Global bifurcation diagrams for coercive third-degree polynomial ordinary differential equations with recurrent nonautonomous coefficients

TL;DR

This paper analyzes global bifurcation diagrams for nonautonomous third-degree polynomial ODEs with recurrent coefficients, providing insights into critical transitions in models like population dynamics with Allee effects.

Contribution

It introduces a comprehensive analysis of bifurcation diagrams for coercive third-degree polynomial nonautonomous ODEs, with applications to population models under migration effects.

Findings

Bifurcation diagrams reveal critical thresholds in population models.

Migration influences the strength of Allee effects.

Global dynamics change significantly with parameter variations.

Abstract

Nonautonomous bifurcation theory is a growing branch of mathematics, for the insight it provides into radical changes in the global dynamics of realistic models for many real-world phenomena, i.e., into the occurrence of critical transitions. This paper describes several global bifurcation diagrams for nonautonomous first order scalar ordinary differential equations generated by coercive third degree polynomials in the state variable. The conclusions are applied to a population dynamics model subject to an Allee effect that is weak in the absence of migration and becomes strong under a migratory phenomenon whose sense and intensity depend on a threshold in the number of individuals in the population.