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Survival analysis of a stochastic cooperation system with functional response in a polluted environment

Abstract

In this paper, we propose and study a stochastic two-species cooperation model with functional response in a polluted environment. We first perform the survival analysis and establish sufficient conditions for extinction, weak persistence, and stochastic permanence. Then we further perform the survival analysis based on the temporal average of population size and derive sufficient conditions for the strong persistence in the mean and weak persistence in the mean. Finally, we present numerical simulations to justify the theoretical results.

Introduction

With rapid development of industries and agriculture, a mass of toxicants has been emitted into the environment, such as the industrial wastewater, domestic sewage, and other contaminants. The presence of a variety of toxicants in the environment not only seriously threatened the survival of the exposed populations but also affected the human life style. Therefore it is important to estimate the environmental toxicity, which requires quantitative estimates for the survival risk of species in a polluted environment. This motivates scholars to utilize the mathematical models to assess the effects of toxicants on various ecosystems. Hallam et al. [13] did pioneering works. It has been a significant topic of considerable researches, and more and more deterministic models have been proposed and analyzed (see [417]). They all provide a great insight into the effects of pollutants. In this paper, we mainly attempt to study a two-species cooperation model with functional response. We assume that the environment is of complete spatial homogeneity and there is no migration. Let \(x_{i}(t)\) represent the population size (population density) of the ith species at time t. We consider the following cooperation model in a polluted environment:

$$ \textstyle\begin{cases} x'_{1}(t)=x_{1}(t) [r_{1}-a_{1}x_{1}(t)+ \frac{c_{2}x_{2}(t)}{1+b_{2}x_{2}(t)} ], \\ x'_{2}(t)=x_{2}(t) [r_{2}-a_{2}x_{2}(t)+ \frac{c_{1}x_{1}(t)}{1+b_{1}x_{1}(t)} ], \end{cases} $$
(1.1)

where the positive coefficients \(r_{1}, r_{2}\) and \(a_{1}, a_{2}\) are the intrinsic growth rates and self-inhibition rates, respectively. In the classical cooperation model the mutualism effects are described by a bilinear function, that is, \(x_{i}\) response to \(x_{j}\) is assumed to be increasingly monotonic, an inherent assumption meaning that the more \(x_{j}\) there exist in the environment, the better off the \(x_{i}\). But here the term \(c_{i}x_{i}(t)/(1+b_{i}x_{i}(t))\) represents the functional response function, and moreover, it is an increasing function with respect to \(x_{i}\) and has a saturation value for large enough \(x_{i}\). The positive coefficients \(c_{j}\) measure the mutualism effects of species \(x_{j}\) on species \(x_{i}\), and \(b_{j}\), \(i, j = 1,2\), \(i\neq j\), are positive control constants. For a relevant ecological model of system (1.1), we refer the readers to [18].

Now we are in the position to describe the dynamics of population in a polluted environment. We assume that the living organisms absorb part of toxicants into their bodies, the dynamics of the population is affected by internal toxicant, and the individuals in two species have identical concentration of organismal toxicant at time t (see [5]). To simplify the mathematical model, we assume that the capacity of the environment is so large that the change of toxicant in the environment that comes from uptake and egestion by the organisms can be neglected (see [3]). Let \(C_{0}(t)\) represent the concentration of toxicant in the organism at time t, and let \(C_{E}(t)\) represent the concentration of toxicant in the environment at time t. A coupling between species and toxicant is formulated by assuming that the intrinsic growth rate of the ith population, \(r_{i0}-r_{i1}C_{0}(t)\), is a linear function of concentration of toxicant present in the organism. Then the population dynamics can be given as follows:

$$ \textstyle\begin{cases} x'_{1}(t)=x_{1}(t) [r_{10}-r_{11}C_{0}(t)-a_{1}x_{1}(t)+ \frac{c_{2}x_{2}(t)}{1+b_{2}x_{2}(t)} ], \\ x'_{2}(t)=x_{2}(t) [r_{20}-r_{21}C_{0}(t)-a_{2}x_{2}(t)+ \frac{c_{1}x_{1}(t)}{1+b_{1}x_{1}(t)} ], \\ \frac{dC_{0}(t)}{dt}=dC_{E}(t)+\eta -(l_{1}+l_{2})C_{0}(t), \\ \frac{dC_{E}(t)}{dt}=-hC_{E}(t)+u(t), \end{cases} $$
(1.2)

where \(r_{i0}\) is the intrinsic growth rate of the ith species in the absence of toxicant, \(r_{i1}\), \(i=1,2\), is the dose-response rate of species i to the toxicant concentration, d represents the uptake rate of toxicant from environment per unit biomass, η represents the intake rate of toxicant from food per unit biomass, \(l_{1}\) represents the organismal net ingestion rates of toxicant, \(l_{2}\) represents the organismal excretion rates of toxicant, and h represents the loss rate of toxicant from the environment due to the processes such as biological transformation, chemical hydrolysis, volatilization, microbial degradation, and photosynthetic degradation. Here \(u(t)\) represents the input rate of exogenous toxin at time t, which is a nonnegative continuous function defined on \([0, \infty )\) with \(u_{1}:=\sup_{t\geq 0}u(t)>0\).

Considering that the fate of young recruits after reproduction is quite sensitive to the environmental fluctuations, so the growth rate of population is inevitably affected. May and Allen [19] have claimed that the growth rates in population systems should be fluctuating around some average values because of the environmental fluctuations. In this sense, it is necessary to incorporate the environmental fluctuations into our model. In practice, we usually estimate the growth rate \(r_{i0}\) by an average value plus an error term; by the central limit theorem the error term follows a normal distribution, and thus we can approximate the error term by \(\sigma _{i}\dot{B}_{i}(t)\), that is,

$$ r_{i0}\rightarrow r_{i0}+\sigma _{i} \dot{B}_{i}(t),\quad i=1, 2, $$

where \(\dot{B}_{i}(t)\) represents a white noise process (i.e., \(B_{i}(t)\) is a standard Brownian motion), and \(\sigma ^{2}_{i}\) is the intensity of the white noise. According to our discussion, a stochastic two-species cooperation model in polluted environment is derived as follows:

$$ \textstyle\begin{cases} dx_{1}(t)=x_{1}(t) [r_{10}-r_{11}C_{0}(t)-a_{1}x_{1}(t)+ \frac{c_{2}x_{2}(t)}{1+b_{2}x_{2}(t)} ]\,dt+\sigma _{1}x_{1}(t)\,dB_{1}(t), \\ dx_{2}(t)=x_{2}(t) [r_{20}-r_{21}C_{0}(t)-a_{2}x_{2}(t)+ \frac{c_{1}x_{1}(t)}{1+b_{1}x_{1}(t)} ]\,dt+\sigma _{2}x_{2}(t)\,dB_{2}(t), \\ \frac{dC_{0}(t)}{dt}=dC_{E}(t)+\eta -(l_{1}+l_{2})C_{0}(t), \\ \frac{dC_{E}(t)}{dt}=-hC_{E}(t)+u(t). \end{cases} $$
(1.3)

The initial conditions satisfy the conditions

$$ x_{1}(0)>0,\qquad x_{2}(0)>0,\qquad 0\leq C_{0}(0)< 1,\qquad 0\leq C_{E}(0)< 1. $$

In recent years, many interesting and important works about the stochastic model in polluted environment have been reported (see [2033]). But to the best of our knowledge, there exist few published papers concerning system (1.3). Noting that the last two equations in system (1.3) can be explicitly solved, so we only need to consider the following subsystem:

$$ \textstyle\begin{cases} dx_{1}(t)=x_{1}(t) [r_{10}-r_{11}C_{0}(t)-a_{1}x_{1}(t)+ \frac{c_{2}x_{2}(t)}{1+b_{2}x_{2}(t)} ]\,dt+\sigma _{1}x_{1}(t)\,dB_{1}(t), \\ dx_{2}(t)=x_{2}(t) [r_{20}-r_{21}C_{0}(t)-a_{2}x_{2}(t)+ \frac{c_{1}x_{1}(t)}{1+b_{1}x_{1}(t)} ]\,dt+\sigma _{2}x_{2}(t)\,dB_{2}(t). \end{cases} $$
(1.4)

Motivated by the existing results, the rest of this paper is arranged as follows. In Sect. 2, we introduce several commonly used basic lemmas and classical definitions. In Sect. 3, we perform the survival analysis and establish sufficient criteria for extinction, weak persistence, and stochastic permanence. In Sect. 4, we further discuss the survival analysis based on the temporal average of population size and derive sufficient conditions for the strong persistence in the mean and weak persistence in the mean. In Sect. 5, we present several numerical simulations to validate our theoretical results. The limitation of the model is also discussed in the last section.

