Stock-recruitment Curve Research Articles

Stock-recruitment curves as used in the stock-reduction analysis model

Stock-recruitment curves as used in the stock-reduction analysis model

Effect of Assumptions about the Stock–Recruitment Relationship on a Lobster (Homarus gammarus) Stock Assessment

Stock assessment of the European lobster (Homarus gammarus) has involved yield per recruit analysis based on the established length cohort methodology of Jones (1974. ICES C.M. 1974/F:33; 1981. FAO Fish. Circ. 734) which assumes that recruitment to the fishery is independent of spawning stock. The Shepherd (1982. J. Cons. Int. Explor. Mer 40: 67–75) model has been used to simulate a range of assumed stock–recruitment relationships, and the resulting sensitivity analysis describes how these affect the relation between yield or biomass and four management variables, namely fishing mortality, minimum carapace length, maximum carapace length, and the capture or noncapture of egg-bearing females. Yield curves show a clear maximum with a marked tendency to stock collapse when fishing effort is high. For the range of simulations considered, the probability of an early recruit failure is greatest for asymptotic stock–recruitment curves, which generate yield curves with maxima at an effort substantially lower than the present level. Only with a highly overcompensatory stock–recruitment curve is there a case for increasing effort to maximise yield, but such a relationship tends to reduce the benefit of increasing minimum carapace length or of setting a maximum carapace length. The model predicts that the assumption made about the stock–recruitment relationship also has a marked effect on the results expected from a ban on the landing of egg-bearing females. Overall the results confirm the unsatisfactory prognosis of the yield per recruit model and emphasise the need to gain an understanding of the biological factors determining the shape of the lobster stock–recruitment curve.

Simplified Calculation of Optimum Spawning Stock Size from Ricker's Stock Recruitment Curve

The optimum spawning stock size for a Ricker stock recruitment curve was shown to be accurately approximated by the equation Ps = Pr(0.5–0.07a) when 0 < a < 3. A simple modification was also shown to incorporate stochastic variation about the stock recruitment curve into calculations of optimum stock size.

Bias in the Estimation of Functional Relationships from Time Series Data

Functional relationships, such as stock–recruitment curves, are generally estimated from time series data where natural "random" factors have generated both deviations from the relationship and also informative variation in the independent variables. Even in the absence of measurement errors, such natural experiments can lead to severely biased parameter estimates. For stock–recruitment models, the bias is misleading for management: the stock will appear too productive when it is low, and too unproductive when it is large. The likely magnitude of such biases can and should be determined for any particular case by Monte Carlo simulations.

Modification of the ricker stock recruitment model to account for environmentally induced variation in recruitment with particular reference to the blue crab fishery in Chesapeake Bay

A modification has been made to the simple Ricker stock recruitment model in order to account for density-independent mortality through fluctuating environmental conditions as well as density-dependent mortality. The modified model is applied to the blue crab fishery data from Chesapeake Bay, Maryland. The model results in a family of stock recruitment curves which assist in the understanding of a complex relationship between spawning stock and recruitment, thereby providing a better basis for recruitment prediction and fishery management. A management strategy for a fishery subject to fluctuating levels of recruitment is also discussed.

Numerical Changes and Population Regulation in Young Migratory Trout Salmo trutta in a Lake District Stream, 1966-83

