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Dynamics of a predator-prey model with impulsive biological control and unilaterally impulsive diffusion
- Jianjun Jiao^{1},
- Shaohong Cai^{1},
- Limei Li^{2} and
- Yujuan Zhang^{3}Email author
https://doi.org/10.1186/s13662-016-0851-1
© Jiao et al. 2016
Received: 27 January 2016
Accepted: 28 April 2016
Published: 8 June 2016
Abstract
In this paper, we establish a predator-prey model with impulsive biological control and unilaterally impulsive diffusion. This predator-prey model for two regions, which are connected by diffusion of predator population, portrays the evolution of the population. We study the model for biological pest control in which a pest population is controlled by a program of periodic releases of a fixed yield of predators that prey on the pest population. We prove that there exists a globally asymptotically stable prey-extinction boundary periodic solution. The condition for permanence is also obtained. Simulations are also employed to verify our results. We conclude that the impulsive diffusion and releasing predator provide reliable tactic basis for pest management.
Keywords
- predator-prey model
- unilaterally impulsive diffusion
- impulsive biological control
- extinction
- permanence
1 Introduction
The warfare between humans and pests has persisted for thousands of years. In the past few decades, man has adopted some advanced and modern weapons for instance chemical pesticides, biological pesticides, remote sensing and measure, computers, atomic energy etc. Some brilliant achievements have been obtained. However, the warfare will never be over. Although a great deal of pesticides were used to control pests, the insect pests impairing crops are increasing because of the resistance to the pesticide. With pesticides employed, the residual pests breed a large number of pests with resistance to pesticides. So the pesticide is invalid in a sense. Moreover, insect pests will remain. 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, we witness environmental contamination and toxicosis of man and animal, 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 which are subject to steep or instantaneous changes. The theories of population dynamical systems and their applications have achieved many good results. However, the oasis vegetation degradation combined with a dynamical system has been considered very little. In this paper, we will investigate an impulsive dispersal on SIR model on restricting infected individuals boarding transports. We expect to obtain some dynamical properties of the investigated system. We also expect that impulsive dispersal will provide a reliable tactic for controlling epidemic.
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 (2.1), and the permanent condition of System (2.1). In Section 4 is a simulation analysis, and a brief discussion are given in the last section to conclude this work.
2 The model
3 The lemmas
The solution of (2.1), denoted by \(X(t)=(x_{1}(t),y_{1}(t),y_{2}(t))^{T}\), is a piecewise continuous function \(X : R_{+}\rightarrow R_{+}^{3}\), \(X(t)\) 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 (2.1) is guaranteed by the smoothness properties of f, which denotes the mapping defined by right side of System (2.1) [21].
- (i)
V is continuous in \((n\tau, (n+l)\tau]\times R_{+}^{3}\) and \(((n+l)\tau, (n+1)\tau]\times R_{+}^{3}\), for each \(z\in R^{3}_{+}\), \(n\in Z_{+}\), \(V(n\tau^{+},z)=\lim_{(t,y)\rightarrow(n\tau^{+},z) }V(t,y)\), \(V((n+l)\tau^{+},z) =\lim_{(t,y)\rightarrow((n+l)\tau^{+},y) }V(t,y)\) 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 \(\triangle y_{1}(t)=\mu>0\), when \(t=(n+1)\tau\), we can easily obtain the following lemma.
Lemma 3.2
Suppose \(X(t)\) is a solution of (2.1) with \(X(0^{+})\geq0\), then \(X(t)\geq0\) for \(t\geq0\). Furthermore, \(X(t)> 0\) (\(t\geq0\)) for \(X(0^{+})> 0\).
Lemma 3.3
[21]
Now, we show that all solutions of (2.1) are uniformly ultimately bounded.
Lemma 3.4
There exists a constant \(M>0\) such that \(x_{1}(t)\leq M\), \(y_{1}(t)\leq M\), \(y_{2}(t)\leq M \) for each solution \((x_{1}(t),y_{1}(t),y_{2}(t))\) of (2.1) with all t large enough.
