192 scales read by both principal scale readers. Ricker (1975) considers an 80-90% agreement to be acceptable. PARAMETER ESTIMATES Scheerer (1982) calculated mean lengths from scales sampled during this study (Table 2). The values for lengths at ages one and two may be unrealistically high according to Scheerer, because of the back calculation technique he used. Therefore, only ages 3-10 were used in the estima- tion procedure. A nonlinear least squares regression method was used to estimate the von Bertalanffy growth parameters. The following values were calculated: 27 Table 2. Mean lengths of lake whitefish in northern Lake Michigan estimated by Scheerer (1982). Length :1: 09 (D *334 millimeters *390 430 455 482 525 570 646 653 666 OOCDNGU'IR’WN—b c-A *Lengths at ages 1 and 2 are probably unrealistic (Scheerer 1982) and are not used in the analysis. 28 Laz713.8 mm and K:0.254 (with to fixed at zero). The length-weight relationship parameters were esti- mated by Scheerer to be a =4.24 X 10"9 and b = 3.12. This converts length in millimeters to weight in kilograms. Scheerer calculated mortality rates by a mark-recap- ture analysis. Instantaneous natural mortality (M) was estimated at 0.368 and instantaneous fishing mortality (all gear types) at 0.861. The trap and gill net components of the fishing mortality can be calculated by determining the relative proportion of fish harvested by each gear type. Trap nets accounted for 65% and gill nets accounted for 35% of the total whitefish harvest in 1981 (unpub. data Michigan DNR). This produces estimates of instantaneous trap net mor- tality of 0.560 (65% of 0.861) and instantaneous gill net mortality of 0.301 (35% of 0.861). The apical instantaneous fishing mortalities associated with these rates were calcu— lated iteratively. Successive one year simulations, using 1980 initial conditions, were made using various combinations of apical trap and gill net mortality rates. These trial sim- ulations continued until a set of apical rates produced in- stantaneous rates that converged on the actual 1981 fishing G The length-girth parameters were estimated to be A : mortality rates. (F; : 0.654 and F* = 0.554). -32.98 and B : 0.5777 by simple linear regression. Initial population levels (Table 3) were calculated by Scheerer. The gill net selectivity curve parameters are directly from McCombie and Fry (1960): 29 Table 3. Population estimates of lake whitefish by age class in November, 1980 in northern Lake Michigan (Scheerer 19 2). Age Number 3 1.272.535 4 365,906 5 16,632 6 0 7 3.326 8 O 9 O 10 3,326 Total 1,663,207 3O 0' = 0.1184 and "G = 0.2303 G (0.0514 and 0.100 when using base 10 logarithms). The parameter CG was a constant multiplier used to adjust the function to a 0.00 to 1.00 scale. The trap net selectivity function parameters were es- timated by non-linear least squares regression from the data presented in Eshenroder et. al. (1980): C1 = 0.9879 C2 = 25.43 The recruitment parameters were perhaps the most diffi- cult to estimate because of a lack of direct measures of spawning stock size and absolute recruitment. These values were estimated by the indirect technique detailed in Appen- dix 13. Briefly, the technique utilizes the principle that stock size is directly proportional to catch per unit effort (CPE) (Ricker 1975). Spawning biomass for each year is cal- culated from the average trap net CPE (kgs/lift) dur— ing that year. The constant that linearly relates the two was estimated from mark-recapture data. This method assumes that only the mature portion of the population is being es- timated. Fortunately, the current trap net minimum size limit of 432 mm is roughly equivalent to the length at ma- turity (Scheerer 1982). Absolute recruitment was estimated by calculating the partial contribution of three-year-olds to the total CPE and then using a.linear constant to calculate the actual number of three-year-old recruits. The actual estimates of recruitment and spawning stock biomass are pre- 31 sented in Appendix 4 and illustrated in Figure 5. The parameters of the recruitment function were estima- ted by the logarithmic transformation technique presented by Ricker (1975). The following is the regression model used: In (R/S) = ln 0 - BS Estimated values for the parameters are: a : 1.409 [3‘ = 3.413 X 10'7. The standard deviation of the error term v was calculated from the standard deviation of the regress- ion residuals at 0.736. The maximum age simulated was set at 15 years. This value was adequate since extremely few whitefish were found to be older than 14 years, even in the most lightly‘exploited stock