Dynamics of a new delayed stage-structured predator-prey model with impulsive diffusion and releasing
- Jianjun Jiao^{1}Email author,
- Shaohong Cai^{1} and
- Limei Li^{2}
https://doi.org/10.1186/s13662-016-1038-5
© The Author(s) 2016
Received: 5 September 2016
Accepted: 11 November 2016
Published: 5 December 2016
Abstract
In this work, we propose a new delayed stage-structured predator-prey model with impulsive diffusion and releasing. By the stroboscopic map of the discrete dynamical system, we obtain a prey-extinction boundary periodic solution. Furthermore, we prove that the prey-extinction boundary periodic solution is globally attractive. We also prove that the investigated system is permanent by the theory on the delay and impulsive differential equations. Our results indicate that time delay, impulsive diffusion, and impulsive releasing have influence to the dynamical behaviors of the investigated system. The results of this paper also provide a tactical basis for pest management.
Keywords
delayed stage structure predator-prey model impulsive diffusion impulsive releasing prey-extinction1 Introduction
Dispersal is a ubiquitous phenomenon in the natural world. It is important for us to understand the ecological and evolutionary dynamics of populations mirrored by the large number of mathematical models devoted to it in the scientific literature [13–24]. If the population dynamics with the effects of spatial heterogeneity is modeled by a diffusion process, most previous papers focused on the population dynamical system modeled by the ordinary differential equations. But in practice, it is often the case that diffusion occurs in regular pulse. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse in other seasons, and the excursion of foliage seeds occurs at a fixed period of time every year. Thus impulsive diffusion provides a more natural description. Lately theories of impulsive differential equations [25, 26] have been introduced into population dynamics. Impulsive differential equations are found in most domains of applied science [16, 17, 20, 24, 27–29].
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 conditions of global attractivity and permanence for system (2.1). In Section 5, a brief discussion is given in the last section to conclude this work.
2 The model
3 The lemmas
The solution of (2.1), denoted by \(X(t)=(x_{1}(t),y_{1}(t),z_{1}(t),x_{2}(t),y_{2}(t),z_{2}(t))\), is a piecewise continuous function \(X: R_{+}\rightarrow R_{+}^{6}\), \(X(t)\) is continuous on \((n\tau,(n+l)\tau]\), \(((n+l)\tau,(n+1)\tau]\), \(n\in Z_{+}\) 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) are guaranteed by the smoothness properties of f, which denotes the mapping defined by the right side of system (2.1) (see Lakshmikantham, [25]). Before we have the the main results, we need to give some lemmas which will be used in the following.
According to the biological meaning, it is assumed that \(x_{i}(t)\geq0\), \(y_{i}(t)\geq0\), and \(z_{i}(t)\geq0\) (\(i=1,2\)).
- (i)
V is continuous in \((n\tau, (n+l)\tau]\times R_{+}^{6}\) and \(((n+l)\tau, (n+1)\tau]\times R_{+}^{6}\), for each \(z\in R^{6}_{+}\), \(n\in Z_{+}\), \(V(n\tau^{+},z)=\lim_{(t,y)\rightarrow(n\tau^{+},z)} V(t,y)\), \(V((n+l)\tau^{+},z) =\lim_{(t,y)\rightarrow((n+l)\tau^{+},y)} V(t,y)\) exist.
- (ii)
V is locally Lipschitzian in z.
Definition 3.1
Lemma 3.2
[26]
Now, we show that all solutions of (2.1) are uniformly ultimately bounded.
Lemma 3.3
There exists a constant \(M>0\) such that \(x_{i}(t)\leq M\), \(y_{i}(t)\leq M\), \(z_{i}(t)\leq M\) (\(i=1,2\)) for each solution \((x_{1}(t),y_{1}(t),z_{1}(t),x_{2}(t),y_{2}(t), z_{2}(t))\) of (2.1) with all t large enough.
Proof
Lemma 3.4
The fixed point \((z_{1}^{\ast}, z_{2}^{\ast})\) of (3.6) is globally asymptotically stable.
