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Stochastic processes in population studies

University of British Columbia

This paper develops a stochastic model for the growth of two interacting populations: when one species preys upon the other. The spatial distribution of the populations is considered, that of the prey being assumed to be clustered and quasi-uniform. This latter distribution is discussed in some detail, and it is found that, although it has been suggested that clustering of the prey may be a protective device against predators, any differences in the stochastic models for clustered and unclustered populations lie only in the constant coefficients involved in the formulation of the model. The approach used in developing the proposed model is that of Chiang. The size of the prey and predator populations; are assumed to be random variables X(t) and Y(t) respectively, and certain assumptions are made concerning the birth-rate and death-rate In either population. These assumptions are based on the deterministic equations (formula omitted). These equations are modifications of equations published by Leslie in 1958. Differential equations are developed for the rate of change of the probability that X(t)=x and Y(t)=y, by giving transition probabilities in some small interval of time and letting the interval shrink to a point. Hence differential equations, for the rate of change of the joint probability generating function of X and Y, and for the rate of change of the joint factorial moments of X and Y are obtained. Because of the complicated nature of these equations, however, no attempt is made to solve them.

Text

2011-11-24

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http://hdl.handle.net/2429/39254

Mathematics

University of British Columbia

Graduate