sampled in this study. The selection of a proper At and simulation run length is important for accurate simulations. A sufficiently small .At will result in accurate numerical integrations. Also, the simulation must run long enough to produce an adequate esti- mate of the mean annual yield from the fluctuating population. Table 4 illustrates the results of an analysis performed to determine an acceptable value for At (Manetsch and Park 1981). Several simulations were made with the controllable input values constant and removing the random recruitment factor. The time step was progressively shrunk with each simulation and the simulated yield converged to a "true" solution. An integration error of approximately 0.7% was associated with a .At of 0.05. It was assumed that any in- tegration error of less than 1% would be negligible. 0 l A fl 0 v I g . y. 2“ _, O 2 . . °._ 0 0 2 . , E 1.1 0 .7 O 0 A ' I o I I Spawning Biomass (Kg 3: 10°) Figure 5. Stock-recruitment curve for lake whitefish in northern Lake Michigan. The shaded area represents :1 stan- dard deviation around the fitted line. 32 33 Table 4. Simulated yield values for different time steps. Controllable inputs set at F : 0.654, F : 0.554, MSL : 432 mm and mG = 228 mm with no randgm recruitmegt factor (v = 0). At ' Yield 0.25 yrs 495,541 kilograms 0.10 482,272 0.05 477,914 0.01 474,953 34 A similar technique was used to determine a suitable simulation run length. With the random recruitment factor operating, sets of five simulations were made at several different run lengths (Table 5). The standard deviation of yield from each set of runs decreased as the run length increased. A coefficient of variation (standard deviation of 5 runs/mean of 5 runs) of less than 2% was assumed to be acceptable. The run length of 1000 years meets this criterion and was used for further simulations. Figure 6a is a time trace of the first one hundred years of annual yield (of a thousand year run) simulated under the present fishing mortality regimes (F% = 0.654 and PG : 0.554). The existence of relatively extended periods of low production, interpersed with periods of relatively high production, was characteristic of many of the simula- tions. The resulting histogram of annual yield frequency is presented in Figure.6b. The skewed distribution of yield was common to all the simulations and is probably a result of the log-normal stochastic recruitment factor. 35 Table 5. Means, standard deviations and coefficients of variation at five separate simulations made*at four different run lengths. Controllable inputs set at FT = 0.654, FG = 0.554, MSL : 432 mm and mG = 228 mm.. Run Length Standard Deviation C of V 100 years 33,414 kgs 0.0556 300 31,450 0.0543 600 20,564 0.0349 1000 8,189 0.0139 12‘ ,1 Annual Yield (Kg x 105) o ( P I I U I r I o - 20 40 80 80 100 Year Simulated 150 100- r. p ""'r-—.. Frequency —" '7 o 1 I I I I I I I I 024081012141eia>2o Annual Yield (Kg 1: 10‘) Figures 6a and 6b. Time trace of the first 100 years of annual yield in a 1000 year simulation (Fig. 6a) and the resulting histogram (Fig. 6b). Controllable inputs were F; - 0.654, F5 : 0.554, MSL : 432 mm and mG : 228 mm. 36 SIMULATION ANALYSIS The future direction of the lake whitefish commercial fishery in northern Lake Michigan is uncertain at this time. The possibilities range from an exclusive trap net fishery to an exclusive gill net fishery. Because of this uncer- tainty, a scenario approach proved valuable in analyzing the possible forms the fishery may take. The simulations were organized into three separate scenarios: Scenario #1 - Multigear fishery with gear-specific fishing mortality as controllable inputs. Scenario #2 — Trap net fishery only with fishing mor- tality and minimum size limits as con- trollable inputs. Scenario #3 - Gill net fishery only with fishing mor- tality and mesh size limitations as con- trollable inputs. The general approach was to identify possible sets of controllable inputs which produce an optimum sustainable yield, OSY being defined as a large sUstainable yield with a relatively low level of variability. SCENARIO #1 - Multigear fishery with gear-specific fishing mortality rates as controllable inputs. One of the key questions concerning the fishery in northern Lake Michigan is whether or not the present level of exploitation is excessive and leading to a collapse of the whitefish stocks. The simulations of the first scenario investigated the effects of the current level of trap and gill net mortality on the long term yield and stability of the population. 