The Monte Carlo Solution of a Competing Species Problem

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Abstract
A simplified stochastic model for interspecies competition is considered for the particular case of competition between the two species of flour beetle, Tribolium castaneum and Tribolium confusum, under specific environmental conditions. The respective birth- and death-rates imply inevitable extinction of one or other of the species, and the behaviour of the system up to extinction of one of the species is examined by consideration of the mathematical properties of the model. The solution is obtained by Monte Carlo methods using a high speed digital computer. Such an approach required the introduction of extensive modifications of the simulation procedure to obtain quantitative results of reasonable accuracy. Tables of probabilities of extinction of Tribolium castaneum are presented for initial population sizes for which the outcome of the competition is uncertain. The general description of the behaviour of the system includes diagramatic representations of some complete competitions and of contours of equal extinction probabilities; and a discussion of the value of the deterministic solution, both as a description of the system for initial population sizes where the extinction probability is close to 0 or 1, and as a means of increasing the efficiency of the Monte Carlo design. Results are compared with empirical and theoretical results from other sources, and the extent to which the model represents the practical situation is discussed.