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- Open Access
Permanence and global attractivity of a discrete pollination mutualism in plant-pollinator system with feedback controls
- Rongyu Han1,
- Xiangdong Xie2 and
- Fengde Chen2Email author
https://doi.org/10.1186/s13662-016-0889-0
© Han et al. 2016
- Received: 30 August 2015
- Accepted: 6 June 2016
- Published: 28 July 2016
Abstract
In this paper, we propose a discrete pollination mutualism in a plant-pollinator system with the Beddington-DeAngelis functional response and feedback controls. By applying the comparison theorem of a difference equation and constructing some suitable Lyapunov functions, sufficient conditions are obtained for the permanence and the extinction of the system. Moreover, under some suitable conditions, we show that the solution of the system is globally attractive. The paper ends by some numerical simulations and a brief discussion.
Keywords
- plant-pollinator system
- feedback controls
- permanence
- extinction
- global attractivity
MSC
- 34C25
- 92D25
- 34D20
- 34D40
1 Introduction
In theoretical ecology, there are several famous functional responses in the ecosystem, which we refer to as Holling type-I, type-II, type-III, type-IV, Monod-Haldane type, and Hassel-Verley type functional response etc. Some authors studied the ecosystem with different types of functional responses. Beddington [1] and DeAngelis et al. [2] first proposed a predator dependent functional response (known as the B-D functional response). After that, a lot of scholars did work on the ecosystems with the Beddington-DeAngelis functional response. Chen and You [3] studied the permanence, extinction, and periodic solution of the periodic predator-prey system with a Beddington-DeAngelis functional response and stage structure for prey. Xiao [4] analyzed the existence and uniqueness of the positive equilibrium and its global asymptotic stability by using the qualitative methods of ordinary differential equation. In [5], the author focused on the uniform persistence, local stability, and global stability for a Beddington-DeAngelis type stage structures predator-prey model. Furthermore, Chen et al. [6] with the help of a fluctuation lemma obtained a set of new conditions on the global asymptotic stability of the boundary solution.
We mention here that, as far as the system (1.4) is concerned, whether the system is to persist is a question that we need to solve. Furthermore whether the feedback-control variables have influence on the extinction and the stability of the system or not is an interesting problem. The aim of this paper is to solve the above questions.
The remainder of the paper is organized as follows: in Section 2, we introduce some useful lemmas and obtain the sufficient conditions to guarantee the permanence and the extinction of system (1.4). In Section 3, a set of sufficient conditions which ensure the stability of the system are obtained. In Section 4, we give some examples to illustrate our results, and we end this paper by a brief discussion.
2 Permanence
In this section, we establish a permanence result for system (1.4). First, let us state several lemmas which will be useful in proving the main results.
Lemma 2.1
[12]
Lemma 2.2
[12]
Lemma 2.3
Lemma 2.4
[13]
Lemma 2.5
[14]
Proposition 2.6
Proof
Theorem 2.7
Proof
Theorem 2.8
Proof
Further, consider the fourth equation of system (1.4). Applying Lemma 2.3 in [15], we easily obtain \(u_{2}(n)\rightarrow0\) as \(n\rightarrow+\infty\). This completes the proof of Theorem 2.8. □
Theorem 2.9
Proof
3 Global attractivity
In this section, we will consider the stability of the system (1.4).
Theorem 3.1
Proof
4 Examples
In the section, we present some examples showing the feasibility of our main results.
Example 4.1
Dynamics behavior of the solution \(\pmb{{(x_{1}(n),x_{2}(n),u_{1}(n),u_{2}(n))}}\) to the system ( 4.1 ) with the initial conditions \(\pmb{(x_{1}(0),x_{2}(0),u_{1}(0),u_{2}(0))= (1.4,0.2,1.2,0.9), (3.2, 2.1,4.6,2.5), \mbox{and }(5.8,3.9,0.8,4.2)}\) , respectively.
Example 4.2
Dynamics behavior of the solution \(\pmb{{(x_{1}(n),x_{2}(n),u_{1}(n),u_{2}(n))}}\) of the system ( 4.2 ) with the initial conditions \(\pmb{(x_{1}(0),x_{2}(0),u_{1}(0),u_{2}(0))= (0.05,0.07,0.01,0.04), (0.06, 0.045,0.032,0.021), \mbox{and } (0.04,0.028,0.042,0.008)}\) , respectively.
5 Conclusion
In this paper, we proposed a discrete non-autonomous plant-pollinator system with the Beddington-DeAngelis functional response and feedback controls. As we see it, plants can build a cooperative interaction with pollinators by providing a reward for the pollinators’ services. From Theorem 2.7, we discover that when \(\alpha_{21}(n)\) is large enough, then (2.10) must hold, that is, the system (1.4) has an upper bound. We know that \(\alpha_{21}(n)\) represents the pollinators’ efficiency in translating plant-pollinator interactions into fitness. In other words, when the efficiency of the pollinators is large enough, the system has an upper bound. What is more, when the coefficient \(e_{1}(n)\) is small enough, then the system has permanence. That is, when the interference of the plant is small, the system is to persist. From Theorem 2.8, it is obvious that when \(r_{2}(n)\) is large enough, \(x_{2}(n)\) will contribute to extinction. That is, if the mortality is large enough, then the population will go to extinction. From Theorem 2.9, we known that when \(e_{1}(n)\) is small enough, then the partial species is globally stable. That is, if the feedback control is small enough, the population may remain stable. Wang et al. [10] have shown that the system (1.3) is globally stable, and our work shows that the feedback controls have no influence on the attractivity of the system. The obtained results may be helpful to maintain the plant-pollinator cooperation and provide insight in the mechanisms by which pollination mutualism could persist and we have global attractivity, which may be helpful for understanding the complexity of these systems.
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. 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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