Permanence of the periodic predator-prey-mutualist system
- Liya Yang^{1}Email author,
- Xiangdong Xie^{2},
- Fengde Chen^{2} and
- Yalong Xue^{1}
https://doi.org/10.1186/s13662-015-0654-9
© Yang et al. 2015
Received: 11 June 2015
Accepted: 30 September 2015
Published: 23 October 2015
Abstract
In this paper, we study the permanence and the periodic solution of the periodic predator-prey-mutualist system. It is well known that mutualist species can reduce the capture rate of the predator species to the prey species. By further developing the analysis technique of Teng, a set of conditions which ensure the permanence of the system are obtained. In addition, sufficient conditions are derived for the existence of positive periodic solutions to the system. An example together with its numerical simulation shows the feasibility of the main results.
Keywords
MSC
1 Introduction
As was pointed out by Berryman [1], the dynamic relationship between predator and prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Already the predator-prey model has been studied by several scholars [2–10]. For example, Das et al. [8] investigated a three species ecosystem consisting of a prey, a predator, and a top predator. They derived the criteria for local and global stability of all the eight equilibrium points by using a Routh-Hurwitz and Lyapunov function. Wu and Li [9] studied the permanence and global attractivity of the discrete predator-prey system with Hassell-Varley-Holling type III functional response. Chen and Chen [10] proposed a ratio-dependent predator-prey model incorporating a prey refuge. They studied the global stability, limit cycle, and Hopf bifurcation of the system.
We arrange the rest of the paper as follows: In Section 2, we introduce one lemma and state the main results of this paper. The results are proved in Section 3. In Section 4, a suitable example together with its numeric simulation is present to show the feasibility of the main results. We end this paper by a briefly conclusion. For more works on the non-autonomous predator-prey system, one could refer to [16–19] and the references cited therein.
2 Statement of the main results
Lemma 2.1
[20]
If \(\beta(t)\geq0\) for all \(t\in R\) and \(\int_{0}^{w}\beta(t)\, dt>0\), then (2.1) has a unique nonnegative w-periodic solution \(u^{*}(t)\) which is globally asymptotically stable, that is, \(u(t)-u^{*}(t)\rightarrow0\) as \(t\rightarrow\infty\) for any positive solution \(u(t)\) of (2.1). Moreover, if \(\int_{0}^{w}\alpha(t)\, dt>0\), then \(u^{*}(t)>0\) for all \(t\in R\) and if \(\int_{0}^{w}\alpha(t)\, dt\leq0\) then \(u^{*}(t)\equiv0\).
Definition 2.2
As concerns the persistent property of the system (1.2), we have the following result.
Theorem 2.3
As a direct corollary of Theorem 2 in [21], from Theorem 2.3, we have the following.
3 Proof of the main result
We need the following propositions to prove Theorem 2.3.
Proposition 3.1
Proof
Proposition 3.2
Proof
Proposition 3.3
Proof
Proposition 3.4
Proof
Proposition 3.5
Proof
Proposition 3.6
Proof
Proposition 3.7
Proof
3.1 Proof of Theorem 2.3
The results of Theorem 2.3 now follow from Propositions 3.1-3.7.
4 Example
In this section, we shall give an example to illustrate the feasibility of the main results.
Example
5 Conclusion
In this paper, we studied a periodic predator-prey-mutualist system. From system (1.2), we see the mutualist species y can reduce the capture rate of the predator species z to the prey species x. By further developing the analysis technique of Teng [20], we obtain a set of conditions which ensure the permanence of system (1.2). Note that \(u_{10}(t)\) and \(u_{20}(t)\) are the globally attractive periodic solution of (2.2) and (2.3), respectively, which, as shown by Lemma 2.1, always exists. Hence, the left side of condition (2.4) implies that if the death rate of the predator species is enough small and the cooperation effect between species x and y is not very strong, the system is permanent.
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. Also, the research was supported by the Natural Science Foundation of Fujian Province (2015J010121, 2015J01019).
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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