Stability analysis of a single species logistic model with Allee effect and feedback control
Abstract A single species logistic model with Allee effect and feedback control dxdt=rx(1−x)xβ+x−axu,dudt=−bu+cx, $$\begin{aligned}& \frac{dx}{dt} = rx(1-x)\frac{x}{\beta+x}-axu, \\& \frac{du}{dt} = -bu+cx, \end{aligned}$$ where β, r, a, b, and c are all positive constants, is for the first...
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doaj-ec3801e8a418444ca983b253b813baf32020-11-24T21:37:59ZengSpringerOpenAdvances in Difference Equations1687-18472018-05-012018111310.1186/s13662-018-1647-2Stability analysis of a single species logistic model with Allee effect and feedback controlQifa Lin0Department of Mathematics, Ningde Normal UniversityAbstract A single species logistic model with Allee effect and feedback control dxdt=rx(1−x)xβ+x−axu,dudt=−bu+cx, $$\begin{aligned}& \frac{dx}{dt} = rx(1-x)\frac{x}{\beta+x}-axu, \\& \frac{du}{dt} = -bu+cx, \end{aligned}$$ where β, r, a, b, and c are all positive constants, is for the first time proposed and studied in this paper. We show that, for the system without Allee effect, the system admits a unique positive equilibrium which is globally attractive. However, for the system with Allee effect, if the Allee effect is limited ( β<b2r2ac(ac+br) $\beta<\frac{b^{2}r^{2}}{ac(ac+br)}$), then the system could admit a unique positive equilibrium which is locally asymptotically stable; if the Allee effect is too large ( β>brac $\beta>\frac{br}{ac}$), the system has no positive equilibrium, which means the extinction of the species. The Allee effect reduces the population density of the species, which increases the extinction property of the species. The Allee effect makes the system “unstable” in the sense that the system could collapse under large perturbation. Numeric simulations are carried out to show the feasibility of the main results.http://link.springer.com/article/10.1186/s13662-018-1647-2Logistic modelAllee effectFeedback controlGlobal stability |
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DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Qifa Lin |
spellingShingle |
Qifa Lin Stability analysis of a single species logistic model with Allee effect and feedback control Advances in Difference Equations Logistic model Allee effect Feedback control Global stability |
author_facet |
Qifa Lin |
author_sort |
Qifa Lin |
title |
Stability analysis of a single species logistic model with Allee effect and feedback control |
title_short |
Stability analysis of a single species logistic model with Allee effect and feedback control |
title_full |
Stability analysis of a single species logistic model with Allee effect and feedback control |
title_fullStr |
Stability analysis of a single species logistic model with Allee effect and feedback control |
title_full_unstemmed |
Stability analysis of a single species logistic model with Allee effect and feedback control |
title_sort |
stability analysis of a single species logistic model with allee effect and feedback control |
publisher |
SpringerOpen |
series |
Advances in Difference Equations |
issn |
1687-1847 |
publishDate |
2018-05-01 |
description |
Abstract A single species logistic model with Allee effect and feedback control dxdt=rx(1−x)xβ+x−axu,dudt=−bu+cx, $$\begin{aligned}& \frac{dx}{dt} = rx(1-x)\frac{x}{\beta+x}-axu, \\& \frac{du}{dt} = -bu+cx, \end{aligned}$$ where β, r, a, b, and c are all positive constants, is for the first time proposed and studied in this paper. We show that, for the system without Allee effect, the system admits a unique positive equilibrium which is globally attractive. However, for the system with Allee effect, if the Allee effect is limited ( β<b2r2ac(ac+br) $\beta<\frac{b^{2}r^{2}}{ac(ac+br)}$), then the system could admit a unique positive equilibrium which is locally asymptotically stable; if the Allee effect is too large ( β>brac $\beta>\frac{br}{ac}$), the system has no positive equilibrium, which means the extinction of the species. The Allee effect reduces the population density of the species, which increases the extinction property of the species. The Allee effect makes the system “unstable” in the sense that the system could collapse under large perturbation. Numeric simulations are carried out to show the feasibility of the main results. |
topic |
Logistic model Allee effect Feedback control Global stability |
url |
http://link.springer.com/article/10.1186/s13662-018-1647-2 |
work_keys_str_mv |
AT qifalin stabilityanalysisofasinglespecieslogisticmodelwithalleeeffectandfeedbackcontrol |
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1725936018133090304 |