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Dynamic behaviors of a non-selective harvesting Lotka–Volterra amensalism model incorporating partial closure for the populations
- Baoguo Chen^{1}Email author
https://doi.org/10.1186/s13662-018-1555-5
© The Author(s) 2018
- Received: 14 January 2018
- Accepted: 12 March 2018
- Published: 27 March 2018
Abstract
A non-selective harvesting Lotka–Volterra amensalism model incorporating partial closure for the populations is proposed and studied in this paper. Local and global stability of the boundary and interior equilibria are investigated. By introducing the harvesting, the dynamic behaviors of the system become complicated. Depending on the fraction of the stock available for harvesting, the system maybe extinction, partial survival or two species may coexist in a stable state. Our results supplement and complement the main results of Xiong, Wang, and Zhang (Adv. Appl. Math. 5(2):255-261, 2016).
Keywords
- Amensalism model
- Lyapunov function
- Global stability
MSC
- 34C25
- 92D25
- 34D20
- 34D40
1 Introduction
Theorem A
- (1)
\(A(0,0)\) is unstable;
- (2)
\(B(P_{1}, 0) \) is a saddle point, thus is unstable;
- (3)
if \(u<\frac{P_{1}}{P_{2}}\), \(C(0, P_{2})\) is a saddle point and consequently unstable; if \(u>\frac{P_{1}}{P_{2}}\), \(C(0, P_{2})\) is a stable node;
- (4)
if \(u<\frac{P_{1}}{P_{2}}\), \(D(P_{1}-uP_{2}, P_{2})\) is a stable node.
On the other hand, as was pointed out by Chakraborty et al. [16], the study of resource management, including fisheries, forestry, and wildlife management, has great importance. They argued that it is necessary to harvest the population, but harvesting should be regulated so that both the ecological sustainability and conservation of the species can be implemented in a long run. Already, they proposed a non-selective harvesting predator–prey system incorporating partial closure for the populations, they investigated the local and global stability property of the system, and some interesting results related to the optimal harvesting were obtained.
As far as system (1.2) is concerned, one interesting issue is the following:
Find out the influence of the parameter m, which reflects the fraction of the stock available for harvesting.
The paper is arranged as follows. We investigate the existence and locally stability property of the equilibrium solutions of system (1.2) in the next section. In Sect. 3, by constructing some suitable Lyapunov function, we investigate the global stability property of the equilibria. The influence of the parameter m is then discussed in Sect. 4. Some examples together with their numeric simulations are presented in Sect. 5 to show the feasibility of the main results. We end this paper with a brief discussion.
2 Local stability of the equilibria
The system always admits the boundary equilibrium \(A(0,0)\).
If \(r_{1}>Emq_{1}\) holds, the system admits the boundary equilibrium \(B(N_{10}, 0)\), where \(N_{10}= \frac{P_{1}(r_{1}-Emq _{1})}{r_{1}}\).
If \(r_{2}>Emq_{2}\) holds, the system admits the boundary equilibrium \(C(0, N_{20})\), where \(N_{20}= \frac{P_{2}(r_{2}-Emq _{2})}{r_{2}}\).
Theorem 2.1
- (1)Assume thatholds, then \(A(0,0)\) is locally stable, otherwise it is unstable;$$ m>\max \biggl\{ \frac{r_{1}}{Eq_{1}}, \frac{r_{2}}{Eq_{2}} \biggr\} $$(2.2)
- (2)Assume thatholds, then \(B(N_{10}, 0)\) is locally stable, otherwise it is unstable;$$ \frac{r_{2}}{Eq_{2}}< m< \frac{r_{1}}{Eq_{1}} $$(2.3)
- (3)Assume thatholds, then \(C(0, N_{20})\) is locally stable, otherwise it is unstable;$$ \frac{r_{1}r_{2}(P_{1}-uP_{2})}{r_{2}q_{1}EP_{1}- r_{1}uEP_{2}q _{2}} < m< \frac{r_{2}}{Eq_{2}} $$(2.4)
- (4)Assume thatholds, then \(D(N_{1}^{*}, N_{2}^{*})\) is locally stable.$$ m< \min \biggl\{ \frac{r_{2}}{Eq_{2}}, \frac{r_{1}r_{2}(P_{1}-uP_{2})}{r _{2}q_{1}EP_{1}- r_{1}uEP_{2}q_{2}} \biggr\} $$(2.5)
Proof
The proof of Theorem 2.1 is finished. □
3 Global stability
One interesting problem is to further investigate the global stability property of the equilibria of system (1.2), since the global one means that despite the random initial condition, the finial dynamic behaviors of the system could be forecasted. In this aspect, we could obtain the following result.
