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.