Preliminaries

From now on, unless otherwise specified, we always work on the complete probability space \((\varOmega,\mathcal{F},P)\) with filtration \(\{\mathcal{F}_{t}\}_{t\geq 0}\) satisfying the usual conditions, that is, it is right continuous, and \(\mathcal{F}_{0}\) contains all P-null sets. Note that both \(C_{0}(t)\) and \(C_{E}(t)\) represent the concentrations of toxicant, so to be realistic, we must have \(0 \leq C_{0}(t) < 1\) and \(0 \leq C_{E}(t) <1\) for all \(t\geq 0\). In fact, we can prove that by solving the last two equations of system (1.3).

For convenience and simplicity, we introduce the following notations:

$$ \begin{aligned} &\bigl\langle f(t)\bigr\rangle \triangleq t^{-1} \int ^{t}_{0}f(s)\,ds, \qquad \langle f\rangle _{*} \triangleq \liminf_{t\rightarrow \infty }t^{-1} \int ^{t}_{0}f(s)\,ds, \\ & \langle f\rangle ^{*} \triangleq \limsup_{t\rightarrow \infty }t^{-1} \int ^{t}_{0}f(s)\,ds,\qquad f_{*}\triangleq \liminf_{t\rightarrow \infty }f(t),\qquad f^{*} \triangleq \limsup_{t\rightarrow \infty }f(t). \end{aligned} $$

Definition 2.1

(See [34])

To state and prove our main results, we recall some classical concepts.

  • Population \(x_{i}\) is said to be extinct if \(\lim_{t\rightarrow \infty }x_{i}(t)=0, i=1,2\), a.s.

  • Population \(x_{i}\) is said to be weakly persistent if \(\limsup_{t\rightarrow \infty }x_{i}(t)>0\) a.s.

  • Population \(x_{i}\) is said to be weakly persistent in the mean if \(\limsup_{t\rightarrow \infty }\langle x_{i}(t) \rangle >0\) a.s.

  • Population \(x_{i}\) is said to be strongly persistent in the mean if \(\liminf_{t\rightarrow \infty }\langle x_{i}(t) \rangle >0\) a.s.

  • System (1.4) is said to be stochastically permanent if for every \(\varepsilon \in (0,1)\), there exists a pair of positive constants \(\alpha =\alpha (\varepsilon ),\beta =\beta (\varepsilon )\) such that for any positive initial condition \(x(0)\), the solution \(x(t)\) satisfies

    $$ \liminf_{t\rightarrow \infty }P\bigl\{ x_{i}(t)\geq \alpha \bigr\} \geq 1- \varepsilon,\qquad \liminf_{t\rightarrow \infty }P\bigl\{ x_{i}(t) \leq \beta \bigr\} \geq 1- \varepsilon. $$
    (2.1)

Survival analysis

In this section, we first study the existence and uniqueness of a globally positive solution for the biological significance and then perform the survival analysis of system (1.4).

Theorem 3.1

For every initial value\(x(0)=(x_{1}(0),x_{2}(0))\in R^{2}_{+}\), system (1.4) has a unique solution\(x(t)\), and\(x(t)\)remains in\(R^{2}_{+}\)for all\(t\geq 0\)with probability one.

Proof

The proof of the theorem is standard. Firstly, let us consider the following stochastic system:

$$ \textstyle\begin{cases} dy_{1}(t)= [r_{10}-0.5\sigma ^{2}_{1}-r_{11}C_{0}(t)-a_{1}e^{y_{1}(t)}- \frac{c_{2}e^{y_{2}(t)}}{1+b_{2}e^{y_{2}(t)}} ]\,dt+ \sigma _{1}\,dB_{1}(t), \\ dy_{2}(t)= [r_{20}-0.5\sigma ^{2}_{2}-r_{21}C_{0}(t)-a_{2}e^{y_{2}(t)}- \frac{c_{1}e^{y_{1}(t)}}{1+b_{1}e^{y_{1}(t)}} ]\,dt+ \sigma _{2}\,dB_{2}(t), \end{cases} $$
(3.1)

with initial condition \(y_{1}(0)=\ln x_{1}(0), y_{2}(0)=\ln x_{2}(0)\). It is easy to verify that the coefficients of system (3.1) satisfy the local Lipschitz condition, and thus system (3.1) has a unique solution \(y(t)\) on \([0, \tau _{e})\), where \(\tau _{e}\) is the explosion time (see [35]). By applying Itô’s formula and a simple calculation, it is easy to derive that \(x_{1}(t)=e^{y_{1}(t)}, x_{2}(t)=e^{y_{2}(t)}\) is the unique positive local solution to system (1.4) with initial value \(x(0)\in R^{2}_{+}\). To show that this solution is global, we only need to show that \(\tau _{e}=\infty \).

Let \(n_{0}\) be sufficiently large such that every component of \(x(0)\) remains in the interval \([\frac{1}{n_{0}}, n_{0}]\). For each integer \(n\geq n_{0}\), we define the stopping time

$$ \tau _{n}=\inf \biggl\{ t\in [0, \tau _{e}): x_{1}(t)\notin \biggl(\frac{1}{n},n\biggr) \text{ or } x_{2}(t)\notin \biggl(\frac{1}{n}, n\biggr) \biggr\} . $$

Here we set \(\inf \emptyset =+\infty \) ( denotes the empty set). Obviously, \(\tau _{n}\) is increasing as \(n\rightarrow \infty \). Let \(\tau _{\infty }=\lim_{n\rightarrow \infty }\tau _{n}\), whence \(\tau _{\infty }\leq \tau _{e}\) almost surely; if we can show that \(\tau _{\infty }=\infty \) almost surely, then \(\tau _{e}=\infty \) almost surely, and the proof is completed.

Define the \(C^{2}\)-function \(\widetilde{V}(x): R^{2}_{+}\rightarrow R_{+}\) as

$$ \widetilde{V}(x_{1}, x_{2})=x_{1}-1-\ln x_{1}+x_{2}-1-\ln x_{2}. $$

The nonnegativity of this function can be seen from \(v-1-\ln v\geq 0\) for \(v>0\). Let \(T > 0\) be an arbitrary constant. Then for \(0 \leq t \leq \tau _{n} \wedge T\), by Itô’s formula we obtain

$$\begin{aligned} &d\widetilde{V}(x_{1},x_{2}) \\ &\quad =\biggl(1- \frac{1}{x_{1}}\biggr)\,dx_{1}+0.5 \frac{1}{x_{1}^{2}}(dx_{1})^{2}+ \biggl(1-\frac{1}{x_{2}}\biggr)\,dx_{1}+0.5 \frac{1}{x_{2}^{2}}(dx_{2})^{2} \\ &\quad=(x_{1}-1) \biggl[r_{10}-r_{11}C_{0}(t)-a_{1}x_{1}+ \frac{c_{2}x_{2}}{1+b_{2}x_{2}} \biggr]\,dt+(x_{1}-1)\sigma _{1} \,dB_{1}(t)+ \frac{1}{2}\sigma ^{2}_{1}\,dt \\ &\qquad{}+(x_{2}-1) \biggl[r_{20}-r_{21}C_{0}(t)-a_{2}x_{2}+ \frac{c_{1}x_{1}}{1+b_{1}x_{1}} \biggr]\,dt+(x_{2}-1)\sigma _{2} \,dB_{2}(t)+ \frac{1}{2}\sigma ^{2}_{2}\,dt \\ &\quad =\biggl\{ -a_{1}x_{1}^{2}+(r_{10}+a_{1})x_{1}-r_{11}C_{0}(t)x_{1}+ \frac{c_{2}x_{1}x_{2}}{1+b_{2}x_{2}}-\frac{c_{2}x_{2}}{1+b_{2}x_{2}}+ \frac{1}{2}\sigma ^{2}_{1}-r_{10} \\ &\qquad{}+r_{11}C_{0}(t)-a_{2}x_{2}^{2}+(r_{20}+a_{2})x_{2}-r_{21}C_{0}(t)x_{2}+ \frac{c_{1}x_{1}x_{2}}{1+b_{1}x_{1}}- \frac{c_{1}x_{1}}{1+b_{1}x_{1}}+ \frac{1}{2}\sigma ^{2}_{2} \\ &\qquad{}-r_{20}+r_{21}C_{0}(t) \biggr\} \,dt+(x_{1}-1)\sigma _{1}\,dB_{1}(t)+(x_{2}-1) \sigma _{2}\,dB_{2}(t) \\ &\quad \leq \biggl\{ -a_{1}x_{1}^{2}+ \biggl(r_{10}+a_{1}+\frac{c_{2}}{b_{2}}\biggr)x_{1}+ \frac{1}{2}\sigma ^{2}_{1}+r_{11}-r_{10}-a_{2}x_{2}^{2}+ \biggl(r_{20}+a_{2}+ \frac{c_{1}}{b_{1}}\biggr)x_{2} \\ &\qquad{}+\frac{1}{2}\sigma ^{2}_{2}+r_{21}-r_{20} \biggr\} \,dt+(x_{1}-1)\sigma _{1}\,dB_{1}(t)+(x_{2}-1) \sigma _{2}\,dB_{2}(t) \\ &\quad =G(x_{1},x_{2})\,dt+(x_{1}-1)\sigma _{1}\,dB_{1}(t)+(x_{2}-1)\sigma _{2} \,dB_{2}(t), \end{aligned}$$
(3.2)

