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Oklahoma Cooperative Fishery Research Unit*, Oklahoma State University, Stillwater, Oklahoma
Two computer simulation models of population dynamics of largemouth bass (Micropterus salmoides) are described. Model I is an agestructured, deterministic model with numbers as the only state vector. Constant agespecific fecundities and survival rates are required inputs. Model I simulates population trends based on an equilibrium population. Sensitivity analysis of this model indicates that density of bass is most sensitive to variations in survival from egg to age I. Multiple regression equations with water level during spawning and water level fluctuation since the end of the previous growing season as predictor variables resolved 88.2% of the observed variation in yearclass strength and 86.7% of the observed variation in mortality from egg to age I of largemouth bass in Lake Carl Blackwell. Model II is similar to Model I except that the effect of reservoir water level and water level fluctuation on survival of the youngoftheyear is included. Predictions of number of age I recruits from Model II agree closely with population estimates for Lake Carl Blackwell.
The largemouth bass, Micropterus salmoides, is one of the most important sport fishes in southern lakes and reservoirs. About 24% of the fishing trips on reservoirs in the southeastern states are specifically for centrarchid basses and another 18% for bass plus other species (1).
The management of largemouth bass in large reservoirs (larger than 200 hectares) is often difficult because reservoir size may prohibit the fishery manager from collecting enough data to allow rational decisions, and because unstable water levels may prevent the fish population of the reservoir system from attaining a stable (or steady) state. Thus, management of largemouth bass in large reservoirs must rely on few studies. What is needed is a reliable method for predicting the consequences of a proposed management action before its implementation.
One approach for development of reliable predictions is the use of models based on systems analysis, computer modeling, and simulation techniques. Mathematical models of fisheries have traditionally been used to assess fish stocks and to predict maximum sustainable yield (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12). The basic weaknesses of these models are that they are deterministic and assume a steadystate fishery. These assumptions may not be unreasonable when one is dealing with a large marine fishery but are unreasonable when largemouth bass fisheries in large fluctuating reservoirs are being considered.
Watt (12) proposed a model which would include the influence of environmental factors on recruitment, growth, and natural mortality in an attempt to avoid the limitation of deterministic models. This model has been applied to a sport fishery for smallmouth bass in South Bay of Lake Huron (13); however, the analysis suffers from too few data.
This paper describes a computer simulation model that includes environmental factors and predicts yearclass strength and population trends for largemouth bass populations. Ultimately, all of our existing knowledge of population dynamics of largemouth bass may be incorporated into such a model. The longrange objective of this research is to develop models of largemouth bass population dynamics that will provide biologists with useful management tools.
Population dynamics of largemouth bass in reservoir environments are complex. Thus, these dynamics may best be studied
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in terms of Holling's (14, 15) experimental components approach, with emphasis on the processes of growth, mortality, reproduction, and yearclass formation. Holling's approach involves determining which components of the reservoir ecosystem are limiting these processes, followed by definition of system variables and parameters and formulation of the mathematical model.
The objective of this study was to construct a model of the bass population in Lake Carl Blackwell, a 1400hectare reservoir in northcentral Oklahoma. This lake was chosen because several previous investigations have been made here on population dynamics of adult bass (16), growth, production and mortality of youngoftheyear bass (17), growth of bass in relation to water level (18), and the relationship between weather and other environmental factors and yearclass strength (19, 20). The philosophy employed in this study was to begin with a simple model (Model I) which included only a few system variables and parameters, and to expand and modify this model so that it would include more of the relevant components. Data collected from Lake Carl Blackwell were analyzed by simple and multiple linear regression techniques (21) to arrive at the mathematical equations used in Model II.