(1) This 18-year study of numerical changes in a trout population is part of a general study on the population dynamics, growth and distribution of migratory trout, Salmo trutta L., in Black Brows Beck, a stony stream in the English Lake District. (2) Earliest and latest spawning dates were 3 November and 7 December; larger females spawned earlier than smaller females. Eggs per redd and mean egg weight increased with female size and decreased with time from the start of the spawning period. Migratory trout produced more and larger eggs per unit length of fish than did resident trout in a neighbouring stream. (3) In laboratory and stream experiments, 50% of eggs hatched and 50% of alevins emerged after 444? days and 852? days from fertilization, respectively. Egg weight positively affected alevin size at hatching and emergence from the gravel, but not development times. Survival times of newly-emerged fish kept in the laboratory without food were positively related to weight at emergence. (4) Population densities of eggs and alevins were very similar within but differed between year-classes (range: 12-133 eggs m-2). The dominant factor affecting population densities of young trout was the population density of eggs, and hence alevins, at the start of each year-class. Alevin emergence time, weight of eggs or alevins, water temperature, numbers of older trout and numbers of other fish species (chiefly Cottus gobio L.) had no obvious effect. The number of recruits to each year-class was related to the initial egg density by the Ricker, dome-shaped, stock-recruitment curve. Maximum recruitment was estimated as 7.2 fish m-2 (95% CL = 6.5-8.0) soon after the fry stage ended in May/early June and 2*0 (? 0-5) fish m-2 in late August/September. Optimum egg densities for these values were estimated as 40.0 m-2 (95% CL = 38.5-41-7) and 62.5 m-2 (55.6-71-4) respectively. (5) Mortality was very low for eggs and alevins but high for young trout trying to establish feeding territories. The instantaneous mortality rate was linearly related to egg density and hence was density-dependent. The mortality rate for June to August was also negatively related to, and appeared to compensate for, the high mortality rate in May/early June.

Recruitment variability and age structure in harvested animal populations

The sensitivity to harvesting in a discrete-time, age-structured model of an animal population is considered. Density dependence and stochasticity are confined to the first year of the life cycle. Sensitivity is characterized by ( a) the characteristic return time and ( b) the relative variance (coefficient of variation) of recruitment and yield. Harvesting affects the return time and relative variance through two mechanisms: (1) a displacement of the equilibrium down the stock-recruitment curve and (2) a change in the shape of the “fecundity profile.” The first mechanism occurs in models without age structure and usually has the effect of increasing both return time and relative variance. The second mechanism has opposite effects on return time and relative variance—reducing the former while in most cases increasing the latter. For this model it can be concluded that the overall effect of harvesting is to increase the relative variance of recruitment and yield. The behavior of the return time is ambiguous and is in poor agreement with that of the relative variance.

A family of general production curves for exploited populations

A form for the growth term in the differential equation for biomass change with time is proposed, which is based on a versatile new family of stock-recruitment curves. The resultant yield curves of a general production model are investigated. The form of the curves is more variable than is usual in such models, but depends systematically on the values adopted for biologically well-defined parameters such as natural mortality and natural rate of increase. The consequences for management are explored by examining the magnitude of the maximum sustainable yield (MSY) and suitable values for the yield⧸biomass ratio. It is concluded that the MSY is of the order of but usually somewhat less than 0.5 MB 0, although variation by a factor of three either way is possible. MSY is usually attained at a yield⧸biomass ratio no more than a few times M (the natural mortality), and at biomasses less than half the pristine biomass ( B 0). However, exploitation of stocks having domed stock-recruitment curves at MSY is dangerous, because the ability of the population to recover from perturbations is impaired. This is associated with an increase in the return time for the population.

A general stock-recruitment curve

A general stock-recruitment curve

Exploiting natural populations in an uncertain world

We consider some aspects of the way random environmental variability can affect fish and other natural populations that are being harvested for sustained yield. A spectrum of 8 stock-recruitment curves, culled from the fisheries literature, is used to study the probability distribution of yield Y as a function of the level of exploitation or “effort” E. If the environmental noise enters via the intrinsic growth rates (“density-independent noise”), the 8 curves are qualitatively in accord in predicting that absolute levels of fluctuation in Y increase as E increases; these trends become strongly marked once exploitation is significantly in excess of the maximum sustained yield (MSY) level. The quantitative details, however, depend on the specifics of the stock-recruitment curve (the CV or relative fluctuation of Y can increase or decrease as E increases toward the neighborhood of the MSY level, and the CV can increase relatively dramatically or relatively gradually as E increases beyond this level). The effects of density dependent noise, and of time delays in population regulatory processes, are also briefly discussed. Broadly, the analysis suggests that environmental unpredictability can pose “portfolio theory” trade-offs between desired levels of average yield and of fluctuations or risk. What seems really needed is not further mathematical refinement, but rather robustly self-correcting strategies that can operate with only fuzzy knowledge about stock levels and recruitment curves.