where

$$\begin{aligned} G(x_{1},x_{2}) ={}& {-}a_{1}x_{1}^{2}+\biggl(r_{10}+a_{1}+ \frac{c_{2}}{b_{2}}\biggr)x_{1}+ \frac{1}{2}\sigma ^{2}_{1}+r_{11}-r_{10} \\ &{} -a_{2}x_{2}^{2}+\biggl(r_{20}+a_{2}+ \frac{c_{1}}{b_{1}}\biggr)x_{2}+\frac{1}{2} \sigma ^{2}_{2}+r_{21}-r_{20}. \end{aligned}$$
(3.3)

Clearly, \(G(x_{1},x_{2})\) is upper bounded, say by \(\mathcal{M}\). So we have

$$ d\widetilde{V}(x_{1},x_{2})\leq \mathcal{M} \,dt+(x_{1}-1)\sigma _{1}\,dB_{1}(t)+(x_{2}-1) \sigma _{2}\,dB_{2}(t). $$
(3.4)

Integrating both sides from 0 to \(\tau _{n} \wedge T\), we have

$$\begin{aligned} \int _{0}^{\tau _{n} \wedge T}\,d\widetilde{V}(x_{1},x_{2}) \leq{}& \int _{0}^{ \tau _{n} \wedge T}\mathcal{M}\,dt+ \int _{0}^{\tau _{n} \wedge T}(x_{1}-1) \sigma _{1}\,dB_{1}(t) \\ &{}+ \int _{0}^{\tau _{n} \wedge T}(x_{2}-1)\sigma _{2}\,dB_{2}(t). \end{aligned}$$
(3.5)

Taking the expectations on both sides yields that

$$\begin{aligned} E\bigl[\widetilde{V} \bigl(x_{1}(\tau _{n} \wedge T),x_{2}(\tau _{n} \wedge T)\bigr)\bigr] & \leq \widetilde{V}\bigl(x_{1}(0),x_{2}(0) \bigr)+\mathcal{M}E(\tau _{n} \wedge T) \\ &\leq \widetilde{V}\bigl(x_{1}(0),x_{2}(0)\bigr)+ \mathcal{M}T. \end{aligned}$$
(3.6)

Let \(\varOmega _{n}=\{\tau _{n}\leq T\}\). For arbitrary \(\omega \in \varOmega _{n}\), there exists some i such that \(x_{i}(\tau _{n},\omega )\) equals either n or \(\frac{1}{n}\), and thus \(\widetilde{V}(x(\tau _{n},\omega ))\) is not less than \((n-1-\ln n)\wedge (\frac{1}{n}-1+\ln n)\triangleq \mu (n)\). It then follows from (3.6) that

$$\begin{aligned} \mu (n)P(\tau _{n}\leq T) & \leq E\bigl[1_{\varOmega _{n}}(\omega ) \widetilde{V}\bigl(x_{1}(\tau _{n},\omega ), x_{2}(\tau _{n},\omega )\bigr) \bigr] \\ &\leq \widetilde{V}\bigl(x_{1}(0),x_{2}(0)\bigr)+ \mathcal{M}T, \end{aligned}$$
(3.7)

where \(1_{\varOmega _{n}}\) is the indicator function of \(\varOmega _{n}\). Let \(n\rightarrow \infty \). Then \(\mu (n)\rightarrow \infty \), and (3.7) leads to \(P(\tau _{\infty }\leq T)=0\). By the arbitrariness of T we have \(P(\tau _{\infty }=\infty )=1\) almost surely. This completes the proof of Theorem 3.1. □

It is well known that the threshold is very important for assessing the risk of extinction for species exposed to toxicant from a biological point of view. In the following, we first show that the pth moment of the solution of system (1.4) is upper bounded and then establish the threshold between weak persistence and extinction for species \(x_{i}\) modeled by (1.4). To begin with, we present the fundamental assumption that \(r_{i0}>0.5\sigma ^{2}_{i}\). Unless otherwise stated, it is always assumed in this paper.

Theorem 3.2

For any\(p>1\), there exists a positive constant\(K=K(p)\)such that the solution\(x(t)\)of system (1.4) has the property that

$$ \limsup_{t\rightarrow \infty }E \bigl\vert x(t) \bigr\vert ^{p}\leq K. $$
(3.8)

Proof

Define \(V(x_{1},x_{2})=x^{p}_{1}+x^{p}_{2}\). By Itô’s formula, we have

$$\begin{aligned} & dV(x_{1},x_{2}) \\ &\quad = px^{p-1}_{1} \,dx_{1}+0.5p(p-1)x_{1}^{p-2}(dx_{1})^{2}+px^{p-1}_{2} \,dx_{2}+0.5p(p-1)x_{2}^{p-2}(dx_{2})^{2} \\ &\quad = px^{p}_{1} \biggl(r_{10}-r_{11}C_{0}(t)-a_{1}x_{1}+ \frac{c_{2}x_{2}}{1+b_{2}x_{2}} \biggr)\,dt+p\sigma _{1}x^{p}_{1} \,dB_{1}(t)+0.5p(p-1) \sigma ^{2}_{1}x_{1}^{p} \,dt \\ &\qquad{} +px^{p}_{2} \biggl(r_{20}-r_{21}C_{0}(t)-a_{2}x_{2}+ \frac{c_{1}x_{1}}{1+b_{1}x_{1}} \biggr)\,dt+p\sigma _{2}x^{p}_{2} \,dB_{2}(t)+0.5p(p-1) \sigma ^{2}_{2}x_{2}^{p} \,dt \\ &\quad = \biggl\{ x^{p}_{1} \biggl(1+r_{10}p-r_{11}pC_{0}(t)-a_{1}px_{1}+p \frac{c_{2}x_{2}}{1+b_{2}x_{2}}+0.5p(p-1)\sigma ^{2}_{1} \biggr)+x^{p}_{2} \biggl(1+r_{20}p \\ &\qquad{} -r_{21}pC_{0}(t)-a_{2}px_{2}+p \frac{c_{1}x_{1}}{1+b_{1}x_{1}}+0.5p(p-1) \sigma ^{2}_{2} \biggr)-V(x_{1},x_{2}) \biggr\} \,dt \\ &\qquad {}+p\sigma _{1}x^{p}_{1}\,dB_{1}(t)+p \sigma _{2}x^{p}_{2}\,dB_{2}(t) \\ &\quad \leq \biggl\{ x^{p}_{1} \biggl(1+r_{10}p+ \frac{c_{2}p}{b_{2}}+0.5p(p-1) \sigma ^{2}_{1}-a_{1}px_{1} \biggr)+ x^{p}_{2} \biggl(1+r_{20}p+ \frac{c_{1}p}{b_{1}} \\ &\qquad{} +0.5p(p-1)\sigma ^{2}_{2}-a_{2}px_{2} \biggr)-V(x_{1},x_{2}) \biggr\} \,dt+p \sigma _{1}x^{p}_{1}\,dB_{1}(t)+p\sigma _{2}x^{p}_{2}\,dB_{2}(t) \\ &\quad= \bigl[L(x_{1}, x_{2})-V(x_{1},x_{2}) \bigr]\,dt+p\sigma _{1}x^{p}_{1} \,dB_{1}(t)+p \sigma _{2}x^{p}_{2} \,dB_{2}(t), \end{aligned}$$
(3.9)

where

$$\begin{aligned} L(x_{1}, x_{2}) ={}& x^{p}_{1}\biggl(1+r_{10}p+ \frac{c_{2}p}{b_{2}}+0.5p(p-1) \sigma ^{2}_{1}-a_{1}px_{1} \biggr) \\ & {}+x^{p}_{2}\biggl(1+r_{20}p+\frac{c_{1}p}{b_{1}}+0.5p(p-1) \sigma ^{2}_{2}-a_{2}px_{2}\biggr). \end{aligned}$$
(3.10)

