Geometric method for global stability of discrete population models

Hou, Zhanyuan (2020) Geometric method for global stability of discrete population models. Discrete and Continuous Dynamical Systems (B), 25 (9). pp. 3305-3334. ISSN 1531-3492


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?

Geometric method for global tability of discrete population models.pdf - Accepted Version

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