37 38 Several simulations were made at various combinations of apical trap and gill net fishing mortality rates (with the trap net minimum size limit and gill net mesh size set at their current values: 432 mm and 114 mm, respectively). Figure 7a is a graphical summary of these many simulation runs. Sustainable yields are presented as a contoured repre- sentation of a topological surface displaying isopleths of equal mean annual yield at various combinations of gear- specific apical fishing mortalities. The general shape of this yield contour diagram shows low yield at low levels of fishing mortality, increasing to a ridge of high yield at moderate fishing levels and then tapering back off to low yields at high fishing mortality rates. The present status of the fishery is indicated in the figure by a black dot. The simulated mean annual yield of 700,000 kilograms is very close to the actual 1981 yield of 792,752 kilograms. This would indicate that the present high levels of yield can be sustained. However, the comparison of simulated yields with actual yields must be made with caution. The exact predictive power and relative accuracy of the model are unknown. A more valid comparison is one between simulated values. The relative position of the current fishery on the yield contour ridge indicates that higher sustainable yields are possible with a, reduction 12 2 3 4 5 O 1 l g 10 11 11 >11 EPOUFETTE BEAVER ISLAND I-H 80-1 F1 00- 00.1 «F ah ah 2s- “ o F I I I’ I I I I {4.111;}..131'112»: 134507a9101112>12 AGE AGE Figure 8. Age compositions of lake whitefish in northern Lake Michigan from commercial catch samples in the spring of 19 1. 41 42 The total catch is broken down by gear types in Figure 9a. As expected, the general trend indicates a higher per- centage of the total yield being harvested by trap nets at high trap net fishing mortalities and vice versa for gill net mortalities. Trap nets currently harvest the majority of the total whitefish catch (65% actual and 68% simulated). Equal allocation is represented in the figure by the 50% isopleth. Isopleths of mean spawning biomass are illustrated in Figure 9b. The largest spawning biomasses are associated with the smallest fishing mortalities. The simulated mean spawning biomass of 1 million kilograms represented by the current status of the fishery agrees reasonably well with the actual 1980 biomass estimate of 1.4 million kilograms. General areas of the parameter space can be identified which meet the requirements of OSY as previously defined. The shaded area of the lower left hand portion of the yield ridge in Figure 7a produces a relatively low level of varia— bility (Figure 7b) and a high sustainable yield. If an equal allocation of harvest between gear types is desired, the intersection of this ridge and the 50% allocation iso- pleth produces the satisfactory combination of mortality rates. For example, an apical trap net mortality of 0.3 and an apical gill net mortality of 0.4 would produce an annual yield of 900,000 kilograms, a coefficient of varia- tion of 0.45 and an equal allocation between gear types. To achieve these mortality rates, the trap net effort would t5 9‘ Trap N Yield m 3.0 70. 11)- 50 0.5 -I 40 1;! l '17 Spawnhg Biomass [Kg 11 1051 . Apical hstantanaoua Trap Nat Mortality , I o 0.5 1.0 1-5 ‘ Apical instantaneous GI Net Mortalty Figures 9a and 9b. Isopleths of percent trap net yield (Fig. 9a) and spawning biomass (Fig. 9b) of lake whitefish in northern Lake Michigan in a multi-gear fishery with MSL = 432 mm and m - 228 mm. Black dots represent the current state of tge fishery (1981). 43 44 have to be reduced by 54 percent, and gill net effort by 28 percent from their 1981 levels (assuming a linear rela- tionship between apical instantaneous fishing mortality and fishing effort). A reduction in effort of this magnitude would have obvious short term detrimental effects on the fishing industry. Although the current level of exploita- tion is somewhat less than optimally efficient, it is probably not exerting a catastrophic impact on the stock. Therefore, a gradual reduction in fishing effOrt may be the most desirable approach to take. SCENARIO #2 - Trap net fishery only with fishing mortality rates and minimum size limits as controllable inputs. The simulation analysis of this scenario identified the combinations of fishing mortality and trap net minimum size limits that produce OSY in an exclusive trap net fishery. The contour diagrams in Figures 10a and 10b represent graph- ical summaries of numerous simulations run with different sets of controllable inputs. The first figure illustrates the isopleths of sustain- able yields associated with the various combinations of trap net mortality and size limits. The diagram is directly ana- logous to a Beverton-Holt (1957) dynamic pool, yield isopleth diagram, except the absolute yield is plotted rather than yield per recruit. The optimal minimum size limit depends heavily on the level of fishing mortality. Considerably more fishing effort can be expended at a 500 mm minimum size limit than a 450 mm limit and still produce relatively large yields. Yield [KO 1: 1031 w 600 .1 .1 500-» q E .I . 