Quantifying stochastic outcomes

Abstract

A system consisting of two species in a fluctuating environment, when the interspecies competition for resources is strong, will have a stochastic outcome: only one of the species will survive, but there is no a priori way of knowing which one this will be. It is natural in such a situation to ask what will be the probability of one or another of the species surviving. This probability is calculated as a function of the average growth rates and the strengths of the interaction between the species and of the randomness. This is an example of a class of stochastic problems in which multiple final states are available for occupation. We refer to the choice of final states as state selection, and the probabilities of final states being occupied as state-selection probabilities. The calculation of these probabilities is carried out in the context of a model of the system which consists of two coupled stochastic differential equations. By reformulating these equations in terms of path integrals, the powerful methods based on the use of optimal paths may be utilized to calculate the probability of one outcome or the other. The analytical results obtained by using this technique agree well with numerical simulations when both species have the same growth rate. Although the method adopted rests on the assumption that the strength of the fluctuations, D , is small, remarkably the analytic results are still found to be in good agreement with the numerical results when D is of order 1.