Survival and Stationary Distribution Analysis of a Stochastic Competitive Model of Three Species in a Polluted Environment

Abstract

In this paper, we develop and study a stochastic model for the competition of three species with a generalized dose–response function in a polluted environment. We first carry out the survival analysis and obtain sufficient conditions for the extinction, non-persistence, weak persistence in the mean, strong persistence in the mean and stochastic permanence. The threshold between weak persistence in the mean and extinction is established for each species. Then, using Hasminskii’s methods and a Lyapunov function, we derive sufficient conditions for the existence of stationary distribution for each population. Numerical simulations are carried out to support our theoretical results, and some biological significance is presented.

Appendix: Hasminskii’s Method

For the readers’ convenience, we briefly introduce the Hasminskii’s methods used to prove the existence of stationary distribution of system (41) in this paper (see Hasminskii 1980). Let $X(t)$ be a homogeneous Markov process defined in the $E_l$ (which is a $l$-dimensional Euclidean space) and be described by the following stochastic differential equation:

$$\begin{aligned} \text {d}X(t)=b(X)\text {d}t+\sum \limits _{r=1}^k f_r(X)\text {d}B_r(t). \end{aligned}$$

(47)

The diffusion matrix is defined as follows:

$$\begin{aligned} D(x)=(a_{\textit{ij}}(x)),\ \ \ \ \ a_{\textit{ij}}=\sum \limits _{r=1}^k f_r^i(x)f_r^j(x). \end{aligned}$$

(48)

Assumption 6.1

There exists a bounded domain $U\in E_l$ with regular boundary $\Gamma$, having the following properties:

• (i) In the domain $U$ and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix $D(x)$ is bounded away from zero.

• (ii) If $x\in E_l\backslash U$, the mean time $\tau$ at which a path emerging from $x$ reaches the set $U$ is finite, and $\sup _{x\in S}E_x \tau <\infty$ for every compact subset $S\in E_l$.

Theorem 6.1

If Assumption 6.1 holds, then the Markov process $X(t)$ has a stationary distribution $\mu (\cdot )$. Let $g(\cdot )$ be a function integrable with respect to the measure $\mu$. Then

$$\begin{aligned} \mathcal {P}_x\big \{ \lim _{T\rightarrow \infty }\frac{1}{T} \int _0^T g(X(t))\text {d}t=\int _{E_l}g(x)\mu (\text {d}x) \big \}=1, \end{aligned}$$

for all $x\in E_l$.

Remark 6.1

The proof of Theorem 6.1 is given in Hasminskii (1980). Exactly, the existence of stationary distribution with density is referred to Theorem 4.1 at p. 119 and Lemma 9.4 at p. 138 in Hasminskii (1980).

To validate Assumption 6.1 (i), it is sufficient to prove that $\mathbb {F}$ is uniformly elliptical in $U$, where $\mathbb {F}u=b(x)\cdot u_x+[tr(D(x)u_{xx})]/2$, that is, there is a positive number $M$ such that

$$\begin{aligned} \sum \limits _{i,j=1}^k a_{\textit{ij}}(x)\xi _i\xi _j\ge M|\xi |^2,\ \ \ x\in U,\ \ \ \xi \in R^k. \end{aligned}$$

(see Gard 1988, Chapter 3, p. 103 and Rayleigh’s principle in Strang 1988, Chapter 6, p. 349). To verify Assumption 6.1 (ii), it is enough to show that there exists some neighborhood $U$ and a nonnegative $C^2$-function $V$ such that for any $E_l\backslash U, LV$ is negative (see p. 1163 of Zhu and Yin 2007 for details).