400 -l .5 m g g Coefficien{ot Variatio‘ 3 000- I! ’0‘ z I g . I- 500d 0.4 .s s 0.1 400 - I 0 0.0 ’ Apical instantaneous Trap Net Mortalty Figures 10a and 10b. Isopleths of simulated mean annual yield (Fig. 10a) and coefficient of variation (Fig. 10b) for lake whitefish in a hypothetical exclusive trap net fishery in northern Lake Michigan with MSL : 432 mm. Shaded area represents possible OSY's. 45 46 The coefficient of variation contour diagram is pre- sented in the second figure. The variability in annual yield ranges from 0.4 at low fishing mortalities to 1.1 at high fishing mortalities. Minimum size limits also affect yield variability with the highest coefficient of variation being associated with the lowest minimum size limits. The shaded area in the figures represent the possible combinations of trap net mortality and minimum size limits which produce OSY. If the fishery does move to an exclusive trap net fishery, it would be advisable to hold the trap net effort at the 1981 level. The current apical fishing mor- tality rate of 0.654 and minimum size limit of 432 mm would produce a yield of 900,000 kilograms. However, if an increse in trap net effort were to be unavoidable, a higher minimum size limit than the present 432 mm would be required to main- tain a relatively high sustainable yield with moderately low variability. SCENARIO #3 - Gill net fishery‘only with fishing mortality and mesh size limitations as controllable inputs. A third possible direction the fishery might take is one towards an exclusive gill net fishery. Management strategies include the regulation of fishing mortality and gill net mesh size. The simulation of this scenario investigates the effects of these two controllable inputs on sustainable yield and variance. The yield contour diagram in Figure 11a is similar to the trap net diagram in Figure 10a with gill net mesh size 175- 3 Yield [Kg 1: 10 l 150 - 125 '- 3 . ('3 100 - —-L '§ 175 - A 2 .- Coefficient £ 01 Variation 0.4 I 1! 150 - 0.5 125 d 100 I 0 015 1A) 115 Apical hatantaneous ea Net Mortalty Figures 11 a and 11b. Isopleths of simulated mean annual yield (Fig. 11a) and coefficient of variation (Fig.11b) for lake whitefish in a hypothetical exclusive gill net fishery in northern Lake Michigan, with mG = 228 mm. Shaded area represents possible OSY's. 47 48 used in place of trap net minimum size limit. Maximum yields occur at apical fishing mortalities of 0.8 to 1.5 and mesh sizes of 110 mm to 130 mm(stretched mesh). Con- siderably more fishing effdrt can be expended at a mesh size of 130 mm than a mesh size of 110 mm and still produce high yields. Isopleths of the coefficients of variation associated with various combinations of controllable inputs are pre- sented in Figure 11b. Variability increases as fishing mortality increases and mesh sizes decrease. The highest variability occurs at the highest fishing mortalities and the smallest mesh sizes. The shaded areas in the figures represent possible sets of controllable inputs which result is OSY. The 1981 apical gill net fishing mortality of 0.554 and mesh size of 114 mm would produce a yield of 750,000 kilograms if the present trap net fishery were entirely eliminated. The stock could also absorb a considerable increase in gill net effort and still maintain relatively high yields. OTHER MODEL APPLICATIONS In addition to the scenarios presented above, there are other interesting combinations of the controllable inputs which can be simulated. For example, the optimal trap net minimum size limit in a multigear fishery is another perti- nent management question. Figure 12 illustrates the sustain- able yields produced from various trap net minimum size limits with gear-specific fishing mortalities held at 1981 levels. 