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

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.