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Dynamics of a stochastic cooperative predator-prey system with Beddington-DeAngelis functional response

Advances in Difference Equations20162016:21

https://doi.org/10.1186/s13662-015-0698-x

  • Received: 11 May 2015
  • Accepted: 16 November 2015
  • Published:

Abstract

In this paper, a stochastic cooperative predator-prey system with Beddington-DeAngelis functional response is proposed. By constructing Lyapunov functions, we demonstrate the existence of global positive solution, which is stochastically bounded and permanent. In addition, under some specific conditions, the solution of this stochastic system is globally asymptotically stable, which means that the properties of the solution will not be changed when the stochastic perturbation is small. Finally, the influence of the stochastic perturbation is demonstrated by simulation.

Keywords

  • stochastic differential equation
  • Beddington-DeAngelis functional response
  • stochastic perturbation
  • global asymptotic stability

1 Introduction

As an important branch of ecology science, population ecology has become a systematic discipline where mathematics is thoroughly applied. The relationship between predator and prey is one of the basic relationships among species. Many significant functional responses are constructed to model various situations. Most of the functional response are prey-dependent, which fail to model the interference among predators. In fact, when predators have to search, share, and compete for food, the functional response should be predator dependent; such a scenario occurs more frequently in nature and laboratory (see [1] and references therein).

In predator-prey systems, there are three classical predator-dependent functional responses: the Hassell-Varley, Beddington-DeAngelis, and Crowley-Martin responses (see [1] and references therein). By comparing statistical evidence from 19 predator-prey systems, Skalski and Gilliam claimed that the above three predator-dependent functional responses provided a better description of a predator feeding over a range of predator-prey abundances [2]. Amongst the Beddington-DeAngelis type functional responses some cases fitted better [2]. Beddington and DeAngelis et al. in 1975 first introduced the Beddington-DeAngelis type predator-prey model (see [3] and references therein), which is
$$ \textstyle\begin{cases} \frac{dx}{dt}=x(r_{1}-a_{11}x-\frac{a_{12}y}{1+\beta x+\gamma y}), \\ \frac{dy}{dt}=y(-r_{2}+\frac{a_{21}x}{1+\beta x+\gamma y}), \end{cases} $$
(1)
where \(x=x(t)\) and \(y=y(t)\) represent prey and predator densities, respectively; \(r_{1}\) stands for the average growth rate of prey, \(r_{2}\) is the death rate of predator, and \(a_{11}\) is the predator density-dependence rate. \(r_{i}\), \(a_{ij}\), β, and γ are positive constants for \(i,j=1,2\). Because of its practical significance, many scholars have studied the system in recent years [412].
In the study of predator-prey systems, the cooperation element is always neglected. Nevertheless, the cooperative system is a sometimes rudimentary and sometimes important ecological system in mathematical biology [13, 14]. To describe the mutual cooperation between two species, May [15] proposed the following equations:
$$ \textstyle\begin{cases} \frac{dx}{dt}=r_{1}x(1-\frac{x}{a_{1}+b_{1}y}-c_{1}x), \\ \frac{dy}{dt}=r_{2}y(1-\frac{y}{a_{2}+b_{2}y}-c_{2}y), \end{cases} $$
(2)
where \(r_{i}\) represents the growth rate, and \(c_{i}=\frac{1}{K_{i}}\) (\(K_{i}\) is the carrying capacity). \(r_{i}\), \(a_{i}\), \(b_{i}\), and \(c_{i}\) are positive constants for \(i,j=1,2\).
For each predator, various species of prey coexist in nature, so the multiple species predator-prey system simulates the actual situation better. However, this has been neglected in many existing studies [2, 16]. In this paper, we investigate the following cooperative predator-prey system of one predator feeding on two prey species with Beddington-DeAngelis functional response:
$$ \textstyle\begin{cases} dx=x(a_{1}-b_{1}x-\frac{h_{1}x}{f_{1}+g_{1}y}-\frac{c_{1}z}{1+\alpha _{1}x+\beta_{1}z})\, dt, \\ dy=y(a_{2}-b_{2}y-\frac{h_{2}y}{f_{2}+g_{2}x}-\frac{c_{2}z}{1+\alpha _{2}y+\beta_{2}z})\, dt, \\ dz=z(a_{3}-b_{3}z+\frac{d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+\frac {d_{2}y}{1+\alpha_{2}y+\beta_{2}z})\, dt, \end{cases} $$
(3)
where the species x, y are the preys of z; x and y are cooperative species. \(a_{i}\) stands for the average growth rate, and \(b_{i}\) is the density-dependence rate. All the parameters in system (3) are positive constants.
In reality, a population system is inevitably affected by environmental perturbation. May [15] have claimed that owing to the environmental perturbation, the birth rates in the system should be stochastic. At the same time, the natural growth of many species vary with t, e.g. owing to the seasonality. However, environmental fluctuations have been neglected in many existing studies. References [1, 3, 17] consider the effect of environmental noise. We introduce stochastic perturbation into the intrinsic growth rate. The intrinsic growth rate can be written as an average growth rate with some small random perturbed terms. In general, by the central limit theorem, the small terms follow some normal distributions, thus we can approximate the error term by a white noise \(\sigma_{i}\dot{B}_{i}(t)\), where \(\sigma_{i}^{2}\) is the intensity of the noise and \(\dot{B}_{i}(t)\) is a standard white noise. \(B_{i}(t)\) is a Brownian motion defined on a complete probability space \((\Omega,F,P)\). The growth rates \(a_{1}\), \(a_{2}\), and \(a_{3}\) are disturbed to \(a_{1}+\sigma_{1}\dot{B}_{1}(t)\), \(a_{2}+\sigma_{2}\dot{B}_{2}(t)\), \(a_{3}+\sigma_{3}\dot{B}_{3}(t)\), respectively. Finally we obtain the following stochastic system:
$$ \textstyle\begin{cases} dx=x(a_{1}-b_{1}x-\frac{h_{1}x}{f_{1}+g_{1}y}-\frac{c_{1}z}{1+\alpha _{1}x+\beta_{1}z})\, dt+\sigma_{1}x\, dB_{1}(t), \\ dy=y(a_{2}-b_{2}y-\frac{h_{2}y}{f_{2}+g_{2}x}-\frac{c_{2}z}{1+\alpha _{2}y+\beta_{2}z})\, dt+\sigma_{2}y\, dB_{2}(t), \\ dz=z(a_{3}-b_{3}z+\frac{d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+\frac {d_{2}y}{1+\alpha_{2}y+\beta_{2}z})\, dt+\sigma_{3}z\, dB_{3}(t), \end{cases} $$
(4)
where \(b_{3}z\) denotes the density dependence of the predator population. All the parameters in system (4) are positive constants.

In this paper, we will study the stochastic predator-prey system of one predator feeding on two prey species with Beddington-DeAngelis functional response. The rest of this paper is organized as follows. Section 2, we show the existence of global positive solution. Section 3, the stochastic boundedness of solution is studied. In Section 4, we prove that the system is stochastically permanent. In Section 5, we investigated the global attractivity of system (4). Finally, in Section 6, we present numerical simulation to verify our analytical results.

2 Global positive solutions

In model (4), as \(x(t)\), \(y(t)\), \(z(t)\) represent predator and prey densities, the solutions of model (4) should be non-negative. In order for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [18]), whereas the coefficients of (4) neither comply with the linear growth condition nor with the local Lipschitz condition. In the following sections, we will construct Lyapunov functions to demonstrate the existence and uniqueness of positive solutions.

Lemma 1

For any initial value \(x_{0}>0\), \(y_{0}>0\), \(z_{0}>0\), system (4) has a unique positive local solution \((x(t),y(t),z(t))\) for \(t\in[0,\tau _{e})\) almost surely (a.s.), where \(\tau_{e}\) is the explosion time.

Proof

Consider the following equations:
$$ \textstyle\begin{cases} du=(a_{1}-\frac{\sigma_{1}^{2}}{2}-b_{1}e^{u}-\frac {h_{1}e^{u}}{f_{1}+g_{1}e^{v}}-\frac{c_{1}e^{w}}{1+\alpha_{1}e^{u}+\beta _{1}e^{w}})\,dt+\sigma_{1}\,dB_{1}(t), \\ dv=(a_{2}-\frac{\sigma_{2}^{2}}{2}-b_{2}e^{v}-\frac {h_{2}e^{v}}{f_{2}+g_{2}e^{u}}-\frac{c_{2}e^{w}}{1+\alpha_{2}e^{v}+\beta _{2}e^{w}})\,dt+\sigma_{2}\,dB_{2}(t), \\ dw=(a_{3}-\frac{\sigma_{3}^{2}}{2}-b_{3}e^{w}+\frac{d_{1}e^{u}}{1+\alpha _{1}e^{u}+\beta_{1}e^{w}}+\frac{d_{2}e^{v}}{1+\alpha_{2}e^{v}+\beta _{2}e^{w}})\,dt+\sigma_{3}\,dB_{3}(t), \end{cases} $$
(5)
on \(t\geq0\) with initial value \(u(0)=\ln x_{0}\), \(v(0)=\ln y_{0}\), \(w(0)=\ln z_{0}\). The coefficients of (5) satisfy the local Lipschitz condition, so there is a unique local solution \((u(t),v(t),z(t))\) on \(t\in[0,\tau_{e})\). By Itô’s formula, we can see that \(x(t)=e^{u(t)}\), \(y(t)=e^{v(t)}\), \(z(t)=e^{w(t)}\) is the unique positive local solution to (5) with initial value \(x_{0}>0\), \(y_{0}>0\), \(z_{0}>0\). □

Theorem 1

Consider system (4), for any given initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\), there is a unique solution \((x(t),y(t),z(t))\) for \(t\ge0\) and the solution will remain in \(R_{+}^{3}\) with probability 1.

