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- Open Access
Dynamic behaviors of a nonlinear amensalism model
- Runxin Wu^{1}Email author
https://doi.org/10.1186/s13662-018-1624-9
© The Author(s) 2018
- Received: 23 March 2018
- Accepted: 26 April 2018
- Published: 18 May 2018
Abstract
Keywords
- Amensalism model
- Differential inequality theory
- Global stability
MSC
- 34C25
- 92D25
- 34D20
- 34D40
1 Introduction
As far as system (1.3) is concerned, one interesting issue is:
Find out the influence of the parameter \(\alpha_{i}\), \(i=1, 2, 3\), which reflects the influence of the nonlinearity.
The paper is arranged as follows. We investigate the existence and local stability property of the equilibrium solutions of system (1.3) in the next section. In Sect. 3, by applying the differential inequality theory, we investigate the global stability property of the equilibria. The influence of the parameter \(\alpha_{i}, i=1, 2, 3\), 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
We shall now investigate the local stability property of the above equilibria.
Theorem 2.1
- (1)
\(A(0,0)\) is unstable;
- (2)
\(B(P_{1}, 0) \) is a saddle point, thus, is unstable;
- (3)
if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}>0\), \(C(0, P_{2})\) is a saddle point and consequently unstable; if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0\), \(C(0, P_{2})\) is a stable node;
- (4)
if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}>0\), \(D(N_{1}^{*},N_{2}^{*})\) is a stable node.
Remark 2.1
If \(\alpha_{i}=1\), \(i=1, 2, 3\), then Theorem 2.1 degenerates to the main result of Xiong et al. [1], hence, we generalize the main result of [1]. Note that the boundary equilibria are independent of \(\alpha_{i}, i=1, 2, 3\), hence, \(\alpha_{i}\), \(i=1, 3\), has no influence on the existence and stability of the boundary equilibria.
Remark 2.2
From (2.1), the second term in \(J(N_{1}, N_{2})\) is \(-\frac{r_{1}uN_{1}\alpha_{2}N_{2}^{\alpha_{2}-1}}{P_{1}^{\alpha _{2}}}\), which means that if \(\alpha_{2}<1\), then at \(N_{2}=0\), the value of this term could not be computed. Hence, for \(0<\alpha_{2}<1\) case, the local stability of the equilibrium \(A(0,0)\) and \(B(P_{1}, 0)\) could not be determined by analyzing the Jacobian matrix.
Proof of Theorem 2.1
The proof of Theorem 2.1 is finished. □
3 Global stability
As was pointed out in the previous section, for the case \(\alpha_{2}<1\), the local stability property of the boundary equilibrium \(A(0,0)\) and \(B(P_{1},0)\) could not be determined by using the Jacobian matrix (2.1). The aim of this section is to try to solve this problem and to further investigate the global stability property of the equilibria of system (1.3).
Lemma 3.1
Proof
- (i)
For all \(N_{2}^{*} > N_{2} > 0, F(N_{2}) > 0\).
- (ii)
For all \(N_{2} > N_{2}^{*} > 0, F(N_{2}) < 0\).
The next lemma is a direct corollary of Lemma 2.2 of Chen [16].
Lemma 3.2
Theorem 3.1
- (a)
Assume that \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0\), then \(C(0, P_{2})\) is globally attractive;
- (b)
Assume that \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}>0\), then \(D(N_{1}^{*},N_{2}^{*})\) is globally attractive.
Proof
Remark 3.1
Theorems 2.1 and 3.1 show that if system (1.3) admits the unique positive equilibrium, then the positive equilibrium is globally attractive.
Remark 3.2
Noting that if \(C(0,P_{2})\) or \(D(N_{1}^{*}, N_{2}^{*})\) is globally attractive, then all the solutions with positive initial conditions will finally asymptotically to the equilibrium, which means that the solutions with positive solution could not be asymptotically to \(A(0,0)\) and \(B(P_{1},0)\), thus, \(A(0,0)\) and \(B(P_{1},0)\) is unstable. Theorem 3.1 shows that for almost all the cases (only \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}=0\) could not be determined), \(A(0,0)\) and \(B(P_{1},0)\) are unstable.
4 The influence of the parameter \(\alpha_{i}\)
- (1)
If \(P_{2} >P_{1} \), then \(\frac{\partial N_{1}^{*}}{\partial\alpha_{2}}<0\), and \(N_{1}^{*}\) is the strictly decreasing function of \(\alpha_{2}\);
- (2)
If \(P_{2} < P_{1}\), then \(\frac{\partial N_{1}^{*}}{\partial\alpha_{2}}>0\), and \(N_{1}^{*}\) is the strictly increasing function of \(\alpha_{2}\).
- (3)
If \(P_{2}=P_{1}\), then \(\frac{\partial N_{1}^{*}}{\partial\alpha_{2}}=0\), and \(N_{1}^{*}\) is independent of \(\alpha_{2}\).
5 Numeric simulations
Example 5.1
Example 5.2
Example 5.3
Example 5.4
- (1)
- (2)
- (3)
- (4)
6 Discussion
Stimulated by the works of Xiong et al. [1] and Chen et al. [14–19], in this paper, we propose the nonlinear amensalism model (1.3).
We first investigated the local stability property of the equilibria, and we found that, by introducing the nonlinear term, the situation became complicated, and only under the assumption \(\alpha _{2}\geq1\) could we give the local stability result of \(A(0,0)\) and \(B(P_{1},0)\). To overcome this difficulty, we used the differential inequality theory, and finally proved the following results: if the positive equilibrium exists, then it is globally attractive.
We also investigated the relationship among \(N_{1}^{*}\) and \(\alpha _{i}\), and found that \(\alpha_{2}\) and the ratio of \(\frac{P_{2}}{P_{1}}\) play most important roles on the final density of the first species.
Declarations
Acknowledgements
The author is grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. The research was supported by the National Natural Science Foundation of China under Grant (11601085) and the Natural Science Foundation of Fujian Province (2017J01400).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The author declares 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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