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Permanence and extinction in nonautonomous logistic system with random perturbation and feedback control
 Hongxiao Hu^{1}Email authorView ORCID ID profile and
 Ling Zhu^{2}
https://doi.org/10.1186/s1366201609045
© Hu and Zhu 2016
 Received: 24 October 2015
 Accepted: 29 February 2016
 Published: 18 July 2016
Abstract
In this paper, we study a stochastic nonautonomous logistic system with feedback control. Sufficient conditions for stochastic asymptotically bounded, extinction, nonpersistence in the mean, weak persistence, and persistence in the mean are established. The critical number between weak persistence and extinction is obtained. A very important fact is found in our results, that is, the feedback control is harmless to the permanence of species under the randomized environment.
Keywords
 nonautonomous logistic model
 feedback control
 stochastic asymptotically bounded
 persistence
 extinction
1 Introduction
 (H_{1}):

There is a positive constant λ such that$$\liminf_{t\rightarrow\infty} \int_{t}^{t+\lambda} a(s)\, \mathrm{d}s>0. $$
 (H_{2}):

There is a positive constant \(\gamma_{1}\) such that$$\liminf_{t\rightarrow\infty} \int_{t}^{t+\gamma_{1}} e(s)\, \mathrm{d}s>0. $$
 (H_{3}):