Proof

Based on Lemma 1, we only need to show the \(\tau_{e}=\infty\). Let \(m_{0}>0\) be sufficiently large for \(x_{0}\), \(y_{0}\), \(z_{0}\) lying within the interval \([\frac{1}{m_{0}},m_{0}]\). For each integer \(m>m_{0}\), define the stopping times
$$ \tau_{m}=\inf\biggl\{ t\in[0,\tau_{e}]:\min \bigl\{ x(t),y(t),z(t)\bigr\} \leq\frac{1}{m} \mbox{ or }\max\bigl\{ x(t),y(t),z(t)\bigr\} \geq m\biggr\} . $$
(6)
Clearly, \(\tau_{m}\) is increasing as \(m\rightarrow\infty\). Set \(\tau _{\infty}=\lim_{m\rightarrow\infty}\tau_{m}\), whence \(\tau_{\infty}\leq \tau_{m}\). If we can prove \(\tau_{\infty}=\infty\), then \(\tau_{e}=\infty \) and \((x_{0},y_{0},z_{0})\in R_{+}^{3}\) a.s. for all \(t\geq0\). If this statement is false, there is a pair of constants \(T>0\) and \(\epsilon\in(0,1)\) such that \(P\{\tau_{\infty}\leq T\}>\epsilon\). So there exists an integer \(m_{1} \geq m_{0}\) such that
$$ P \{\tau_{m}\leq T\}\geq\epsilon,\quad m \geq m_{1}. $$
(7)
Define \(V(x,y,z)=(x-1-\ln x)+(y-1-\ln y)+(z-1-\ln z)\). Since \(u-1-\ln u\geq0 \) for all \(u\geq0\), we see that \(V(x,y,z)\) is non-negative. Applying Itô’s formula, we have
$$\begin{aligned} dV =&(x-1) \biggl(a_{1}-b_{1}x- \frac{h_{1}x}{f_{1}+g_{1}y}-\frac {c_{1}z}{1+\alpha_{1}x+\beta_{1}z}\biggr)\,dt+\frac{\sigma_{1}^{2}}{2}\,dt \\ &{}+(y-1) \biggl(a_{2}-b_{2}y-\frac{h_{2}y}{f_{2}+g_{2}x}- \frac{c_{2}z}{1+\alpha _{2}y+\beta_{2}z}\biggr)\,dt+\frac{\sigma_{2}^{2}}{2}\,dt \\ &{}+(z-1) \biggl(a_{3}-b_{3}z+\frac{d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+ \frac {d_{2}y}{1+\alpha_{2}y+\beta_{2}z}\biggr)\,dt+\frac{\sigma_{3}^{2}}{2}\,dt \\ &{}+\sigma_{1}(x-1)\,dB_{1}(t)+\sigma_{2}(y-1) \,dB_{2}(t)+\sigma_{3}(z-1)\,dB_{3}(t), \end{aligned}$$
(8)
then
$$\begin{aligned} LV =&(x-1) \biggl(a_{1}-b_{1}x- \frac{h_{1}x}{f_{1}+g_{1}y}-\frac {c_{1}z}{1+\alpha_{1}x+\beta_{1}z}\biggr) \\ &{}+(y-1) \biggl(a_{2}-b_{2}y- \frac {h_{2}y}{f_{2}+g_{2}x}-\frac{c_{2}z}{1+\alpha_{2}y+\beta_{2}z}\biggr) \\ &{}+(z-1) \biggl(a_{3}-b_{3}z+\frac{d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+ \frac {d_{2}y}{1+\alpha_{2}y+\beta_{2}z}\biggr)+\frac{\sigma_{1}^{2}+\sigma _{2}^{2}+\sigma_{3}^{2}}{2} \\ \le& x(a_{1}-b_{1}x)+y(a_{2}-b_{2}y)+z \biggl(a_{3}-b_{3}z+\frac {d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+\frac{d_{2}y}{1+\alpha_{2}y+\beta _{2}z} \biggr) \\ &{}+\frac{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}}{2} \\ \le& x(a_{1}-b_{1}x)+y(a_{2}-b_{2}y) +z\biggl(a_{3}-b_{3}z+\frac {d_{1}}{\alpha_{1}}+\frac{d_{2}}{\alpha_{2}} \biggr) +\frac{\sigma _{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}}{2} \\ \le& K, \end{aligned}$$
(9)
where K is a positive number. Consequently,
$$ dV\leq K\,dt+(x-1)\sigma_{1}\,dB_{1}(t)+(y-1) \sigma_{2}\,dB_{2}(t)+(z-1)\sigma _{3} \,dB_{3}(t). $$
(10)
Integrating both sides of the inequality from 0 to \(\tau_{m}\wedge T \) and then taking the expectations, we get
$$ EV\bigl(x(\tau_{m}\wedge T),y(\tau_{m}\wedge T),z(\tau_{m}\wedge T)\bigr) \leq V(x_{0},y_{0},z_{0})+KT. $$
(11)
Set \(\Omega_{m}=\{\tau_{m}\leq T\}\) for \(m \geq m_{1}\), then we get \(P(\Omega_{m})\geq\epsilon\) by (7). For every \(\omega\in\Omega_{m}\), there is at least one of \(x(\tau_{m},\omega)\), \(y(\tau_{m},\omega)\), \(z(\tau _{m},\omega)\), which equals either m or \(\frac{1}{m}\), therefore \(V(x(\tau_{m}\wedge T), y(\tau_{m}\wedge T), z(\tau_{m}\wedge T))\) is no less than \(m-1-\ln m \) or \(\frac{1}{m}-1-\ln\frac{1}{m}\). Consequently, we have
$$ V\bigl(x(\tau_{m}\wedge T),y(\tau_{m}\wedge T),z(\tau_{m}\wedge T)\bigr)\geq (m-1-\ln m)\wedge\biggl( \frac{1}{m}-1-\ln\frac{1}{m}\biggr). $$
(12)
It follows from (11) that
$$\begin{aligned} V(x_{0},y_{0},z_{0})+KT \geq& E \bigl[I_{\Omega_{m}}V\bigl(x(\tau_{m}\wedge T),y(\tau_{m} \wedge T),z(\tau_{m}\wedge T)\bigr)\bigr] \\ \geq& \epsilon(m-1-\ln m)\wedge\biggl(\frac{1}{m}-1-\ln\frac{1}{m} \biggr), \end{aligned}$$
(13)
where \(I_{\Omega_{m}}\) is the indicator function of \(\Omega_{m}\). Letting \(m\rightarrow\infty\), it results in the contradiction that \(\infty>V(x_{0},y_{0},z_{0})+KT=\infty\), thus we draw a conclusion that \(\tau_{\infty}=\infty\) a.s. □

Theorem 1 only tells us the solution of model (4) will remain in \(R_{+}^{3}\) with probability 1. Next, we will discuss how the solution varies in \(R_{+}^{3}\) more detail.

3 Stochastic boundedness

Definition 1

(see [3] and references therein)

The solution \((x(t),y(t),z(t))\) of system (4) is said to be stochastically ultimately bounded, if for any \(\varepsilon\in(0,1)\), there is a positive constant \(\delta=\delta(\varepsilon)\). Such that for any initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\), the solution \((x(t),y(t),z(t))\) of system (4) has the property that
$$ \limsup_{t\rightarrow\infty}P\bigl\{ \bigl\vert x(t),y(t),z(t) \bigr\vert =\sqrt {x^{2}(t)+y^{2}(t)+z^{2}(t)}> \delta\bigr\} < \varepsilon. $$
(14)

Assumption 1

For any initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\), there exists \(p\geq1 \) such that
$$\begin{aligned}& x_{0} < \frac{a_{1}+\frac{p-1}{2}\sigma_{1}^{2}}{b_{1}}, \end{aligned}$$
(15)
$$\begin{aligned}& y_{0} < \frac{a_{2}+\frac{p-1}{2}\sigma_{2}^{2}}{b_{2}}, \end{aligned}$$
(16)
$$\begin{aligned}& z_{0} < \frac{a_{3}+\frac{d_{1}}{\alpha_{1}}+\frac{d_{2}}{\alpha _{2}}+\frac{p-1}{2}\sigma_{3}^{2}}{b_{3}}. \end{aligned}$$
(17)

Lemma 2

If Assumption  1 holds, let \((x(t),y(t),z(t))\) be a solution to system (4) with initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\). Then, for all \(p>1\),
$$\begin{aligned}& E\bigl[x^{p}(t)\bigr] \leq K_{1}(p), \end{aligned}$$
(18)
$$\begin{aligned}& E\bigl[y^{p}(t)\bigr] \leq K_{2}(p), \end{aligned}$$
(19)
$$\begin{aligned}& E\bigl[z^{p}(t)\bigr] \leq K_{3}(p), \end{aligned}$$
(20)
where
$$\begin{aligned}& K_{1}(p) := \biggl(\frac{a_{1}+\frac{p-1}{2}\sigma_{1}^{2}}{b_{1}}\biggr)^{p}, \end{aligned}$$
(21)
$$\begin{aligned}& K_{2}(p) := \biggl(\frac{a_{2}+\frac{p-1}{2}\sigma_{2}^{2}}{b_{2}}\biggr)^{p}, \end{aligned}$$
(22)
$$\begin{aligned}& K_{3}(p) := \biggl(\frac{a_{3}+\frac{d_{1}}{\alpha_{1}}+\frac{d_{2}}{\alpha _{2}}+\frac{p-1}{2}\sigma_{3}^{2}}{b_{3}}\biggr)^{p}. \end{aligned}$$
(23)

