Analysis of a predator-prey model with impulsive diffusion and releasing on predator population
- Airen Zhou^{1, 2},
- Pairote Sattayatham^{1} and
- Jianjun Jiao^{2}Email author
https://doi.org/10.1186/s13662-016-0834-2
© Zhou et al. 2016
Received: 19 January 2016
Accepted: 6 April 2016
Published: 20 April 2016
Abstract
In this paper, we establish a predator-prey model with impulsive diffusion and releasing on predator population. This predator-prey model for two regions, which are connected by diffusion of predator population, portrays the evolvement of population. We prove that all solutions of the investigated system are uniformly ultimately bounded. We also prove that there exists globally asymptotically stable prey-extinction boundary periodic solution. The condition for permanence is obtained. Simulations are also employed to verify our results. It is discovered that the increasing diffusive rate of predator population will count against the pest management. We conclude that the impulsive diffusion and releasing predator provide reliable tactic basis for pest management.
Keywords
predator-prey model impulsive diffusion impulsive releasing extinction permanence1 Introduction
The warfare between human and pests has sustained for thousands of years. In the past few decades, man have adopted some advanced and modern weapons for instance chemical pesticides, biological pesticides, remote sensing and measure, computers, atomic energy, et cetera. Some brilliant achievements have been obtained. However, the warfare will never be over. Although a great number of pesticides were used to control pests, the insect pests impairing crops are increasing for the resistance to pesticides. With pesticides employed, the residual pests breed a large number of pests with resistance to pesticides. So the pesticide is invalid in some sense. Moreover, insect pests will continue. On the other hand, the chemical pesticides kill not only pests but also their natural enemies. Therefore, insect pests are rampant again. Then the effect of chemical control was challenged. Furthermore, the practice proves that long-term adopting chemical control may give rise to disastrous results, for example, environmental contamination and toxicosis of the man and animals and so on.
Theories of impulsive differential equations have been introduced into population dynamics lately [23–28]. Impulsive equations are found in almost every domain of applied science and have been studied in many investigations [28–33]; they generally describe phenomena that are subject to steep or instantaneous changes. The theories of population dynamical systems and their applications have achieved many good results. In this paper, we investigate a predator-prey model with impulsive diffusion and releasing on predator population. We expect to obtain some dynamical properties of the investigated system. We also expect that the impulsive diffusion and releasing predator will provide reliable tactic basis for pest management.
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. We give the globally asymptotically stable conditions of the prey-extinction boundary periodic solution of system (3) and the permanent condition of system (3) in Section 4. Simulation analysis and brief discussion are given in the last section to conclude this work.
2 The model
3 The lemmas
The solution of (3), denoted by \(X(t)=(x_{1}(t),y_{1}(t),x_{2}(t),y_{2}(t))^{T}\), is a piecewise continuous function \(X:R_{+}\rightarrow R_{+}^{4}\), \(X(t)\) is continuous on \((n\tau,(n+l)\tau]\) and \(((n+l)\tau,(n+1)\tau ]\), \(n\in Z_{+}\), and \(X(n\tau^{+})=\lim_{t\rightarrow n\tau^{+} }X(t)\), \(X((n+l)\tau ^{+})=\lim_{t\rightarrow(n+l)\tau^{+} }X(t)\) exist. Obviously, the global existence and uniqueness of solutions of (3) is guaranteed by the smoothness properties of f, the mapping defined by the right side of system (3) [21].
- (i)
V is continuous in \((n\tau, (n+l)\tau]\times R_{+}^{4}\) and \(((n+l)\tau, (n+1)\tau]\times R_{+}^{4}\) for all \(z\in R^{4}_{+}\), \(n\in Z_{+}\), and \(V(n\tau^{+},z)=\lim_{(t,y)\rightarrow(n\tau^{+},z) }V(t,y)\) and \(V((n+l)\tau^{+},z) =\lim_{(t,y)\rightarrow((n+l)\tau^{+},y) }V(t,y)\) exist;
- (ii)
V is locally Lipschitzian in z.
Definition 3.1
Since \(\frac{dx_{i}(t)}{dt}=0\) when \(x_{i}(t)=0\), \(\frac {dy_{i}(t)}{dt}=0\) when \(y_{i}(t)=0\), and \(\Delta y_{i}(t)=\mu_{i}>0\) when \(t=(n+1)\tau\), we easily obtain the following lemma.
Lemma 3.2
Suppose that \(X(t)\) is a solution of (3) with \(X(0^{+})\geq0\). Then \(X(t)\geq0\) for \(t\geq0\), and further \(X(t)> 0\) (\(t\geq0\)) for \(X(0^{+})> 0\).
Lemma 3.3
[21]
Now, we show that all solutions of (3) are uniformly ultimately bounded.
Lemma 3.4
There exists a constant \(M>0\) such that \(x_{i}(t)\leq M\), \(y_{i}(t)\leq M\) (\(i=1,2\)) for each solution \((x_{1}(t),y_{1}(t),x_{2}(t),y_{2}(t))\) of (3) with all t large enough.
Proof
Lemma 3.5
The fixed point \((y_{1}^{\ast}, y_{2}^{\ast})\) of (9) is globally asymptotically stable.
Proof
Lemma 3.6
4 The dynamics
Theorem 4.1
Proof
The next work is to investigate the permanence of system (3).
Definition 4.2
System (3) 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 \((x_{1}(t), y_{1}(t), x_{2}(t), y_{2}(t))\) with any initial values \(x_{1}(0^{+})>0\), \(y_{1}(0^{+})>0\), \(x_{2}(0^{+})>0\), \(y_{2}(0^{+})>0\), we have \(m\leq x_{1}(t)\leq M\), \(m\leq y_{1}(t)\leq M\), \(m\leq x_{2}(t)\leq M\), \(m\leq y_{2}(t)\leq M\) for all \(t\geq T_{0}\). Here \(T_{0}\) may depend on the initial values \((x_{1}(0^{+}),y_{1}(0^{+}), x_{2}(0^{+}),y_{2}(0^{+}))\).
Theorem 4.3
Proof
5 Simulation analysis and discussion
5.1 The dynamical behaviors influenced by parameter D
5.2 The dynamical behaviors influenced by parameters \(\mu_{1}\) and \(\mu_{2}\)
From the simulations we discover that the increasing diffusive rate of predator population will count against the pest management. We conclude that the impulsive diffusion and releasing predator provide reliable tactic basis for pest management.
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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