Obviously, \(L(x_{1}, x_{2})\) is upper bounded; we denote it by \(\mathcal{L}\), that is,

$$ \mathcal{L}\triangleq \sum_{i=1}^{2} \frac{1+r_{i0}p+\frac{c_{j}p}{b_{j}}+0.5p(p-1)\sigma ^{2}_{i}}{(p+1)a_{i}}>0. $$
(3.11)

Applying Itô’s formula to \(e^{t}V(x_{1},x_{2})\) yields that

$$\begin{aligned} d\bigl[e^{t}V(x_{1},x_{2}) \bigr] = {}& e^{t}V(x_{1},x_{2})\,dt+e^{t} \,dV(x_{1},x_{2}) \\ \leq{} & \mathcal{L}e^{t}\,dt+p\sigma _{1}x^{p}_{1} \,dB_{1}(t)+p\sigma _{2}x^{p}_{2} \,dB_{2}(t). \end{aligned}$$
(3.12)

Integrating from 0 to t and then taking the expectation of both sides yield that

$$ E\bigl[V\bigl(x_{1}(t),x_{2}(t)\bigr)\bigr] \leq \exp (-t)\bigl[V\bigl(x_{1}(0),x_{2}(0)\bigr)+ \mathcal{L}\bigl(\exp (t)-1\bigr)\bigr]. $$
(3.13)

This gives that

$$ \limsup_{t\rightarrow \infty }E \bigl\vert x(t) \bigr\vert ^{p}\leq 2^{p-1}\limsup_{t \rightarrow \infty }E\bigl[V \bigl(x_{1}(t),x_{2}(t)\bigr)\bigr]\leq 2^{p-1} \mathcal{L} \triangleq K(p). $$
(3.14)

This completes the proof of Theorem 3.2. □

Theorem 3.3

If\(C^{*}_{0}<(r_{i0}-0.5\sigma ^{2}_{i})/r_{i1}\), then system (1.4) is stochastically permanent.

Proof

Taking arbitrary \(0<\varepsilon <1\), we first claim that there is a constant \(\alpha > 0\) such that

$$ \liminf_{t\rightarrow \infty }P\bigl\{ x_{i}(t)\geq \alpha \bigr\} \geq 1- \varepsilon. $$
(3.15)

Since \(C^{*}_{0}<(r_{i0}-0.5\sigma ^{2}_{i})/r_{i1}\), we can choose a constant \(m>0\) such that

$$ r_{i0}-0.5\sigma _{i}^{2}-r_{i1}C^{*}_{0}-0.5m \sigma _{i}^{2}>0,\quad i=1,2. $$
(3.16)

Define

$$ V_{1}(x_{1}, x_{2})=m^{-1} \sum_{i=1}^{2}\bigl(1+x_{i}^{-1} \bigr)^{m},\quad (x_{1}, x_{2}) \in R^{2}_{+}. $$
(3.17)

By Itô’s formula we derive that

$$\begin{aligned} &dV_{1} (x_{1}, x_{2}) \\ &\quad = \bigl(1+x_{1}^{-1}\bigr)^{m-1}\bigl[-x_{1}^{-2} \,dx_{1}+x_{1}^{-3}(dx_{1})^{2} \bigr]+0.5(m-1) \bigl(1+x_{1}^{-1}\bigr)^{m-2} \sigma _{1}^{2}x_{1}^{-2}\,dt \\ & \qquad{}+\bigl(1+x_{2}^{-1}\bigr)^{m-1} \bigl[-x_{2}^{-2}\,dx_{2}+x_{2}^{-3}(dx_{2})^{2} \bigr]+0.5(m-1) \bigl(1+x_{2}^{-1}\bigr)^{m-2} \sigma _{1}^{2}x_{2}^{-2}\,dt \\ &\quad = \bigl(1+x_{1}^{-1}\bigr)^{m-2} \biggl\{ \bigl(1+x_{1}^{-1}\bigr) \biggl[-x_{1}^{-1} \biggl(r_{10}-r_{11}C_{0}(t)-a_{1}x_{1}+ \frac{c_{2}x_{2}}{1+b_{2}x_{2}} \biggr)\,dt -\sigma _{1}x_{1}^{-1} \,dB_{1}(t) \\ &\qquad{} +\sigma _{1}^{2}x_{1}^{-1}\,dt \biggr]+0.5(m-1)\sigma _{1}^{2}x_{1}^{-2} \,dt \biggr\} +\bigl(1+x_{2}^{-1}\bigr)^{m-2} \biggl\{ \bigl(1+x_{2}^{-1}\bigr) \biggl[-x_{2}^{-1} \biggl(r_{20} -r_{21}C_{0}(t) \\ &\qquad{} -a_{2}x_{2}+\frac{c_{1}x_{1}}{1+b_{1}x_{1}} \biggr)\,dt-\sigma _{2}x_{2}^{-1}\,dB_{2}(t)+ \sigma _{2}^{2}x_{2}^{-1}\,dt \biggr] +0.5(m-1) \sigma _{2}^{2}x_{2}^{-2}\,dt \biggr\} \\ &\quad= \bigl(1+x_{1}^{-1}\bigr)^{m-2} \biggl\{ -x_{1}^{-2} \biggl(r_{10}-r_{11}C_{0}(t)+ \frac{c_{2}x_{2}}{1+b_{2}x_{2}} -0.5\sigma _{1}^{2}-0.5m\sigma _{1}^{2} \biggr)+x_{1}^{-1} \biggl(-r_{10} \\ &\qquad{} +r_{11}C_{0}(t)-\frac{c_{2}x_{2}}{1+b_{2}x_{2}}+\sigma _{1}^{2}+a_{1} \biggr)+a_{1} \biggr\} \,dt+\bigl(1+x_{2}^{-1}\bigr)^{m-2} \biggl\{ -x_{2}^{-2} \biggl(r_{20}-r_{21}C_{0}(t) \\ &\qquad{} +\frac{c_{1}x_{1}}{1+b_{1}x_{1}}-0.5\sigma _{2}^{2}-0.5m\sigma _{2}^{2} \biggr)+x_{2}^{-1} \biggl(-r_{20}+r_{21}C_{0}(t)- \frac{c_{1}x_{1}}{1+b_{1}x_{1}} + \sigma _{2}^{2}+a_{2} \biggr) \\ & \qquad{}+a_{2} \biggr\} \,dt-\bigl(1+x_{1}^{-1} \bigr)^{m-1}\sigma _{1}x_{1}^{-1} \,dB_{1}(t)-\bigl(1+x_{2}^{-1}\bigr)^{m-1} \sigma _{2}x_{2}^{-1}\,dB_{2}(t) \\ &\quad \leq \bigl(1+x_{1}^{-1}\bigr)^{m-2} \bigl\{ -x_{1}^{-2} \bigl(r_{10}-r_{11}C^{*}_{0}- \varepsilon -0.5\sigma _{1}^{2}-0.5m\sigma _{1}^{2} \bigr)+ x_{1}^{-1} \bigl(-r_{10}+r_{11}C^{*}_{0} \\ &\qquad{} +\varepsilon +\sigma _{1}^{2}+a_{1} \bigr)+a_{1} \bigr\} \,dt+\bigl(1+x_{2}^{-1} \bigr)^{m-2} \bigl\{ -x_{2}^{-2} \bigl(r_{20}-r_{21}C^{*}_{0}- \varepsilon -0.5\sigma _{2}^{2}-0.5m \sigma _{2}^{2} \bigr) \\ &\qquad{} +x_{2}^{-1} \bigl(-r_{20}+r_{21}C^{*}_{0}+ \varepsilon +\sigma _{2}^{2}+a_{2} \bigr)+a_{2} \bigr\} \,dt-\bigl(1+x_{1}^{-1} \bigr)^{m-1}\sigma _{1}x_{1}^{-1} \,dB_{1}(t) \\ &\qquad{} -\bigl(1+x_{2}^{-1}\bigr)^{m-1}\sigma _{2}x_{2}^{-1}\,dB_{2}(t). \end{aligned}$$
(3.18)

Let k be sufficiently small to satisfy

$$ 0< \frac{k}{m}< r_{i0}-r_{i1}C^{*}_{0}- \varepsilon -0.5\sigma _{i}^{2}-0.5m \sigma _{i}^{2},\quad i=1,2. $$
(3.19)

We define

$$ V_{2}(t, x_{1}, x_{2})=e^{kt}V_{1}(x_{1}, x_{2}). $$
(3.20)

