Stability of a stochastic one-predator-two-prey population model with time delays

This paper is concerned with the stability in distribution of a delay stochastic population model with two competing preys (X-1 and X-2) and one predator (X-3). Under some assumptions we prove that there are three numbers gamma(1) > gamma(2) > gamma(3) which have the following properties: if gamma(1) < 1, then all the populations go to extinction almost surely (a.s.), i.e., lim(t ->+infinity)X(i)(t) = 0 a.s., i = 1, 2,3; If gamma(i) > 1 > gamma(i+1) i = 1,2, then the distribution of (X-1(t),...,X-i(t))(T) converges weakly to a unique ergodic invariant distribution and lim(t ->+infinity)X(j)(t)= 0 a.s., j = i+1,...,3; If gamma(3) > 1, then the distribution of (X-1(t), X-2(t), X-3(t))(T) converges weakly to a unique ergodic invariant distribution a.s.. The influence of random perturbations on the stability are discussed and some numerical simulations are given to illustrate the main results. (C) 2017 Elsevier B.V. All rights reserved.