Proof

Define \(V_{1}=x^{p}\) for \(x\in R_{+}\) and \(p>0\). By Itô’s formula, we have
$$\begin{aligned} dV_{1} =&px^{p-1}\, dx+\frac{1}{2}p(p-1)x^{p-2}(dx)^{2} \\ =&px^{p-1}\biggl[x\biggl(a_{1}-b_{1}x- \frac{h_{1}x}{f_{1}+g_{1}y}-\frac {c_{1}z}{1+\alpha_{1}x+\beta_{1}z}\biggr)\,dt+\sigma_{1}x \,dB_{1}(t)\biggr] \\ &{}+\frac{1}{2}p(p-1)x^{p}\sigma_{1}^{2} \,dt \\ =&px^{p}\biggl\{ \biggl[a_{1}-b_{1}x- \frac{h_{1}x}{f_{1}+g_{1}y}-\frac {c_{1}z}{1+\alpha_{1}x+\beta_{1}z}+\frac{1}{2}(p-1)\sigma _{1}^{2}\biggr]\,dt+\sigma_{1} \,dB_{1}(t)\biggr\} . \end{aligned}$$
(24)
Integrating both sides of the equality from 0 to t and then taking the expectations, we get
$$\begin{aligned}& E\bigl[x^{p}(t)\bigr]-E\bigl[x^{p}(0)\bigr] \\& \quad = \int^{t}_{0}pE\biggl\{ x^{p} \biggl[a_{1}-b_{1}x-\frac {h_{1}x}{f_{1}+g_{1}y}-\frac{c_{1}z}{1+\alpha_{1}x+\beta_{1}z}+ \frac {1}{2}(p-1)\sigma_{1}^{2}\biggr]\biggr\} \, ds; \end{aligned}$$
(25)
thus
$$\begin{aligned} \frac{dE[x^{p}(t)]}{dt} =&pE\biggl\{ x^{p} \biggl[a_{1}-b_{1}x-\frac {h_{1}x}{f_{1}+g_{1}y}-\frac{c_{1}z}{1+\alpha_{1}x+\beta_{1}z}+ \frac {1}{2}(p-1)\sigma_{1}^{2}\biggr]\biggr\} \\ \leq& pE\biggl[a_{1}x^{p}-b_{1}x^{p+1}+ \frac{1}{2}(p-1)\sigma_{1}^{2}x^{p}\biggr] \\ =&p\biggl\{ \biggl[a_{1}+\frac{1}{2}(p-1)\sigma _{1}^{2}\biggr]E\bigl[x^{p}(t) \bigr]-b_{1}E\bigl[x^{p+1}(t)\bigr]\biggr\} \\ =&pE\bigl[x^{p}(t)\bigr]\biggl\{ a_{1}+ \frac{1}{2}(p-1)\sigma _{1}^{2}-b_{1}E \bigl[x^{p}(t)\bigr]^{\frac{1}{p}}\biggr\} . \end{aligned}$$
(26)
Let \(X(t)=E[x^{p}(t)]\), then
$$ \frac{dX(t)}{dt}=pX(t)\biggl\{ a_{1}+ \frac{1}{2}(p-1)\sigma _{1}^{2}-b_{1}X^{\frac{1}{p}}(t) \biggr\} . $$
(27)
By Assumption 1, we know \(0< b_{1}X^{\frac {1}{p}}(0)=b_{1}x(0)< a_{1}+\frac{p-1}{2}\sigma_{1}^{2}\). Applying the standard comparison argument,
$$ E\bigl[x^{p}(t)\bigr]^{\frac{1}{p}}< \frac{a_{1}+\frac{p-1}{2}\sigma_{1}^{2}}{b_{1}} ; $$
(28)
thus
$$ E\bigl[x^{p}(t)\bigr]\leq K_{1}(p)=\biggl( \frac{a_{1}+\frac{p-1}{2}\sigma_{1}^{2}}{b_{1}}\biggr)^{p}. $$
(29)
Similarly, we can prove that
$$\begin{aligned}& E\bigl[y^{p}(t)\bigr] \leq \biggl(\frac{a_{2}+\frac{p-1}{2}\sigma _{2}^{2}}{b_{2}} \biggr)^{p}, \end{aligned}$$
(30)
$$\begin{aligned}& E\bigl[z^{p}(t)\bigr] \leq \biggl(\frac{a_{3}+\frac{d_{1}}{\alpha_{1}}+\frac {d_{2}}{\alpha_{2}}+\frac{p-1}{2}\sigma_{3}^{2}}{b_{3}} \biggr)^{p}. \end{aligned}$$
(31)
 □

Theorem 2

Assume that Assumption  1 holds, the solutions of system (4) with initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\) are stochastically ultimately bounded.

Proof

If \((x(t),y(t),z(t))\in R_{+}^{3}\), its norm here is denoted by
$$\bigl|x(t),y(t),z(t)\bigr|=\sqrt{x^{2}(t)+y^{2}(t)+z^{2}(t)}. $$
Then \(|x(t),y(t),z(t)|^{p}=3^{\frac {p}{2}}(|x^{p}(t)|+|y^{p}(t)|+|z^{p}(t)|)\). By Lemma 2, \(E[|x(t),y(t),z(t)|^{p}]\leq L(p)\), \(t>0\). \(L(p)\) is dependent on \((x_{0},y_{0},z_{0})\in R_{+}^{3}\) and defined by \(L(p)=3^{\frac{p}{2}}[K_{1}(p)+K_{2}(p)+K_{3}(p)]\). By Chebyshev’s inequality, the above result is straightforward. □

4 Stochastic permanence

Definition 2

(see [3] and references therein)

The solution \((x(t),y(t),z(t))\) of system (4) is said to be stochastically permanent, if for any \(\varepsilon\in(0,1)\), there is a positive constant \(\delta=\delta(\varepsilon)\) and \(\chi=\chi (\varepsilon)\), such that for any initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\), system (4) has the properties that
$$\begin{aligned}& \liminf_{t\rightarrow\infty}P\bigl\{ \bigl\vert x(t),y(t),z(t)\bigr\vert \geq\delta\bigr\} \geq 1-\varepsilon, \end{aligned}$$
(32)
$$\begin{aligned}& \liminf_{t\rightarrow\infty}P\bigl\{ \bigl\vert x(t),y(t),z(t)\bigr\vert \leq\chi\bigr\} \geq 1-\varepsilon. \end{aligned}$$
(33)

Assumption 2

$$ \frac{1}{2}\max\bigl\{ \sigma_{1}^{2}, \sigma_{2}^{2},\sigma_{3}^{2}\bigr\} < \min \biggl\{ a_{1},a_{2},a_{3}+\frac{c_{1}}{\alpha_{1}}+ \frac{c_{2}}{\alpha_{2}}\biggr\} . $$
(34)

Theorem 3

If Assumption  2 holds, for any initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\), the solution \((x(t),y(t),z(t))\) satisfies \(\limsup_{t\rightarrow\infty}E[\frac{1}{|x(t),y(t),z(t)|^{\theta}}]\leq H\), where θ is an arbitrary positive constant satisfying
$$ \frac{\theta+3}{2}\max\bigl\{ \sigma_{1}^{2}, \sigma_{2}^{2},\sigma_{3}^{2}\bigr\} < \min\biggl\{ a_{1},a_{2},a_{3}+\frac{c_{1}}{\alpha_{1}}+ \frac{c_{2}}{\alpha _{2}}\biggr\} , $$
(35)
where k is an arbitrary positive constant satisfying
$$ k+\frac{\theta(\theta+3)}{2}\max\bigl\{ \sigma_{1}^{2}, \sigma_{2}^{2},\sigma _{3}^{2}\bigr\} < \theta\min\biggl\{ a_{1},a_{2},a_{3}+ \frac{c_{1}}{\alpha_{1}}+\frac {c_{2}}{\alpha_{2}}\biggr\} . $$
(36)