By Itô’s formula again,

$$\begin{aligned} &dV_{2}(x_{1}, x_{2}) \\ &\quad = ke^{kt}V_{1} \,dt+e^{kt}\,dV_{1} \\ &\quad\leq e^{kt}\bigl(1+x_{1}^{-1} \bigr)^{m-2} \bigl\{ km^{-1}\bigl(1+x_{1}^{-1} \bigr)^{2}-x_{1}^{-2}\bigl(r_{10}-r_{11}C^{*}_{0}- \varepsilon -0.5\sigma _{1}^{2}-0.5m\sigma _{1}^{2}\bigr) \\ & \qquad{}+x_{1}^{-1}\bigl(-r_{10}+r_{11}C^{*}_{0}+ \varepsilon +\sigma _{1}^{2}+a_{1} \bigr)+a_{1} \bigr\} \,dt +e^{kt}\bigl(1+x_{2}^{-1} \bigr)^{m-2} \bigl\{ km^{-1}\bigl(1+x_{2}^{-1} \bigr)^{2} \\ & \qquad{}-x_{2}^{-2}\bigl(r_{20}-r_{21}C^{*}_{0}- \varepsilon -0.5\sigma _{2}^{2}-0.5m \sigma _{2}^{2}\bigr)+x_{2}^{-1} \bigl(-r_{20}+r_{21}C^{*}_{0} + \varepsilon + \sigma _{2}^{2}+a_{2}\bigr) \\ & \qquad{}+a_{2} \bigr\} \,dt-\bigl(1+x_{1}^{-1} \bigr)^{m-1}\sigma _{1}x_{1}^{-1}e^{kt} \,dB_{1}(t)- \bigl(1+x_{2}^{-1}\bigr)^{m-1} \sigma _{2}x_{2}^{-1}e^{kt} \,dB_{2}(t) \\ &\quad = e^{kt}\bigl(1+x_{1}^{-1}\bigr)^{m-2} \biggl\{ -x_{1}^{-2}\biggl(r_{10}-r_{11}C^{*}_{0}- \varepsilon -0.5\sigma _{1}^{2} -0.5m\sigma _{1}^{2}-\frac{k}{m}\biggr)+x_{1}^{-1} \biggl(-r_{10} \\ &\qquad{} +r_{11}C^{*}_{0}+\varepsilon +\sigma _{1}^{2}+a_{1}+\frac{2k}{m} \biggr)+a_{1}+ \frac{k}{m} \biggr\} \,dt +e^{kt} \bigl(1+x_{2}^{-1}\bigr)^{m-2} \biggl\{ -x_{2}^{-2}\biggl(r_{20}-r_{21}C^{*}_{0} \\ & \qquad{}-\varepsilon -0.5\sigma _{2}^{2}-0.5m\sigma _{2}^{2}-\frac{k}{m}\biggr)+x_{2}^{-1} \biggl(-r_{20}+r_{21}C^{*}_{0} + \varepsilon +\sigma _{2}^{2}+a_{2}+ \frac{2k}{m}\biggr)+a_{2} \\ & \qquad{}+\frac{k}{m} \biggr\} \,dt-\bigl(1+x_{1}^{-1} \bigr)^{m-1}\sigma _{1}x_{1}^{-1}e^{kt} \,dB_{1}(t)- \bigl(1+x_{2}^{-1}\bigr)^{m-1} \sigma _{2}x_{2}^{-1}e^{kt} \,dB_{2}(t) \\ &\quad = e^{kt}J(x_{1}, x_{2})\,dt- \bigl(1+x_{1}^{-1}\bigr)^{m-1}\sigma _{1}x_{1}^{-1}e^{kt} \,dB_{1}(t)-\bigl(1+x_{2}^{-1}\bigr)^{m-1} \sigma _{2}x_{2}^{-1}e^{kt} \,dB_{2}(t). \end{aligned}$$
(3.21)

Clearly, \(J(x_{1}, x_{2})\) is upper bounded for \((x_{1}, x_{2})\in R^{2}_{+}\), that is, \(\mathcal{J}\triangleq \sup_{x\in R^{2}_{+}}J(x_{1}, x_{2})<\infty \). As a result, we have

$$ dV_{2}(x_{1}, x_{2})\leq \mathcal{J}e^{kt}\,dt-\bigl(1+x_{1}^{-1} \bigr)^{m-1} \sigma _{1}x_{1}^{-1}e^{kt} \,dB_{1}(t) - \bigl(1+x_{2}^{-1} \bigr)^{m-1}\sigma _{2}x_{2}^{-1}e^{kt} \,dB_{2}(t). $$
(3.22)

Integrating both sides and then taking the expectations, we obtain that

$$ E\bigl[V_{2}(x_{1}, x_{2}) \bigr]=e^{\mathit{kt}}E\bigl[V_{1}(x_{1}, x_{2}) \bigr]\leq V_{1}\bigl[x_{1}(0), x_{2}(0)\bigr]+ \frac{\mathcal{J}}{k}\bigl(e^{kt}-1\bigr), $$
(3.23)

that is,

$$ \limsup_{t\rightarrow \infty }E\bigl[V_{1}(x_{1}, x_{2})\bigr]\leq \frac{\mathcal{J}}{k}. $$
(3.24)

On the other hand,

$$\begin{aligned} \limsup_{t\rightarrow \infty }E \bigl[x^{-m}_{i}(t)\bigr] &\leq \limsup_{t \rightarrow \infty }E \biggl[ \biggl(1+\frac{1}{x_{i}} \biggr)^{m} \biggr] \\ &\leq m\limsup_{t\rightarrow \infty }E\bigl[V_{1}(x_{1}, x_{2})\bigr] \\ &\leq \frac{m\mathcal{J}}{k}\triangleq \delta. \end{aligned}$$
(3.25)

For arbitrary \(\varepsilon \in (0, 1)\), let \(\alpha =(\frac{\varepsilon }{\delta })^{\frac{1}{m}}\). By Chebyshev’s inequality we have

$$ P\bigl\{ x_{i}(t)< \alpha \bigr\} =P\bigl\{ x^{-m}_{i}(t)>\alpha ^{-m}\bigr\} \leq \frac{E[x^{-m}_{i}(t)]}{\alpha ^{-m}}=\alpha ^{m}E\bigl[x^{-m}_{i}(t) \bigr], $$
(3.26)

which gives that

$$ \limsup_{t\rightarrow \infty }P\bigl\{ x_{i}(t)< \alpha \bigr\} \leq \alpha ^{m} \delta =\varepsilon, $$
(3.27)

that is,

$$ \liminf_{t\rightarrow \infty }P\bigl\{ x_{i}(t)\geq \alpha \bigr\} \geq 1- \varepsilon. $$
(3.28)

In the following, we prove that for arbitrary \(\varepsilon \in (0,1)\), there is a constant \(\beta > 0\) such that

$$ \liminf_{t\rightarrow \infty }P\bigl\{ x_{i}(t)\leq \beta \bigr\} \geq 1- \varepsilon. $$
(3.29)

Let \(\beta = [\frac{K}{\varepsilon } ]^{\frac{1}{p}}\). Then by Chebyshev’s inequality and Theorem 3.2 we have

$$ P\bigl\{ x_{i}(t)>\beta \bigr\} =P\bigl\{ x^{p}_{i}(t)>\beta ^{p}\bigr\} \leq \frac{E[x^{p}_{i}(t)]}{\beta ^{p}}, $$
(3.30)

which implies that

$$ \limsup_{t\rightarrow \infty }P\bigl\{ x_{i}(t)>\beta \bigr\} \leq \frac{K}{\beta ^{p}}=\varepsilon. $$
(3.31)

Consequently,

$$ \liminf_{t\rightarrow \infty }P\bigl\{ x_{i}(t)\leq \beta \bigr\} \geq 1- \varepsilon. $$
(3.32)

This completes the proof of Theorem 3.3. □

Remark 3.1

Theorem 3.3, which directly measures the population size \(x_{i}(t)\), indicates that the population size will neither too small nor too large with large probability when the time is sufficiently large.

Theorem 3.4

If\(\langle C_{0} \rangle _{*}>(r_{i0}-0.5\sigma ^{2}_{i}+ \frac{c_{j}}{b_{j}})/r_{i1}\), \(i,j=1,2\), \(i\neq j\), then population\(x_{i}\)goes to extinction with probability one.

