Allee effect increasing the final density of the species subject to the Allee effect in a Lotka–Volterra commensal symbiosis model
- Qifa Lin^{1}Email author
https://doi.org/10.1186/s13662-018-1646-3
© The Author(s) 2018
Received: 1 April 2018
Accepted: 9 May 2018
Published: 22 May 2018
Abstract
A Lotka–Volterra commensal symbiosis model with first species subject to the Allee effect is proposed and studied in this paper. Local and global stability property of the equilibria are investigated. An amazing finding is that with increasing Allee effect, the final density of the species subject to the Allee effect is also increased. Such a phenomenon is different from the known results, and it is the first time to be observed. Numeric simulations are carried out to show the feasibility of the main results.
Keywords
MSC
1 Introduction
- (1)
\(F^{'}(x)= \frac{\beta }{(\beta +x)^{2}}>0\) for all \(x\in (0,+ \infty )\), that is, the Allee effect decreases as density increases;
- (2)
\(\lim_{x\rightarrow +\infty }F(x)=1\), that is, the Allee effect vanishes at high densities.
Theorem A
The positive equilibrium \(P_{0}(x_{0},y_{0})\) of system (1.2) is globally stable.
It came to our attention that the Allee effect has different influence on systems (1.4) and (1.5). The Allee effect reduces the density of the species in system (1.4), while it has no influence on the final density of the species in system (1.5). Maybe the reason is that the authors made different assumptions: in system (1.4), the authors assumed that the first species (prey species) admits the Allee effect, while in system (1.5), the authors assumed that the second species is subjected to the Allee effect. One of the interesting issues proposed is as follows: Noting that [37] studied the influence of the Allee effect on the commensalism model, it is natural to ask: what would happen if we assumed the first species suffers to the Allee effect in a commensalism model? This leads us to proposing system (1.1).
We arrange the paper as follows. In the next section, we investigate the existence and local stability property of the equilibria of system (1.1). In Sect. 3, the Dulac criterion is applied to investigate the global stability property of positive equilibrium of system (1.1). In Sect. 4, an example together with its numeric simulations is presented to show the feasibility of the main results. We end this paper with a brief discussion.
2 Local stability
Theorem 2.1
\(D_{1}(x^{*}; y^{*})\) is locally asymptotically stable; \(A_{1}(0,0),B_{1}(\frac{b_{1}}{a_{11}},0)\) and \(C_{1}(0,\frac{b _{2}}{a_{22}})\) are all unstable.
Proof
This ends the proof of Theorem 2.1. □
3 Global stability
Previously, we have shown that three boundary equilibria of system (1.1) are all unstable, and the positive equilibrium is locally asymptotically stable. In this section, we will investigate the global dynamic behaviors of system (1.1). Indeed, we have the following result.
Theorem 3.1
The positive equilibrium \(D_{1}(x^{*},y^{*})\) of system (1.1) is globally asymptotically stable.
Proof
Remark 3.1
Compared with Theorem 3.1 and Theorem A, one could easily see that the Allee effect has no influence on the stability property of the positive equilibrium, that is, for the system with or without Allee effect, the system always admits a unique positive equilibrium, which is globally asymptotically stable.
Remark 3.2
Noting that \(D_{1}(x^{*},y^{*})\) is the unique positive equilibrium of system (1.1), which is locally asymptotically stable and globally asymptotically stable, and \(x^{*}\) depends on the parameter β, it means that the Allee effect could have influence on final density of the species. However, it could not lead to the extinction of the species, since the positive equilibrium is always globally attractive. Such a finding is very different to that of the Allee effect to the predator–prey system as shown by Hüseyin Merdan [36], where with increase in the Allee effect, the species may be driven to extinction, and the commensalism model with Allee effect on the second species as shown by Wu et al. [37], where the Allee effect has no influence on the final density of the species.
4 Numeric simulations
Example 4.1
5 Discussion
Many scholars incorporated the Allee effect to the ecosystem and considered the dynamic behaviors of the system, see [28–37]. Some of them [29, 36, 37] focused on the stability property of the positive equilibrium. Çelik and Duman [29] showed that Allee effects have a stabilizing role in the discrete-time predator–prey model, while Merdan [36] showed that the continuous predator–prey system subject to an Allee effect takes a much longer time to reach its stable steady-state solution. Such a phenomenon also was observed in the commensalism model [37]. It came to our attention that with increasing Allee effect, the density of the predator and prey species both decrease [36], and the density of the second species has no change for the commensalism model [37]. In this paper, we showed that the density of the first species is the increasing function of the Allee effect. Such a phenomenon is observed for the first time. Note that with increasing final density, the chance for the extinction of the species will be reduced. It seems that the cooperation between the species plays a crucial role in the persistence property of the endangered species. Also, the more endangered species could have more benefit from cooperation with other 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 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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