Open Access

Asymptotic properties of a stochastic nonautonomous competitive system with impulsive perturbations

Advances in Difference Equations20172017:201

https://doi.org/10.1186/s13662-017-1256-5

Received: 10 March 2017

Accepted: 27 June 2017

Published: 21 July 2017

Abstract

In this paper, a generalized nonautonomous stochastic competitive system with impulsive perturbations is studied. By the theories of impulsive differential equations and stochastic differential equations, we have established some asymptotic properties of the system, such as the extinction, nonpersistence and persistence in the mean, weak persistence and stochastic permanence and so on. In order to show the correctness and feasibility of the theoretical results, several numerical examples are presented. Finally, the effects of different white noise perturbations and different impulsive perturbations are discussed and illustrated.

Keywords

competitive system impulsive perturbations stochastic perturbations extinction stochastic permanence

1 Introduction

It is well known that there are four kinds of relationships between the species in the population ecological systems, that is, competition, predation, mutualism and parasitism. Among these relationships, competition can always ensure the survival of species and make effective use of resources, maintain the permanence of a ecological system and keep the healthy development of the population. Thus, a competitive system has received great interest by many mathematical and ecological researchers in the last decades (see [110]). As far as the competition is concerned, there are usually two kinds of competitive relationship, i.e. one is the interspecific competition and the other is the intraspecific competition.

The basic two-species competitive system is governed by the following coupled differential equations:
$$ \left \{ \textstyle\begin{array}{l} \frac{dN_{1}(t)}{dt}=N_{1}(t) [r_{1}-a_{11}N_{1}(t)- a_{12}N_{2}(t) ],\\ \frac{dN_{2}(t)}{dt}=N_{2}(t) [r_{2}-a_{21}N_{1}(t)- a_{22}N_{2}(t) ], \end{array}\displaystyle \right . $$
(1)
where \(a_{ii}\) is the intraspecific competition coefficient, while \(a_{ij}\) (\(i\neq j\), \(i,j=1,2\)) is the interspecific competition coefficient.
Based on the classic competitive system, Gopalsamy proposed a series of generalized competitive systems in the monograph [11], and one of the generalized competitive systems is as follows (see p. 168, [11]):
$$ \left \{ \textstyle\begin{array}{l} \frac{dN_{1}(t)}{dt}=N_{1}(t) [r_{1}-a_{11}N_{1}(t-\tau)- \frac {a_{12}N_{2}(t-\tau)}{1+N_{2}(t-\tau)} ],\\ \frac{dN_{2}(t)}{dt}=N_{2}(t) [r_{2}-a_{22}N_{2}(t-\tau)- \frac {a_{21}N_{1}(t-\tau)}{1+N_{1}(t-\tau)} ], \end{array}\displaystyle \right . $$
(2)
which means two species are allowed to cohabit in a common community, and each species inhibits the average growth rate of the other.
Recently, Wang and Liu (see [12]) studied the following nonautonomous competitive system:
$$ \left \{ \textstyle\begin{array}{l} \frac{dx_{1}(t)}{dt}=x_{1}(t) [r_{1}(t)-a_{1}(t)x_{1}(t)- \frac {b_{1}(t)x_{2}(t)}{1+x_{2}(t)} ],\\ \frac{dx_{2}(t)}{dt}=x_{2}(t) [r_{2}(t)-a_{2}(t)x_{2}(t)- \frac {b_{2}(t)x_{1}(t)}{1+x_{1}(t)} ], \end{array}\displaystyle \right . $$
(3)
in which the existence and global asymptotic stability of positive almost periodic solutions is obtained. More references related to these generalized competitive systems can be also seen in [11, 13, 14].

However, most of the above mentioned references focused on the deterministic models, while the growth of the species is often affected by the interferences of the environmental noises in the real world. Thus, it is more reasonable to study ecological models. The dynamical behavior of the ecological system, and whether it will make a change to the existing results, has received wide attention in the recent several years (see references [4, 1520] etc.).

Enlightened by the above mentioned references, we suppose that the random fluctuations of the environment will mainly affect the intrinsic growth rate \(r_{i}(t)\) of the species, and they are estimated by the following form:
$$r_{i}(t) \rightarrow r_{i}(t)+\sigma_{i}(t)\,dB_{i}(t), $$
where \(B_{i}(t)\) is Brownian motion, \(\sigma_{i}(t)\) is a continuous and bounded function on \(t\geq0\) and \(\sigma^{2}_{i}(t)\) represents the intensity of the white noise, \(i=1,2\).
On the other hand, many natural or man-made factors, such as crop-dusting, planting, hunting, harvesting, drought, flooding and so on, will lead to sudden changes to the system. From the viewpoint of mathematical modeling, these sudden changes could be described by impulsive effects or perturbations to the models (see [21, 22]). Thus, if we introduce both impulsive perturbations and stochastic perturbations of white noises on the previous system (3), we can obtain the following system:
$$ \left\{ \textstyle\begin{array}{l} \left. \textstyle\begin{array}{l} dx_{1}(t)=x_{1}(t) [r_{1}(t)-a_{1}(t)x_{1}(t)- \frac {c_{1}(t)x_{2}(t)}{1+x_{2}(t)} ]\,dt+\sigma_{1}(t)x_{1}(t)\,dB_{1}(t)\\ dx_{2}(t)=x_{2}(t) [r_{2}(t)-a_{2}(t)x_{2}(t)- \frac {c_{2}(t)x_{1}(t)}{1+x_{1}(t)} ]\,dt+\sigma_{2}(t)x_{2}(t)\,dB_{2}(t) \end{array}\displaystyle \right\},\quad t\neq t_{k},k\in N, \hspace{-12pt} \\ \left. \textstyle\begin{array}{l} x_{1}(t^{+}_{k})=(1+h_{1k})x_{1}(t_{k})\\ x_{2}(t^{+}_{k})=(1+h_{2k})x_{2}(t_{k}) \end{array}\displaystyle \right\},\quad t=t_{k},k\in N, \end{array}\displaystyle \right. $$
(4)
where \(x_{i}(t)\) is the population density of the ith population, \(r_{i}(t)\) and \(a_{i}(t)\) are the intrinsic growth rate and the intraspecific competing rate, respectively, and \(c_{i}(t)\) represents the interspecific competing rate. \(r_{i}(t)\), \(a_{i}(t)\), \(c_{i}(t)\), \(t\in R^{+}=[0,\infty)\) are positive, continuous and bounded. \(0< t_{1}< t_{2}<\cdots\), \(\lim_{k\rightarrow+\infty}t_{k}=+\infty\). \(h_{ik}>-1\), \(i = 1,2\), \(k\in N\), when \(h_{ik}>0\), the impulsive effects represent planting, while \(h_{ik}<0\) denote harvesting.
Throughout the present paper, we denote
$$f^{l}=\inf_{t\in R^{+}}f(t), \qquad f^{u}=\sup _{t\in R^{+}}f(t) $$
for any positive, bounded function \(f(t)\) defined on \(R^{+}=[0,+\infty)\).

The rest of this paper is organized as follows. In Section 2 we demonstrate and prove the main results of the paper, such as the existence of a unique positive solution of the system, sufficient conditions for the extinction, nonpersistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the system. In Section 3, several numerical examples are presented to support the theoretical results. Moreover, effects on the impulsive and stochastic perturbations are also analyzed and discussed at the end of the paper.