Proof
Lemma 3.5
Lemma 3.6
[31]
- (i)
if \(a_{1}< a_{2} \), then, \(\lim_{t\rightarrow\infty}x(t)=0\),
- (ii)
if \(a_{1}>a_{2} \), then, \(\lim_{t\rightarrow\infty}x(t)=+\infty\).
4 The dynamics
From the above discussion, we know there exists a prey-extinction boundary periodic solution \((0,\widetilde{z_{1}(t)}, 0, \widetilde{z_{2}(t)})\) of system (2.2). In this section, we will prove that the prey-extinction boundary periodic solution \((0,\widetilde{z_{1}(t)}, 0, \widetilde{z_{2}(t)})\) of system (2.2) is globally attractive.
Theorem 4.1
Proof
The next work is to investigate the permanence of system (2.2). Before starting our theorem, we give the following definition.
Definition 4.2
System (2.2) 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 \((y_{1}(t), z_{1}(t), y_{2}(t), z_{2}(t))\) with all initial values \(y_{i}(0^{+})>0\), \(z_{i}(0^{+})>0\) (\(i=1,2\)), \(m\leq y_{i}(t)\leq M\), \(m\leq z_{i}(t)\leq M\) (\(i=1,2\)) hold for all \(t\geq T_{0}\). Here \(T_{0}\) may depend on the initial values \((y_{1}(0^{+}), z_{1}(0^{+}),y_{2}(0^{+}), z_{2}(0^{+}))\).
Theorem 4.3
Proof
By the claim, we are left to consider two cases. First, \(y_{i}(t)\geq y_{i}^{\ast}\) (\(i=1,2\)) for all t large enough. Second, \(y_{i}(t)\) (\(i=1,2\)) oscillates about \(y_{i}^{\ast}\) (\(i=1,2\)) for t large enough.
Theorem 4.4
Proof
5 Discussion
In this paper, we investigate a new delayed stage-structured predator-prey model with impulsive diffusion and releasing. We analyze that the prey-extinction boundary periodic solution of system (2.2) is globally attractive, and we also obtain the permanent condition of system (2.2). From Theorem 4.1 and Theorem 4.4, we can easily guess that there must exist a threshold \(\mu^{\ast}\) (\(\mu^{\ast}=\max_{i=1,2}\{\mu^{\ast}_{i}\}\) and \(\mu^{\ast}_{i}\) (\(i=1,2\))) is determined by the condition of Theorem 4.1), if \(\mu>\mu^{\ast}\), the prey-extinction boundary periodic solution \((0,\widetilde{z_{1}(t)},0,\widetilde{z_{2}(t)})\) of (2.2) is globally attractive. If \(\mu<\mu^{\ast\ast}\) (\(\mu ^{\ast\ast}=\min_{i=1,2}\{\mu^{\ast\ast}_{i}\}\) and \(\mu^{\ast\ast }_{i}\) (\(i=1,2\)) is determined by the condition of Theorem 4.4), system (2.2) is permanent. From Theorem 4.1 and Theorem 4.4, we can also easily guess that there must exist a threshold \(D^{\ast}\) (\(0< D^{\ast}<1\)). If \(D < D^{\ast}\), the prey-extinction boundary periodic solution \((0,\widetilde{z_{1}(t)},0,\widetilde{z_{2}(t)})\) of (2.2) is globally attractive. If \(D > D^{\ast}\), system (2.2) is permanent. This indicates that impulsive diffusion and impulsive releasing can affect the dynamical behaviors of the investigated system (2.2). That is to say, impulsive diffusion and impulsive releasing of the predator population play important roles for the prey-extinction of system (2.2). The parameters as \(\tau_{i}\) (\(i=1,2\)) and τ can also be discussed, its change also affects the dynamical system of (2.2). The results of this paper provide a tactical basis for pest management.
Declarations
Acknowledgements
Supported by National Natural Science Foundation of China (11361014, 10961008), Natural Science Foundation of Guizhou Education Department (2008038), the Project of High Level Creative Talents in Guizhou Province (No. 20164035), and the Science Technology Foundation of Guizhou Province (2008J2250).
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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