Proof

Define \(U(x,y,z)=x+y+z\). Applying Itô’s formula, we have
$$\begin{aligned} dU(x,y,z) =&x\biggl(a_{1}-b_{1}x- \frac{h_{1}x}{f_{1}+g_{1}y}-\frac {c_{1}z}{1+\alpha_{1}x+\beta_{1}z}\biggr)\,dt+\sigma_{1}x \,dB_{1}(t) \\ &{}+y\biggl(a_{2}-b_{2}y-\frac{h_{2}y}{f_{2}+g_{2}x}- \frac{c_{2}z}{1+\alpha _{2}y+\beta_{2}z}\biggr)\,dt+\sigma_{2}y\,dB_{2}(t) \\ &{}+z\biggl(a_{3}-b_{3}z+\frac{d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+ \frac {d_{2}y}{1+\alpha_{2}y+\beta_{2}z}\biggr)\,dt+\sigma_{3}z\,dB_{3}(t). \end{aligned}$$
(37)
Define \(V(x,y,z)=\frac{1}{U(x,y,z)}\). Using Itô’s formula, we obtain
$$\begin{aligned} dV(x,y,z) =&-V^{2}\biggl[x\biggl(a_{1}-b_{1}x- \frac{h_{1}x}{f_{1}+g_{1}y}-\frac {c_{1}z}{1+\alpha_{1}x+\beta_{1}z}\biggr) \\ &{}+y\biggl(a_{2}-b_{2}y-\frac{h_{2}y}{f_{2}+g_{1}x}- \frac{c_{2}z}{1+\alpha _{2}y+\beta_{2}z}\biggr) \\ &{}+z\biggl(a_{3}-b_{3}z+\frac{d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+ \frac {d_{2}y}{1+\alpha_{2}y+\beta_{2}z}\biggr)\biggr]\,dt \\ &{}+V^{3}\bigl(\sigma_{1}^{2}x^{2}+ \sigma_{2}^{2}y^{2}+\sigma_{3}^{2}z^{2} \bigr)\,dt \\ &{}-V^{2}\bigl(\sigma_{1}x\,dB_{1}(t)+ \sigma_{2}y\,dB_{2}(t)+\sigma_{3}z \,dB_{3}(t)\bigr), \end{aligned}$$
(38)
where
$$\begin{aligned} LV =&-V^{2}\biggl[x\biggl(a_{1}-b_{1}x- \frac {h_{1}x}{f_{1}+g_{1}y}-\frac{c_{1}z}{1+\alpha_{1}x+\beta_{1}z}\biggr) \\ &{}+y\biggl(a_{2}-b_{2}y-\frac{h_{2}y}{f_{2}+g_{1}x}- \frac{c_{2}z}{1+\alpha _{2}y+\beta_{2}z}\biggr) \\ &{}+z\biggl(a_{3}-b_{3}z+\frac{d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+ \frac {d_{2}y}{1+\alpha_{2}y+\beta_{2}z}\biggr)\biggr] \\ &{}+V^{3}\bigl(\sigma_{1}^{2}x^{2}+ \sigma_{2}^{2}y^{2}+\sigma_{3}^{2}z^{2} \bigr). \end{aligned}$$
(39)
Under Assumption 2, choosing a positive constant θ such that it satisfies (35), then by Itô’s formula, we have
$$\begin{aligned} d(1+V)^{\theta} =&\theta(1+V)^{\theta-1}\, dV+ \frac{1}{2}\theta(\theta -1) (1+V)^{\theta-2}(dV)^{2} \\ =&\biggl\{ \theta(1+V)^{\theta-1}LV+\frac{1}{2}\theta(\theta-1) (1+V)^{\theta -2}V^{4}\bigl(\sigma_{1}^{2}x^{2}+ \sigma_{2}^{2}y^{2}+\sigma_{3}^{2}z^{2} \bigr)\biggr\} \,dt \\ &{}-\theta V^{2}(1+V)^{\theta-1}\bigl(\sigma_{1}x \,dB_{1}(t)+\sigma _{2}y\,dB_{2}(t)+ \sigma_{3}z\,dB_{3}(t)\bigr). \end{aligned}$$
(40)
Thus
$$ L(1+V)^{\theta}=\theta(1+V)^{\theta-1}LV+ \frac{1}{2}\theta(\theta -1) (1+V)^{\theta-2}V^{4}\bigl( \sigma_{1}^{2}x^{2}+\sigma_{2}^{2}y^{2}+ \sigma _{3}^{2}z^{2}\bigr). $$
(41)
Then choosing a positive constant k such that it satisfies (36). By Itô’s formula, we get
$$\begin{aligned}& de^{kt}(1+V)^{\theta}= e^{kt}d(1+V)^{\theta}+ke^{kt}(1+V)^{\theta} \,dt \\& \hphantom{de^{kt}(1+V)^{\theta}}= e^{kt}L(1+V)^{\theta}\,dt+ke^{kt}(1+V)^{\theta} \,dt \\& \hphantom{de^{kt}(1+V)^{\theta}={}}{}-e^{kt}\theta V^{2}(1+V)^{\theta-1} \bigl[\sigma_{1}x\,dB_{1}(t)+\sigma _{2}y \,dB_{2}(t)+\sigma_{3}z\,dB_{3}(t)\bigr], \\& Le^{kt}(1+V)^{\theta} \\& \quad = e^{kt}(1+V)^{\theta-2}\biggl\{ k(1+V)^{2}+ \frac{1}{2}\theta(\theta -1)V^{4}\bigl(\sigma_{1}^{2}x^{2}+ \sigma_{2}^{2}y^{2}+\sigma_{3}^{2}z^{2} \bigr) \\& \qquad {} +\theta(1+V)V^{3}\bigl(\sigma_{1}^{2}x^{2}+ \sigma_{2}^{2}y^{2}+\sigma_{3}^{2}z^{2} \bigr) \\& \qquad {}-\theta(1+V)V^{2}\biggl[x\biggl(a_{1}-b_{1}x- \frac{h_{1}x}{f_{1}+g_{1}y}-\frac {c_{1}z}{1+\alpha_{1}x+\beta_{1}z}\biggr) \\& \qquad {} +y\biggl(a_{2}-b_{2}y-\frac{h_{2}y}{f_{2}+g_{2}x}- \frac{c_{2}z}{1+\alpha _{2}y+\beta_{2}z}\biggr) \\& \qquad {}+z\biggl(a_{3}-b_{3}z+ \frac{d_{1}x}{1+\alpha_{1}x+\beta_{1}z}+\frac {d_{2}y}{1+\alpha_{2}y+\beta_{2}z}\biggr)\biggr]\biggr\} \\& \quad \leq e^{kt}(1+V)^{\theta-2}\biggl\{ k(1+V)^{2}+ \frac{1}{2}\theta(\theta +1)V^{4}\max\bigl\{ \sigma_{1}^{2},\sigma_{2}^{2}, \sigma_{3}^{2}\bigr\} \bigl(x^{2}+y^{2}+z^{2} \bigr) \\& \qquad {} +\theta(1+V)V^{3}\max\bigl\{ \sigma_{1}^{2}, \sigma_{2}^{2},\sigma_{3}^{2}\bigr\} \bigl(x^{2}+y^{2}+z^{2}\bigr) \\& \qquad {}-\theta(1+V)V^{2} \biggl[x\biggl(a_{1}-b_{1}x-\frac{h_{1}x}{f_{1}}- \frac {c_{1}z}{\alpha_{1}x}\biggr) \\& \qquad {} +y\biggl(a_{2}-b_{2}y-\frac{h_{2}y}{f_{2}}- \frac{c_{2}z}{\alpha_{2}y}\biggr) +z(a_{3}-b_{3}z)\biggr]\biggr\} \\& \quad \leq e^{kt}(1+V)^{\theta-2}\biggl\{ k(1+V)^{2}+ \frac{1}{2}\theta(\theta +1)V^{4}\max\bigl\{ \sigma_{1}^{2},\sigma_{2}^{2}, \sigma_{3}^{2}\bigr\} \bigl(x^{2}+y^{2}+z^{2} \bigr) \\& \qquad {} +\theta(1+V)V^{3}\max\bigl\{ \sigma_{1}^{2}, \sigma_{2}^{2},\sigma_{3}^{2}\bigr\} \bigl(x^{2}+y^{2}+z^{2}\bigr) \\& \qquad {}-\theta(1+V)V^{2} \biggl[a_{1}x+a_{2}y+\biggl[a_{3}- \frac{c_{1}}{\alpha_{1}}-\frac {c_{2}}{\alpha_{2}}\biggr]z \\& \qquad {} -\biggl(b_{1}+\frac{h_{1}}{f_{1}}\biggr)x^{2}- \biggl(b_{2}+\frac {h_{2}}{f_{2}}\biggr)y^{2}-b_{3}z^{2} \biggr]\biggr\} \\& \quad \leq e^{kt}(1+V)^{\theta-2}\biggl\{ k(1+V)^{2}+ \frac{1}{2}\theta(\theta +1)V^{4}\max\bigl\{ \sigma_{1}^{2},\sigma_{2}^{2}, \sigma_{3}^{2}\bigr\} \frac {1}{V^{2}} \\& \qquad {} +\theta(1+V)V^{3}\max\bigl\{ \sigma_{1}^{2}, \sigma_{2}^{2},\sigma_{3}^{2}\bigr\} \frac{1}{V^{2}} -\theta(1+V)V^{2}\biggl[\min\biggl\{ a_{1},a_{2},a_{3}-\frac{c_{1}}{\alpha_{1}}- \frac {c_{2}}{\alpha_{2}}\biggr\} \frac{1}{V} \\& \qquad {} -\max\biggl\{ b_{1}+\frac{h_{1}}{f_{1}},b_{2}+ \frac{h_{2}}{f_{2}},b_{3}\biggr\} \frac {1}{V^{2}}\biggr]\biggr\} \\& \quad \leq e^{kt}(1+V)^{\theta-2}\biggl\{ \biggl[k+\theta\max\biggl\{ b_{1}+\frac {h_{1}}{f_{1}},b_{2}+\frac{h_{2}}{f_{2}},b_{3} \biggr\} \biggr] \\& \qquad {} +\biggl[2k+\theta\max\bigl\{ \sigma_{1}^{2}, \sigma_{2}^{2},\sigma_{3}^{2}\bigr\} - \theta \min\biggl\{ a_{1},a_{2},a_{3}- \frac{c_{1}}{\alpha_{1}}-\frac{c_{2}}{\alpha _{2}}\biggr\} \\& \qquad {}+\theta\max\biggl\{ b_{1}+ \frac{h_{1}}{f_{1}},b_{2}+\frac{h_{2}}{f_{2}},b_{3}\biggr\} \biggr]V \\& \qquad {} +\biggl[k+\frac{1}{2}\theta(\theta+3)\max\bigl\{ \sigma_{1}^{2},\sigma _{2}^{2}, \sigma_{3}^{2}\bigr\} -\theta\min\biggl\{ a_{1},a_{2},a_{3}- \frac{c_{1}}{\alpha_{1}}-\frac {c_{2}}{\alpha_{2}}\biggr\} \biggr]V^{2}\biggr\} , \end{aligned}$$
(42)
where
$$\begin{aligned}& V^{4}\bigl(\sigma_{1}^{2}x^{2}+ \sigma_{2}^{2}y^{2}+\sigma_{3}^{2}z^{2} \bigr) \leq V^{2}\max\bigl\{ \sigma_{1}^{2}, \sigma_{2}^{2},\sigma_{3}^{2}\bigr\} , \end{aligned}$$
(43)
$$\begin{aligned}& a_{1}x+a_{2}y+\biggl(a_{3}-\frac{c_{1}}{\alpha_{1}}- \frac{c_{2}}{\alpha _{2}}\biggr)z \geq \min\biggl\{ a_{1},a_{2},a_{3}- \frac{c_{1}}{\alpha_{1}}-\frac {c_{2}}{\alpha_{2}}\biggr\} \frac{1}{V}. \end{aligned}$$
(44)
Hence, we get \(Le^{kt}(1+V)^{\theta}\leq K_{4}e^{kt}\). Then
$$ de^{kt}(1+V)^{\theta}\leq K_{4}e^{kt} \,dt-\theta e^{kt}V^{2}(1+V)^{\theta -1}\bigl[ \sigma_{1}x\,dB_{1}(t)+\sigma_{2}y \,dB_{2}(t)+\sigma_{3}z\,dB_{3}(t)\bigr]. $$
(45)
Integrating both sides of the inequality from 0 to t and then taking the expectations, we can see that
$$ Ee^{kt}(1+V)^{\theta}\leq\bigl(1+V(0) \bigr)^{\theta}+\frac {K_{4}}{k}e^{kt}=\bigl(1+V(0) \bigr)^{\theta}+K_{5}e^{kt}, $$
(46)
where \(K_{5}=\frac{K_{4}}{k}\), so
$$ \limsup_{t\rightarrow\infty}E\bigl[V(t)\bigr]^{\theta} \leq\limsup_{t\rightarrow \infty}E\bigl[1+V(t)\bigr]^{\theta}\leq K_{5}. $$
(47)
Since \((x+y+z)^{\theta}\leq3^{\theta}(x^{2}+y^{2}+z^{2})^{\frac{\theta }{2}}=3^{\theta}|x+y+z|^{\theta}\) and \(V(x,y,z)=\frac{1}{x+y+z}\),
$$ \frac{1}{|x+y+z|^{\theta}}\leq3^{\theta}\frac{1}{(x+y+z)^{\theta }}=3^{\theta}V^{\theta}(t). $$
(48)
Clearly,
$$ \limsup_{t\rightarrow\infty}E\biggl[\frac{1}{|x+y+z|^{\theta}}\biggr] \leq3^{\theta}\limsup_{t\rightarrow\infty}E\bigl[V^{\theta}(t) \bigr] \leq3^{\theta} K_{5}=K_{6}, $$
(49)
which is the desired assertion. □