2 Preliminaries

In this section, based on the methods proposed by Yan and Zhao (see [23]), the corresponding stochastic differential equations without impulses are studied, and we will discuss the existence of a positive solution of above system (4) firstly. Further, by the definitions proposed by Liu and Wang (see [18]), we will derive some asymptotic behavior of this system, such as the extinction, nonpersistence and persistence in the mean, weak persistence and stochastic permanence and so on.

Theorem 2.1

For any initial conditions \((x_{10}, x_{20})^{T}\in R_{+}^{2}=\{(x,y)^{T}\in R^{2} |x>0,y>0\}\), system (4) has a unique positive solution \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) on \([0, +\infty)\), and the solution will remain in \(R^{2}_{+}\) almost surely.

Proof

Consider the following stochastic differential equations (SDEs) without impulses:
$$ \left \{ \textstyle\begin{array}{ll} dy_{1}(t)=&y_{1}(t) [r_{1}(t)-a_{1}(t) y_{1}(t) \prod_{0< t_{k}< t}(1+h_{1k})y_{1}(t) - \frac{c_{1}(t) \prod_{0< t_{k}< t}(1+h_{2k})y_{2}(t)}{1+ \prod_{0< t_{k}< t}(1+h_{2k})y_{2}(t)} ]\,dt\\ &{}+\sigma_{1}(t)y_{1}(t)\,dB_{1}(t),\\ dy_{2}(t)=&y_{2}(t) [r_{2}(t)-a_{2}(t) y_{2}(t) \prod_{0< t_{k}< t}(1+h_{2k})y_{2}(t) - \frac{c_{2}(t) \prod_{0< t_{k}< t}(1+h_{1k})y_{1}(t)}{1+ \prod_{0< t_{k}< t} (1+h_{1k} )y_{1}(t)} ]\,dt \\ &{}+\sigma_{2}(t) y_{2}(t)\,dB_{2}(t) \end{array}\displaystyle \right . $$
(5)
with the initial value \((y_{10}, y_{20})^{T}=(x_{10}, x_{20})^{T}\).

It is easy to prove that there is a unique global positive solution \(y(t)=(y_{1}(t),y_{2}(t))^{T}\) of system (5) by the theory of non-impulsive stochastic differential equations (see [18]).

Denote \(x_{i}(t)= \prod_{0< t_{k}< t}(1+h_{1k})y_{i}(t)\) (\(i=1,2\)), then we claim that \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is the solution of system (4) with the initial data \((x_{10}, x_{20})^{T}\).

In fact, since \(x_{1}(t)\) is continuous on \((0,t_{1})\) and each interval \((t_{k},t_{k+1})\subset[0,+\infty)\) and for \(t\neq t_{k}\), \(k\in N\),
$$\begin{aligned} dx_{1}(t)={}&\prod _{0< t_{k}< t}(1+h_{1k})\,dy_{1}(t) \\ ={}&\prod_{0< t_{k}< t}(1+h_{1k})y_{1}(t)\\ &\times \biggl[r_{1}(t)-a_{1}(t)\prod_{0< t_{k}< t}(1+h_{1k})y_{1}(t) -\frac{c_{1}(t)\prod_{0< t_{k}< t}(1+h_{2k})y_{2}(t)}{1+ \prod_{0< t_{k}< t}(1+h_{2k})y_{2}(t)} \biggr]\,dt \\ &+\sigma_{1}(t)\prod_{0< t_{k}< t}(1+h_{1k})y_{1}(t)\,dB_{1}(t) \\ ={}&x_{1}(t) \biggl[r_{1}(t)-a_{1}(t)x_{1}(t)- \frac{c_{1}(t)x_{2}(t)}{1+x_{2}(t)} \biggr]\,dt+\sigma_{1}(t)x_{1}(t)\,dB_{1}(t). \end{aligned} $$
Similarly, we can check that
$$ dx_{2}(t)=x_{2}(t) \biggl[r_{2}(t)-a_{2}(t)x_{1}(t)- \frac {c_{2}(t)x_{1}(t)}{1+x_{1}(t)} \biggr]\,dt +\sigma_{2}(t)x_{2}(t)\,dB_{2}(t). $$
(6)
And for every \(t_{k} \in R^{+}\), \(k\in N\),
$$\begin{aligned}& \begin{aligned}[b] x_{i}\bigl(t^{+}_{k} \bigr)&= \lim_{t\rightarrow t^{+}_{k}}x_{i}(t)= \prod _{0< t_{j}\leq t_{k}}(1+h_{ij})y_{i}(t_{k}) \\&=(1+h_{ik}) \prod_{0< t_{j}< t_{k}}(1+h_{ij})y_{i}(t_{k})=(1+h_{ik})x_{i}(t_{k}). \end{aligned} \end{aligned}$$
(7)
$$\begin{aligned}& \begin{aligned}[b] x_{i}\bigl(t^{-}_{k}\bigr)&= \lim _{t\rightarrow t^{-}_{k}}x_{i}(t)= \lim_{t\rightarrow t^{-}_{k}} \prod _{0< t_{j}< t_{k}}(1+h_{ij})y_{i} \bigl(t^{-}_{k}\bigr) \\ &= \prod_{0< t_{j}< t_{k}}(1+h_{ij})y_{i}(t_{k})=x_{i}(t_{k}). \end{aligned} \end{aligned}$$
(8)

These mean that \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is the unique global positive solution of system (4), so we complete the proof of this theorem. □

In Theorem 2.1, we can see that solutions of system (4) will remain in the first quadrant, but how do they vary in this quadrant? In the following part, we will discuss the sufficient conditions for several cases, such as extinction and weak persistence, nonpersistence and persistence in the mean and so on.

Theorem 2.2

Denote by \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) a solution of system (4), then
$$\limsup_{t\rightarrow+\infty}\frac{\ln x_{i}(t)}{t}\leq\limsup_{t\rightarrow+\infty} \frac{1}{t} \biggl[ \sum_{0< t_{k}< t} \ln(1+h_{ik})+ \int^{t}_{0}b_{i}(s)\,ds \biggr]:=b^{*}_{i}, \quad\textit{a.s.}, $$
where \(b_{i}(t)=r_{i}(t)-0.5\sigma^{2}_{i}(t)\), \(i=1,2\).

