Asymptotic properties of a stochastic nonautonomous competitive system with impulsive perturbations
- Liu Yang^{1} and
- Baodan Tian^{2}Email author
https://doi.org/10.1186/s13662-017-1256-5
© The Author(s) 2017
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
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 [1–10]). 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.
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, 15–20] etc.).
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
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 \((x_{10}, x_{20})^{T}\).
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
Proof
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
Proof
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
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 \(\omega\in S\), \(\lim_{t\rightarrow+\infty}x_{i}(t,\omega)=0\).
Theorem 2.5
Proof
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
- (H1)
there exist positive constants \(m_{i}\) and \(M_{i}\) such that \(m_{i}< \prod_{0<t_{k}<t}(1+h_{ik})<M_{i}\);
- (H2)
\((\sigma^{u}_{i})^{2}<\bar{b}^{l}_{i}\);
Proof
Now we will prove that for \(\forall\xi_{i}>0\), \(\exists \eta_{i}>0\), s.t. \(\liminf_{t\rightarrow+\infty} \mathscr{P}\{x_{i}(t)\geq\eta_{i}\}\geq1-\xi_{i}\).
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
Example 3.2
Example 3.3
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
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