Proof
Lemma 3.5
Proof
- (i)If \(1-e^{-a_{2}\tau}< D<1\) holds, that is, \(\frac{B_{2}}{a_{2}}<1\), we can easily see that \((y_{1}^{\ast},0)\) is a unique fixed point of (3.6) or (3.7), andFor$$ \left . \textstyle\begin{array}{@{}l@{}} M= \left( \textstyle\begin{array}{@{}c@{\quad}c@{}} A&\frac{B_{1}}{a_{2}} \\ 0&\frac{B_{2}}{a_{2}} \end{array}\displaystyle \right). \end{array}\displaystyle \right . $$(3.11)From the Jury criterion, \((y_{1}^{\ast}, 0)\) is locally stable, then it is globally asymptotically stable.$$\begin{aligned} 1- \operatorname {tr}M+ \det M&=1-\biggl(A+\frac{B_{2}}{a_{2}}\biggr)+\biggl(A \times \frac{B_{2}}{a_{2}}\biggr) \\ &=(1-A) \biggl(1-\frac{B_{2}}{a_{2}}\biggr)>0. \end{aligned}$$
- (ii)If \(0< D<1-e^{-a_{2}\tau}\) holds, that is, \(\frac {a_{2}}{B_{2}}<1\), we can easily see that \((y_{1}^{\ast\ast},y_{2}^{\ast \ast})\) is a unique fixed point of (3.6) or (3.7), andFor$$ \left . \textstyle\begin{array}{@{}l@{}} M= \left( \textstyle\begin{array}{@{}c@{\quad}c@{}} A&\frac{B_{1}a_{2}}{(a_{2}+C_{1}y_{2}^{\ast\ast})^{2}} \\ 0&\frac{B_{2}a_{2}}{(a_{2}+C_{2}y_{2}^{\ast\ast})^{2}} \end{array}\displaystyle \right). \end{array}\displaystyle \right . $$(3.12)From the Jury criterion, \((y_{1}^{\ast\ast},y_{2}^{\ast\ast})\) is locally stable, then it is globally asymptotically stable. This completes the proof.$$\begin{aligned} 1- \operatorname {tr}M+ \det M &=1-\biggl(A+\frac{B_{2}a_{2}}{(a_{2}+C_{2}y_{2}^{\ast\ast })^{2}}\biggr)+\biggl(A \times \frac{B_{2}a_{2}}{(a_{2}+C_{2}y_{2}^{\ast\ast})^{2}}\biggr) \\ &=(1-A)\biggl[1-\frac{B_{2}a_{2}}{(a_{2}+C_{2}y_{2}^{\ast\ast})^{2}}\biggr] \\ &=(1-A) \biggl(1-\frac{a_{2}}{B_{2}}\biggr)>0. \end{aligned}$$
Lemma 3.6
- (i)If \(0< D<1-e^{-a_{2}\tau}\) holds, the periodic solution \((\widetilde{y_{1}(t)},\widetilde{y_{2}(t)} )\) of System (3.2) is globally asymptotically stable, wherewhere \(y^{\ast\ast}_{1}\) and \(y^{\ast\ast}_{2}\) are determined as (3.8), \(y_{1}^{\ast\ast\ast}\) and \(y_{2}^{\ast\ast\ast}\) are defined as$$ \left \{ \textstyle\begin{array}{@{}l@{}} \left . \textstyle\begin{array}{@{}l@{}} \widetilde{y_{1}(t)}= \end{array}\displaystyle \right . \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} y_{1}^{\ast\ast}e^{-d_{1}(t-n\tau)}, & t\in[n\tau,(n+l)\tau), \\ y_{1}^{\ast\ast\ast}e^{-d_{1}(t-(n+l)\tau)}, &t\in[(n+l)\tau,(n+1)\tau), \end{array}\displaystyle \right . \\ \left . \textstyle\begin{array}{@{}l@{}} \widetilde{ y_{2}(t)}= \end{array}\displaystyle \right . \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} \frac{a_{2}y_{2}^{\ast\ast}e^{a_{2}(t-n\tau)}}{a_{2}+b_{2}y_{2}^{\ast \ast}(e^{a_{2}(t-n\tau)}-1)}, & t\in[n\tau,(n+l)\tau), \\ \frac{a_{2}y_{2}^{\ast\ast\ast}e^{a_{2}(t-(n+l)\tau )}}{a_{2}+b_{2}y_{2}^{\ast\ast\ast}(e^{a_{2}(t-(n+l)\tau)}-1)}, &t\in[(n+l)\tau,(n+1)\tau), \end{array}\displaystyle \right . \end{array}\displaystyle \right . $$(3.13)$$ \left \{ \textstyle\begin{array}{@{}l@{}} y_{1}^{\ast\ast\ast}=e^{-d_{1}l\tau}y_{1}^{\ast\ast}+D\times\frac {a_{2}y_{2}^{\ast\ast}e^{a_{2}l\tau}}{a_{2}+b_{2}y_{2}^{\ast\ast }(e^{a_{2}l\tau}-1)}, \\ y_{2}^{\ast\ast\ast}=(1-D)\times\frac{a_{2}y_{2}^{\ast\ast}e^{a_{2}l\tau }}{a_{2}+b_{2}y_{2}^{\ast\ast}(e^{a_{2}l\tau}-1)}. \end{array}\displaystyle \right . $$(3.14)
