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1 INTRODUCTION Consider the stochastic control system of the form

A stochastic discrete Fredholm system in the form xn(ω) = hn(ω) + ∞Σj=1 Cn,j(ω)ƒj(xj(ω)), n = 1,2,…, is studied, where ω ∈ Ω, the supporting set of a complete probability measure space (Ω,A,P). A random solution of the discrete system is defined to be a discrete parameter second order stochastic process xn(ω) that satisfies the equation almost surely. The stochastic geometric stability of xn(ω) is defined, and conditions are given under which the random solution has this property. A finite system which approximates the infinite system is given, and it is shown that under certain conditions the unique random solution of the approximating system converges to the unique random solution of the infinite system.

The interaction of two species can be considered to be either mutually competitive, the “co‐predator” case, or dominated by one species, the “predator‐prey” model. The study of stochastic versions of these models has been impeded by the occurrence of nonlinear expressions in the characterization of the resulting probability structure. Generating function methods are used to study the bivariate and conditional probability distributions and their moments. The results are illustrated using a Laplace transform technique for obtaining numerical values. These methods can be used in the development of ecological prediction techniques and can be extended easily to the bivariate logistic model.

The aim of the present paper is to present certain existence theorems for stochastic control systems whose state variables, χ(t;ω), are continuous functions from the set R+ = {t;t ≥ 0} into the space L2(Ω, A, μ). That is, for each t R+, χ(t;ω) is a vector‐valued random variable whose second absolute moment exists. U = μ(t), the admissible controls, are taken as measurable functions of t only. It is assumed that the initial time is fixed but allow the terminal time tf(ω) to vary with ω∈Ω. The usual space constraints and boundary conditions are also allowed to vary with ω∈Ω. The cost functional is taken to be a continuous functional over a suitable class of continuous functions.