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Dynamics of a periodic impulsive switched predator-prey system with hibernation and birth pulse
- Jianjun Jiao^{1}Email author,
- Shaohong Cai^{1}Email author,
- Limei Li^{2} and
- Lansun Chen^{3}
https://doi.org/10.1186/s13662-015-0460-4
© Jiao et al.; licensee Springer. 2015
Received: 16 October 2014
Accepted: 6 April 2015
Published: 10 June 2015
Abstract
By hibernating, animals can reduce their energy requirements by at least ninety percent and survive for many months while slowly catabolizing body lipid reserves. Hibernation constitutes an effective strategy of animals in order to survive cold environments and limited availability of food. In this work, we investigate a periodic impulsive switched predator-prey system with hibernation and birth pulse. We firstly obtain the conditions of the globally asymptotically stable prey-extinction boundary periodic solution of the investigated system. Secondly, we obtain the permanent conditions of the investigated system. Finally, numerical analysis is presented to illustrate the results. Our results provide reliable tactic basis for the practical biological economics management.
Keywords
- hibernation
- periodic switched systems
- birth pulse
- prey-extinction
- permanence
1 Introduction
Hibernation allows small mammals to minimize metabolic energy costs at a time when a scarcity of food and cold environmental temperatures endanger normal life. By hibernating, animals can reduce their energy requirements by at least ninety percent and survive for many months while slowly catabolizing body lipid reserves [1]. Hibernation constitutes an effective strategy of animals in order to survive cold environments and limited availability of food [2].
The hibernation constitutes an effective strategy of animals in order to survive cold environments and limited availability of food, it is a universal phenomenon in biological world. However, there are few papers considering and investigating mathematical models with winter hibernation. In this paper, we introduce the phenomenon of hibernation and focus on a periodic impulsive switched predator-prey system with hibernation and birth pulse.
The organization of this paper is as follows. In the next section, we introduce the model and background concepts. In Section 3, some important lemmas are presented. In Section 4, we give the globally asymptotically stable conditions of a prey-extinction periodic solution of system (2.1) and the permanent condition of system (2.1). In Section 5, a brief discussion and the simulations are given to conclude this work.
2 The model
3 Some lemmas
The solution of system (2.1), denoted by \(X(t)=(x(t),y(t))^{T}\), is a non-smooth function \(X: R_{+}\rightarrow R_{+}^{2}\), \(X(t)\) is continuous on \((n\tau,(n+l)\tau]\) and \(((n+l)\tau,(n+1)\tau ]\), \(n\in Z_{+}\). \(X(n\tau^{+})=\lim_{t\rightarrow n\tau^{+} }X(t)\) and \(X((n+l)\tau ^{+})=\lim_{t\rightarrow(n+l)\tau^{+} }X(t)\) exist. Obviously the global existence and uniqueness of solutions of system (2.1) are guaranteed by the smoothness properties of f, which denotes the mapping defined by the right-hand side of system (2.1) (see Lakshmikantham et al. [23]).
Lemma 3.1
For each solution \((x(t),y(t))\) of system (2.1), there exists a constant \(M>0\) such that \(x(t)\leq M\), \(y(t)\leq M \) with all t large enough.
Proof
Lemma 3.2
[25]
Lemma 3.3
Proof
From Lemma 3.2, we obtain that the fixed points \(P(0)\) and \(P(y^{\ast })\) of (3.3) are stable, and then they are globally asymptotically stable. □
It is well known that the following lemma can easily be proved.
Lemma 3.4
- (i)
If \((1-\mu_{1})(a_{1}+1)e^{-[d_{1}l+d_{3}(1-l)]\tau}<1\), the triviality periodic solution of system (3.1) is globally asymptotically stable.
- (ii)If \((1-\mu_{1})(a_{1}+1)e^{-[d_{1}l+d_{3}(1-l)]\tau}>1\), the periodic solution \(\widetilde{y(t)}\) of system (3.1) is globally asymptotically stable, where \(\widetilde{y(t)}\) is defined asand \(y^{\ast}\) is defined as (3.4).$$ \widetilde{y(t)}= \left \{ \textstyle\begin{array}{l@{\quad}l} y^{\ast}e^{-d_{1}(t-n\tau)},& t\in(n\tau,(n+l)\tau], \\ (e^{-d_{1}l\tau}y^{\ast})e^{-d_{3}(t-(n+l)\tau)},&t\in((n+l)\tau ,(n+1)\tau], \end{array}\displaystyle \right . $$(3.11)
4 Dynamics for system (2.1)
Theorem 4.1
Proof
The next work is the investigation of permanence of system (2.1). Before starting our theorem, we give the following definition.
Definition 4.2
System (2.1) is said to be permanent if there are constants \(m,M >0 \) (independent of initial value) and a finite time \(T_{0}\) such that for all solutions \((x(t), y(t))\) with all initial values \(x(0^{+})>0\), \(y(0^{+})>0\), \(m\leq x(t)\leq M\), \(m\leq y(t)\leq M\) hold for all \(t\geq T_{0}\). Here \(T_{0}\) may depend on the initial values \((x(0^{+}), y(0^{+}))\).
Theorem 4.3
Proof
Suppose \((x(t), y(t))\) is a solution of (2.1) with \(x(0)>0\), \(y(0)>0\). By Lemma 3.1, we have proved that there exists a constant \(M >0\) such that \(x(t)\leq M\), \(y(t)\leq M\) for t large enough, we may assume \(x(t)\leq M\), \(y(t)\leq M\) for \(t\geq0\). From Theorem 4.1, we know \(y(t)>\widetilde{y(t)}-\varepsilon_{2}\) for all t large enough and \(\varepsilon_{2}> 0\), so \(y(t)\geq e^{-d_{1}l\tau}y^{\ast }(1+e^{-d_{3}(1-l)\tau})-\varepsilon_{2}=m_{2}\) for t large enough. Thus, we only need to find \(m_{1}>0\) such that \(x(t)\geq m_{1}\) for t large enough, we will do it in what follows.
5 Discussion
From the simulation experiment of Figures 1 and 2, the parameter μ affects the dynamical behaviors of system (2.1). If all parameters of system (2.1) are fixed, when \(\mu=0.7\), the prey population of system (2.1) goes extinct; when \(\mu=0.5\), system (2.1) is permanent. From Theorem 4.1 and Theorem 4.3, we can easily deduce that there must exist a threshold \(\mu^{\ast}\). If \(\mu >\mu^{\ast}\), the prey-extinction periodic solution \((0,\widetilde {y(t)})\) of system (2.1) is globally asymptotically stable. If \(\mu<\mu^{\ast}\), system (2.1) is permanent. That is to say, impulsive harvesting rate of the prey population plays an important role in system (2.1). The impulsive harvesting rate of the prey population will also reduce the predator population. It tells us that destroying or excessive exploiting of the prey population will cause extinction of the predator population. Our results also provide reliable tactic basis for the practical biological economics management and the protection of biodiversity.
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (11361014, 10961008).
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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