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UBC Theses and Dissertations

Some stochastic models in animal resource management Reed, William John


This thesis is concerned with mathematical models for the management of renewable animal resources. Discrete-time Markov processes are used to model the dynamics of populations living in fluctuating environments, and control models are developed on this basis. Of particular interest is the way in which the results of this stochastic analysis compare with known results of the analysis based on similar deterministic population models. In the principal model used in this thesis it is assumed that successive annual levels of population form a discrete-time Markov process with transitions governed by the equation [the equation is not included]. This deterministic process is a special case of the stochastic model, and holds in expectation. Comparison of results derived from the two models are made, thus revealing the adequacy or otherwise of existing theory based on the deterministic model. The steady-state of the stochastic model is discussed for various forms of the function f , and this discussion is extended to the case when the population is subjected to regular harvesting, under a stationary harvest policy. Rates of harvest which would lead to the eventual extinction of the population are investigated, and it is shown how, in some cases, rates of harvest which may not appear critical in an analysis of the deterministic model, can be critical for a population satisfying the stochastic model. The yield in steady-state for various stationary harvest policies is discussed and comparisons are made with steady-state yield estimates from the equivalent deterministic model. It is shown how the long-run average yield from any stationary policy cannot exceed the maximum sustainable yield as computed from the deterministic model. A dynamic economic control model with the objective of maximizing the expected discounted revenue that can be earned from the resource is developed. Under certain conditions, optimal policies are characterized qualitatively. When there is a positive 'mobilization cost' associated with harvesting, it is shown that an optimal policy is of the (S,s) type. When there is no mobilization cost a policy of 'stabilization-of-escapement' is optimal. In the latter case comparisons are made with the optimal level of escapement as determined from the equivalent deterministic model. On the basis of the deterministic model it is shown how the presence of a positive mobilization cost can lead to the optimality of a policy of pulse harvesting. Bio-economic conditions which determine the optimality of a policy of conservation or extinction are discussed, and results similar to the known results obtained from a deterministic analysis are obtained. It is shown how the presence of a positive mobilization cost can lower the critical discount rate. Control models based on multi-dimensional dynamic models for populations with age-structure are discussed, although very few definitive results are obtained. Also two-sex models are discussed, and a control model for the sex-specific harvesting of Pacific Salmon is developed. In the literature, theoretical reproduction functions have been derived by modelling various stages of the life-history of a species. In this thesis some stochastic models have been used for this, and reproduction functions, based on expected values, have been obtained, which are qualitatively similar to those derived from analagous deterministic models. All the results of this thesis are derived analytically and are not based on data analysis or simulation techniques.

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