Proof

For the non-impulsive system (5), by Itô’s formula, we obtain
$$ \begin{aligned}[b] d\ln y_{i}(t)={}& \frac{dy_{i}(t)}{y_{i}(t)}-\frac{(dy_{i}(t))^{2}}{2y^{2}_{i}(t)} \\={}& \biggl[r_{i}(t)-0.5\sigma^{2}_{i}(t)-a_{i}(t) \prod_{0< t_{k}< t}(1+h_{ik})y_{i}(t)- \frac{c_{i}(t) \prod_{0< t_{k}< t}(1+h_{jk})y_{j}(t)}{1+ \prod_{0< t_{k}< t}(1+h_{jk})y_{j}(t)} \biggr]\,dt \\ &+\sigma_{i}(t)\,dB_{i}(t), \end{aligned} $$
(9)
where \(j=1,2\), \(j\neq i\), and this leads to
$$ d\ln y_{i}(t)\leq \bigl[b_{i}(t)-a_{i}(t)x_{i}(t) \bigr]\,dt+\sigma _{i}(t)\,dB_{i}(t). $$
(10)
Integrating both sides of inequality (10) on the interval \([0,t]\) yields
$$ \ln y_{i}(t)-\ln y_{i}(0) \leq \int^{t}_{0} b_{i}(s)\,ds- \int^{t}_{0} a_{i}(s)x_{i}(s)\,ds+M_{i}(t), $$
(11)
where \(M_{i}(t)= \int^{t}_{0} \sigma_{i}(s)\,dB_{i}(s)\).
Thus,
$$ \begin{aligned}[b] &\sum_{0< t_{k}< t}\ln(1+h_{ik})+\ln y_{i}(t)-\ln y_{i}(0) \\&\quad\leq \sum _{0< t_{k}< t}\ln(1+h_{ik})+ \int^{t}_{0} b_{i}(s)\,ds- \int^{t}_{0} a_{i}(s)x_{i}(s)\,ds+M_{i}(t), \end{aligned}$$
(12)
which yields
$$ \ln x_{i}(t) \leq\ln y_{i0}+ \sum _{0< t_{k}< t}\ln(1+h_{ik})+ \int^{t}_{0} b_{i}(s)\,ds+M_{i}(t) - \int^{t}_{0} a_{i}(s)x_{i}(s)\,ds. $$
(13)
Note that \(M_{i}(t)\) is a local martingale whose quadratic variation is
$$\bigl\langle M_{i}(t),M_{i}(t)\bigr\rangle = \int^{t}_{0} \sigma^{2}_{i}(s)\,ds \leq\bigl(\sigma^{u}_{i}\bigr)^{2}t. $$
By the strong law of large numbers for local martingale (see [24]), we have
$$\lim_{t\rightarrow+\infty}\frac{M_{i}(t)}{t}=0, \quad\text{a.s.} $$
If we multiply \(\frac{1}{t}\) on each side of inequality (13) and take superior limit on both sides of it, we can obtain
$$\limsup_{t\rightarrow+\infty}\frac{\ln x_{i}(t)}{t}\leq\limsup_{t\rightarrow+\infty} \frac{1}{t} \biggl[ \sum_{0< t_{k}< t} \ln(1+h_{ik})+ \int^{t}_{0}b_{i}(s)\,ds \biggr]:=b^{*}_{i}, \quad\text{a.s.} $$
 □

Corollary 2.1

If \(b^{*}_{i}= \limsup_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t}\ln(1+h_{ik})+\int^{t}_{0}b_{i}(s)\,ds ]<0\), then the ith species of system (4) is extinct.

Theorem 2.3

Suppose that \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is a solution of system (4), then
$$\limsup_{t\rightarrow+\infty} \frac{\int^{t}_{0}x_{i}(s)\,ds}{t}\leq \frac{b^{*}_{i}}{a^{l}_{i}}, \quad\textit{a.s.} $$

Proof

By the definition of the limit, for \(\forall\varepsilon_{i}>0\), there exists \(T_{1}>0\) such that
$$\begin{gathered} \frac{\ln y_{i}(0)}{t}=\frac{\ln y_{i0}}{t}\leq\varepsilon_{i}/3,\quad\quad \frac{M_{i}(t)}{t}\leq \varepsilon_{i}/3, \\\frac{1}{t} \biggl[ \sum _{0< t_{k}< t}\ln(1+h_{ik})+ \int^{t}_{0}b_{i}(s)\,ds \biggr]\leq b^{*}_{i}+\varepsilon_{i}/3 \end{gathered}$$
for \(t>T_{1}\).
Combining inequality (13) and the above inequality, we have
$$ \ln x_{i}(t)\leq\bigl( b^{*}_{i}+ \varepsilon_{i}\bigr)t- \int^{t}_{0} a_{i}(s)x_{i}(s)\,ds \leq \lambda_{i}t-a^{l}_{i} \int^{t}_{0} x_{i}(s)\,ds $$
(14)
for \(\forall t>T_{1}\) a.s., where \(\lambda_{i}=b^{*}_{i}+\varepsilon_{i}\).
If we denote \(h_{i}(t)= \int^{t}_{0} x_{i}(s)\,ds\), then \(h_{i}'(t)=x_{i}(t)\). Then it follows from inequality (14) that
$$ e^{a^{l}_{i}h_{i}(t)}\frac{dh_{i}(t)}{dt}\leq e^{\lambda_{i}t}. $$
(15)
Integrating inequality (15) from \(T_{1}\) to t, we have
$$ e^{a^{l}_{i}h_{i}(t)}\leq\frac{a^{l}_{i}}{\lambda_{i}}e^{\lambda _{i}t}+e^{a^{l}_{i}h_{i}(T_{1})}- \frac{a^{l}_{i}}{\lambda_{i}}e^{\lambda_{i}T_{1}}. $$
(16)
Thus,
$$ \int^{t}_{0} x_{i}(s)\,ds \leq \frac{1}{a^{l}_{i}}\ln \biggl[\frac{a^{l}_{i}}{\lambda _{i}}e^{\lambda_{i}t}+e^{a^{l}_{i}h_{i}(T_{1})}- \frac{a^{l}_{i}}{\lambda_{i}}e^{\lambda _{i}T_{1}} \biggr]. $$
(17)
If we multiply \(\frac{1}{t}\) on each side of inequality (17) and take superior limit on both sides of it, we can obtain
$$ \limsup_{t\rightarrow+\infty} \frac{\int ^{t}_{0} x_{i}(s)\,ds}{t} \leq \limsup _{t\rightarrow+\infty} \frac{1}{a^{l}_{i}t}\ln \biggl[\frac{a^{l}_{i}}{\lambda_{i}}e^{\lambda_{i}t}+e^{a^{l}_{i}h_{i}(T_{1})}- \frac {a^{l}_{i}}{\lambda_{i}}e^{\lambda_{i}T_{1}} \biggr]. $$
(18)
By L’Hospital’s rule we have
$$ \limsup_{t\rightarrow+\infty} \frac{\int ^{t}_{0} x_{i}(s)\,ds}{t} \leq \limsup _{t\rightarrow+\infty}\frac{\lambda_{i}}{a^{l}_{i}}=\frac{b^{*}_{i}}{a^{l}_{i}}. $$
(19)

This means that we have completed the proof. □

If \(b^{*}_{i}=0\), it is easy to obtain \(\lim_{t\rightarrow+\infty} \frac{\int^{t}_{0} x_{i}(s)\,ds}{t}=0\), and we can obtain the following Corollary 2.2.

Corollary 2.2

If \(b^{*}_{i}= \limsup_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t}\ln(1+h_{ik})+\int^{t}_{0}b_{i}(s)\,ds ]=0\), then system (4) is nonpersistent in the mean.

Theorem 2.4

If \(b^{*}_{i}= \limsup_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t}\ln(1+h_{ik})+\int^{t}_{0}b_{i}(s)\,ds ]>0\), then at least one of the species in system (4) is weakly persistent.

Proof

It follows from (9) that
$$d\ln y_{i}(t)= \biggl[b_{i}(t)-a_{i}(t)x_{i}(t)- \frac{c_{i}(t)x_{j}(t)}{1+ x_{j}(t)} \biggr]\,dt+\sigma_{i}(t)\,dB_{i}(t),\quad i=1,2,j \neq i. $$
If we integrate on each side of the above equation, we have
$$\ln x_{i}(t)-\ln x_{i0}= \sum_{0< t_{k}< t} \ln(1+h_{ik})+ \int^{t}_{0}b_{i}(s)\,ds- \int ^{t}_{0}a_{i}(s)x_{i}(s)\,ds - \int^{t}_{0}\frac{c_{i}(s)x_{j}(s)}{1+ x_{j}(s)}\,ds+M_{i}(t). $$

Set \(S= \{ \lim_{t\rightarrow+\infty}\sup x_{i}(t)=0 \}\), if the assertion of this theorem is not true, then \(\mathscr{P}(S)>0\), and for \(\omega\in S\), \(\lim_{t\rightarrow+\infty}x_{i}(t,\omega)=0\).

