Geometric ideas on global stability and monotonicity for discrete systems

Abstract: It is an important problem in discrete dynamics to determine when local stability of fixed points implies global stability. We will focus on the planar Ricker competition model and introduce ideas from singularity theory to describe the dynamics of the images of the critical curves to show that local stability of the coexistence (positive) fixed point implies global stability. The introduction of geometric methods will allow us to revisit the notion of monotonicity and develop a geometric generalization for the notion of monotonicity (or competitive) maps in higher dimensions. We show that this definition is equivalent for known results for planar maps and provide analytic conditions to check for geometric monotonicity and global stability. We illustrate our results with the Beverton-Holt and Ricker competition map.

dynamical systems

Audience: researchers in the topic

Little school dynamics

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