The steady state of a stochastic harvesting model

A stochastic harvesting model based on a discrete-time Markov population model is discussed. The population model generalizes the well-known deterministic stock-recruitment model, and can describe the dynamics of a population living in a fluctuating environment. Indeed, its form is the one often tacitly adopted by fisheries scientists in the estimation of stock-recruitment curves. The steady state of the model is discussed both in the case when it is unexploited and in the case when it is subject to a regular exploitation. Conditions on the average intrinsic growth rate which determine the eventual survival or extinction of the population are derived. These results are used to determine the maximum rate of exploitation that can be sustained by the population in the long run. It is shown how this maximum rate of exploitation decreases with the degree of environmental fluctuation present, and is always less than the deterministic maximum sustainable harvest rate. The long-run average yield of various harvest policies is compared with their nominal yield as determined from the equivalent deterministic model. It is shown that the long-run average yield of a stationary policy cannot exceed the deterministic maximum sustainable yield.

Behavior of Age-Specific, Density-Dependent Models and the Northern California Dungeness Crab (Cancer magister) Fishery

A continuous time and age model which reflects the effect of older age-groups on mortality in young is presented. The conditions under which this model is unstable and the characteristics of solutions under various conditions are examined. Results obtained are related to results obtained by others from stock–recruitment models. Analytical expressions for the period of oscillations, the effect of fishing all age-classes, and the effect of the slope of the stock–recruitment curve at the replacement point support earlier simulation results. The effect on the model of size-selective fishing is shown to be a decrease in the stability of the population in some cases. These results are demonstrated with a model that approximately reflects the Dungeness crab (Cancer magister) population in northern California. The cyclic behavior of this population is similar to an unstable mode of the model. Conditions under which this mode arises are derived and potential causes of decreased stability, among which is fishing, are examined. Key words: fishery, model, stability, Dungeness crab, age-specific, density-dependent, stock–recruitment, cannibalism, Carcinonemertes

Relationship Between Two Stock–Recruitment Curves

The Beverton and Holt and Ricker stock–recruitment curves can be used to generate population growth curves. The Beverton and Holt curve is then identical to a difference equation model for the logistic growth curve, and may be derived in terms of equations for linearly density-dependent population regulation. The same equations lead to the Ricker curve if the density-regulating effect is assumed to depend only on population size at the beginning of the interval between generations. At low rates of population growth, the Ricker curve approaches that of Beverton and Holt. The two curves appear to represent certain concepts known in population biology as "r and K selection."

Optimal Exploitation of Multiple Stocks by a Common Fishery: A New Methodology

Optimal harvest rates for mixed stocks of fish are calculated using stochastic dynamic programming. This technique is shown to be superior to the best methods currently described in the literature. The Ricker stock recruitment curve is assumed for two stocks harvested by the same fishery. The optimal harvest rates are calculated as a function of the size of each stock, for a series of possible parameter values. The dynamic programming solution is similar to the fixed escapement policy only when the two stocks have similar Ricker parameters, or when the two stocks are of equal size. Normally, one should harvest harder than calculated from fixed escapement analysis.

Analog Computer Simulation and Catch Forecasting in Commercially Fished Populations

Abstract An analog computer was programmed and used to derive data on population dynamics of fish and whales. The types of mathematical models that were applied and the results of the studies, several of which have been published, are briefly reviewed. Applications included two types of mathematical models—models of populations for studying and forecasting yields and models of competing populations. The yield models combined fishing and natural mortality rates with a Gompertz growth curve in a differential equation. When integrated by the computer, this equation produced survival curves for successive year classes, whose annual values were summed graphically. Recruitment was determined from a stock-recruitment curve. Yields for each season were calculated from stock weight by using known rate of exploitation and were compared with actual yields to test the validity of the models. The technique has been demonstrated for Pacific sardine (Sardinops sagax), haddock (Melanogrammus aeglefinus), Atlantic cod (Ga...