Dynamics of a Stochastic Delayed Competitive Model with Impulsive Toxicant Input in Polluted Environments

and Applied Analysis 3 Remark 4. In comparison with most of the existing results, our key contributions in this paper are as follows. (i) To the best of our knowledge, this paper is the first attempt to consider stochastic delay competitive model in polluted environments. (ii) Our conditions are much weaker. For example, the authors [15] supposed Γ 1 > 0 and Γ 2 > 0 which are dropped in this paper. (iii) Our results improve some recent works. For example, Lemma 1 indicates that the superior limit is positive while Theorem 2 proves that the limit exists and establishes the explicit form of the limit. 2. Proof For simplicity, define R 2 + = {a = (a 1 , a 2 ) ∈ R 2 | a i > 0, i = 1, 2} ,


Introduction
In this paper, we consider the following stochastic delay competitive model in polluted environments with impulsive toxicant input: where   ≥ 0,  = max{ 1 ,  2 }, and   () is a continuous function on [−, 0].All coefficients in model (1) are positive.Δ() = ( + ) − () and  + = {1, 2, . ..};   () is the size of the th population,  = 1, 2;  0 is the growth rate of the th population;  1 is the response to the pollutant present in the organism of the th population;  0 () is the toxicant concentration in the organism;   () is the toxicant concentration in the environment;   () is the organism's net uptake of toxicant from the environment;  0 ()+ 0 () is the egestion and depuration rates of the toxicant in the organism; ℎ  () is the toxicant loss from the environment itself;  is the period of the impulsive effect about the exogenous input of toxicant;  is the amount of toxicant input at every time.  () is a standard Brownian motion defined on a complete probability space (Ω, F, P);  2  is the intensity of the environmental noise.
From the work of Liu and Zhang [15], some important and interesting questions arise naturally.
(Q1) In the real world, the growth of population is inevitably affected by random environmental fluctuations.May [19] have claimed that population systems should be stochastic.
, then  2 goes to extinction a.s. and  1 is stable in the mean a.s.
and  2 are stable in the mean a.s.
Remark 4. In comparison with most of the existing results, our key contributions in this paper are as follows.
(i) To the best of our knowledge, this paper is the first attempt to consider stochastic delay competitive model in polluted environments.(ii) Our conditions are much weaker.For example, the authors [15] supposed Γ 1 > 0 and Γ 2 > 0 which are dropped in this paper.(iii) Our results improve some recent works.For example, Lemma 1 indicates that the superior limit is positive while Theorem 2 proves that the limit exists and establishes the explicit form of the limit.

Proof
For simplicity, define Lemma 5.For any given initial data Proof.The proof is a special case of Theorems 5.1 and 5.2 in Liu and Wang [25] and hence is omitted.

Numerical Simulations
In this section, using the classical Milstein method (see, e.g., [31]), we work out some numerical figures to support the analytical results.In Figure 1