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

Abstract 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 lim t → + ∞ X i ( t ) = 0 a.s., i = 1 , 2 , 3 ; If γ i > 1 > γ 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 → + ∞ X j ( 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.

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