Existence and stability of a unique almost periodic solution for a prey-predator system with impulsive effects and multiple delays
- Baodan Tian^{1, 2}Email author,
- Shouming Zhong^{2} and
- Ning Chen^{1}
https://doi.org/10.1186/s13662-016-0915-2
© Tian et al. 2016
Received: 19 January 2016
Accepted: 28 June 2016
Published: 8 July 2016
Abstract
In this paper, a nonautonomous almost periodic prey-predator system with impulsive effects and multiple delays is considered. By the mean-value theorem of multiple variables, integral inequalities, differential inequalities, and other mathematical analysis skills, sufficient conditions which guarantee the permanence of the system are obtained. Furthermore, by constructing a series of Lyapunov functionals, we derive that there exists a unique almost periodic solution of the system which is uniformly asymptotically stable. Finally, a numerical example and some simulations are presented to support our theoretical results.
Keywords
1 Introduction
It is widely known that predation is a very common ecological phenomenon in the natural world, and research on the dynamics of the prey-predator system is extremely meaningful and important in many fields, such as protecting varieties of the biological species, maintaining ecological balance, agricultural pests control and management, etc. As research of the prey-predator system is concerned, the earliest work dates back to the great contribution of Lotka (1925) and Volterra (1926). However, based on the classic Lotka-Volterra model, many ecologists found that the predation rate was not just simply proportional to the product of the density of the predator and prey. It was always different for different systems. Then the conception of the functional response was proposed. Functional response refers to the predation rate per predator with a response to changes in the prey density, i.e., predation effects of predators on the prey.
Since then, studies depending on all kinds of functional responses sprang up quickly, such as of Holling type [1–3], Michaelis-Menton type [4], Beddington-DeAngelis type [5], Ivlev type [6], Hassell-Varley type [7], and so on. In 1989, Crowley and Martin [8] proposed a new functional response that can accommodate interference between predators. It is similar to the well-known Beddington-DeAngelis functional response but has an additional term in the first right term equation modeling mutual interference between the predator terms.
Here m, α, β are positive parameters that describe the effects of capture rate handling time and the magnitude of interference among predators, respectively, on the feeding rate.
On the one hand, since models with Crowly-Martin functional response can be generated by a number of natural mechanisms and admits rich but biological dynamics. However, to the best of our knowledge, references reported on this functional response seems to much less until recent several years (see [9–13]) for the form of the functional response is relatively complex, for the form of this functional response is relatively complex, and it is worthwhile to further study the models with it.
On the other hand, it is well known that there are many natural or man-made factors in the real world, such as earthquake, flooding, drought, crop-dusting, planting, hunting and harvesting. These kinds of factors can bring sudden changes to an ecological system, and the intrinsic discipline of the environment or the species in the system will often undergoes these changes in a relatively short interval, usually we could regard it happens at some fixed times. From the viewpoint of mathematics, such sudden changes could be described by the impulsive effects to the system. In addition, when a prey-predator system is studied, it is more reasonable to consider time delay during the predation, because there is often a digest time instead of transforming food into intrinsic growth rate of the predator. As far as the impulsive system is concerned, there have appeared much excellent work in the last decades, such as impulsive effects in ecological systems (see [14–21]), in epidemic systems (see [22]) and in the neural network models (see [23–25]). Besides, there are many important monographs in this field (see [26–28]).
Particularly, research on the impulsive system with delays still seems to be a hot issue (see [29–37]), and these kinds of hybrid systems are usually called impulsive functional systems. When an impulsive functional system is concerned, the difficulty both in the impulsive differential equation and in the functional differential equation will occur at the same time, then the dynamical behaviors, such as permanence, periodic solution, almost periodic solution, and its asymptotical stability properties, as well as bifurcations and chaotic behaviors etc., might be richer, more complex, and more interesting.
- (C1)
\(r_{i}(t)\), \(c_{i}(t)\), \(d_{i}(t)\) (\(i=1,2\)), \(\alpha(t)\) and \(\beta(t)\) are all bounded and positive almost periodic functions;
- (C2)
\(H_{i}(t)= \prod_{0< t_{k}< t}(1+h_{ik})\), \(i=1,2\), \(k\in N\) is almost period functions and there exist positive constants \(H^{u}_{i}\) and \(H^{l}_{i}\) such that \(H^{l}_{i}\leq H_{i}(t)\leq H^{u}_{i}\).