There is a positive constant \(\gamma_{2}\) such that$$\liminf_{t\rightarrow\infty} \int_{t}^{t+\gamma_{2}} f(s)\, \mathrm{d}s>0. $$
In this work, our purpose is to establish the sufficient conditions for asymptotically bounded, extinction, nonpersistence in the mean, weak persistence and persistence in the mean of system (1.3). We will find that, in our results, the feedback control is harmless to the permanence of species with stochastic perturbation.
2 Preliminaries
Lemma 2.1
Remark 2.1
In [24], the authors obtained the same results as Lemma 2.1 with conditions \(m(t), n(t), \alpha(t)>0\). But checking the proof in Theorem 2.2 in [24], we can obtain the same results in Lemma 2.1, only \(n(t)\) needs to be nonnegative.
Lemma 2.2
 (a)
\(\alpha=1\) and \(n(t)\) is nonnegative;
 (b)
\(\alpha+\beta=1\), \(\alpha\ge0\), and \(m(t)\) is nonnegative.
 (i)
for any given initial value \(y_{0}>0\), there is a unique solution \(y(t)\) of (2.2) which is global positive;
 (ii)there exist positive constants l and L such thatfor any positive solution \(y(t)\) of equation (2.2);$$l\le\liminf_{t\rightarrow\infty}y(t)\le\limsup_{t\rightarrow \infty}y(t)\le L $$
 (iii)for any two positive solutions \(x(t)\) and \(y(t)\) of system (2.2) we have$$\lim_{t\rightarrow\infty}\bigl(x(t)y(t)\bigr)=0. $$
Proof
 (i)there exist positive constants l and L such thatfor any positive solution \(w(t)\) of equation (2.4);$$l\le\liminf_{t\rightarrow\infty}w(t)\le\limsup_{t\rightarrow \infty}w(t)\le L $$
 (ii)for any two positive solutions \(w_{1}(t)\) and \(w_{2}(t)\) of system (2.4) we have$$\lim_{t\rightarrow\infty}\bigl(w_{1}(t)w_{2}(t) \bigr)=0. $$
Remark 2.2
In [25], the authors considered the case \(\alpha=\beta=1\) of system (2.2), and obtained the same conclusions with this lemma. Hence, their results are generalized by Lemma 2.2.
Remark 2.3
Lemma 2.3
Lemma 2.4
Proof
Remark 2.4
In Lemma 2.3, the authors discussed the case \(\alpha=0\) and \(\beta=1\) of this lemma. Hence, their results are extended by this lemma.
3 Asymptotically bounded of the global positive solution
In system (1.3), \(x(t)\) is the size of the species and \(u(t)\) is the regulator, thus we are only interested in the positive solutions. Moreover, in order for a stochastic differential equation to have a unique global (i.e. no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (cf. Mao [27]). However, the coefficients of system (1.3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous. In this section, using the comparison theorem of stochastic equations (see [28]) we will show there is a unique positive solution with positive initial value of system (1.3).
Theorem 3.1
For any given initial value \((x_{0},u_{0})\in R_{+}^{2}\), there is a unique solution \((x(t),u(t))\) to system (1.3) on \(t\ge0\) and the solution will remain in \(R_{+}^{2}\) with probability one, namely \((x(t),u(t))\in R_{+}^{2}\) for all \(t\ge0\) almost surely.
Proof
Now, we will discuss the asymptotically bounded property of the unique global positive solution of system (1.3). To be precise, let us now give the definition of asymptotically bounded.
Definition 3.1
Theorem 3.2
Proof
In the following, we denote \(q(t)=r(t)+0.5(p1)\sigma^{2}(t)\).
Remark 3.1
Corollary 3.1
Remark 3.2
Definition 3.2
Theorem 3.3
Proof
This can easily be verified by Chebyshev’s inequality and Theorem 3.2. □
Corollary 3.2
Suppose \(a_{l}\) and \(e_{l}\) are positive, and for some \(p\ge1\) such that \(q_{u}>0\). Then solution of system (1.3) are stochastically ultimately bounded.
4 Extinction and persistence in time average
Now, we will discuss extinction and persistence of system (1.3). For any positive solution \((x(t),u(t))\) of system (1.3) we first introduce some useful definitions.
Definition 4.1
Theorem 4.1
If (H_{2}) holds and \(\langle b\rangle^{\ast}<0\), then system (1.3) will go to extinction almost surely.
Proof
Remark 4.1
If \(c(t)\equiv0\), we can obtain system (3.8). In Theorem 7 in [22], the authors obtained the extinction of system (3.8) under the same conditions with Theorem 4.1. Hence, if \(\langle b\rangle^{\ast}<0\), the feedback control cannot change the extinction of the species x.
Theorem 4.2
 (i)
if \(\langle a\rangle_{\ast}>0\), \(\langle c\rangle_{\ast}>0\), and (H_{2}) hold, then \(\liminf_{t\rightarrow\infty}x(t)=0\) and \(\liminf_{t\rightarrow\infty}u(t)=0\) a.s.;
 (ii)
if \(a_{l}, e_{l}>0\), then system (1.3) will be nonpersistent in the mean a.s.
Proof
Theorem 4.3
If \(e_{l}>0\) and \(\langle b\rangle^{\ast}>0\), then species x will be weakly persistent in the mean a.s., i.e. \(\langle x\rangle ^{\ast}>0\) a.s.
Proof
Remark 4.2
In Theorem 9 in [22], the authors studied the weakly persistent in the mean of system (3.8) with the conditions \(a_{l}>0\) and \(\langle b\rangle^{\ast}>0\). Obviously, from Theorem 4.3 we can obtain the same result with [22] only under the condition \(\langle b\rangle^{\ast}>0\). Therefore, the result in [22] is improved by Theorem 4.3.
Remark 4.3
In this theorem, due to shortage of the analysis techniques on the stochastic model, the weakly persistent in the mean of u case has not been studied. But we can see that the feedback control does not affect the persistence property of the species x under the conditions in this theorem.
Theorem 4.4
Proof
5 Numerical simulation
6 Future directions
Recently, some scholars studied some interesting problems, such as model with jumps (see [30, 31]) and model with time delay (see [32, 33]). It is an interesting question to investigate the dynamics property of the stochastic species systems with feedback control, jumps, and time delay. This will be our future work.
Declarations
Acknowledgements
We thank the National Natural Science Foundation of China (grant number: 11401382) and Hujiang Foundation of China (grant number: B14005).
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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