Note that \(\limsup_{t\rightarrow+\infty}\frac{M_{i}(t)}{t} =0\). Further, it follows from the boundedness of \(a_{i}(t)\) and \(c_{i}(t)\) that
$$\begin{gathered} \limsup_{t\rightarrow+\infty} \frac{1}{t} \int^{t}_{0}a_{i}(s)x_{i}(s)\,ds= 0, \\\limsup_{t\rightarrow+\infty} \frac{1}{t} \int^{t}_{0}\frac{c_{i}(s)x_{j}(s)}{1+ x_{j}(s)}\,ds= 0, \\\limsup_{t\rightarrow+\infty} \frac{1}{t} \bigl[\ln x_{i}(t,w)- \ln x_{i0} \bigr]\leq0. \end{gathered}$$
Thus,
$$0\geq \limsup_{t\rightarrow+\infty} \frac{1}{t} \bigl[\ln x_{i}(t,w)-\ln x_{i0} \bigr]=b^{*}_{i}>0, $$
which is a contradiction, and this completes the proof of this theorem. □

Theorem 2.5

Denote \(b_{i*}= \liminf_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t}\ln (1+h_{ik})+ \int^{t}_{0} \bar{b_{i}}(s)\,ds ]\), then the solution of system (4) satisfies
$$\liminf_{t\rightarrow+\infty} \frac{1}{t} \int^{t}_{0}x_{i}(s)\,ds \geq \frac{b_{i*}}{a^{u}_{i}},\quad \textit {a.s.}, $$
where \(\bar{b_{i}}(t)=r_{i}(t)-c_{i}(t)-0.5\sigma^{2}_{i}(t)\), \(i=1,2\).

Proof

It follows from (9) again that
$$ d\ln y_{i}(t)\geq \bigl[r_{i}(t)-c_{i}(t)-0.5 \sigma^{2}_{i}(t)-a_{i}(t)x_{i}(t) \bigr]\,dt+\sigma _{i}(t)\,dB_{i}(t). $$
(20)
Integrating both sides of inequality (20) from 0 to t yields
$$ \ln y_{i}(t)-\ln y_{i}(0)\geq \int^{t}_{0} \bar {b_{i}}(s)\,ds - \int^{t}_{0}a _{i}(s)x_{i}(s)\,ds+M_{i}(t). $$
(21)
Then
$$ \ln x_{i}(t) \geq\ln x_{i0}+ \sum _{0< t_{k}< t}\ln(1+h_{ik})+ \int^{t}_{0} \bar{b_{i}}(s)\,ds+M_{i}(t)-a^{u}_{i} \int^{t}_{0} x_{i}(s)\,ds. $$
(22)
Note the definition of \(M_{i}(t)\) and \(b_{i*}\), according to the property of the limit again, for any \(\epsilon_{i}>0\), \(i=1,2\), there exists \(T_{2}>0\) such that, for \(t>T_{2}\),
$$\begin{gathered} \frac{\ln x_{i0}}{t}\geq-\epsilon_{i}/3,\qquad \frac{M_{i}(t)}{t}\geq-\epsilon _{i}/3, \\\frac{1}{t} \biggl[ \sum_{0< t_{k}< t} \ln(1+h_{ik})+ \int^{t}_{0}\bar {b_{i}}(s)\,ds \biggr]\geq \bar{b_{i}}-\epsilon_{i}/3. \end{gathered}$$
Substituting above inequalities into (22) yields
$$ \ln x_{i}(t) \geq\mu_{i}t-a^{u}_{i} \int^{t}_{0} x_{i}(s)\,ds $$
(23)
for all \(t>T_{1}\) almost surely, where \(\mu_{i} =b_{i*}-\epsilon_{i}\).
Note that \(h_{i}(t)= \int^{t}_{0} x_{i}(s)\,ds\) and \(\frac {dh_{i}(t)}{dt}=x_{i}(t)\), then from (23) we have
$$ e^{a^{u}_{i}h_{i}(t)}\frac{dh_{i}(t)}{dt}\geq e^{\mu_{i}t}. $$
(24)
If we integrate the above inequality on the interval \([T_{2},t]\), then we have
$$ e^{a^{u}_{i}h_{i}(t)}\geq\frac{a^{u}_{i}}{\mu _{i}}e^{\mu_{i}t}+e^{a^{u}_{i}h_{i}(T_{2})}- \frac{a^{u}_{i}}{\mu_{i}}e^{\mu_{i}T_{2}}. $$
(25)
Thus,
$$ \int^{t}_{0} x_{i}(s)\,ds \geq \frac{1}{a^{u}_{i}}\ln \biggl[\frac{a^{u}_{i}}{\mu_{i}}e^{\mu_{i}t}+e^{a^{u}_{i}h_{i}(T_{2})}- \frac{a^{u}_{i}}{\mu _{i}}e^{\mu_{i}T_{2}} \biggr]. $$
(26)
Taking inferior limit on (26) yields
$$ \liminf_{t\rightarrow+\infty} \frac{\int ^{t}_{0} x_{i}(s)\,ds}{t} \geq \liminf _{t\rightarrow+\infty} \frac{1}{a^{u}_{i}t}\ln \biggl[\frac{a^{u}_{i}}{\mu_{i}}e^{\mu_{i}t} +e^{a^{u}_{i}h_{i}(T_{2})}-\frac{a^{u}_{i}}{\mu_{i}}e^{\mu_{i}T_{2}} \biggr]. $$
(27)
By L’Hospital’s rule again, we have
$$ \liminf_{t\rightarrow+\infty} \frac{\int ^{t}_{0} x_{i}(s)\,ds}{t} \geq \liminf _{t\rightarrow+\infty} \frac{\mu_{i}}{a^{u}_{i}}=\frac{b_{i*}}{a^{u}_{i}}. $$
(28)

Thus, we complete the proof of the above theorem. □

Further, if \(b_{i*}>0\), we have the following Corollary 2.3.

Corollary 2.3

If \(b_{i*}= \liminf_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t} \ln(1+h_{ik})+ \int^{t}_{0}\bar{b_{i}}(s)\,ds ]>0\), then system (4) is persistent in the mean a.s.

Theorem 2.6

If system (4) satisfies the following two conditions:
  1. (H1)

    there exist positive constants \(m_{i}\) and \(M_{i}\) such that \(m_{i}< \prod_{0<t_{k}<t}(1+h_{ik})<M_{i}\);

     
  2. (H2)

    \((\sigma^{u}_{i})^{2}<\bar{b}^{l}_{i}\);

     
then system (4) is stochastically permanent, where \(\bar{b}_{i}(t)=r_{i}(t)-c_{i}(t)-0.5\sigma^{2}_{i}(t)\), \(i=1,2\).