Yearclass strength of bass in Lake Carl Blackwell was estimated for 1965 through 1975, using ecological density on 13 August (20), and an additional estimate was made by the author in 1976, using the same sampling techniques (application of rotenone to coves). I estimated survival from egg to age I for 7 consecutive years, from estimates of the egg potential and number of age I recruits the following spring. Estimates of agespecific fecundities were based on the mean lengths presented in Zweiacker et al. (18) and a lengthfecundity regression of logtransformed data from Kelley (22) and Coomer (23). Egg potential was calculated by
number of eggs  Eq. 1 
where N_{i} = number of fish of age i and m_{i} = number of eggs per female of age i, assuming a 1: 1 sex ratio and maturity at age III. Zweiacker et al. (18) noted that in Lake Carl Blackwell a few age II bass spawned but most did not spawn until age III. Population estimates from Zweiacker (16) and Shirley (17, and unpubl. data) were adjusted to 15 May (the approximate midpoint of the spawning period) by assuming a constant exponential mortality and using agespecific rates (17, 18). Zweiacker's estimates were stratified by single age groups using percentage age composition in his samples.
I adapted the mathematical models for computer simulation, using FORTRAN IV programming language after the forms of the equations were specified. The FORTRAN computer programs were written with flexible input requirements to allow for manipulation of the simulated fish population and to accommodate different parameter values. Programs were run on the IBM System 370/Model 158 digital computer at the Oklahoma State University Computer Center. Program listing and sample outputs are available from the author.
Sensitivity analysis was performed to determine how varying the input parameters affected population size (density) after a 20year simulation. Net sensitivity of the population side to a 10% change in any given input parameter was computed according to the formula given by Francis (24):
Eq. 2 
where S (x, y, D x) is the net sensitivity of y to a change, D, in x. The relative sensitivity was then obtained by dividing net sensitivity by the largest net sensitivity value.
Both models were agestructured, based on agespecific fecundities and survival rates, and similar to the Leslie matrix algorithm (25). The notation is as follows: N_{i} (t) = number of individuals of age i at time t, m_{i} = fecundity per individual of age i, and S_{i} = probability that an individual of age i will survive to age i+1. Fecundity per individual, m_{i}, would equal fecundity per female multiplied by 0.5, assuming a 1:1 sex ratio. The basic time unit is a year which commences at the time eggs are laid (approximately 15 May for Lake Carl Blackwell). The number of eggs produced is calculated by
Eq. 3 
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and a new age distribution is obtained by
Eq. 4 
for all i = 1, 2, 3,..., k, where k = maximum age.
Reliable estimates of S_{o}, survival from egg to age I, are difficult to obtain for natural populations. For this reason, S_{o} is estimated by assuming an equilibrium population and using agespecific fecundity and survival data. Vaughan and Saila (26) derive this estimation procedure on the basis of the Leslie matrix algorithm:
Eq. 5 
Model I
I made a simulation run using average agespecific survival rates from Zweiacker et al. (18), average agespecific fecundities from Kelley (22) (Table 1), and an initial population of 1000 age I fish (Figure 1). The simulated population initially oscillated due to the timelag for the fish to reach maturity, and finally stabilized at about simulation year 18.
Simulation of Model I with a stable age structure and parameter values in Table 1 resulted in a total population of 2396 after a 20year simulation. The effect of varying the survival rate of age 0 fish is shown in Figure 2. Results of the sensitivity analysis (Table 2) indicated that density was most responsive to changes in survival rates of age 0, age I, and age II bass, in that order. Therefore, on the basis of this model, it is important that one have accurate estimates of survival rates of these age groups to simulate population trends. Horst (27) also found that population growth of the cunner (Tautogolabrus adspersus) was most sensitive to changes in survivorship of younger ages, from a sensitivity analysis of the Leslie matrix model.
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In Model I, I assumed that the population operated in a deterministic fashion, with constant agespecific survival rates and fecundity. Consequently the simulated population trends beginning with 1000 age I fish (Figure 1) do not mimic the situation in new reservoirs, where in the first years of impoundment, large yearclasses of bass are produced and the population exhibits a "boom and bust" phenomenon. Also, the effects of density and environmental factors were ignored in Model I. Since this omission was unrealistic, further development of this model should involve varying the agespecific survival rates or fecundity, or both, based on density or other environmental conditions. Also there is evidence for higher mortality of males than females among older bass (28, 29), which tends to shift the sex ratio away from unity. In many reservoirs this shift may be negligible but inclusion of an appropriate parameter would increase the flexibility of the model.
