Global stability of the predator-prey model with a sigmoid functional response

Yinshu Wu; Wenzhang Huang

doi:10.3934/dcdsb.2019214

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2020, Volume 25, Issue 3: 1159-1167. Doi: 10.3934/dcdsb.2019214

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Global stability of the predator-prey model with a sigmoid functional response

Yinshu Wu^1, , and
Wenzhang Huang^2,

1.
Department of Mathematics, Alabama A & M University, Normal, AL 35762, USA
2.
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA

^* Corresponding author: Yinshu Wu
^* Corresponding author: Yinshu Wu

Received: November 30, 2018

Revised: April 30, 2019

Early access: September 2019

Published: March 2020

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Abstract

A predator-prey model with Sigmoid functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recent developed approach shows that, associated with the model, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing an analytic and geometrical analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium. With this crucial property we are able to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional response.

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Mathematics Subject Classification: Primary: 37N25, 34D23; Secondary: 37D35.

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