10" a. s _ Total Yield /’,__. _.1 9 / x ,l’ a? / Gill N01 Yield :9 0-1 //' g / c CO- ’/ ‘ .1 ”a"‘ r .ys ‘ ’ 1’ “~‘ / ~ ./ ‘\ Trap Net Yield a” "s 2 ‘f/ \\ \ ~\~‘ i ‘\‘ 0 “‘ I l I 400 432 489 500 800 Trap Net Mhinum Size Limit [mm] Figure 12. Simulated mean annual yield at different trap net minimum size limits for lake whitefish in a multi-gear fishery in northern Lake Michigan. Other controllable in- puts set at their 1981 values: F; = 0.654, F5 = 0.554 and mG = 228 mm. 49 50 The maximum sustainable yield occurs at a size limit of 550 mm and the harvest would be done primarily by gill nets. A size limit of 480 mm produces a yield allocated equally to each gear type, but has a slightly lower sustainable yield. There is currently discussion within the regulatory agencies to raise the trap net minimum size limit to 489 mm from the current 432 mm. Although the total sustainable yield would rise slightly, the major effect would be a shift from a harvest dominated by trap nets (68%) to a har- vest dominated by gill nets (53%). The 114 mm stretched mesh gill nets would be harvesting cohorts considerably earlier than would trap nets. Another interesting aspect of a multigear fishery con- cerns the inherent stability characteristics of each gear type. Does one gear produce a more stable yield than the other? Figure 13 illustrates the cgefficients of varia- tion at different levels of simulated sustainable yield for each gear type when fished exclusively. A trap net has a slightly lower associated level of variability than a gill net does for any given yield. This is due primarily to the inherent selectivity of each gear. The trap net harvests fish over a much broader range of fish lengths than does a single-sized mesh gill net. However, if mesh size limita- tions are in the form of minimum size mesh restrictions and fishermen are not required to use a single mesh size this may not be the case. Knowledgeable gill net fishermen are t4 \ L2 4 to d ‘\ \\ Gill Nate 5 \ 3 \ b 1; Trap Nets ,_ (L8 d e E .9 .9 3: 3 o 0.0 J OAl-' 0 I I I I I 2 4 8 8 10 Annual Yield (Kg It 105) Figure 13. Simulated coefficients of variation in yield of lake whitefish in northern Lake Michigan at different mean annual yields when each gear is fished exclusively. Controllable inputs of MSL : 432 mm and mG = 228 mm. 51 52 able to "track" strong cohorts by using nets of different mesh size as the year class grows which reduces the overall variability. But with a single-size mesh restriction, trap nets do produce a slightly more stable yield than do gill nets. MODEL LIMITATIONS While a simulation analysis is a valuable approach to this type of problem, certain inherent limitations must be kept in mind when interpreting the results. One potential weakness involves the possible inaccurate estimates of sensitive key parameters which can significantly distort the model behavior. Table 6 contains the results of a sensitivity analysis of the model parameters. Each para- meter value was increased 10% and then decreased 10%, holding the others constant and noting the deviation in simulated yield for each change (with the random recruit- ment factor not operating). This identified the relative sensitivity of each parameter. The length-weight relation- ship (b) was the most sensitive.- In fact, the model output was not stable (fluctuated chaotically) for either a 10% in— crease or decrease in this parameter. It enters the model in an exponential fashion, which greatly magnifies a 10% change in its value. Fortunately, the length-weight para- meters were among the most accurately estimated, being de- rived from a sample of 1577 fish. The recruitment function parameters were also identi- fied as being relatively sensitive. Unfortunately, the 53 Table 6. Results of the sensitivity analysis performed on the model parameters. The deviation in simulated yield which resulted from a +10% change in each parameter is listed. ' DeViation from Deviation from Parameter a -10% change _ a +10% change a b0.0936 +0.0847 b - - a -0.0936 +0.0847 B +0.1111 -0.0909 L -0.2298 +0.3007 K -0.0802 +0.1613 A -0.0074 +0.0060 B -0.1027 +0.0086 M +0.1105 -0.1051 C -0.0020 +0.0015 C +0.0248 —0.0571 +0.0013 -0.0038 +0.0055 -0.0060 -0.0258 +0.0250 Deviation : [(simulated yield from +10% change in parameter value) - (simulated yield from no cfiange in parameter value)]/ (simulated yield from no change in parameter value) 54 accuracy of these two important parameters is unknown. There are several problems associated with the estimation procedures (Appendix 3) used to calculate spawning biomass and recruitment. Most serious is the possibility that CPE may not be directly proportional to stock size (either spawning biomass or number of three-year-olds). Trap net effort is measured by kilograms per lift, but soak time (number of nights out) is not considered. If fishermen tend to leave their nets out longer when there are fewer fish present, the reported CPE would overestimate fish a- bundance. Improvements in the gear or the