Optimal Exploitation Strategies for an Animal Population in a Markovian Environment: A Theory and an Example

Abstract

Optimal exploitation strategies were studied for an animal population in a Markovian (stochastic, serially correlated) environment. This is a general case and encompasses a number of important special cases as simplifications. Extensive empirical data on the Mallard (Anas platyrhynchos) were used as an example of general theory. The number of small ponds on the central breeding grounds was used as an index to the state of the environment. A general mathematical model was formulated to provide a synthesis of the existing literature, estimates of parameters developed from an analysis of data, and hypotheses regarding the specific effect of exploitation on total survival. The literature and analysis of data were inconclusive concerning the effect of exploitation on survival. Therefore, two hypotheses were explored: (1) exploitation mortality represents a largely additive form of mortality, and (2) exploitation mortality is compensatory with other forms of mortality, at least to some threshold level. Models incorporating these two hypotheses were formulated as stochastic dynamic programming models and optimal exploitation strategies were derived numerically on a digital computer. Optimal exploitation strategies were found to exist under the rather general conditions. Direct feedback control was an integral component in the optimal decision—making process. Optimal exploitation was found to be substantially different depending upon the hypothesis regarding the effect of exploitation on the population. If we assume that exploitation is largely an additive force of mortality in Mallards, then optimal exploitation decisions are a convex function of the size of the breeding population and a linear or slight concave function of the environmental conditions. Under the hypothesis of compensatory mortality forces, optimal exploitation decisions are approximately linearly related to the size of the Mallard breeding population. Dynamic programming is suggested as a very general formulation for realistic solutions to the general optimal exploitation problem. The concepts of state vectors and stage transformations are completely general. Populations can be modeled stochastically and the objective function can include extra—biological factors. The optimal level of exploitation in year t must be based on the observed size of the population and the state of the environment in year t unless the dynamics of the population, the state of the environment, and the result of the exploitation decisions are completely deterministic. Exploitation based on an average harvest, or harvest rate, or designed to maintain a constant breeding population size is inefficient.

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