Proof

Applying Itô’s integration by parts formula, we can derive that
$$\begin{aligned} d\bigl(e^{t}y^{2}_{i}(t) \bigr)={}&e^{t}y^{2}_{i}(t)\,dt+2e^{t}y_{i}(t)dy_{i}(t)+e^{t} \bigl(dy_{i}(t)\bigr)^{2} \\ ={}&e^{t}y^{2}_{i}(t) \biggl[1+2 \biggl(r_{i}(t)-a_{i}(t)\prod_{0< t_{k}< t}(1+h_{ik})y_{i}(t)- \frac{c_{i}(t) \prod_{0< t_{k}< t}(1+h_{jk})y_{j}(t)}{1+ \prod_{0< t_{k}< t}(1+h_{jk}) y_{j}(t)} \biggr)\\ &+\sigma^{2}_{i}(t) \biggr]\,dt \\ &+2e^{t}y^{2}_{i}(t)\sigma_{i}(t)\,dB_{i}(t) \\ \leq{}& e^{t}y^{2}_{i}(t) \biggl[1+2r_{i}(t)+ \sigma^{2}_{i}(t)-2a_{i}(t)\prod _{0< t_{k}< t}(1+h_{ik})y_{i}(t) \biggr]\,dt\\ &+2e^{t}y^{2}_{i}(t)\sigma_{i}(t)\,dB_{i}(t). \end{aligned} $$
Integrating the above inequality on the interval \([0, t]\), we have
$$ \begin{aligned}[b]e^{t}y^{2}_{i}(t)-y^{2}_{i}(0) \leq{}& \int^{t}_{0} e^{s}y^{2}_{i}(s) \bigl[1+2r_{i}(s)+\sigma^{2}_{i}(s)-2m_{i}a_{i}(s)y_{i}(s) \bigr]\,dt\\ &+2 \int^{t}_{0} e^{s}y^{2}_{i}(s) \sigma_{i}(s)\,dB_{i}(s). \end{aligned}$$
(29)
Taking expectations on both sides of (29) and making some estimations lead to
$$\begin{aligned} E\bigl[e^{t}y^{2}_{i}(t) \bigr] \leq{}& y^{2}_{i0}+ E \biggl[ \int^{t}_{0} e^{s}y^{2}_{i}(s) \bigl[1+2r_{i}(s)+\sigma^{2}_{i}(s) -2m_{i}a_{i}(s)y_{i}(s) \bigr]\,ds \biggr] \\ \leq{}& y^{2}_{i0}+E \biggl[ \int^{t}_{0} e^{s}y^{2}_{i}(s) \bigl[1+2r^{u}_{i}+\bigl(\sigma ^{u}_{i} \bigr)^{2}-2m_{i}a^{l}_{i}y_{i}(s) \bigr]\,ds \biggr]. \end{aligned} $$
Thus, by the maximum principle, we have
$$ E\bigl[e^{t}y^{2}_{i}(t)\bigr] \leq y^{2}_{i0}+L_{i}E \biggl[ \int^{t}_{0} e^{s}ds \biggr]=y^{2}_{i0}+L_{i} \bigl(e^{t}-1\bigr), $$
(30)
where \(L_{i}= \frac{(1+2r^{u}_{i}+(\sigma^{u}_{i})^{2})^{3}}{27(m_{i}a^{l}_{i})^{2}}\), \(i=1,2\).
Thus,
$$ \limsup_{t\rightarrow+\infty} E\bigl[y^{2}_{i}(t) \bigr] \leq \limsup_{t\rightarrow+\infty} \frac{y^{2}_{i0}+L_{i}(e^{t}-1)}{t}=L_{i}, $$
(31)
which yields
$$ \limsup_{t\rightarrow+\infty} E\bigl[x^{2}_{i}(t) \bigr]= \limsup_{t\rightarrow+\infty} E \biggl[\prod _{0< t_{k}< t}(1+h_{ik})^{2}y^{2}_{i}(t) \biggr]\leq L_{i}M^{2}_{i}. $$
(32)
Then, for any \(\xi_{i}>0\), set \(\beta_{i}=M_{i}\sqrt{\frac{L_{i}}{\xi_{i}}}\), and by Chebyshev’s inequality, we have
$$ \limsup_{t\rightarrow+\infty} \mathscr{P}\bigl\{ x_{i}(t)>\beta_{i}\bigr\} = \limsup_{t\rightarrow+\infty} \mathscr {P}\bigl\{ x^{2}_{i}(t)>\beta^{2}_{i} \bigr\} \leq \limsup_{t\rightarrow+\infty}\frac{E [x^{2}_{i}(t) ]}{\beta^{2}_{i}}\leq \frac{L_{i}M^{2}_{i}}{\beta^{2}_{i}}=\xi_{i}. $$
(33)
In other words,
$$ \liminf_{t\rightarrow+\infty} \mathscr{P}\bigl\{ x_{i}(t)\leq\beta_{i}\bigr\} \geq1-\xi_{i}. $$
(34)

Now we will prove that for \(\forall\xi_{i}>0\), \(\exists \eta_{i}>0\), s.t. \(\liminf_{t\rightarrow+\infty} \mathscr{P}\{x_{i}(t)\geq\eta_{i}\}\geq1-\xi_{i}\).