fishermen's ex- pertise over time would also complicate the relationship. A problem specifically related to the recruitment estima- tion procedure is that the relationship may be complicated by density dependent body growth. The first significantly vulnerable age group is the three-year-olds. Therefore, the rate of growth may be important in determining the por- tion of the cohort entering the fishery. For example, a large cohort might be underrepresented because of slower body growth. The spawning biomass and recruitment esti- mates are, admittedly, the least accurate of all the data used to estimate the parameters. Another problem with the stock-recruitment relation— ship is that the behavior of the function at high stock levels comes entirely from the extrapolation of a statis- tically fitted line. However, the fit appears to be bio- logically reasonable for this population. The curve is 55 consistent with the observation that the recent large re- cruitments of 2 million three-year-olds are probably near the maximum possible for this stock. . A serious limitation of the simulation model is that no mechanism for density-dependent body growth was incor- porated. 'Several authors have reported relatively slower rates of growth associated with higher densities of lake whitefish. Healey (1979), noted this phenomonon in several northern Canadian lakes and Jensen (1982) inferred, by a mathematical analysis, the importance of density-dependent body growth in the population regulation in lake whitefish. The potential effects of density-dependent body growth could be very important if the north shore population bio- mass has the real capability of increasing to the simulated 4 million kilograms with very light fishing pressure (Fig- ure 9b). The model behavior would be affected at high stock densities (extreme lower left-hand corners of Figures 7a, 7b, 9a and 9b). However, the extent to which density— dependent growth is important to the whitefish population in northern Lake Michigan is not known. Even with these possible inherent weaknesses, it is felt that the model is biologically sound and realistically simulates the population dynamics of this stock over rela- tively wide ranges of population size. CONCLUSION The whitefish stock in northern Lake Michigan is pro- bably not in imminent danger of collapse, but is operating at a point somewhat less than optimal efficiency. Although a reduction in fishing effort would eventually produce a larger yield, the short term effects would be detrimental to the fishing industry. An abrupt reduction in fishing effort would be undesirable since the present fishery is capable of producing large, sustainable annual yields of 700,000 kilograms. 56 APPENDICES 57 Appendix 1. Glossary of variable names used in the text. length-weight parameter length-girth parameter stock-recruitment parameter length-weight parameter length-girth parameter stock-recruitment parameter trap net selectivity constant trap net selectivity constant coefficientxfl‘variation of yield instantaneous gill net fishing mortality rate operating on cohort i at time t instantaneous trap net fishing mortality rate operating on cohort i at time t apical instantaneous gill net fishing mortality rate apical instantaneous trap net fishing mortality rate girth of fish in cohort i at time t cohort integer age maximum age attained von Bertalanffy growth parameter length of fish in cohort i at time t length at maturity InntBertalanffyasymptotic length parameter gill net mesh perimeter (equal to 2x stretched mesh measurement) trap net mesh perimeter (equal to 2x stretched mesh measurement) YG i(t) Popm T im rm1000000. PUP(2)=1000000. POP(3)=400000. P0P<4I=1272535. PUP(S)=365906. PUP(6)=26632. P0P(7):3326. INITIALTXE RATE VARIABLES DO 22 IP$0115 AGE$1P+T LEN(IP)3LINF*(1o00‘EXP(*K*AUE)) GIRTH2A6+BU*LEN(TP) IF((3IRTT1.LT§¢O.())lTfliN GIRTHflooOOOT ELSE ENUIF GPRTN=GIRTH/PTN 13F'R13NI3131RTH/PC‘IN PETNxGPRTN**TNN/(TNKXXTNN+GPRTNX$TNN) PEGNxfiNK/(2.506628XUNSUXOPRUN)*EXP(*(LOG(OPRGN)~GNMU)*KR/(2.0*UNSO +tra)> F'i'N( IF' ) ==AF'FTN>1+M(IF))XPUP(IP) CONTINUE STYLD30.0 SUMSGYz0.0 STNY=0.0 SSUTN=0.0 SGNY=0.0 SSGGNw0.0 88820.0 SSQSBz0.0 SR330 00 SSGR20.0 ‘ START YEARLY ITERQTIUNS no 300 NnOyNYITT ISTnRT wITHIN YEAR ITERRTIUNE D0 200 .J='-‘19III|T II=T+I|T START [MM CULfiIJIHMS EUR EMWJIIHJHURT III!) 100 I 33357155 CEHICLHIEIIE l EXQGIIIS (\NII MKHRITNLICIY IRQIIES {IE KJUEHJRII AGEzI+ ' LENxLINF*(1.00~EXP<~K*AGE)) GIRTHxAG+BfifiLEN(I) IF(GIRTH.LE.0.0)THEN GIRTHm0.0001 ELSE ENDIF BPRTNmGIHTH/PTN GPRuNmGIRrH/PUN PETNmGPRTNXXTNN/(TNKXXTNN+UPRTN$$TNNJ PE GN==GNN/ ( 132 . 1506 61-3 8 I~’.