Theorem 4

Under Assumption  2, system (4) is stochastically permanent.

Proof

By Theorem 2, we know that \(\limsup_{t\rightarrow\infty }E|x+y+z|^{p}\leq L(p)\). Let \(\chi=(\frac{L(p)}{\varepsilon})^{\frac {1}{p}}\), for all \(\varepsilon>0\). By Chebyshev’s inequality, we can obtain the required assertion. □

5 Global asymptotic stability

Definition 3

(see [3] and references therein)

Let \((x_{1}(t),y_{1}(t),z_{1}(t))\) be a positive solution of system (4). If we say that \((x_{1}(t),y_{1}(t),z_{1}(t))\) is globally asymptotically stable in expectation, it means that any other solution \((x_{2}(t),y_{2}(t),z_{2}(t))\) of system (4) has \(t\geq0\) and that we have initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\). That is,
$$ P\Bigl\{ \lim_{t\rightarrow\infty}E\bigl[\bigl\vert \bigl(x_{1}(t),y_{1}(t),z_{1}(t)\bigr)- \bigl(x_{2}(t),y_{2}(t),z_{2}(t)\bigr)\bigr\vert \bigr]=0\Bigr\} =1. $$
(50)

Lemma 3

[19]

Suppose that an n-dimensional stochastic process \(X(t)\) on \(t\geq0\) satisfies the condition
$$ E\bigl\vert X(t)-X(s)\bigr\vert ^{\alpha}\leq c\vert t-s\vert ^{1+\beta},\quad 0\leq s,t< \infty, $$
(51)
for some positive constants α, β, and c. There exists a continuous modification \(\widetilde{X}(t)\) of \(X(t)\) which has the property that for every \(\vartheta\in(0,\frac{\beta}{\alpha})\) there is a positive random variable \(h(\omega)\) such that
$$ P\biggl\{ \omega:\sup_{0< |t-s|< h(\omega), 0\leq s, t< \infty} \frac{\widetilde {X}(t,\omega)-\widetilde{X}(s,\omega)}{|t-s|^{\vartheta}} \leq\frac {2}{1-2^{-\vartheta}}\biggr\} =1. $$
(52)
In other words, almost every sample path of \(\widetilde{X}(t)\) is locally but uniformly Hölder continuous with exponent ϑ.

Lemma 4

Let \((x(t),y(t),z(t))\) be a positive solution of system (4) on \(t\geq 0\) with initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\). Then almost every sample path of \((x(t),y(t),z(t))\) is uniformly continuous on \(t\geq0\).

Proof

The first equation of system (4) is equivalent to the following stochastic integral equation:
$$\begin{aligned} x(t) =& x(0)+ \int^{t}_{0}x(s) \biggl( a_{1}-b_{1}x(s)- \frac {h_{1}x(s)}{f_{1}+g_{1}y(s)}-\frac{c_{1}z(s)}{1+\alpha_{1}x(s)+\beta _{1}z(s)} \biggr) \, ds \\ &{}+ \int^{t}_{0}\sigma_{1}x(s) \,dB_{1}(s). \end{aligned}$$
(53)
We estimate
$$\begin{aligned}& E\biggl\vert x\biggl(a_{1}-b_{1}x- \frac{h_{1}x}{f_{1}+g_{1}y}-\frac{c_{1}z}{1+\alpha _{1}x+\beta_{1}z}\biggr)\biggr\vert ^{p} \\& \quad \leq \frac{1}{2}E\vert x\vert ^{2p}+ \frac{1}{2}E\biggl\vert a_{1}-b_{1}x- \frac {h_{1}x}{f_{1}+g_{1}y}-\frac{c_{1}z}{1+\alpha_{1}x+\beta_{1}z}\biggr\vert ^{2p} \\& \quad \leq \frac{1}{2}E\vert x\vert ^{2p}+ \frac{1}{2}E\biggl\vert a_{1}+b_{1}x+ \frac {h_{1}x}{f_{1}}+\frac{c_{1}}{\beta_{1}}\biggr\vert ^{2p} \\& \quad \leq \frac{1}{2}E\vert x\vert ^{2p}+ \frac{3^{2p-1}}{2}\biggl\{ a_{1}^{2p}+\biggl(b_{1}+ \frac {h_{1}}{f_{1}}\biggr)^{2p}E\vert x\vert ^{2p}+\biggl( \frac{c_{1}}{\beta_{1}}\biggr)^{2p}\biggr\} \\& \quad \leq \frac{1}{2}K_{1}(2p)+\frac{3^{2p-1}}{2}\biggl\{ a_{1}^{2p}+\biggl(b_{1}+\frac {h_{1}}{f_{1}} \biggr)^{2p}K_{1}(2p)+\biggl(\frac{c_{1}}{\beta_{1}} \biggr)^{2p}\biggr\} \\& \quad =: K_{5}(p). \end{aligned}$$
(54)
In addition, applying the moment inequality for stochastic integrals [18], we have, for \(0< t_{1}\leq t_{2}<\infty\) and \(p>2\),
$$\begin{aligned} E\biggl[\biggl\vert \int^{t_{2}}_{t_{1}}\sigma_{1}x(s) \,dB_{1}(s)\biggr\vert ^{p}\biggr]&\leq\bigl(\sigma _{1}^{2}\bigr)^{p}\biggl[\frac{p(p-1)}{2} \biggr]^{\frac{p}{2}}(t_{2}-t_{1})^{\frac {p-2}{2}} \int^{t_{2}}_{t_{1}}E\bigl\vert x(s)\bigr\vert ^{p}\, ds \\ &\leq\bigl(\sigma_{1}^{2}\bigr)^{p}\biggl[ \frac{p(p-1)}{2}\biggr]^{\frac {p}{2}}(t_{2}-t_{1})^{\frac{p}{2}}K_{1}(p). \end{aligned}$$
(55)
Let \(t_{2}-t_{1}\leq1\) and \(\frac{1}{p}+\frac{1}{q}=1\), we obtain
$$\begin{aligned}& E\bigl\vert x(t_{2})-x(t_{1})\bigr\vert ^{p} \\& \quad = E\biggl\vert \int^{t_{2}}_{t_{1}}x(s) \biggl(a_{1}-b_{1}x(s)- \frac {h_{1}x(s)}{f_{1}+g_{1}y(s)}-\frac{c_{1}z(s)}{1+\alpha_{1}x(s)+\beta _{1}z(s)}\biggr)\,ds+ \int^{t_{2}}_{t_{1}}\sigma_{1}x(s) \,dB_{1}(s)\biggr\vert ^{p} \\& \quad \leq 2^{p-1}E\biggl\vert \int^{t_{2}}_{t_{1}}x(s) \biggl(a_{1}-b_{1}x(s)- \frac {h_{1}x(s)}{f_{1}+g_{1}y(s)}-\frac{c_{1}z(s)}{1+\alpha_{1}x(s)+\beta _{1}z(s)}\biggr)\,ds\biggr\vert ^{p} \\& \qquad {} +2^{p-1}E\biggl[\biggl\vert \int^{t_{2}}_{t_{1}}\sigma_{1}x(s) \,dB_{1}(s)\biggr\vert ^{p}\biggr] \\& \quad \leq 2^{p-1}(t_{2}-t_{1})^{\frac{p}{q}} \int ^{t_{2}}_{t_{1}}E\biggl\vert x(s) \biggl(a_{1}-b_{1}x(s)-\frac {h_{1}x(s)}{f_{1}+g_{1}y(s)}-\frac{c_{1}z(s)}{1+\alpha_{1}x(s)+\beta _{1}z(s)} \biggr)\biggr\vert ^{p}\,ds \\& \qquad {} +2^{p-1}\bigl(\sigma_{1}^{2} \bigr)^{p}\biggl[\frac{p(p-1)}{2}\biggr]^{\frac {p}{2}}(t_{2}-t_{1})^{\frac{p}{2}}K_{1}(p) \\& \quad \leq 2^{p-1}(t_{2}-t_{1})^{p}K_{5}(p)+2^{p-1}(t_{2}-t_{1})^{\frac {p}{2}} \bigl(\sigma_{1}^{2}\bigr)^{p}\biggl[ \frac{p(p-1)}{2}\biggr]^{\frac{p}{2}}K_{1}(p) \\& \quad \leq 2^{p-1}(t_{2}-t_{1})^{\frac{p}{2}} \biggl\{ 1+\biggl[\frac{p(p-1)}{2}\biggr]^{\frac {p}{2}}\biggr\} \max\bigl\{ K_{5}(p),\bigl(\sigma_{1}^{2} \bigr)^{p}K_{1}(p)\bigr\} \\& \quad \leq 2^{p-1}(t_{2}-t_{1})^{\frac{p}{2}} \biggl\{ 1+\biggl[\frac{p(p-1)}{2}\biggr]^{\frac {p}{2}}\biggr\} K_{6}(p), \end{aligned}$$
(56)
where \(K_{6}(p)=\max\{{K_{5}(p),(\sigma_{1}^{2})^{p}K_{1}(p)}\}\). Then according to Lemma 3, we know that almost every sample path of \(x(t)\) is locally but uniformly Hölder continuous with exponent ϑ for every \(\vartheta\in(0,\frac{p-2}{2p})\). Therefore almost every sample path of \(x(t)\) is uniformly continuous on \(t\geq0\). In the same way, we can demonstrate that almost every sample path of \(y(t)\), \(z(t)\) is uniformly continuous on \(t\geq0\). □