In fact, it follows from condition (H2) that \(b^{l}_{i}>(\sigma^{u}_{i})^{2}\). If we define
$$V_{1}(y_{i})=y^{-1}_{i},\qquad V_{2}(y_{i})=e^{kt}\bigl(1+V_{1}(y_{i}) \bigr)^{2}, \quad i=1,2, $$
where \(0< k<2[b^{l}_{i}-(\sigma^{u}_{i})^{2}]\).
Then it follows from Itô’s formula again that
$$\begin{aligned} dV_{1}(y_{i})={}&{-}y^{-2}_{i}dy_{i}(t)+y^{-3}_{i} \bigl(dy_{i}(t)\bigr)^{2} \\ ={}&{-}V_{1}(y_{i}) \biggl[r_{i}(t)-a_{i}(t) \prod_{0< t_{k}< t}(1+h_{ik})y_{i}(t) - \frac{c_{i}(t)\prod_{0< t_{k}< t}(1+h_{jk})y_{j}(t)}{1+\prod_{0< t_{k}< t}(1+h_{jk})y_{j}(t)} \biggr]\,dt \\ &-\sigma_{i}(t)V_{1}(y_{i})\,dB_{i}(t)+ \sigma^{2}_{i}(t)V_{1}(y_{i})\,dt, \end{aligned} $$
which yields
$$ dV_{1}(y_{i})\leq-V_{1}(y_{i}) \bigl[r_{i}(t)-c_{i}(t)-\sigma^{2}_{i}(t)-M_{i}a_{i}(t)y_{i}(t) \bigr]\,dt -\sigma_{i}(t)V_{1}(y_{i})\,dB_{i}(t). $$
(35)
Similarity, if we apply Itô’s integration by parts formula on \(V_{2}(t)\), then
$$\begin{aligned} d\bigl(V_{2}(y_{i}) \bigr)={}&ke^{kt}\bigl(1+V_{1}(y_{i}) \bigr)^{2}\,dt+2e^{kt}\bigl(1+V_{1}(y_{i}) \bigr)dV_{1}(y_{i})+e^{kt}\bigl(dV_{1}(y_{i}) \bigr)^{2} \\ \leq{}&ke^{kt}\bigl(1+V_{1}(y_{i}) \bigr)^{2}\,dt\\ &-2e^{kt}V_{1}(y_{i}) \bigl(1+V_{1}(y_{i})\bigr) \bigl[r_{i}(t)-c_{i}(t)- \sigma^{2}_{i}(t)-M_{i}a_{i}(t)y_{i}(t) \bigr]\,dt \\ &-2e^{kt}\sigma_{i}(t) \bigl(1+V_{1}(y_{i}) \bigr)V_{1}(y_{i})\,dB_{i}(t)+e^{kt} \sigma ^{2}_{i}(t)V^{2}_{1}(y_{i})\,dt \\ \leq{}& e^{kt} \bigl[-2\bigl(b_{i}(t)-\sigma ^{2}_{i}(t)-0.5k\bigr)V^{2}_{1}(y_{i})+2 \bigl(M_{i}a_{i}(t)-b_{i}(t)+0.5 \sigma^{2}_{i}(t)+k\bigr)V_{1}(y_{i}) \\ &+\bigl(2M^{u}_{i}a^{u}_{i}+k\bigr) \bigr]\,dt-2e^{kt}\sigma_{i}(t) \bigl(1+V_{1}(y_{i}) \bigr)V_{1}(y_{i})\,dB_{i}(t), \end{aligned} $$
which yields
$$ d\bigl(V_{2}(y_{i})\bigr)\leq e^{kt}J(y_{i})\,dt-2e^{kt}\sigma _{i}(t)V_{1}(y_{i}) \bigl(1+V_{1}(y_{i}) \bigr)\,dB_{i}(t), $$
(36)
where
$$J(y_{i})=-2\bigl(b^{l}_{i}-\bigl(\sigma ^{u}_{i}\bigr)^{2}-0.5k\bigr)V^{2}_{1}(y_{i})+2 \bigl(M_{i}a^{u}_{i}-b^{l}_{i}+0.5 \bigl(\sigma ^{u}_{i}\bigr)^{2}+k \bigr)V_{1}(y_{i})+\bigl(2M^{u}_{i}a^{u}_{i}+k \bigr). $$
Since \(0< k<2[b^{l}_{i}-(\sigma^{u}_{i})^{2}]\), then \(J(y_{i})\) is upper bounded, and we denote \(J_{i}:= \sup_{y_{i}\in R^{+}}J(y_{i})<+\infty\), then
$$ d\bigl(V_{2}(y_{i})\bigr)\leq J_{i}e^{kt}\,dt-2e^{kt}\sigma _{i}(t)V_{1}(y_{i}) \bigl(1+V_{1}(y_{i})\bigr)\,dB_{i}(t). $$
(37)
Integrating (37) from 0 to t, then multiplying \(e^{-kt}\) and taking expectations on each side of it, we can obtain
$$ E\bigl[\bigl(1+V_{1}(y_{i}) \bigr)^{2}\bigr]\leq V_{2}(y_{i0})e^{-kt}+ \frac{J_{i}}{k}\bigl(1-e^{-kt}\bigr), $$
(38)
which yields
$$ \limsup_{t\rightarrow\infty} E\bigl[y^{-2}_{i}(t) \bigr]\leq \limsup_{t\rightarrow\infty} E\bigl[\bigl(1+V_{1}(y_{i}) \bigr)^{2}\bigr]\leq \limsup_{t\rightarrow\infty} \biggl[ \frac{V_{2}(y_{i0})}{e^{kt}}+\frac{J_{i}(1-e^{-kt})}{k} \biggr]=\frac{J_{i}}{k}. $$
(39)
Thus,
$$ \limsup_{t\rightarrow+\infty} E\bigl[x^{-2}_{i}(t) \bigr]= \limsup_{t\rightarrow+\infty} E \biggl[\prod _{0< t_{k}< t}(1+h_{ik})^{-2}y^{-2}_{i}(t) \biggr]\leq \frac{J_{i}}{km^{2}_{i}}. $$
(40)
Then, for any \(\xi_{i}>0\), set \(\eta_{i}=m_{i} \sqrt{\frac{k\xi_{i}}{J_{i}}}\), applying Chebyshev’s inequality again, we have
$$ \limsup_{t\rightarrow+\infty}\mathscr{P}\bigl\{ x_{i}(t)< \eta_{i}\bigr\} = \limsup_{t\rightarrow+\infty} \mathscr {P}\bigl\{ x^{-2}_{i}(t)>\eta^{-2}_{i} \bigr\} \leq \lim_{t\rightarrow+\infty}\frac{E [x^{-2}_{i}(t) ]}{\eta^{-2}_{i}}\leq \xi_{i}. $$
(41)
In other words,
$$ \liminf_{t\rightarrow+\infty} \mathscr{P}\bigl\{ x_{i}(t)\geq\eta_{i}\bigr\} \geq1-\xi_{i}. $$
(42)

From (34) and (42), the stochastic permanence of system (4) is obtained. This completes the proof of this theorem. □

Remark

In fact, ‘persistence in the mean’ in this section is not a good definition of persistence for stochastic population models. Some authors have introduced some more appropriate definitions of permanence for stochastic population models. For example, stochastic persistence in probability (see [25, 26]) or a new definition of stochastic permanence (see [27]).

3 Numerical simulations and discussions

In this paper, a stochastic nonautonomous competitive system with impulsive perturbations is proposed and studied. We establish sufficient conditions for the extinction, nonpersistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the system. Furthermore, the critical value between extinction, nonpersistence and weak persistence of at least one species in the system is obtained.

In order to verify the correctness and the feasibility of the derived conditions in the theoretical results, we will give a series of numerical examples to illustrate them by using the extension of Milstein’s method (see [28]) in this section. Furthermore, we will show the effects of different white noises or impulsive perturbations to the dynamics of the system, and by the figures of corresponding simulations, one can observe the population fluctuation of the species in the competitive system more intuitively.

In the following, we choose the same initial value \((x_{10},x_{20})=(0.5,0.2)\) and parameters \(a_{1}(t)=0.1+0.01\sin(t)\), \(a_{2}(t)=0.1+0.01\cos(t)\), \(c_{1}(t)=0.22+0.02\sin(t)\), \(c_{2}(t)=0.22+0.02\cos(t)\), \(\Delta t=0.01\).

Example 3.1

For system (4), we set the following choice of parameters:
$$\begin{gathered} r_{1}(t)=0.2+0.01\sin(t),\qquad r_{2}(t)=0.1+0.01\cos(t); \\ \sigma^{2}_{1}(t)=0.5+0.04\sin(t),\qquad \sigma^{2}_{2}(t)=0.4+0.04\cos(t); \\ h_{1k}=h_{2k}=e^{\frac{(-1)^{k}}{k^{2}}}-1,\quad t_{k}=k, k=1,2,\ldots. \end{gathered} $$
By a simple computation, we have \({\bar{b}}_{1}(t)=-0.05-0.01\sin(t)\), \({\bar{b}}_{2}(t)=-0.1-0.01\cos(t)\), then \(b^{*}_{1}=-0.04<0\), \(b^{*}_{2}=-0.09<0\), which satisfies the condition of Corollary 2.1, then both of the species are extinct (see Figure 1).
Figure 1

The time series of system ( 4 ) with \(\pmb{\sigma ^{2}_{1}(t)=0.5+0.04\sin(t)}\) , \(\pmb{\sigma^{2}_{2}(t)=0.4+0.04\cos(t)}\) , \(\pmb{h_{1k}=h_{2k}=-1+e^{\frac{(-1)^{k}}{k^{2}}}}\) , \(\pmb{\Delta t=0.01}\) , \(\pmb{k=1,2,\ldots }\) .

