- Research
- Open Access
Stability analysis of a single species logistic model with Allee effect and feedback control
- Qifa Lin^{1}Email author
https://doi.org/10.1186/s13662-018-1647-2
© The Author(s) 2018
- Received: 21 March 2018
- Accepted: 3 May 2018
- Published: 18 May 2018
Abstract
Keywords
- Logistic model
- Allee effect
- Feedback control
- Global stability
MSC
- 34C25
- 92D25
- 34D20
- 34D40
1 Introduction
It came to our attention that, to this day, still no scholars have investigated the ecosystem with both Allee effect and feedback control. As we all know, the Allee effect is one of the most frequently seen phenomena since more and more species become endangered, and such kind of species have difficulties in finding mates, social dysfunction is present at small population sizes. On the other hand, the feedback control variable represents the harvesting of the human beings [1], which is one of the most important factors that leads to reduction of the amount of the species. Stimulated by the works mentioned above, in this paper, we propose and study the dynamic behaviors of system (1.1).
The paper is arranged as follows. In Sect. 2, we investigate the dynamic behaviors of system (1.1) without feedback control; and system (1.1) without Allee effect is studied in Sect. 3. We investigate the stability property of the equilibria of system (1.1) in Sect. 4. Section 5 presents some numerical simulations to show the feasibility of the main results. We end this paper by a brief discussion.
2 Dynamic behaviors of system (1.1) without feedback control
As far as system (2.1) is concerned, we have the following result.
Theorem 2.1
The unique positive equilibrium \(x^{**}=1\) of system (2.1) is globally attractive.
Proof
It is a direct corollary of Lemma 2.1 of Wu et al. [28], and we omit the detailed proof here. □
3 Dynamic behaviors of system (1.1) without Allee effect
Theorem 3.1
\(B(x^{*},u^{*})\) is locally asymptotically stable, \(A(0,0)\) is unstable.
Proof
This ends the proof of Theorem 3.1. □
Theorem 3.1 shows that the positive equilibrium is locally asymptotically stable. One interesting issue is whether it is a globally stable one, we give an affirmative answer to this issue. Indeed, we have the following.
Theorem 3.2
The unique positive equilibrium \(B(x^{*},u^{*})\) of system (3.1) is globally asymptotically stable.
Proof
4 Dynamic behaviors of system (1.1)
Now let us consider the dynamic behaviors of system (1.1).
Concerned with the local stability property of the above two equilibria, we have the following.
Theorem 4.1
- (1)$$ \beta< \frac{b^{2}r^{2}}{ac(ac+2br)}; $$(4.4)
- (2)$$ \frac{b^{2}r^{2}}{ac(ac+2br)}< \beta< \frac{br}{ac} \quad \textit{and} \quad b>r; $$(4.5)
Proof
This ends the proof of Theorem 4.1. □
5 Numerical simulations
Now let us consider the following two examples.
Example 5.1
Example 5.2
6 Discussion
- (1)To ensure that system (1.1) has positive steady-state, the Allee effect should be restricted so that the inequalityholds. If$$ \beta< \frac{br}{ac} $$(6.3)holds, then system (1.1) has no positive equilibrium and, as it was shown in Fig. 4, the species will be driven to extinction.$$ \beta>\frac{br}{ac} $$(6.4)
- (2)
By introducing the Allee effect, the boundary equilibrium \(A_{1}(0,0)\) of system (4.1) becomes non-hyperbolic. We could not judge its stability property by using the Jacobian matrix, and we have to develop some new analysis technique. Here, by transforming the system to the standard form, we could judge the stability property of the equilibrium by using Theorem 7.1 in Zhang et al. [33].
- (3)Forwe showed that system (1.1) also admits a unique positive equilibrium which is locally asymptotically stable. However, with the increase in the Allee effect, if the inequality$$ \beta< \frac{b^{2}r^{2}}{ac(ac+2br)}, $$(6.5)holds, to ensure the positive equilibrium is locally asymptotically stable, we have to make some restriction on the feedback control variable, i.e., the inequality$$ \frac{b^{2}r^{2}}{ac(ac+2br)}< \beta< \frac{br}{ac} $$(6.6)holds.$$ b>r $$(6.7)
- (4)Note that the positive equilibrium \(B_{1}(x_{1}^{*}, u_{1}^{*})\) of system (1.1) takes the formSince$$ x_{1}^{*}=\frac{br-a\beta c}{ac+br},\qquad u_{1}^{*}= \frac {c(br-a\beta c)}{b(ac+br)}. $$(6.8)it follows that \(x_{1}^{*}\) is the strictly decreasing function of β, that is, the Allee effect reduces the population densities.$$ \frac{dx_{1}^{*}}{d\beta}=-\frac{ac}{ac+br}< 0, $$(6.9)
To sum up, the system incorporating the Allee effect becomes “unstable”: it becomes weak in the sense that it could not endure the large disturbance and the density of the species decreases with the Allee effect, which may accelerate the extinction of the species.
Declarations
Acknowledgements
The author is grateful to anonymous referees for their excellent comments. This work is 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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