Proof

Applying Itô’s formula yields that

$$\begin{aligned} d\ln x_{i} &= \frac{1}{x_{i}}\,dx_{i}-\frac{1}{2x^{2}_{i}}(dx_{i})^{2} \\ &= \biggl[r_{i0}-0.5\sigma ^{2}_{i}-r_{i1}C_{0}(t)-a_{i}x_{i}+ \frac{c_{j}x_{j}}{1+b_{j}x_{j}} \biggr]\,dt+\sigma _{i}\,dB_{i}(t). \end{aligned}$$
(3.33)

Integrating both sides yields that

$$ \frac{\ln x_{i}(t)-\ln x_{i}(0)}{t}=\bigl(r_{i0}-0.5\sigma ^{2}_{i}\bigr) -r_{i1} \bigl\langle C_{0}(t) \bigr\rangle -a_{i}\bigl\langle x_{i}(t) \bigr\rangle +\biggl\langle \frac{c_{j}x_{j}(t)}{1+b_{j}x_{j}(t)} \biggr\rangle +\sigma _{i} \frac{B_{i}(t)}{t}. $$
(3.34)

Letting \(t\rightarrow \infty \) and applying the strong law of large numbers, we obtain that

$$ \limsup_{t\rightarrow \infty }\frac{\ln x_{i}(t)}{t}\leq r_{i0}-0.5 \sigma ^{2}_{i}+\frac{c_{j}}{b_{j}}-r_{i1} \langle C_{0} \rangle _{*}< 0. $$
(3.35)

In other words, \(\lim_{t\rightarrow \infty }x_{i}(t) = 0\). This completes the proof of Theorem 3.4. □

Remark 3.2

Theorem 3.4 indicates the worst case that the population will go to extinction almost surely.

Theorem 3.5

If\(\langle C_{0} \rangle _{*}<(r_{i0}-0.5\sigma ^{2}_{i}+ \frac{c_{j}}{b_{j}})/r_{i1}\), \(i,j=1,2\), \(i\neq j\), then population\(x_{i}\)is weakly persistent almost surely.

Proof

We denote \(S:=\{\omega: \limsup_{t\rightarrow \infty }x_{i}(t,\omega )=0\}\) and assume that \(P(S)>0\). Then for all \(\omega \in S\), we have \(\limsup_{t\rightarrow \infty }x_{i}(t, \omega )=0\). For arbitrary small ε satisfying \(0<\varepsilon <1\), there exists \(T(\omega )\) such that

$$ x_{i}(t, \omega )\leq \varepsilon \quad\text{for } t\geq T( \omega ). $$
(3.36)

It then follows that

$$ \limsup_{t\rightarrow \infty }\frac{\ln x_{i}(t)-\ln x_{i}(0)}{t} \leq 0. $$
(3.37)

By the continuity of \(x_{i}(t,\omega )\) there must be a constant such that \(x_{i}(t,\omega )\leq \tilde{K}\) for \(0\leq t\leq T(\omega )\). On the other hand,

$$\begin{aligned} \frac{1}{t} \int _{0}^{t}x_{i}(s,\omega )\,ds &= \frac{1}{t} \int _{0}^{T( \omega )}x_{i}(s,\omega )\,ds+ \frac{1}{t} \int _{T(\omega )}^{t}x_{i}(s, \omega )\,ds \\ &\leq \tilde{K}\frac{T(\omega )}{t}+\varepsilon \frac{t-T(\omega )}{t} \end{aligned}$$
(3.38)

for sufficient large t. Since ε is arbitrarily small, we obtain that

$$ \limsup_{t\rightarrow \infty }\frac{1}{t} \int _{0}^{t}x_{i}(s,\omega )\,ds \leq 0. $$
(3.39)

Since \(x_{i}(t)>0\), we have \(\liminf_{t\rightarrow \infty }\frac{1}{t}\int _{0}^{t}x_{i}(s, \omega )\,ds\geq 0\), and thus

$$ \lim_{t\rightarrow \infty }\frac{1}{t} \int _{0}^{t}x_{i}(s,\omega )\,ds=0. $$
(3.40)

Substituting (3.40) into (3.34) and using the strong law of large numbers, we deduce the contradiction

$$ \limsup_{t\rightarrow \infty }\frac{\ln x_{i}(t)-\ln x_{i}(0)}{t} \leq r_{i}-0.5\sigma ^{2}_{i}-r_{i1}\langle C_{0} \rangle _{*}+ \frac{c_{j}}{b_{j}}< 0. $$
(3.41)

This completes the proof of Theorem 3.5. □

Remark 3.3

Theorem 3.5 admit the case that the population size is close to zero even if the time is sufficiently large. In this case the survival of species can be dangerous in reality. In addition, Theorems 3.4 and 3.5 reveal that \(\langle C_{0} \rangle _{*}=(r_{i0}-0.5\sigma ^{2}_{i}+ \frac{c_{j}}{b_{j}})/r_{i1}\) is the threshold between extinction and weak persistence.

The estimation of temporal averages

In this section, we further discuss the survival analysis of system (1.4) based on the temporal average of the population size \(x_{i}(t)\).

Theorem 4.1

The solution of system (1.4) has the property that

$$ \limsup_{t\rightarrow \infty }\frac{\ln x_{i}(t)}{\ln t}\leq 1\quad \textit{a.s.} $$
(4.1)

Proof

By Itô’s formula,

$$\begin{aligned} d\bigl(e^{t}\ln x_{i}\bigr) & = e^{t}\ln x_{i} \,dt+e^{t}\,d\ln x_{i} \\ & = e^{t} \biggl[\ln x_{i}+r_{i0}-0.5\sigma ^{2}_{i}-r_{i1}C_{0}(t)-a_{i}x_{i}+ \frac{c_{j}x_{j}}{1+b_{j}x_{j}} \biggr]\,dt+\sigma _{i}e^{t} \,dB_{i}(t). \end{aligned}$$
(4.2)

Integrating both sides yields that

$$ e^{t}\ln x_{i}(t)-\ln x_{i}(0)= \int _{0}^{t}e^{s} \biggl[\ln x_{i}+r_{i0}-0.5 \sigma ^{2}_{i}-r_{i1}C_{0}(s)-a_{i}x_{i}+ \frac{c_{j}x_{j}}{1+b_{j}x_{j}} \biggr]\,ds+N(t), $$
(4.3)

where \(N(t)=\int _{0}^{t}\sigma _{i}e^{s}\,dB_{i}(s)\). Note that \(N(t)\) is a local martingale with quadratic variation

$$ \langle N, N\rangle = \int _{0}^{t}\sigma ^{2}_{i}e^{2s} \,ds. $$
(4.4)

By the exponential martingale inequality we have

$$ P \Bigl\{ \sup_{0\leq t \leq \gamma n} \bigl[N(t)-0.5e^{-\gamma n} \langle N, N\rangle \bigr]>\theta e^{\gamma n}\ln n \Bigr\} \leq n^{- \theta }, $$
(4.5)

where \(\theta >1\), and \(\gamma >0\) is arbitrary. By the Borel–Cantelli lemma, for almost all \(\omega \in \varOmega \), there is a random integer \(n_{0}(\omega )\) such that for all \(n \geq n_{0}(\omega )\) and \(0\leq t \leq \gamma n\),

$$ N(t)\leq 0.5e^{-\gamma n} \langle N, N\rangle + \theta e^{\gamma n} \ln n. $$
(4.6)

Substituting (4.6) into (4.3) yields that

$$\begin{aligned} e^{t}\ln x_{i}(t)-\ln x_{i}(0) \leq{}& \int _{0}^{t}e^{s} \biggl[\ln x_{i}+r_{i0}-0.5 \sigma ^{2}_{i}-r_{i1}C_{0}(s)-a_{i}x_{i}+ \frac{c_{j}}{b_{j}} \biggr]\,ds \\ &{} +0.5e^{-\gamma n} \int _{0}^{t}\sigma ^{2}_{i}e^{2s} \,ds + \theta e^{\gamma n}\ln n \\ ={}& \int _{0}^{t}e^{s} \biggl[\ln x_{i}+r_{i0}-0.5\sigma ^{2}_{i}-r_{i1}C_{0}(s)-a_{i}x_{i}+ \frac{c_{j}}{b_{j}} \\ &{} +0.5\sigma ^{2}_{i}e^{s-\gamma n} \biggr]\,ds + \theta e^{\gamma n} \ln n. \end{aligned}$$
(4.7)

Since \(0\leq t \leq \gamma n\) and \(x_{i}(t)>0\), there is a constant ϱ such that

$$ \ln x_{i}+r_{i0}-0.5\sigma ^{2}_{i}-r_{i1}C_{0}(t)-a_{i}x_{i}+ \frac{c_{j}}{b_{j}}+0.5\sigma ^{2}_{i}e^{t-\gamma n} \leq \varrho, $$
(4.8)

that is, for \(0\leq t \leq \gamma n\), we have

$$ e^{t}\ln x_{i}(t)-\ln x_{i}(0) \leq \varrho \bigl(e^{t}-1\bigr) + \theta e^{ \gamma n}\ln n. $$
(4.9)