If we decrease the white noises of the species \(x_{1}\) and let \(\sigma ^{2}_{1}(t)=0.2+0.04\sin(t)\), while the values of other parameters are the same as above, then \(b^{*}_{1}=0.11>0\), at the moment the species \(x_{2}\) will still be extinct, while the species \(x_{1}\) can survive (see Figure 2).
Figure 2

The time series of system ( 4 ) with \(\pmb{\sigma ^{2}_{1}(t)=0.2+0.04\sin(t)}\) , \(\pmb{\sigma^{2}_{2}(t)=0.4+0.04\cos(t)}\) , \(\pmb{h_{1k}=h_{2k}=-1+e^{\frac{(-1)^{k}}{k^{2}}}}\) , \(\pmb{\Delta t=0.01}\) , \(\pmb{k=1,2,\ldots }\) .

Example 3.2

For system (4), if we set the following choice of parameters:
$$\begin{gathered} r_{1}(t)=0.16+0.02\sin(t),\qquad r_{2}(t)=0.24+0.03\cos(t); \\ \sigma^{2}_{1}(t)=0.32+0.04\sin(t),\qquad \sigma^{2}_{2}(t)=0.48+0.06\cos(t); \\ h_{1k}=h_{2k}=e^{\frac{(-1)^{k}}{k^{2}}}-1,\quad t_{k}=k, k=1,2,\ldots. \end{gathered}$$
It is easy to calculate that \({\bar{b}}_{1}(t)={\bar{b}}_{2}(t)=0\), then \(b^{*}_{1}=b^{*}_{2}=0\), which satisfies the condition of Corollary 2.2, then the species of system (4) is nonpersistent in the mean (see Figure 3).
Figure 3

The time series of system ( 4 ) with \(\pmb{\sigma ^{2}_{1}(t)=0.32+0.04\sin(t)}\) , \(\pmb{\sigma^{2}_{2}(t)=0.48+0.06\cos(t)}\) , \(\pmb{h_{1k}=h_{2k}=-1+e^{\frac{(-1)^{k}}{k^{2}}}}\) , \(\pmb{\Delta t=0.01}\) , \(\pmb{k=1,2,\ldots }\) .

If we increase the intrinsic growth rate of the species as \(r_{1}(t)=0.18+0.02\sin(t)\), \(r_{2}(t)=0.28+0.03\cos(t)\), while the values of other parameters are the same as above, then one can calculate \(b^{*}_{1}=0.02 >0\), \(b^{*}_{2}=0.04>0\) at this time, which satisfies the condition of Theorem 2.3. According to the theorem, at least one of the two species will be weakly persistent in the mean. On the other hand, from the stochastic simulation of this case (see Figure 4), we can observe that \(x_{1}\) is weakly persistent.
Figure 4

The time series of system ( 4 ) with \(\pmb{\sigma ^{2}_{1}(t)=0.32+0.04\sin(t)}\) , \(\pmb{\sigma^{2}_{2}(t)=0.48+0.06\cos(t)}\) , \(\pmb{h_{1k}=h_{2k}=-1+e^{\frac{(-1)^{k}}{k^{2}}}}\) , \(\pmb{\Delta t=0.01}\) , \(\pmb{k=1,2,\ldots }\) .

If we go on increasing the intrinsic growth rate of the species as \(r_{1}(t)=0.40+0.02\sin(t)\), \(r_{2}(t)=0.48+0.03\cos(t)\), while the values of other parameters are the same as above, then one can calculate that \(b_{1*}= b_{2*}=0.01>0\) at this time, which satisfies the condition of Corollary 2.3, then system (4) should be persistent in the mean almost surely by this corollary. And this is also proved by our stochastic numerical simulation (see Figure 5).
Figure 5

The time series of system ( 4 ) with \(\pmb{\sigma ^{2}_{1}(t)=0.32+0.04\sin(t)}\) , \(\pmb{\sigma^{2}_{2}(t)=0.48+0.06\cos(t)}\) , \(\pmb{h_{1k}=h_{2k}=-1+e^{\frac{(-1)^{k}}{k^{2}}}}\) , \(\pmb{\Delta t=0.01}\) , \(\pmb{k=1,2,\ldots }\) .

Example 3.3

For system (4), we set another series of parameters as follows:
$$\begin{gathered} r_{1}(t)=0.80+0.01\sin(t),\qquad r_{2}(t)=0.70+0.05\cos(t); \\ \sigma^{2}_{1}(t)=0.12+0.02\sin(t),\qquad \sigma^{2}_{2}(t)=0.10+0.02\cos(t); \\ h_{1k}=h_{2k}=e^{\frac{(-1)^{k}}{k^{2}}}-1,\quad t_{k}=k, k=1,2, \ldots . \end{gathered}$$
Further, by a simple computation, we can verify that \(e^{-1}< \prod_{0<t_{k}<t}(1+h_{ik})<e^{-0.75}\) and \(b_{1}(t)=r_{1}(t)-c_{1}(t)-0.5\sigma^{2}_{1}(t)=0.52-0.02\sin t\), \(b_{2}(t)=r_{2}(t)-c_{2}(t)-0.5\sigma^{2}_{2}(t)=0.64-0.03\cos t\), which leads to \((\sigma^{u}_{1})^{2}=0.14< b^{l}_{1}=0.5\), \((\sigma^{u}_{2})^{2}=0.12< b^{l}_{2}=0.61\). That is to say, conditions (H1) and (H2) in Theorem 2.5 hold, then the system is stochastic permanent, and this is also proved by our stochastic numerical simulation (see Figure 6).
Figure 6

The time series of system ( 4 ) with \(\pmb{\sigma ^{2}_{1}(t)=0.12+0.02\sin(t)}\) , \(\pmb{\sigma^{2}_{2}(t)=0.1+0.02\cos(t)}\) , \(\pmb{h_{1k}=h_{2k}=-1+e^{\frac{(-1)^{k}}{k^{2}}}}\) , \(\pmb{\Delta t=0.01}\) , \(\pmb{k=1,2,\ldots }\) .

However, if we suppose that the white noises are increased as \(\sigma ^{2}_{1}(t)=2.12+0.02\sin(t)\), \(\sigma^{2}_{2}(t)=1.92+0.02\cos(t)\), then it is obvious that both species of system (4) will be extinct rapidly by our stochastic simulation (see Figure 7). This means that the species in the ecological system might become extinct as the white noises increase.
Figure 7

The time series of system ( 4 ) with \(\pmb{\sigma ^{2}_{1}(t)=2.12+0.02\sin(t)}\) , \(\pmb{\sigma^{2}_{2}(t)=1.92+0.02\cos(t)}\) , \(\pmb{h_{1k}=h_{2k}=-1+e^{\frac{(-1)^{k}}{k^{2}}}}\) , \(\pmb{\Delta t=0.01}\) , \(\pmb{k=1,2,\ldots }\) .

On the other hand, in order to see how the impulsive perturbations will affect the system, we choose the same parameters as those in Example 3.2, but only change the intensity of the impulses to \(h_{1k}=h_{2k}=e^{-0.2}-1\), \(t_{k}=k=1,2,\ldots \) , then the condition \(b^{*}_{i}>0\) does not hold any more. At the moment, we can see that both species become extinct instead of being persistent in the mean by the stochastic simulation (see Figure 8).
Figure 8

The time series of system ( 4 ) with \(\pmb{\sigma ^{2}_{1}(t)=0.32+0.04\sin(t)}\) , \(\pmb{\sigma^{2}_{2}(t)=0.48+0.06\cos(t)}\) , \(\pmb{h_{1k}=h_{2k}=e^{-0.2}-1}\) , \(\pmb{\Delta t=0.01}\) , \(\pmb{k=1,2,\ldots }\) .