Model II
Sensitivity analysis of Model I showed that the population size was most sensitive to changes in survival from egg to age I. This conclusion is also supported by field studies in which the major factors influencing this mortality were environmental factors such as wave action, wind, and temperature (19, 30, 31, 32). In many reservoirs large yearclasses of largemouth bass are produced in years of stable or rising water levels during spawning (33, 34, 35, 36), probably because the flooding of shoreline areas increases cover for nest sites and for shelter from predation, and releases nutrients into the littoral zone. Also, the increased depth of water over the nests decreases the effects of wind, wave action, and fluctuations in temperature.
The relation between environmental factors and yearclass strength of largemouth bass in Lake Carl Blackwell was studied by Summerfelt and Shirley (20), who correlated yearclass strength with a series of biotic and abiotic environmental factors. They showed that yearclass strength was positively correlated with water level, change in water level, and turbidity; negatively correlated with hardness, alkalinity, and pH; and uncorrelated with wind, air and water temperature, and size of the spawning population. They concluded that the fluctuations in yearclass size were due primarily to the water level and its effect on food and cover for youngoftheyear bass. Other significant correlations were attributed to the effects of changing water levels on the physical and chemical composition of the water.
Model II represents an attempt to include the effects of environmental factors on reproduction and yearclass formation within the framework of Model I. Yearclass strength of largemouth bass varied considerably in Lake Carl Blackwell, as indicated by ecological densities on 13 August (Figure 3). Regression analysis was performed to determine which of the variables in Table 3 would be useful in predicting yearclass strength. The correlation coefficient between water level fluctuation and yearclass strength (Figure 4) of 0.8779 was highly significant (P = 0.0002), and water
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level fluctuation accounted for 77.07% (R^{2} = 0.7707) of the observed variation in density. The residuals were greatest for 1970, 1971, 1972, when water level was extremely low, and 1975, when water level was at or near spillway level all year. These data indicate the importance of water level itself, in addition to water level fluctuation, in determining the strength of yearclasses.
The correlation between water level during the spawning season and yearclass strength (Figure 5) was not significant (r = 0.5337; P = 0.0739). However, analysis without the 1973 data yielded a highly significant correlation coefficient (r = 0.7828; P = 0.0044). Even though the water level during the 1973 spawning season was more than 2m below spillway level, it was nevertheless rising, and this condition resulted in very successful largemouth bass reproduction and survival and growth of youngoftheyear. Thus, there is apparently an important interaction between water level during spawning and water level fluctuation with respect to yearclass formation.
Results of the multiple regression in which these two variables are used as predictor variables are summarized in Table 4. The equation for predicting density of youngoftheyear bass on 13 August (Y) is
Eq. 6 
where X_{1} = water level fluctuation from 1 October of the previous fall to 15 May (m), and X_{2}) = mean water level during May, M.S.L.). This relationship is
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highly significant because the calculated F = 33.6198 for regression has an associated probability of a greater Fvalue of 0.0002. Furthermore, these two variables account for 88.20% (R^{2} = 0.8820) of the observed variation in density. This R^{2} value is substantially higher than that observed for the regression with either water level fluctuation alone (R^{2} = 0.7707), or water level during May alone (R^{2} = 0.2848). Also, the addition of the second variable, water level during spawning, to the model was significant as evidenced by the sequential Ftest (F = 8.4835; 1, 9; P = 0.0172). Water level fluctuation is more than twice as important as water level during May in predicting density of young of the year bass; the ratio of standardized regression coefficients was 2.317 (0.7959/0.3435).
For inclusion of this relationship in the population dynamics model, it was necessary to relate survival from egg to age I to these two variables. I therefore used estimates of agespecific fecundities for Lake Carl Blackwell (Table 5) and estimates of number of fish per age group in the spring, and total egg potential of the stock to calculate annual instantaneous mortality rates (Z_{o}) from egg stage to age I (Table 6). Annual instantaneous mortality rate (Z) is related to survival rate (S) by
Eq. 7 
where e = the base of natural logarithm^{5} (7).
