American Institute of Mathematical Sciences

September  2020, 25(9): 3305-3334. doi: 10.3934/dcdsb.2020063

Geometric method for global stability of discrete population models

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

* Corresponding author: Zhanyuan Hou

Received  February 2019 Revised  November 2019 Published  September 2020 Early access  April 2020

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?

Citation: Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063
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References:
A stability region for $(\alpha, \beta)$ obtained by Theorems 3.3 and 3.5
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) $a_{ji} = 0$
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) $u_j = 0$
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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