Population dynamical behavior of a two-predator one-prey stochastic model with time delay

Abstract

In this paper, the convergence of the distributions of the solutions (CDS) of a stochastic two-predator one-prey model with time delay is considered. Some traditional methods that are used to study the CDS of stochastic population models without delay can not be applied to investigate the CDS of stochastic population models with delay. In this paper, we use an asymptotic approach to study the problem. By taking advantage of this approach, we show that under some simple conditions, there exist three numbers ρ 1 > ρ 2 > ρ 3 , which are represented by the coefficients of the model, closely related to the CDS of our model. We prove that if ρ 1 < 1, then lim t→+∞ N i (t) = 0 almost surely, i = 1, 2, 3; If ρ i > 1 > ρ i+1 , i = 1, 2, then lim t→+∞ N j (t) = 0 almost surely, j = i + 1, ..., 3, and the distributions of (N 1 (t), ..., N i (t)) T converge to a unique ergodic invariant distribution (UEID); If ρ 3 > 1, then the distributions of (N 1 (t), N 2 (t), N 3 (t)) T converge to a UEID. We also discuss the effects of stochastic noises on the CDS and introduce several numerical examples to illustrate the theoretical results.

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