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Dynamic behaviors of a commensal symbiosis model involving Allee effect and one party can not survive independently
- Baoguo Chen^{1}Email author
https://doi.org/10.1186/s13662-018-1663-2
© The Author(s) 2018
- Received: 17 February 2018
- Accepted: 11 May 2018
- Published: 19 June 2018
Abstract
A two-species commensal symbiosis model involving Allee effect and one party can not survive independently is proposed and studied in this paper. Sufficient conditions which ensure the local and global stability of the boundary equilibrium and the positive equilibrium are obtained, respectively. Numeric simulations show that with the increasing of Alee effect, the system takes much longer time to reach its stable steady-state solution, though the Allee effect has no influence on the final density of the species. The Allee effect has instable effect on the system, however, such effect is controllable.
Keywords
- Commensal symbiosis model
- Stability
- Ratio-dependent
- Allee effect
MSC
- 34C25
- 92D25
- 34D20
- 34D40
1 Introduction
- 1.
The second species is favorable to the first species, while the first species has no influence on the second species;
- 2.
Without the help of the second species, the first species will be driven to extinction, i.e., the first species could not survive independently;
- 3.
We use the ratio-dependent functional response \(\frac{c_{1}y}{x+y}\) to describe the influence of the second species on the first species, where \(c_{1}\) describes the intensity of the cooperative effect of the second species on the first species;
- 4.We incorporate the Allee effect \(\alpha(y) = \frac{y}{u+y}\) on the second species, such an Allee effect describes the fact of limitations in finding mates, which is also called weak Allee effect function. \(\alpha(y)\) is the probability of finding a mate where u is the individuals searching efficiency [1]. The bigger u is the stronger Allee effect is. By introducing the Allee effect, the per capita growth rate of the second species is reduced fromto$$y (a_{2}-b_{2}y ) $$$$y (a_{2}-b_{2}y )\frac{y}{u+y}. $$
During the last decades, many scholars investigated the dynamic behaviors of the mutualism model or commensalism model, see [2–33] and the references cited therein. Such topics as the stability of the positive equilibrium [2, 4, 5, 11, 12, 14, 17, 19, 24, 27, 28], the persistence of the feedback control cooperative system [2, 6, 8–10, 20], the existence of the positive periodic solutions [15, 21, 23, 26], the existence of almost periodic solutions [22], etc. have been extensively investigated. However, only recently did scholars pay attention to the commensal symbiosis model with one party can not survive independently [26–31].
The Allee effect, which describes the fact that the phenomenon of reduced per-capita population growth rate at low densities can be caused by difficulties in finding a mate or predator, avoid danger or defense. As was pointed out by Wang, Zhang, and Liu [34], “The population goes extinct below a threshold and increases above the threshold of population density. The Allee effect is therefore important in conservation of endangered and exploited species.” During the last decades, many scholars studied the influence of Allee effect, see [1, 34–42] and the references cited therein. Kang and Yakuba [37] studied the population dynamics of a two-species discrete-time competition model where each species suffers from either predator saturation induced Allee effects or mate limitation induced Allee effects. Kang and Udiani [42] studied the dynamic behaviors of a single species with Allee effect. Some scholars [1, 40, 41] also considered the influence of Allee effect on the epidemic ecosystem.
Stimulated by the works of [1, 26–42], we propose system (1.1), where the second species is subjected to the Allee effect from the mating limitation.
We arrange the paper as follows. In the next section, we investigate the existence and local stability of the equilibria. In Sect. 3, we investigate the global stability property of boundary equilibrium and positive equilibrium of the system. In Sect. 4, an example together with its numeric simulations is presented to show the feasibility of our main results. We end this paper with a brief discussion.
2 The existence and local stability of the equilibria
Concerned with the local stability property of the above three equilibria, we have the following.
Theorem 2.1
\(A_{1} (0, \frac{a_{2}}{b_{2}} )\) is unstable if \(a_{1}< c_{1}\) and locally stable if \(a_{1}>c_{1}\); \(A_{2} (x^{*},y^{*} )\) is locally stable if \(a_{1}< c_{1}\).
Proof
This ends the proof of Theorem 2.1. □
3 Global stability of the equilibria
This section will further investigate the global stability property of the equilibria.
Lemma 3.1
([1])
Theorem 3.1
Assume that \(a_{1}>c_{1}\) holds, then \(A_{1} (0, \frac{a_{2}}{b_{2}} )\) is globally stable.
Proof
Theorem 3.2
Assume that \(a_{1}< c_{1}\) holds, then \(A_{2} (x^{*}, y^{*} )\) is globally stable.
Proof
4 Numeric simulations
Now let us consider the following example, which is the modification of Example 5.1 of Wu and Lin [27].
Example 4.1
- (1)
Now let us consider the influence of the parameter u. Let us take \(u=0.1, 0.5, 1\), respectively. Figure 5 and Fig. 6 show that in this case, parameter u has no influence on the final density of both species.
5 Discussion
Based on a commensalism model proposed by Wu and Lin [27], we propose a two-species commensal symbiosis model involving Allee effect and one party can not survive independently. We show that the dynamic behaviors of system (1.1) coincide with those of system (1.5), i.e., if \(a_{1}>c_{1}\), that is, the intrinsic death rate of the first species is larger than the commensalism effect between the species, then the first species will be driven to extinction; and if \(a_{1}< c_{1}\), that is, the cooperative effect between two species is larger than the intrinsic death rate of the first species, then two species could coexist in a stable state.
For the case \(a_{1}>c_{1}\), the boundary equilibrium is globally stable, which means the extinction of the first species and the global attractivity of the second species (see Fig. 1). Numeric simulations (Fig. 2, Fig. 3) show that with the increasing of Allee effect (the increasing of u), the first species almost takes the same time to drive to extinction, while the second species needs much more time to reach its stable state.
We mention here that a suitable system should incorporate some past state of the species, and this leads to a system with time delay, generally speaking, time delay may lead to the Hopf bifurcation or some other new phenomenon, we will leave this for future investigation.
Declarations
Acknowledgements
The author is grateful to three anonymous referees for their excellent suggestions, which have greatly improved the presentation of the paper.
Funding
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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