Theorem 3.1
- (1)Assume thatholds, then \(A(0,0)\) is globally asymptotically stable;$$ m>\max \biggl\{ \frac{r_{1}}{Eq_{1}}, \frac{r_{2}}{Eq_{2}} \biggr\} $$(3.1)
- (2)Assume thatholds, then \(B(N_{10}, 0)\) is globally asymptotically stable;$$ \frac{r_{2}}{Eq_{2}}< m< \frac{r_{1}}{Eq_{1}} $$(3.2)
- (3)Assume thatholds, then \(C(0, N_{20})\) is globally asymptotically stable;$$ \frac{r_{2}}{Eq_{2}}>m> \frac{r_{1}}{Eq_{1}} $$(3.3)
- (4)Assume thatholds, then \(D(N_{1}^{*}, N_{2}^{*})\) is globally asymptotically stable.$$ m< \min \biggl\{ \frac{r_{2}}{Eq_{2}}, \frac{r_{1}r_{2}(P_{1}-uP_{2})}{r _{2}q_{1}EP_{1}- r_{1}uEP_{2}q_{2}} \biggr\} $$(3.4)
Proof
We will prove Theorem 3.1 by constructing some suitable Lyapunov functions.
Remark 3.1
Theorems 2.1 and 3.1 show that if system (1.2) admits the unique positive equilibrium, then the positive equilibrium is globally asymptotically stable.
Remark 3.2
Compared with Theorems 2.1 and 3.1, one could see that in three cases, the local stability of the equilibrium also implies the global one. However, to ensure \(C(0,N_{20})\) is globally stable, we need assumption (3.3) since our condition is a set of sufficient conditions, maybe it is not the necessary one. Whether (2.4) is enough to ensure the globally attractivity of \(C(0,N_{20})\) or not is still unknown. Obviously, we could not deal with this problem by constructing a suitable Lyapunov function.
Remark 3.3
From Theorem 3.1(4) and the biological meaning of the parameter m, we can draw the conclusion: if the fraction of the stock available for harvesting is limited, then two species could coexist in the long run, despite the initial state.
4 The influence of the parameter m
- (1)
If \(P_{2}q_{2}r_{1}u>P_{1}q_{1}r_{2}\), then \(\frac{dN_{1}^{*}}{dt}>0\), and \(N_{1}^{*}\) is the strictly increasing function of m;
- (2)
If \(P_{2}q_{2}r_{1}u< P_{1}q_{1}r_{2}\), then \(\frac{dN_{1}^{*}}{dt}<0\), and \(N_{1}^{*}\) is the strictly decreasing function of m.
5 Numerical simulations
Example 5.1
- (1)
- (2)
- (3)
- (4)
6 Discussion
With the aim of the ecological sustainability and conservation of the species to be implemented in a long run, in this paper, we propose a non-selective harvesting Lotka–Volterra amensalism model incorporating partial closure for the populations, i.e., system (1.2), which can be seen as the generalization of system (1.1), and the model is more suitable for the real situation.
With the introducing of harvesting, the dynamic behaviors of the system become very complicated. Depending on the fraction of the stock that could be harvested, the system may have positive equilibrium, which is globally asymptotically stable, which means that two species could coexist in a stable state; or one of the species will be driven to extinction, or both of the species could be driven to extinction.
To sum up, to ensure the conservation of the species, we need to restrict the harvesting to a limited area. Otherwise, although we can afford the area which could not be harvested, the species may still be driven to extinction. Theorem 2.1 and 3.1 give some threshold on m, which ensures the coexistence of the two species. The results obtained in this paper maybe useful in designing the natural protection area.
Declarations
Acknowledgements
The author is grateful to anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. This work is supported by the National Social Science Foundation of China (16BKS132), Humanities and Social Science Research Project of Ministry of Education Fund (15YJA710002) and the Natural Science Foundation of Fujian Province (2015J01283).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests.
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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