Lemma 5

[1]

Let f be a non-negative function defined on \(R_{+}\) such that f is integrable on \(R_{+}\) and is uniformly continuous on \(t\geq0\). Then \(\lim_{t\rightarrow\infty}f(t)=0\).

Theorem 5

If
$$ \textstyle\begin{cases} A = b_{1}-h_{1}-\frac{h_{1}}{f_{1}}-\frac{c_{1}\alpha_{1}}{\beta _{1}}-2d_{1}-\frac{h_{2}g_{2}[a_{2}+\frac{p-1}{2}\sigma _{2}^{2}]}{b_{2}f_{2}^{2}}>0, \\ B = b_{2}-h_{2}-\frac{h_{2}}{f_{2}}-\frac{c_{2}\alpha_{2}}{\beta _{2}}-2d_{2}-\frac{h_{1}g_{1}[a_{1}+\frac{p-1}{2}\sigma _{1}^{2}]}{b_{1}f_{1}^{2}}>0, \\ C = b_{3}-2c_{1}-2c_{2}-\frac{\beta_{1}d_{1}}{\alpha_{1}}-\frac{\beta _{2}d_{2}}{\alpha_{2}}>0, \end{cases} $$
(57)
then system (4) is globally attractive.

Proof

Define \(W(t)=|\ln x_{1}(t)-\ln x_{2}(t)|+|\ln y_{1}(t)-\ln y_{2}(t)|+|\ln z_{1}(t)-\ln z_{2}(t)|\). We can prove \(W(t)\) is continuous positive function on \(t\geq0\). By a direct calculation of the right differential \(d^{+}W(t)\) of \(W(t)\), and then applying Itô’s formula, we get
$$\begin{aligned} d^{+}W(t) =&\operatorname{sgn}(x_{1}-x_{2}) \biggl\{ \biggl[\frac{dx_{1}}{x_{1}}-\frac {(dx_{1})^{2}}{2x_{1}^{2}}\biggr]-\biggl[ \frac{dx_{2}}{x_{2}}-\frac {(dx_{2})^{2}}{2x_{2}^{2}}\biggr]\biggr\} \\ &{}+\operatorname{sgn}(y_{1}-y_{2})\biggl\{ \biggl[ \frac{dy_{1}}{y_{1}}-\frac {(dy_{1})^{2}}{2y_{1}^{2}}\biggr]-\biggl[\frac{dy_{2}}{y_{2}}- \frac {(dy_{2})^{2}}{2y_{2}^{2}}\biggr]\biggr\} \\ &{}+\operatorname{sgn}(z_{1}-z_{2})\biggl\{ \biggl[ \frac{dz_{1}}{z_{1}}-\frac {(dz_{1})^{2}}{2z_{1}^{2}}\biggr]-\biggl[\frac{dz_{2}}{z_{2}}- \frac {(dz_{2})^{2}}{2z_{2}^{2}}\biggr]\biggr\} \\ =& \operatorname{sgn}(x_{1}-x_{2})\biggl\{ -b_{1}(x_{1}-x_{2})-\biggl(\frac {h_{1}x_{1}}{f_{1}+g_{1}y_{1}}- \frac {h_{1}x_{2}}{f_{1}+g_{1}y_{2}}\biggr) \\ &{}-\biggl(\frac{c_{1}z_{1}}{1+\alpha _{1}x_{1}+\beta_{1}z_{1}}- \frac{c_{1}z_{2}}{1+\alpha_{1}x_{2}+\beta_{1}z_{2}}\biggr) \biggr\} \,dt \\ &{}+\operatorname{sgn}(y_{1}-y_{2})\biggl\{ -b_{1}(y_{1}-y_{2})-\biggl(\frac {h_{2}y_{1}}{f_{2}+g_{2}x_{1}}- \frac {h_{2}y_{2}}{f_{2}+g_{2}x_{2}}\biggr) \\ &{}-\biggl(\frac{c_{2}z_{1}}{1+\alpha _{2}y_{1}+\beta_{2}z_{1}}- \frac{c_{2}z_{2}}{1+\alpha_{2}y_{2}+\beta_{2}z_{2}}\biggr) \biggr\} \,dt \\ &{}+\operatorname{sgn}(z_{1}-z_{2})\biggl\{ -b_{3}(z_{1}-z_{2}) +\biggl(\frac{d_{1}x_{1}}{1+\alpha_{1}x_{1}+\beta_{1}z_{1}}- \frac {d_{1}x_{2}}{1+\alpha_{1}x_{2}+\beta_{1}z_{2}}\biggr) \\ &{}+\biggl(\frac{d_{2}y_{1}}{1+\alpha_{2}y_{1}+\beta_{2}z_{1}}-\frac {d_{2}y_{2}}{1+\alpha_{2}y_{2}+\beta_{2}z_{2}}\biggr)\biggr\} \,dt. \end{aligned}$$
(58)
Integrating both sides of the equality from 0 to t and then taking expectations
$$\begin{aligned}& E\bigl[W(t)-W(0)\bigr] \\& \quad = E\biggl\{ \int^{t}_{0}\biggl[\operatorname{sgn} \bigl(x_{1}(s)-x_{2}(s)\bigr)\biggl\{ -b_{1} \bigl(x_{1}(s)-x_{2}(s)\bigr)-\biggl(\frac{h_{1}x_{1}(s)}{f_{1}+g_{1}y_{1}(s)}- \frac{h_{1}x_{2}(s)}{f_{1}+g_{1}y_{2}(s)}\biggr) \\& \qquad {} -\biggl(\frac{c_{1}z_{1}(s)}{1+\alpha_{1}x_{1}(s)+\beta_{1}z_{1}(s)}- \frac{c_{1}z_{2}(s)}{1+\alpha_{1}x_{2}(s)+\beta_{1}z_{2}(s)}\biggr)\biggr\} \\& \qquad {} +\operatorname{sgn}\bigl(y_{1}(s)-y_{2}(s)\bigr) \biggl\{ -b_{1}\bigl(y_{1}(s)-y_{2}(s)\bigr)- \biggl(\frac {h_{2}y_{1}(s)}{f_{2}+g_{2}x_{1}(s)}-\frac {h_{2}y_{2}(s)}{f_{2}+g_{2}x_{2}(s)}\biggr) \\& \qquad {} -\biggl(\frac{c_{2}z_{1}(s)}{1+\alpha_{2}y_{1}(s)+\beta_{2}z_{1}(s)}- \frac{c_{2}z_{2}(s)}{1+\alpha_{2}y_{2}(s)+\beta_{2}z_{2}(s)}\biggr)\biggr\} \\& \qquad {} +\operatorname{sgn}\bigl(z_{1}(s)-z_{2}(s)\bigr) \biggl\{ -b_{3}\bigl(z_{1}(s)-z_{2}(s)\bigr) \\& \qquad {}+ \biggl(\frac{d_{1}x_{1}(s)}{1+\alpha_{1}x_{1}(s)+\beta_{1}z_{1}(s)}-\frac {d_{1}x_{2}(s)}{1+\alpha_{1}x_{2}(s)+\beta_{1}z_{2}(s)}\biggr) \\& \qquad {} +\biggl(\frac{d_{2}y_{1}(s)}{1+\alpha_{2}y_{1}(s)+\beta_{2}z_{1}(s)}-\frac {d_{2}y_{2}(s)}{1+\alpha_{2}y_{2}(s)+\beta_{2}z_{2}(s)}\biggr)\biggr\} \,ds \biggr]\biggr\} . \end{aligned}$$
(59)
So
$$\begin{aligned}& \frac{dE[W(t)]}{dt}= E\biggl[\operatorname{sgn} \bigl(x_{1}(t)-x_{2}(t)\bigr)\biggl\{ -b_{1} \bigl(x_{1}(t)-x_{2}(t)\bigr)-\biggl(\frac {h_{1}x_{1}(t)}{f_{1}+g_{1}y_{1}(t)}- \frac{h_{1}x_{2}(t)}{f_{1}+g_{1}y_{2}(t)}\biggr) \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} -\biggl(\frac{c_{1}z_{1}(t)}{1+\alpha_{1}x_{1}(t)+\beta_{1}z_{1}(t)}-\frac {c_{1}z_{2}(t)}{1+\alpha_{1}x_{2}(t)+\beta_{1}z_{2}(t)}\biggr)\biggr\} \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} +\operatorname{sgn}\bigl(y_{1}(t)-y_{2}(t) \bigr)\biggl\{ -b_{2}\bigl(y_{1}(t)-y_{2}(t)\bigr)- \biggl(\frac {h_{2}y_{1}(t)}{f_{2}+g_{2}x_{1}(t)}-\frac {h_{2}y_{2}(t)}{f_{2}+g_{2}x_{2}(t)}\biggr) \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} -\biggl(\frac{c_{2}z_{1}(t)}{1+\alpha_{2}y_{1}(t)+\beta_{2}z_{1}(t)}-\frac {c_{2}z_{2}(t)}{1+\alpha_{2}y_{2}(t)+\beta_{2}z_{2}(t)}\biggr)\biggr\} \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} +\operatorname{sgn}\bigl(z_{1}(t)-z_{2}(t) \bigr)\biggl\{ -b_{3}\bigl(z_{1}(t)-z_{2}(t)\bigr) \\& \hphantom{\frac{dE[W(t)]}{dt}=}{}+\biggl(\frac{d_{1}x_{1}(t)}{1+\alpha_{1}x_{1}(t)+\beta_{1}z_{1}(t)}-\frac {d_{1}x_{2}(t)}{1+\alpha_{1}x_{2}(t)+\beta_{1}z_{2}(t)}\biggr) \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} +\biggl(\frac{d_{2}y_{1}(t)}{1+\alpha_{2}y_{1}(t)+\beta_{2}z_{1}(t)}-\frac {d_{2}y_{2}(t)}{1+\alpha_{2}y_{2}(t)+\beta_{2}z_{2}(t)}\biggr)\biggr\} \biggr] \\& \hphantom{\frac{dE[W(t)]}{dt}}\leq -b_{1}E\bigl\vert x_{1}(t)-x_{2}(t) \bigr\vert +E\biggl\vert \frac {h_{1}x_{1}(t)}{f_{1}+g_{1}y_{1}(t)}-\frac {h_{1}x_{2}(t)}{f_{1}+g_{1}y_{2}(t)}\biggr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} +E\biggl\vert \frac{c_{1}z_{1}(t)}{1+\alpha_{1}x_{1}(t)+\beta_{1}z_{1}(t)}-\frac {c_{1}z_{2}(t)}{1+\alpha_{1}x_{2}(t)+\beta _{1}z_{2}(t)}\biggr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{}-b_{2}E\bigl\vert