Geometric method for global stability of discrete population models

Zhanyuan Hou

doi:10.3934/dcdsb.2020063

Article Contents

2020, Volume 25, Issue 9: 3305-3334. Doi: 10.3934/dcdsb.2020063

This issue Previous Article Next Article Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms

Geometric method for global stability of discrete population models

Zhanyuan Hou^,

School of Computing & Digital Media, London Metropolitan University, 166–220 Holloway Road, London N7 8DB, UK

^* Corresponding author: Zhanyuan Hou
^* Corresponding author: Zhanyuan Hou

Received: February 2019

Revised: November 2019

Early access: April 2020

Published: September 2020

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Abstract

A class of autonomous discrete dynamical systems as population models for competing species are considered when each nullcline surface is a hyperplane. Criteria are established for global attraction of an interior or a boundary fixed point by a geometric method utilising the relative position of these nullcline planes only, independent of the growth rate function. These criteria are universal for a broad class of systems, so they can be applied directly to some known models appearing in the literature including Ricker competition models, Leslie-Gower models, Atkinson-Allen models, and generalised Atkinson-Allen models. Then global asymptotic stability is obtained by finding the eigenvalues of the Jacobian matrix at the fixed point. An intriguing question is proposed: Can a globally attracting fixed point induce a homoclinic cycle?

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Mathematics Subject Classification: Primary: 37B25; Secondary: 37N25, 39A30, 92D25.

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Figure 1. A stability region for $ (\alpha, \beta) $ obtained by Theorems 3.3 and 3.5

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Figure 2. Configuration of $ h([u, v]\cap(\cup_{k\in \{i,j\}}(\Gamma_k\cup\Gamma^-_k))) $ for $ a_{ij} = 0 $, (a) $ a_{ji}>0 $, (b) $ a_{ji} = 0 $

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Figure 3. Configuration of $ h([u, v]\cap(\cup_{k\in \{i,j\}}(\Gamma_k\cup\Gamma^-_k))) $ when $ a_{ij}>0, a_{ji}>0 $ and $ [u, v]\cap\Gamma_j $ is on or below $ \Gamma_i $. (a) $ \Gamma_i = \Gamma_j $, (b) $ u_j\geq 0 $, (c) $ u_j = 0 $

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Figure 4. Configuration of $ h([u, v]\cap(\cup_{k\in \{i,j\}}(\Gamma_k\cup\Gamma^-_k))) $ when there is a fixed point $ x^* $ in the interior of $ [u, v] $

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