A Degenerate Bifurcation Perspective on High Sensitivity in a Modified Gower-Leslie Model with Additive Allee Effect

TL;DR

This paper analyzes a modified ecological model revealing extreme sensitivity to parameters and initial conditions, linked to complex bifurcations, including high-codimension bifurcations and multiple limit cycles.

Contribution

It identifies and characterizes high-codimension bifurcations in a population model, explaining the origins of high sensitivity and multistability in ecological dynamics.

Findings

Existence of a codimension 4 nilpotent cusp and degenerate Bogdanov-Takens bifurcation.

Hopf bifurcation can produce up to five limit cycles.

Numerical simulations confirm heteroclinic loops and multiple limit cycles.

Abstract

The population dynamics in a modified Leslie-Gower model with an additive Allee effect are highly sensitive to both parameters and initial population densities, leading to outcomes ranging from coextinction to sustained multistable steady states. This work links this sensitivity to complicated bifurcations. We establish the existence of a codimension 4 nilpotent cusp and a corresponding degenerate Bogdanov-Takens bifurcation with codimension 4, which critically shape the system's response to parameter changes. Most significantly, we prove that the Hopf bifurcation occurring at a center-type equilibrium can give rise to up to five limit cycles-a phenomenon scarcely documented in previous ecological studies-thereby inducing a pronounced dependence of oscillatory regimes on initial conditions. Numerical simulations confirming heteroclinic loops and multiple limit cycles provide consistent support for the theoretical analysis.