y_{1}(t)-y_{2}(t)\bigr\vert -E\biggl\vert \frac{h_{2}y_{1}(t)}{f_{2}+g_{2}x_{1}(t)}-\frac {h_{2}y_{2}(t)}{f_{2}+g_{2}x_{2}(t)}\biggr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} -E\biggl\vert \frac{c_{2}z_{1}(t)}{1+\alpha _{2}y_{1}(t)+\beta_{2}z_{1}(t)}-\frac{c_{2}z_{2}(t)}{1+\alpha _{2}y_{2}(t)+\beta_{2}z_{2}(t)}\biggr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} -b_{3}E\bigl\vert z_{1}(t)-z_{2}(t) \bigr\vert +E\biggl\vert \frac{d_{1}x_{1}(t)}{1+\alpha _{1}x_{1}(t)+\beta_{1}z_{1}(t)}-\frac{d_{1}x_{2}(t)}{1+\alpha _{1}x_{2}(t)+\beta_{1}z_{2}(t)}\biggr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} +E\biggl\vert \frac{d_{2}y_{1}(t)}{1+\alpha_{2}y_{1}(t)+\beta_{2}z_{1}(t)}-\frac {d_{2}y_{2}(t)}{1+\alpha_{2}y_{2}(t)+\beta_{2}z_{2}(t)}\biggr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}}\leq -b_{1}E\bigl\vert x_{1}(t)-x_{2}(t) \bigr\vert +h_{1}\biggl(1+\frac {1}{f_{1}}\biggr)E\bigl\vert x_{1}(t)-x_{2}(t)\bigr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{}+\frac {h_{1}g_{1}}{f_{1}^{2}}E\bigl\vert x_{1}(t)\bigr\vert E\bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert +2c_{1}E\bigl\vert z_{1}(t)-z_{2}(t) \bigr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} +\frac{c_{1}\alpha_{1}}{\beta _{1}}E\bigl\vert x_{1}(t)-x_{2}(t) \bigr\vert -b_{2}E\bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} +h_{2}\biggl(1+\frac{1}{f_{2}}\biggr)E\bigl\vert y_{1}(t)-y_{2}(t)\bigr\vert +\frac {h_{2}g_{2}}{f_{2}^{2}}E \bigl\vert y_{1}(t)\bigr\vert E\bigl\vert x_{1}(t)-x_{2}(t) \bigr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{}+2c_{2}E\bigl\vert z_{1}(t)-z_{2}(t) \bigr\vert +\frac{c_{2}\alpha_{2}}{\beta _{2}}E\bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} -b_{3}E\bigl\vert z_{1}(t)-z_{2}(t) \bigr\vert +2d_{1}E\bigl\vert x_{1}(t)-x_{2}(t) \bigr\vert \\& \hphantom{\frac{dE[W(t)]}{dt}=}{} +\frac{\beta_{1}d_{1}}{\alpha_{1}}E\bigl\vert z_{1}(t)-z_{2}(t) \bigr\vert +2d_{2}E\bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert +\frac{\beta_{2}d_{2}}{\alpha _{2}}E\bigl\vert z_{1}(t)-z_{2}(t) \bigr\vert , \end{aligned}$$
(60)
$$\begin{aligned}& \frac{dE[W(t)]}{dt} = -\biggl[b_{1}-h_{1}- \frac{h_{1}}{f_{1}}-\frac {c_{1}\alpha_{1}}{\beta_{1}}-2d_{1}-\frac {h_{2}g_{2}}{f_{2}^{2}}E\bigl\vert y_{1}(t)\bigr\vert \biggr]E\bigl\vert x_{1}(t)-x_{2}(t) \bigr\vert \\& \hphantom{\frac{dE[W(t)]}{dt} ={}}{} -\biggl[b_{2}-h_{2}-\frac{h_{2}}{f_{2}}- \frac{c_{2}\alpha_{2}}{\beta _{2}}-2d_{2}-\frac {h_{1}g_{1}}{f_{1}^{2}}E\bigl\vert x_{1}(t)\bigr\vert \biggr]E\bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert \\& \hphantom{\frac{dE[W(t)]}{dt} ={}}{} -\biggl[b_{3}-2c_{1}-2c_{2}- \frac{\beta_{1}d_{1}}{\alpha_{1}}-\frac{\beta _{2}d_{2}}{\alpha_{2}}\biggr]E\bigl\vert z_{1}(t)-z_{2}(t) \bigr\vert . \end{aligned}$$
(61)
By Lemma 2,
$$\begin{aligned}& E\bigl\vert x_{1}(t)\bigr\vert = E\bigl\vert x^{3}_{1}(t)\bigr\vert ^{\frac{1}{3}}= \bigl\{ E \bigl[x^{3}_{1}(t)\bigr]\bigr\} ^{\frac{1}{3}}\leq \frac{a_{1}+\frac{p-1}{2}\sigma_{1}^{2}}{b_{1}}, \end{aligned}$$
(62)
$$\begin{aligned}& E\bigl\vert y_{1}(t)\bigr\vert = E\bigl\vert y^{3}_{1}(t)\bigr\vert ^{\frac{1}{3}}= \bigl\{ E \bigl[y^{3}_{1}(t)\bigr]\bigr\} ^{\frac{1}{3}}\leq \frac{a_{2}+\frac{p-1}{2}\sigma_{2}^{2}}{b_{2}}, \end{aligned}$$
(63)
$$\begin{aligned}& E\bigl\vert z_{1}(t)\bigr\vert = E\bigl\vert z^{3}_{1}(t)\bigr\vert ^{\frac{1}{3}}=\bigl\{ E \bigl[z^{3}_{1}(t)\bigr]\bigr\} ^{\frac {1}{3}}\leq \frac{a_{3}+\frac{d_{1}}{\alpha_{1}}+\frac{d_{2}}{\alpha _{2}}+\frac{p-1}{2}\sigma_{3}^{2}}{b_{3}}. \end{aligned}$$
(64)
Thus
$$\begin{aligned} \frac{dE[W(t)]}{dt} \leq&-\biggl[b_{1}-h_{1}- \frac{h_{1}}{f_{1}}-\frac {c_{1}\alpha_{1}}{\beta_{1}}-2d_{1}-\frac{h_{2}g_{2}(a_{2} +\frac{p-1}{2}\sigma_{2}^{2})}{b_{2}f_{2}^{2}} \biggr]E\bigl\vert x_{1}(t)-x_{2}(t)\bigr\vert \\ &{}-\biggl[b_{2}-h_{2}-\frac{h_{2}}{f_{2}}- \frac{c_{2}\alpha_{2}}{\beta _{2}}-2d_{2}-\frac{h_{1}g_{1}({a_{1} +\frac{p-1}{2}\sigma_{1}^{2}})}{b_{1}f_{1}^{2}}\biggr]E\bigl\vert y_{1}(t)-y_{2}(t)\bigr\vert \\ &{}-\biggl[b_{3}-2c_{1}-2c_{2}- \frac{\beta_{1}d_{1}}{\alpha_{1}}-\frac{\beta _{2}d_{2}}{\alpha_{2}}\biggr]E\bigl\vert z_{1}(t)-z_{2}(t) \bigr\vert \\ \leq&-AE\bigl\vert x_{1}(t)-x_{2}(t)\bigr\vert -BE \bigl\vert y_{1}(t)-y_{2}(t)\bigr\vert -CE\bigl\vert z_{1}(t)-z_{2}(t)\bigr\vert , \end{aligned}$$
(65)
where
$$\begin{aligned}& A = b_{1}-h_{1}-\frac{h_{1}}{f_{1}}-\frac{c_{1}\alpha_{1}}{\beta _{1}}-2d_{1}- \frac{h_{2}g_{2}(a_{2}+\frac{p-1}{2}\sigma _{2}^{2})}{b_{2}f_{2}^{2}}, \end{aligned}$$
(66)
$$\begin{aligned}& B = b_{2}-h_{2}-\frac{h_{2}}{f_{2}}-\frac{c_{2}\alpha_{2}}{\beta _{2}}-2d_{2}- \frac{h_{1}g_{1}(a_{1}+\frac{p-1}{2}\sigma _{1}^{2})}{b_{1}f_{1}^{2}}, \end{aligned}$$
(67)
$$\begin{aligned}& C = b_{3}-2c_{1}-2c_{2}-\frac{\beta_{1}d_{1}}{\alpha_{1}}- \frac{\beta _{2}d_{2}}{\alpha_{2}}. \end{aligned}$$
(68)
Integrating both sides, we get
$$\begin{aligned} E\bigl[W(t)\bigr] \leq& E\bigl[W(0)\bigr] \\ &{}- \int ^{t}_{0}\bigl[AE\bigl\vert x_{1}(s)-x_{2}(s)\bigr\vert +BE\bigl\vert y_{1}(s)-y_{2}(s)\bigr\vert +CE\bigl\vert z_{1}(s)-z_{2}(s)\bigr\vert \bigr]\,ds. \end{aligned}$$
(69)
Consequently,
$$ W(t)+ \int ^{t}_{0}\bigl[AE\bigl\vert x_{1}(s)-x_{2}(s)\bigr\vert +BE\bigl\vert y_{1}(s)-y_{2}(s)\bigr\vert +CE\bigl\vert z_{1}(s)-z_{2}(s)\bigr\vert \bigr]\,ds\leq W(0)< \infty. $$
(70)
Recall that \(W(t)\geq0\) and \(A>0\), \(B>0\), \(C>0\), yielding
$$ \begin{aligned} &\bigl\vert x_{1}(t)-x_{2}(t)\bigr\vert \in L^{1}[0,+\infty) ,\qquad \bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert \in L^{1}[0,+\infty) , \\ &\bigl\vert z_{1}(t)-z_{2}(t)\bigr\vert \in L^{1}[0,+ \infty). \end{aligned} $$
(71)
Then from Lemmas 4 and 5, we get the desired assertion. □

