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

•A stochastic one-predator-two-prey model with delays is proposed.•An asymptotic approach is used to study the stability in distributions of the model.•Sharp sufficient criteria for the stability in distributions are established.

This paper is concerned with the stability in distribution of a delay stochastic population model with two competing preys (X1 and X2) and one predator (X3). Under some assumptions we prove that there are three numbers γ1 > γ2 > γ3 which have the following properties: if γ1 < 1, then all the populations go to extinction almost surely (a.s.), i.e., limt→+∞Xi(t)=0 a.s., i=1,2,3; If γi>1>γi+1,i=1,2, then the distribution of (X1(t),…,Xi(t))T converges weakly to a unique ergodic invariant distribution and limt→+∞Xj(t)=0 a.s., j=i+1,…,3; If γ3 > 1, then the distribution of (X1(t), X2(t), X3(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.