Results of the multiple regression analysis in which water level fluctuation from 1 October of the previous fall to 15 May (m) (X_{1}) and mean water level during May (m, M.S.L.) (X_{2}) are used to predict
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the annual instantaneous mortality rate Z_{o} ) from egg to age I are significant (Table 7). The equation for predicting Z_{o} is
Eq. 8 
This regression equation is significant F = 13.1073; 2, 4; P = 0.0193) and accounts for 86.76% (R^{2} = 0.8676) of the observed variation in Z_{o}. The observed and predicted values coincided well (Figure 6).
Model II is essentially the same as Model I, except that Model II includes the equation to predict the survival from egg to age I, parameters to account for the percentages for each age group that are mature and female (incorporated into equation 3), and the use of fecundity per female rather than fecundity per individual.
Simulation of the Lake Carl Blackell population was made using initial age structure for spring 1968 (Table 6), agespecific survival rates (Table 1) from Zweiacker et al. (18), agespecific fecundity from Table 5, and water level data for Lake Carl Blackwell from 1968 to 1977 (Table 3). Inasmuch as confidence limits on some of the input data vary widely, discrepancies between simulated and observed number of age I recruits could be attributed to either errors in the population estimates or errors in the model.
Comparison of the simulated predictions of yearclass strength, as indicated by the number of age I recruits, with the observed number of age I recruits (Figure 7) show that precision of the estimates is high; however, the model cannot be soundly validated with data that were used for its derivation.
Model II should prove to be of value in largemouth bass fishery management by enabling fishery biologists to quickly and easily predict yearclass strength for any given year, and hence the future population size and structure. With this information at hand, fishery managers can make better decisions on stocking recommendations and creel limits.
It is unlikely, though, that the parameters derived in this study for the relationship between yearclass strength and water level fluctuation and water level during the spawning season will be the same for all reservoirs. Therefore research must be done on other reservoirs to evaluate the generality of this relationship and to determine the appropriate parameter values for individual reservoirs.
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Funds for this study were provided by Federal Aid to Fish and Wildlife Restoration, Oklahoma DJ Project F36R, Job 3. This paper was condensed from a portion of a M.S. thesis by the author (37). I thank Drs. O. Eugene Maughan, Michael D. Clady, Robert J. Mulholland, and Ronald W. McNew for critically reviewing the manuscript, and Dr. Robert C. Summerfelt for his encouragement and guidance during the initial phase of this research. I also thank all Unit personnel who were involved in field studies on Lake Carl Blackwell, especially Mr. Kenneth E. Shirley and Dr. Paul L. Zweiacker.
1. J. C. HORVATH, Southeastern Economic Survey of Wildlife Recreation, Executive Summary, Environmental Research Group, Georgia State Univ., Atlanta, 1974, 100 pp.
2. R. J. H. BEVERTON and S. J. HOLT, On the Dynamics of Exploited Fish Populations, Min. Agr. Fish. Food (U. K.), Fish Invest. (Ser. II) 19, 1957, 533 pp.
3. L. M. DICKIE, in: Economic Effects of Fishery Regulation, FAO Fisheries Rep. 5, 1962, pp. 102133.
4. W. W. FOX, JR., Trans. Am. Fish. Soc. 99: 8088 (1970).
5. J. A. GULLAND, in: J. B. CRAGG (ed.), Advances in Ecological Research, Vol. 7, Academic Press, London and New York, 1971, pp. 115175.
6. J. J. PELLA and P. K. TOMLINSON, Bull. InterAm. Trop. Tuna Comm. 13: 421496 (1969).
7. W. E. RICKER, Bull. Fish. Res. Board Can. No. 191, 382 pp. (1975).
8. M. B. SCHAEFER, Bull. InterAm. Trop. Tuna Comm. 1: 2756 (1954).
9. M. B. SCHAEFER, Ibid. 2: 245285 (1957).
10. M. B. SCHAEFER and R. J. H. BEVERTON, in: M. N. HILL, (ed.), The Sea, Vol. 2, John Wiley & Sons, Inc., New York, 1963, pp. 464483.