6 Numerical simulations

In this section, to substantiate the analytical results; the dynamics of system (4) with and without environmental noises are illustrated by the Milstein method [20].

Consider the discrete equations:
$$ \begin{aligned} &x_{k+1} = x_{k}+x_{k} \biggl(a_{1}-b_{1}x_{k}-\frac {h_{1}x_{k}}{f_{1}+g_{1}y_{k}}- \frac{c_{1}z_{k}}{1+\alpha_{1}x_{k}+\beta _{1}z_{k}}\biggr)\Delta t \\ &\hphantom{x_{k+1} ={}}{}+\sigma_{1}x_{k}\xi_{k}\sqrt{ \Delta t}+\frac{1}{2}\sigma _{1}^{2}x_{k} \bigl(\xi_{k}^{2}-1\bigr)\Delta t, \\ &y_{k+1} = y_{k}+y_{k}\biggl(a_{2}-b_{2}y_{k}- \frac {h_{2}y_{k}}{f_{2}+g_{2}x_{k}}-\frac{c_{2}z_{k}}{1+\alpha_{2}x_{k}+\beta _{2}z_{k}}\biggr)\Delta t \\ &\hphantom{y_{k+1} ={}}{}+\sigma_{2}y_{k}\eta_{k}\sqrt{ \Delta t}+\frac{1}{2}\sigma _{2}^{2}y_{k} \bigl(\eta_{k}^{2}-1\bigr)\Delta t, \\ &z_{k+1} = z_{k}+z_{k}\biggl(a_{3}-b_{3}z_{k}+ \frac{d_{1}x_{k}}{1+\alpha _{1}x_{k}+\beta_{1}z_{k}}+\frac{d_{2}y_{k}}{1+\alpha_{2}x_{k}+\beta _{2}z_{k}}\biggr)\Delta t \\ &\hphantom{z_{k+1} ={}}{}+\sigma_{3}z_{k}\zeta_{k} \sqrt{\Delta t}+\frac{1}{2}\sigma _{3}^{2}z_{k} \bigl(\zeta_{k}^{2}-1\bigr)\Delta t, \end{aligned} $$
(72)
where \(\xi_{k}\), \(\eta_{k}\), and \(\zeta_{k}\) are Gaussian random variables that follow \(N(0,1)\).
In Figure 1, we choose the initial value \((x_{0},y_{0},z_{0})=(0.5,0.5,0.5)\) and all the parameters satisfying the conditions of Theorem 3, the system (4) is stochastically permanent.
Figure 1
Figure 1

Solutions of system ( 4 ) for \(\pmb{(x_{0},y_{0},z_{0})=(0.5,0.5,0.5)}\) , \(\pmb{a_{1}=2}\) , \(\pmb{a_{2}=1}\) , \(\pmb{a_{3}=1.5}\) , \(\pmb{b_{1}=2}\) , \(\pmb{b_{2}=2}\) , \(\pmb{b_{3}=3}\) , \(\pmb{c_{1}=0.2}\) , \(\pmb{c_{2}=0.1}\) , \(\pmb{d_{1}=1}\) , \(\pmb{d_{2}=1}\) , \(\pmb{\alpha_{1}=1}\) , \(\pmb{\beta_{1}=0.5}\) , \(\pmb{\alpha_{2}=0.6}\) , \(\pmb{\beta _{2}=0.8}\) , \(\pmb{h_{1}=0.5}\) , \(\pmb{h_{2}=0.8}\) , \(\pmb{f_{1}=1}\) , \(\pmb{f_{2}=0.3}\) , \(\pmb{g_{1}=0.5}\) , \(\pmb{g_{2}=1}\) , \(\pmb{\sigma_{1}=0.05}\) , \(\pmb{\sigma_{2}=0.05}\) , \(\pmb{\sigma _{2}=0.05}\) .

In Figure 2, we choose the initial value \((x_{0},y_{0},z_{0})=(0.5,0.5,0.5)\) and all the parameters satisfying conditions of Theorem 5. Using Matlab, we see that, under a small perturbation, the solution of system (4) is fluctuating in a small neighborhood, at this time, the stochastic system is getting more similar to the deterministic.
Figure 2
Figure 2

Solutions of system ( 4 ) for \(\pmb{(x_{0},y_{0},z_{0})=(0.5,0.5,0.4)}\) , \(\pmb{a_{1}=2}\) , \(\pmb{a_{2}=2}\) , \(\pmb{a_{3}=1}\) , \(\pmb{b_{1}=2.5}\) , \(\pmb{b_{2}=2.5}\) , \(\pmb{b_{3}=2.6}\) , \(\pmb{c_{1}=0.1}\) , \(\pmb{c_{2}=0.15}\) , \(\pmb{d_{1}=0.8}\) , \(\pmb{d_{2}=0.8}\) , \(\pmb{\alpha_{1}=1}\) , \(\pmb{\beta_{1}=0.8}\) , \(\pmb{\alpha _{2}=1}\) , \(\pmb{\beta_{2}=0.8}\) , \(\pmb{h_{1}=0.25}\) , \(\pmb{h_{2}=0.3}\) , \(\pmb{f_{1}=1}\) , \(\pmb{f_{2}=0.8}\) , \(\pmb{g_{1}=0.5}\) , \(\pmb{g_{2}=0.6}\) , \(\pmb{\sigma_{1}=0.05}\) , \(\pmb{\sigma _{2}=0.05}\) , \(\pmb{\sigma_{2}=0.05}\) .

In Figure 3, suffering sufficiently large white noise, system (4) gets extinct, while none of the species in the deterministic system die out.
Figure 3
Figure 3

Solutions of system ( 4 ) for \(\pmb{(x_{0},y_{0},z_{0})=(1.5,1.2,1.0)}\) , \(\pmb{a_{1}=1.1}\) , \(\pmb{a_{2}=1.1}\) , \(\pmb{a_{3}=1}\) , \(\pmb{b_{1}=0.8}\) , \(\pmb{b_{2}=0.9}\) , \(\pmb{b_{3}=1.1}\) , \(\pmb{c_{1}=0.02}\) , \(\pmb{c_{2}=0.01}\) , \(\pmb{d_{1}=1.2}\) , \(\pmb{d_{2}=1}\) , \(\pmb{\alpha_{1}=0.8}\) , \(\pmb{\beta _{1}=0.5}\) , \(\pmb{\alpha_{2}=0.7}\) , \(\pmb{\beta_{2}=0.5}\) , \(\pmb{h_{1}=0.5}\) , \(\pmb{h_{2}=0.3}\) , \(\pmb{f_{1}=1}\) , \(\pmb{f_{2}=0.5}\) , \(\pmb{g_{1}=0.5}\) , \(\pmb{g_{2}=0.6}\) , \(\pmb{\sigma_{1}=1.8}\) , \(\pmb{\sigma_{2}=1.8}\) , \(\pmb{\sigma_{2}=2}\) .

7 Conclusion

In this paper, we studied a stochastic cooperative predator-prey system with Beddington-DeAngelis functional response. We first demonstrate the existence and uniqueness of global positive solution. We then investigate that the solution is stochastically bounded and permanent. Under some specific conditions, the solution of this stochastic system is globally asymptotically stable, which is useful to estimate the risk of extinction of species in the system. Some interesting topics deserve further study. For example, we can try to study three or more trophic levels, which simulates the actual situation better. Moreover, we can consider the extinction of system (4).

Declarations

Acknowledgements

This work was supported by Shanxi Scholarship Council of China (No. 800104-02090206).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics, Taiyuan University of Technology, 79 Yingze W St., Taiyuan, Shanxi, 030024, China

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