Therefore, if \(\gamma (n-1)\leq t \leq \gamma n\) and \(n \geq n_{0}(\omega )\), then we obtain that

$$ \frac{\ln x_{i}(t)}{\ln t} \leq \frac{e^{-t}\ln x_{i}(0)}{\ln t}+ \frac{\varrho (1-e^{-t})}{\ln t}+ \frac{\theta e^{-\gamma (n-1)} e^{\gamma n}\ln n}{\ln t}, $$
(4.10)

which implies that

$$ \limsup_{t\rightarrow \infty } \frac{\ln x_{i}(t)}{\ln t} \leq \theta e^{\gamma }. $$
(4.11)

Then letting \(\theta \rightarrow 1\) and \(\gamma \rightarrow 0\) gives the required assertion (4.1). This completes the proof of Theorem 4.1. □

Theorem 4.2

Let\(x(t)\)be a solution of system (1.4). If\(r_{i0}-0.5\sigma ^{2}_{i}-r_{i1}\langle C_{0} \rangle ^{*}>0\), then the component\(x_{i}(t)\)satisfies

$$ \lim_{t\rightarrow \infty }\frac{\ln x_{i}(t)}{t}=0\quad \textit{a.s.} $$
(4.12)

Proof

We first show that \(\limsup_{t\rightarrow \infty }\frac{\ln x_{i}(t)}{t} \leq 0\). From Theorem 4.1 we have that

$$\begin{aligned} \limsup_{t\rightarrow \infty } \frac{\ln x_{i}(t)}{t} & = \limsup_{t \rightarrow \infty }\frac{\ln x_{i}(t)}{\ln t}\limsup _{t\rightarrow \infty }\frac{\ln t}{t} \\ & \leq \limsup_{t\rightarrow \infty }\frac{\ln t}{t} \\ & = 0. \end{aligned}$$
(4.13)

In the following, we show that \(\liminf_{t\rightarrow \infty }\frac{\ln x_{i}(t)}{t} \geq 0\). Since \(\lim_{t\rightarrow \infty } \frac{\int _{0}^{t}\sigma _{i}\,dB_{i}(s)}{t}=0\) a.s., for any \(\varepsilon \in (0,1)\), there exists a positive constant T such that

$$ \biggl\vert \int _{s}^{t}\sigma _{i} \,dB_{i}(\tau ) \biggr\vert \leq \biggl\vert \int _{0}^{t} \sigma _{i} \,dB_{i}(\tau ) \biggr\vert + \biggl\vert \int _{0}^{s}\sigma _{i} \,dB_{i}( \tau ) \biggr\vert \leq \varepsilon (t+s) \quad\text{a.s.} $$
(4.14)

for \(t>s\geq T\). From equations (3.38) and (4.14) we have

$$\begin{aligned} x_{i}^{-1}(t) \leq {}& x_{i}^{-1}(T)\exp \biggl\{ - \bigl(r_{i0}-0.5 \sigma ^{2}_{i} \bigr) (t-T)+r_{i1} \int ^{t}_{T}C_{0}(s)\,ds- \int _{T}^{t}\sigma _{i} \,dB_{i}( \tau ) \biggr\} \\ &{}+ a_{i} \int _{T}^{t}\exp \biggl\{ \bigl(r_{i0}-0.5 \sigma ^{2}_{i} \bigr) (s-t)+r_{i1} \int ^{t}_{s}C_{0}(\tau )\,d\tau - \int _{s}^{t}\sigma _{i} \,dB_{i}(\tau ) \biggr\} \,ds \\ \leq{} & x_{i}^{-1}(T)\exp \bigl\{ - \bigl(r_{i0}-0.5 \sigma ^{2}_{i}-r_{i1} \langle C_{0} \rangle ^{*}-r_{i1}\varepsilon \bigr) (t-T)+\varepsilon (t+T) \bigr\} \\ &{} +a_{i} \int _{T}^{t}\exp \bigl\{ \bigl(r_{i0}-0.5 \sigma ^{2}_{i}-r_{i1} \langle C_{0} \rangle ^{*}-r_{i1}\varepsilon \bigr) (s-t)+\varepsilon (t+s) \bigr\} \,ds. \end{aligned}$$
(4.15)

This gives that

$$\begin{aligned} e^{-2\varepsilon (t+T)}x_{i}^{-1}(t) \leq{} & x_{i}^{-1}(T)\exp \bigl\{ - \bigl(r_{i0}-0.5 \sigma ^{2}_{i}-r_{i1}\langle C_{0} \rangle ^{*}-r_{i1} \varepsilon \bigr) (t-T)-\varepsilon (t+T) \bigr\} \\ & {}+a_{i} \int _{T}^{t}\exp \bigl\{ \bigl(r_{i0}-0.5 \sigma ^{2}_{i}-r_{i1} \langle C_{0} \rangle ^{*}-r_{i1}\varepsilon \bigr) (s-t)-\varepsilon (t-s)-2 \varepsilon T \bigr\} \,ds \\ \leq {}& \bar{K}, \end{aligned}$$
(4.16)

that is, \(x_{i}^{-1}(t) \leq \bar{K}e^{2\varepsilon (t+T)}\) almost surely. Therefore we obtain that

$$ \frac{\ln x_{i}^{-1}(t)}{t}\leq \frac{1}{t}\bigl[\ln \bar{K}+2 \varepsilon (t+T)\bigr] \quad\text{a.s.}, $$
(4.17)

which yields that

$$ \limsup_{t\rightarrow \infty }\frac{\ln x_{i}^{-1}(t)}{t}\leq 2 \varepsilon \quad\text{a.s.} $$
(4.18)

In other words,

$$ \liminf_{t\rightarrow \infty }\frac{\ln x_{i}(t)}{t}\geq -2 \varepsilon \quad\text{a.s.} $$
(4.19)

Then from the arbitrariness of ε it follows that

$$ \liminf_{t\rightarrow \infty }\frac{\ln x_{i}(t)}{t}\geq 0 \quad\text{a.s.} $$
(4.20)

This completes the proof of Theorem 4.2. □

Theorem 4.3

Let\(x(t)\)be a solution of system (1.4). If\(r_{i0}-0.5\sigma ^{2}_{i}-r_{i1} \langle C_{0} \rangle ^{*}>0\), then the component\(x_{i}(t)\)has the following property:

$$ \limsup_{t\rightarrow \infty }\frac{\int _{0}^{t}x_{i}(s)\,ds}{t}\leq \frac{r_{i0}-0.5\sigma ^{2}_{i}+\frac{c_{j}}{b_{j}}-r_{i1} \langle C_{0} \rangle _{*}}{a_{i}} \quad\textit{a.s.} $$
(4.21)

Moreover,

$$ \liminf_{t\rightarrow \infty }\frac{\int _{0}^{t}x_{i}(s)\,ds}{t}\geq \frac{r_{i0}-0.5\sigma ^{2}_{i}-r_{i1} \langle C_{0} \rangle ^{*}}{a_{i}}, \quad i=1, 2, \textit{ a.s.} $$
(4.22)

that is, population\(x_{i}\)is strongly persistent in the mean almost surely.

Proof

Recalling (3.34), we obtain that

$$ \frac{\ln x_{i}(t)-\ln x_{i}(0)}{t}+a_{i}\bigl\langle x_{i}(t) \bigr\rangle \leq \bigl(r_{i0}-0.5\sigma ^{2}_{i}\bigr) -r_{i1}\bigl\langle C_{0}(t) \bigr\rangle + \frac{c_{j}}{b_{j}}+\sigma _{i} \frac{B_{i}(t)}{t}. $$
(4.23)

Then from Theorem 4.2 it follows that

$$ \bigl\langle x_{i}(t) \bigr\rangle ^{*}\leq \frac{r_{i0}-0.5\sigma ^{2}_{i}+\frac{c_{j}}{b_{j}}-r_{i1} \langle C_{0} \rangle _{*}}{a_{i}}. $$
(4.24)

On the other hand,

$$ \frac{\ln x_{i}(t)-\ln x_{i}(0)}{t}+a_{i}\bigl\langle x_{i}(t) \bigr\rangle \geq \bigl(r_{i0}-0.5\sigma ^{2}_{i}\bigr) -r_{i1}\langle C_{0} \rangle + \sigma _{i}\frac{B_{i}(t)}{t}. $$
(4.25)

Similarly, we obtain that

$$ \bigl\langle x_{i}(t) \bigr\rangle _{*} \geq \frac{r_{i0}-0.5\sigma ^{2}_{i}-r_{i1} \langle C_{0} \rangle ^{*}}{a_{i}}. $$
(4.26)

This completes the proof of Theorem 4.3. □

Theorem 4.4

If\(\langle C_{0} \rangle _{*}<(r_{i0}-0.5\sigma ^{2}_{i})/r_{i1}\), then the component\(x_{i}(t)\)satisfies

$$ \limsup_{t\rightarrow \infty }\bigl\langle x_{i}(t) \bigr\rangle >0. $$
(4.27)

In other words, population\(x_{i}\)is weakly persistent in the mean almost surely.