The rest of this article is organized as follows: In Section 2, we will give some definitions and several useful lemmas for the proof of our main results. In Section 3, we will state and prove our main results such as permanence of the system, existence, and the uniqueness of an almost periodic solution which is uniformly asymptotically stable by constructing a series of Lyapunov functionals. In the last section, we give a numerical examples to support our theoretical results, then provide a brief discussion and summary of our main results.
2 Preliminaries
In this section, we will state the following definitions and lemmas, which will be used in the next sections.
We have \(K= \{{t_{k}}\in R |t_{k}< t_{k+1}, k\in N, \lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty \}\), and we thus denote the set of all sequences that are unbounded and increasing. Let \(D\subset\Omega\), \(\Omega\neq\Phi\), \(\tau= \max_{1\leq i\leq4}\{2\tau_{i}\}\), \(\xi_{0}\in R\). Also, we denote \(\operatorname{PC}(\xi_{0})\) is the space of all functions \(\phi:[\xi_{0}-\tau,\xi_{0}]\rightarrow\Omega\) having points of discontinuity at \(\mu_{1},\mu_{2},\ldots\in [\xi_{0}-\tau,\xi_{0}]\) of the first kind and left continuous at these points.
For \(J\subset R\), \(\operatorname{PC}(J,R)\) is the space of all piecewise continuous functions from J to R with points of discontinuity of the first kind \(t_{k}\), at which we have left continuity.
Definition 2.1
(see [38])
The set of sequence \(\{t_{k}^{j}\}\), \(t^{k}_{j}=t_{k+j}-t_{k}\), \(k,j\in N\), \({t_{k}}\in K\) is said to be uniformly almost periodic, if for arbitrary \(\varepsilon>0\) there exists a relatively dense set of ε-almost periodic common for any sequences.
Definition 2.2
(see [38])
- (1)
The set of sequences \(\{t_{k}^{j}\}\), \(t^{k}_{j}=t_{k+j}-t_{k}\), \(k,j\in N\), \({t_{k}}\in K\) is uniformly almost periodic.
- (2)
For any \(\varepsilon>0\), there exists a real number \(\delta>0\), such that if the points \(t_{1}\) and \(t_{2}\) belong to one and the same interval of continuity of \(\varphi(t)\) and satisfy the inequality \(| t_{1}-t_{2} |<\delta\), then \(|\varphi(t_{1})-\varphi(t_{2}) |< \varepsilon\).
- (3)
For any \(\varepsilon>0\), there exists a relatively dense set T such that if \(\eta\in T\), then \(|\varphi(t+\eta)-\varphi(t) | <\varepsilon\) for all \(t\in R\) satisfying the condition \(| t-t_{k} |> \varepsilon\), \(k\in N\). The elements of T are called ε-almost periods.
In the following, we will give some useful lemmas, which will be used in the next sections.
Lemma 2.1
Assume that \((u(t),v(t))^{T}\) is any solution of system (6) with the initial conditions (7), then \(u(t)>0\), \(v(t)>0\) for all \(t\in R^{+}\).
Proof
This completes the proof of this lemma. □
Proof
Combined (10) with (11) and (12), we can see that \((x(t),y(t))^{T}\) is the solution of the impulsive system (4).
(2) Because \(u(t)\) and \(v(t)\) are continuous on each interval \((t_{k},t_{k+1}]\), we only need to check the continuity of \(u(t)\) and \(v(t)\) at the impulsive point \(t=t_{k}\), \(k\in N\).
Thus, \(u(t)\) and \(v(t)\) are continuous on \([0,+\infty)\).
On the other hand, similar to the above proof of step (1), we can verify that \((u(t),v(t))^{T}\) satisfies system (6).
Then \((u(t),v(t))^{T}\) is a solution of the non-impulsive system (6).