- (ii)If \(1-e^{-a_{2}\tau}< D<1\) holds, the periodic solution \((\widehat{y_{1}(t)},0 )\) of System (3.2) is globally asymptotically stable, wherewhere \(y_{1}^{\ast\ast\ast\ast}=e^{-d_{1}\tau}y^{\ast}_{1}\), and \(y^{\ast}_{1}\) is determined as (3.8).$$ \left . \textstyle\begin{array}{@{}l@{}} \left . \textstyle\begin{array}{@{}l@{}} \widehat{y_{1}(t)}= \end{array}\displaystyle \right . \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} y_{1}^{\ast}e^{-d_{1}(t-n\tau)},& t\in[n\tau,(n+l)\tau), \\ y_{1}^{\ast\ast\ast\ast}e^{-d_{1}(t-(n+l)\tau)},& t\in[(n+l)\tau ,(n+1)\tau), \end{array}\displaystyle \right . \end{array}\displaystyle \right . $$(3.15)
4 The dynamics
Theorem 4.1
Proof
We can easily prove Theorem 4.2 similar to Theorem 4.1.
Theorem 4.2
The next task is to investigate the permanence of System (2.1).
Definition 4.3
System (2.1) is said to be permanent if there are constants \({m,M >0} \) (independent of the initial value) and a finite time \(T_{0}\) such that for all solutions \((x_{1}(t), y_{1}(t), y_{2}(t))\) with all initial values \(x_{1}(0^{+})>0\), \(y_{1}(0^{+})>0\), \(y_{2}(0^{+})>0\), \(m\leq x_{1}(t)\leq M\), \(m\leq y_{1}(t)\leq M\), \(m\leq y_{2}(t)\leq M\) hold for all \(t\geq T_{0}\). Here \(T_{0}\) may depend on the initial values \((x_{1}(0^{+}),y_{1}(0^{+}), y_{2}(0^{+}))\).
Theorem 4.4
Proof
5 Discussion
From the simulations, we can guess that there exist three controlling thresholds with D. It is always assumed that \(0< D^{\ast}< D^{\ast\ast }< D^{\ast\ast\ast}<1\). If \(0< D< D^{\ast}\) holds, System (2.1) is permanent. If \(D^{\ast }< D< D^{\ast\ast}\) holds, the prey-extinction periodic solution \((0,\widetilde{y_{1}(t)},\widetilde {y_{2}(t)})\) of System (2.1) is globally asymptotically stable. If \(D^{\ast\ast}< D< D^{\ast\ast\ast}\) holds, the prey-extinction periodic solution \((0,\widehat{y_{1}(t)},0)\) of System (2.1) is globally asymptotically stable. If \(D^{\ast\ast\ast}< D<1\) holds, the predator \(y_{2}(t)\) will go into extinction, prey \(x_{1}(t)\) and predator \(y_{1}(t)\) will be permanent. We can discuss parameter μ similar to parameter D. We discover that the diffusive rate of the predator population plays an important role in pest management. We conclude that the impulsive diffusion and the released predator provide reliable tactic bases for pest management.
Declarations
Acknowledgements
Article is supported by the National Natural Science Foundation of China (11361014, 10961008), and the project of high level creative talents in Guizhou province (No. 20164035).
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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