11. R. P. SILLIMAN, J. Fish Res. Board Can. 28: 12111214 (1971).
12. K. E. F. WATT, Ibid. 13: 613645 (1956).
13. K. E. F. WATT, Ecol. Monogr. 29: 367392 (1959).
14. C. S. HOLLING, Memoirs Entomol. Soc. Can. 32: 2232 (1963).
15. C. S. HOLLING, in: K. E. F. WATT (ed.), Systems Analysis in Ecology, Academic Press, London and New York, 1966, pp. 195214.
16. P. L. ZWEIACKER, Population Dynamics of Largemouth Bass in an 808Hectare Reservoir, Ph.D. Thesis, Okla. State Univ., Stillwater, 1972.
17. K. E. SHIRLEY, Growth, Production and Mortality of Largemouth Bass During their First Year of Life in Lake Carl Blackwell, Oklahoma, M. S. Thesis, Okla. State Univ., Stillwater, 1975.
18. P. L. ZWEIACKER, R. C. SUMMERFELT and J. N. JOHNSON, Proc. 26th Ann. Conf., Southeast. Assoc. Game Fish Comm. (1972): 530540 (1973).
19. R. C. SUMMERFELT, in: R. H. STROUD and H. CLEPPER (eds.), Black Bass Biology and Management, Sport Fishing Inst., Washington, D.C., 1975, pp. 166174.
20. R. C. SUMMERFELT and K. E. SHIRLEY, Relation Between Environmental Factors and YearClass Strength of Largemouth Bass in Lake Carl Blackwell, Oklahoma. Unpubl. Manuscr., Okla. Coop. Fish. Res. Unit, Stillwater, 1976.
21. N. R. DRAPER and H. SMITH, Applied Regression Analysis, John Wiley & Sons, Inc., New York, 1966.
22. J. W. KELLEY, Trans. Am. Fish. Soc. 91: 2328 (1962).
23. C. E. COOMER, JR., Population Dynamics of Black Bass in Center Hill Reservoir, Tennessee, M. S. Thesis, Tenn. Tech. Univ., Cookeville, 1976.
24. R. C. FRANCIS, Bull. InterAm. Trop. Tuna Comm. 16: 235279 (197475).
25. P. H. LESLIE, Biometrika 33: 183212 (1945).
26. D. S. VAUGHAN and S. B. SAILA, Trans. Am. Fish. Soc. 105: 380383 (1976).
27. T. J. HORST, Trans. Am. Fish. Soc. 106: 253257 (1977).
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28. H. E. BRYANT and A. HOUSER, in: G. E. HALL (ed.), Reservoir Fisheries and Limnology, Am. Fish. Soc., Washington, D.C., 1971, pp. 349357.
29. W. A. HUBERT, Proc. 29th Ann. Conf., Southeast. Assoc. Game Fish Comm. (1975): 126134 (1976).
30. R. C. SUMMERFELT and K. E. SHIRLEY, Methods for Early Estimates of Largemouth Bass YearClass Strength, Final Rep., Coop. Study, Okla. State Univ. and Okla. Dept. Wildl. Conserv., 1975.
31. R. H. KRAMER and L. L. SMITH, JR., Trans. Am. Fish. Soc. 91: 2941 (1962).
32. A. W. EIPPER, in., R. H. STROUD and H. CLEPPER (eds.), Black Bass Biology and Management, Sport Fishing Inst., Washington, D.C., 1975, pp. 295305.
33. L. R. AGGUS and G. V. ELLIOT, Ibid. pp. 317322.
34. W. E. KEITH, Ibid. pp. 489497.
35. M. G. BROSS, Proc. Okla. Acad. Sci. 48: 194199 (1969).
36. C. E. VON GELDERN, JR., Calif. Fish Game 57: 228245 (1971).
37. D. J. ORTH, Development of a Computer Simulation Model of Largemouth Bass Population Dynamics, M. S. Thesis, Okla. State Univ., Stillwater, 1977.