This completes the proof of Lemma 2.2. □
Lemma 2.3
(see [39])
- (1)Assume that for \(w(t)>0\), \(t\geq0\), we havewith initial conditions \(w(s)=\phi(s) \geq0\), \(s\in[-\tau,0]\), where a, b are positive constants, then there exists a positive constant \(w^{*}\) such that$$w'(t) \leq w(t) \bigl(a-bw(t-\tau) \bigr) $$$$\lim_{t\rightarrow\infty}\sup w(t) \leq w^{*}:= \frac{ae^{a\tau}}{b}. $$
- (2)Assume that for \(w(t)>0\), \(t\geq0\), we havewith initial conditions \(w(s)=\phi(s) \geq0\), \(s\in[-\tau,0]\), where c, d are positive constants, then there exists a positive constant \(w_{*}\) such that$$w'(t) \geq w(t) \bigl(c-dw(t-\tau) \bigr) $$$$\lim_{t\rightarrow\infty}\inf w(t) \geq w_{*}:= \frac{ce^{(c-dw^{*})\tau}}{d}. $$
Let \(R^{2}\) be the plane Euclidean space with element \(X=(x,y)^{T}\) and norm \(|X|_{0}=|x|+|y|\), \(C=C([-\tau,0],R^{2})\), \(B\in R^{+}\), and denote \(C_{B}= \{\varphi=(\varphi_{1}(s),\varphi_{2}(s))^{T} \in C| \|\varphi \|\leq B \}\) with \(\|\varphi\|= \sup_{s\in[-\tau ,0]}|\varphi(s)|_{0}= \sup_{s\in[-\tau,0]} (|\varphi_{1}(s)|+|\varphi_{2}(s)| )\).
Then by the conclusions of [38, 40], we have Lemma 2.4.
Lemma 2.4
(see [40])
- (1)
\(u (\|\phi-\psi\| ) \leq V(t,\phi,\psi) \leq v (\|\phi-\psi\| )\), where \(u,v\in\mathcal{P}\) = {\(u:R^{+}\rightarrow R^{+}| u\) is continuous increasing function and \(u(s)\rightarrow0\), as \(s\rightarrow0\)};
- (2)
\(|V(t,\bar{\phi},\bar{\psi})-V(t,\hat{\phi}, \hat {\psi})|\leq L (\|\bar{\phi}-\hat{\phi}\|+\|\bar{\psi }-\hat{\psi}\| )\), where L is a positive constant;
- (3)
\(D^{+}V(t,\phi,\psi)|_{\text{(17)}}\leq-\gamma V(t,\phi,\psi)\), where γ is a positive constant.
Lemma 2.5
(see [36])
3 Main results
Theorem 3.1
- (C3)
\(C^{u}_{1}< B^{l}r^{l}_{1}\),
Proof
Therefore, there exists a \(T_{1}>0\), such that \(v(t)\leq v^{*}\) when \(t>T_{1}\).
Thus, we complete the proof of this theorem by combining (19), (20), (21), and (22). □
Theorem 3.2
Proof
This completes the proof of this theorem. □
Remark 3.1
Suppose that (C1)-(C3) hold, then system (6) is permanent.
Theorem 3.3
- (C4)
positive \(\lambda_{1}\) and \(\lambda_{2}\) exist, such that \(\frac{\alpha_{1}}{\beta_{1}}<\frac{\lambda_{1}}{\lambda _{2}}<\frac{\alpha_{2}}{\beta_{2}}\),
Proof
At first, we prove that system (6) has a unique uniformly asymptotically stable almost periodic solution.
This means the second condition of Lemma 2.4 is satisfied.
Finally, we will prove the last condition of Lemma 2.4.
If we denote \(\delta=\min \{\beta_{1}-\frac{\lambda _{2}}{\lambda_{1}}\alpha_{1}, \alpha_{2}-\frac{\lambda_{1}}{\lambda_{2}}\beta _{2} \}\), then \(\delta>0\) by the condition (C4).
This means the last condition of Lemma 2.4 is satisfied. By Lemma 2.4, the system admits a unique uniformly asymptotically stable almost periodic solution \((x(t),y(t))^{T}\).
Thus, by the transformation (28), we can conclude that system (6) admits a unique uniformly asymptotically stable almost periodic solution \((u(t),v(t))^{T}=(e^{x(t)},e^{y(t)})^{T}\).
Finally, we will explain that system (4) has a unique uniformly asymptotically stable almost periodic solution.
This completes the proof of this theorem. □
4 Numerical simulations and discussions
In this section, we will give a numerical example to illustrate the feasibility of our analytical results, then some discussions of the effects of impulsive perturbations and time delays to the system are referred to in the end of the paper.
Example 4.1
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11372294, 51349011), Scientific Research Fund of Sichuan Provincial Education Department (14ZB0115, 15ZB0113), and Doctoral Research Fund of Southwest University of Science and Technology (15zx7138).
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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