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- Open Access
The influence of partial closure for the populations to a non-selective harvesting Lotka–Volterra discrete amensalism model
- Qianqian Su^{1}Email author and
- Fengde Chen^{2}
https://doi.org/10.1186/s13662-019-2209-y
© The Author(s) 2019
- Received: 3 January 2019
- Accepted: 20 June 2019
- Published: 11 July 2019
Abstract
In this paper, a non-selective harvesting Lotka–Volterra amensalism discrete model incorporating partial closure for the populations is proposed and studied. By applying the relevant conclusions of difference inequality and some calculation technique, sufficient conditions are obtained to ensure the permanence and extinction of the system. By constructing a suitable Lyapunov function, sufficient conditions that ensure the global attractivity of the system are obtained. Finally, numerical simulations show the feasibility of our results.
Keywords
- Amensalism
- Harvesting
- Permanence
- Extinction
- Globally attractive
1 Introduction
As we all know, though most dynamic behaviors of population models are based on the continuous models governed by differential equations, the discrete time models governed by difference equation are more appropriate than the continuous ones when the size of the population is rarely small or the population has non-overlapping generations. It has been found that the dynamic behaviors of the discrete system is rather complex and contains richer dynamics than the continuous ones [14]. Recently, more and more scholars pay attention to studying the discrete population models (see [14–19] and the references cited therein).
Here, for any bounded sequence \(\{a(n) \}\), \(a^{u}= \sup_{n\in N} \{a(n) \}\), \(a^{l}=\inf_{n\in N} \{a(n) \}\).
From the point of view of biology, we assumed that \(x_{i}(0)>0\), (\(i=1,2\)). Then it is easy to see that the solutions of (1.3) with the above initial condition remain positive for all \(n\in N^{+}= \{0,1,2,\ldots \}\).
The organization of this paper is as follows. In Sect. 2, we give some useful lemmas. Sufficient conditions for the permanence and extinction of (1.3) are given in Sect. 3 and Sect. 4. Then, in Sect. 5, we establish sufficient conditions for the global attractivity of (1.3). Some examples together with their numeric simulations are presented in Sect. 6. We end this paper with a brief discussion.
2 Preliminaries
In this section, we will introduce several useful lemmas.
Lemma 2.1
([21])
Lemma 2.2
([22])
3 Permanence
Theorem 3.1
Proof
So the proof of Theorem 3.1 is completed. □
4 Extinction
Theorem 4.1
Proof
In this section, we will use the analysis technique of [14].
Theorem 4.2
Proof
The proof of Theorem 4.2 is completed. □
5 Globally attractive
Theorem 5.1
Proof
The proof of Theorem 5.1 is completed. □
Similarly, we can get the following theorem.
Theorem 5.2
6 Examples and numeric simulations
The following examples lend credence to the plausibility of the main results.
Example 6.1
Example 6.2
Example 6.3
Example 6.4
7 Discussion
With the aim of the ecological sustainability and conservation of the species to be implemented in a long run, in this paper, we have attempted to study the dynamic behaviors of a non-selective harvesting Lotka–Volterra discrete amensalism model. We have proved that if \((H_{1})\) holds, then the system is permanent, which means that if m, which is the fraction of the stock available for harvesting, is small enough, the system will coexist. Theorem 4.1 implies that if m is large enough, then the system will be driven to extinction. Theorem 4.2 gives some threshold on m, which ensures that the species \(x_{2}\) is permanent while \(x_{1}\) will be driven to extinction. In Sect. 5, sufficient conditions for the global attractivity of (1.3) are given, which means that if m is larger than a certain value and satisfies \((H_{5})\), then the species \(x_{1}\) is globally attractive while \(x_{2}\) will be driven to extinction. The results obtained in this paper maybe useful in designing the natural protection area.
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Funding
This work is supported by the Natural Science Project of Fujian province (2015 J 01012, 2015 J 01019) and the Funding of the Young Key Teachers Training Program of Zhengzhou Chenggong University of Finance and Economics.
Authors’ contributions
QQS is a major contributor in writing the manuscript. FDC is responsible for numerical simulation and drawing. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
We agree to publish in your journal.
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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