Proof

Recalling (4.25), we obtain that

$$ \limsup_{t\rightarrow \infty }\frac{\ln x_{i}(t)-\ln x_{i}(0)}{t}+a_{i} \bigl\langle x_{i}(t) \bigr\rangle ^{*} \geq \bigl(r_{i0}-0.5\sigma ^{2}_{i}\bigr) -r_{i1} \langle C_{0} \rangle _{*}. $$
(4.28)

For all \(\omega \in \{\omega: \limsup_{t\rightarrow \infty }\langle x_{i}(t, \omega ) \rangle =0\}\), we have \(\langle x_{i}(t,\omega ) \rangle ^{*}=0\). Then from Theorem 4.2 and (4.28) it follows that

$$ 0\geq \bigl(r_{i0}-0.5\sigma ^{2}_{i} \bigr) -r_{i1}\langle C_{0} \rangle _{*}, $$
(4.29)

which contradicts the assumption of Theorem 4.4. So we must have \(\limsup_{t\rightarrow \infty }\langle x_{i}(t) \rangle >0\). This completes the proof of Theorem 4.4. □

Numerical simulations

In this section, we present several specific examples to justify our theoretical results based on the Milstein method, which is mentioned by Higham [36]. For system (1.4), we assign \(r_{10} = 1.6, r_{11} = 0.8, a_{1} = 0.5, b_{2} = 0.4, c_{2} = 0.2, \sigma _{1}=0.6, r_{20} = 1.4, r_{21} = 0.7, a_{2} = 0.4, b_{1} = 0.5, c_{1} = 0.4,\sigma _{2}=0.7\), and the initial value \((x_{1}(0), x_{2}(0)) =(0.8, 0.6)\) and then consider the following discrete version:

$$ \textstyle\begin{cases} x_{1}(k+1)=x_{1}(k)+x_{1}(k) [r_{10}-r_{11}C_{0}(k\Delta t)-a_{1}x_{1}(k)+ \frac{c_{2}x_{2}(k)}{1+b_{2}x_{2}(k)} ]\Delta t \\ \phantom{x_{1}(k+1)=} + \sigma _{1}x_{1}(k)\sqrt{\Delta t}\xi _{1}(k)+0.5 \sigma ^{2}_{1}x_{1}(k)[\xi ^{2}_{1}(k)-1]\Delta t, \\ x_{2}(k+1)=x_{2}(k)+x_{2}(k) [r_{20}-r_{21}C_{0}(k\Delta t)-a_{2}x_{2}(k)+ \frac{c_{1}x_{1}(k)}{1+b_{1}x_{1}(k)} ]\Delta t \\ \phantom{x_{2}(k+1)=}+ \sigma _{2}x_{2}(k)\sqrt{\Delta t}\xi _{2}(k)+0.5 \sigma ^{2}_{2}x_{2}(k)[\xi ^{2}_{2}(k)-1]\Delta t, \end{cases} $$
(5.1)

where \(\xi _{1}(k)\) and \(\xi _{2}(k)\) are independent standard Gaussian random variables. To reveal the effects of toxicant, we need to choose \(C_{0}(t)=0\) for comparison with system (5.1). By a simple calculation, \(r_{10}=1.6>0.5\sigma _{1}^{2}=0.18, r_{20}=1.4>0.5\sigma _{2}^{2}=0.245\). Then each species can live alone in a nonpolluted environment; see Fig. 1. Theorem 3.1 shows that for any positive initial value, the solution of system (5.1) remains in \(R^{2}_{+}\) almost surely. The nice property provides us a great opportunity to discuss the survival analysis of system (5.1), and so it is the fundamental conclusion in this paper.

Figure 1
figure1

Stochastic permanence of system (5.1) with \(C_{0}(t)=0, \Delta t=0.01\)

Example 5.1

Theorem 3.3 shows that a tiny amount of toxicant cannot disrupt the stochastic permanence; in other words, the population size will be neither too small nor too large with large probability if the time is sufficiently large. Let \(C_{0}(t)=0.4+0.1\sin (t), \Delta t=0.01\); by a simple calculation, \(C_{0}^{*}=0.5, (r_{10}-0.5\sigma _{1}^{2})/r_{11}=1.775, (r_{20}-0.5 \sigma _{2}^{2})/r_{21}=1.65\), that is, \(C_{0}^{*}<(r_{i0}-0.5\sigma _{i}^{2})/r_{i1}\). So population \(x_{i}\) is stochastically permanent by Theorem 3.3; see Fig. 2.

Figure 2
figure2

Stochastic permanence of system (5.1) with \(C_{0}(t)=0.4+0.1\sin (t), \Delta t=0.01\)

Example 5.2

Theorem 3.4 shows that a large amount of toxicant will force the species incline to extinction with probability one. Let \(C_{0}(t)=3.1+0.1\sin (t), \Delta t=0.01\). It is easy to compute that \(\langle C_{0} \rangle _{*}=3.1, (r_{10}-0.5\sigma _{1}^{2}+ \frac{c_{1}}{b_{1}})/r_{11}=2.4, (r_{20}-0.5\sigma _{2}^{2}+ \frac{c_{2}}{b_{2}})/r_{21}=2.79\), that is, \(\langle C_{0}\rangle _{*}>(r_{i0}-0.5\sigma _{i}^{2}+ \frac{c_{j}}{b_{j}})/r_{i1}\). So population \(x_{i}\) goes to extinction almost surely; see Fig. 3.

Figure 3
figure3

Extinction of system (5.1) with \(C_{0}(t)=3.1+0.1\sin (t), \Delta t=0.01\)

Example 5.3

Theorem 3.5 admits the case that the population size is closed to zero even if the time is sufficiently large. Let \(C_{0}(t)=1.78+0.1\sin (t), \Delta t=0.01\). Then by a straightforward calculation we have \(\langle C_{0} \rangle _{*}=1.78\) and \(\langle C_{0} \rangle _{*}<(r_{i0}-0.5\sigma _{i}^{2}+ \frac{c_{j}}{b_{j}})/r_{i1}\). By Theorem 3.5 population \(x_{i}\) is weakly persistent almost surely; see Fig. 4.

Figure 4
figure4

Weak persistence of system (5.1) with \(C_{0}(t)=1.78+0.1\sin (t), \Delta t=0.01\)

Discussion

In this paper, considering the fact of polluted environment, we propose and study a stochastic two-species cooperation model with functional response. However, we only consider the case that a coupling between species and toxicant is a linear function of concentration of toxicant present in the organism, that is, \(r_{i0}-r_{i1}H_{i}(C_{0})=r_{i0}-r_{i1}C_{0}(t)\), where \(H_{i}(C_{0})\) is the dose-response function of species \(x_{i}\) to the toxin. However, someone suggested that \(H_{i}(C_{0})\) should be nonlinear in many cases, such as a sigmoid dose-response curve (see [37, 38]). Liu and Wang [24] also introduced a more general case: \(H_{i}(C_{0})\) is a nondecreasing continuous function of \(C_{0}\) with \(H_{i}(0) = 0\). Clearly, our assumption is a particular case of this generalized condition. Moreover, the exogenous toxin cannot be continuous but rather of the impulse form, the growth rate of population is also inevitably affected by other environment noise such as Lévy jumps, and the regime-switching is other common random perturbation in the environment (see [3941]). All these questions associated with the polluted environment are interesting topics to deserve further investigation, and we leave them for our future works.

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Acknowledgements

The author thanks the anonymous reviewers for their careful reading of the manuscript and valuable comments and suggestions.

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Data sharing is not applicable to this paper as no datasets were generated or analyzed during the current study.

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The work is supported by the Scientific Research Foundation of Chongqing Technology and Business University (No. 1856048).

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Guo, S. Survival analysis of a stochastic cooperation system with functional response in a polluted environment. Adv Differ Equ 2020, 354 (2020). https://doi.org/10.1186/s13662-020-02816-5

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Keywords

  • Cooperation system
  • White noise
  • Polluted environment
  • Survival analysis
  • Numerical simulations