3.1 Conclusions

From the above numerical simulations and discussions, we can conclude that both heavy intensity of environmental noises and large impulsive perturbations to the ecological system will lead to the extinction of the species. And this shows that the departments of environment protection should control the environmental noises and impulsive disturbance reasonably to protect the ecological balance.

In addition, as far as the study of population models is concerned, stability of the positive equilibrium state is one of the most interesting topics. For example, models with noise, some of the stochastic models do not keep the positive equilibrium state of the corresponding deterministic systems. And many authors have studied stability in distribution of several stochastic population models in recent years (see [29, 30] etc.). Thus, we could try to consider these aspects and get much more interesting results in the future investigation.

Declarations

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11372294), Applied Basic Research Program of Sichuan Provincial Science and Technology Department in 2017, Scientific Research Fund of Sichuan Provincial Education Department (11ZB192, 14ZB0115) and the Doctorial Research Fund of Southwest University of Science and Technology (15zx7138).

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)
School of Mathematics and Computational Science, Hunan University of Arts and Science
(2)
School of Science, Southwest University of Science and Technology

References

  1. Ahmad, S, Stamova, IM: Asymptotic stability of competitive systems with delays and impulsive perturbations. J. Math. Anal. Appl. 334, 686-700 (2007) MathSciNetView ArticleMATHGoogle Scholar
  2. Hou, ZY: Permanence and extinction in competitive Lotka-Volterra systems with delays. Nonlinear Anal., Real World Appl. 12, 2130-2141 (2011) MathSciNetView ArticleMATHGoogle Scholar
  3. Lisena, B: Competitive exclusion in a periodic Lotka-Volterra system. Appl. Math. Comput. 177, 761-768 (2006) MathSciNetMATHGoogle Scholar
  4. Liu, M, Wang, K: Asymptotic behavior of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations. Math. Comput. Model. 57, 909-925 (2013) MathSciNetView ArticleMATHGoogle Scholar
  5. Qiu, JL, Cao, JD: Exponential stability of a competitive Lotka-Volterra system with delays. Appl. Math. Comput. 201, 819-829 (2008) MathSciNetMATHGoogle Scholar
  6. Samanta, GP: A two-species competitive system under the influence of toxic substances. Appl. Math. Comput. 216, 291-299 (2010) MathSciNetMATHGoogle Scholar
  7. Song, XY, Chen, LS: Global asymptotic stability of a two species competitive system with stage structure and harvesting. Commun. Nonlinear Sci. Numer. Simul. 6, 81-87 (2001) MathSciNetView ArticleMATHGoogle Scholar
  8. Wei, FY, Lin, YR, et al.: Periodic solution and global stability for a nonautonomous competitive Lotka-Volterra diffusion system. Appl. Math. Comput. 216, 3097-3104 (2010) MathSciNetMATHGoogle Scholar
  9. Zhao, JD, Zhang, ZC, Ju, J: Necessary and sufficient conditions for permanence and extinction in a three dimensional competitive Lotka-Volterra system. Appl. Math. Comput. 230, 587-596 (2014) MathSciNetGoogle Scholar
  10. Zhang, L, Teng, ZD: Permanence for a class of periodic time-dependent competitive system with delays and dispersal in a patchy-environment. Appl. Math. Comput. 188, 855-864 (2007) MathSciNetMATHGoogle Scholar
  11. Gopalsamy, K: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and Its Applications. Kluwer Academic, Dordrecht (1992) View ArticleMATHGoogle Scholar
  12. Wang, QL, Liu, ZJ: Existence and global asymptotic stability of positive almost periodic solutions of a two-species competitive system. Int. J. Biomath. 7, Article ID 1450040 (2014) MathSciNetView ArticleMATHGoogle Scholar
  13. Chen, YP, Tan, RH, Liu, ZJ: Global attractivity in a periodic delayed competitive system. Appl. Math. Sci. 34, 1675-1684 (2007) MathSciNetMATHGoogle Scholar
  14. Qin, WJ, Liu, ZJ: Permanence and positive periodic solutions of a discrete delay competitive system. Discrete Dyn. Nat. Soc. 2010, Article ID 381750 (2010) MathSciNetView ArticleMATHGoogle Scholar
  15. Liu, M, Wang, K: Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response. Commun. Nonlinear Sci. Numer. Simul. 16, 1114-1121 (2011) MathSciNetView ArticleMATHGoogle Scholar
  16. Liu, M, Wang, K: Persistence and extinction in stochastic non-autonomous logistic systems. J. Math. Anal. Appl. 375, 443-457 (2011) MathSciNetView ArticleMATHGoogle Scholar
  17. Liu, M, Wang, K: Persistence and extinction of a single-species population system in a polluted environment with random perturbations and impulsive toxicant input. Chaos Solitons Fractals 45, 1541-1550 (2012) View ArticleGoogle Scholar
  18. Liu, M, Wang, K: On a stochastic logistic equation with impulsive perturbations. Comput. Math. Appl. 63, 871-886 (2012) MathSciNetView ArticleMATHGoogle Scholar
  19. Wu, RH, Zou, XL, Wang, K: Asymptotic properties of a stochastic Lotka-Volterra cooperative system with impulsive perturbations. Nonlinear Dyn. 77, 807-817 (2014) MathSciNetView ArticleMATHGoogle Scholar
  20. Wu, RH, Zou, XL, Wang, K: Asymptotic behavior of a stochastic non-autonomous predator-prey model with impulsive perturbations. Commun. Nonlinear Sci. Numer. Simul. 20, 965-974 (2015) MathSciNetView ArticleMATHGoogle Scholar
  21. Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow (1993) MATHGoogle Scholar
  22. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) View ArticleMATHGoogle Scholar
  23. Yan, JR, Zhao, AM: Oscillation and stability of linear impulsive delay differential equations. J. Math. Anal. Appl. 227, 187-194 (1998) MathSciNetView ArticleMATHGoogle Scholar
  24. Mao, XR: Stochastic Differential Equations and Applications, 2nd edn. Horwood, Chichester (2007) MATHGoogle Scholar
  25. Schreiber, SJ, Benam, M, Atchadé, KAS: Persistence in fluctuating environments. J. Math. Biol. 62, 655-683 (2011) MathSciNetView ArticleMATHGoogle Scholar
  26. Liu, M, Bai, CZ: Analysis of a stochastic tri-trophic food-chain model with harvesting. J. Math. Biol. 73, 597-625 (2016) MathSciNetView ArticleMATHGoogle Scholar
  27. Zhu, Y, Liu, M: Permanence and extinction in a stochastic service-resource mutualism model. Appl. Math. Lett. 69, 1-7 (2017) MathSciNetView ArticleGoogle Scholar
  28. Higham, DJ: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525-546 (2001) MathSciNetView ArticleMATHGoogle Scholar
  29. Liu, M, Bai, CZ: Dynamics of a stochastic one-prey two-predator model with Lévy jumps. Appl. Math. Comput. 284, 308-321 (2016) MathSciNetGoogle Scholar
  30. Liu, M, Bai, CZ: Optimal harvesting of a stochastic delay competitive model. Discrete Contin. Dyn. Syst., Ser. B 22, 1493-1508 (2017) MathSciNetView ArticleMATHGoogle Scholar

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