Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments

TL;DR

This paper investigates the existence, multiplicity, and chaotic behavior of subharmonic solutions in predator-prey models with periodic coefficients, revealing conditions under which complex dynamics can occur.

Contribution

It introduces a combined topological and dynamical systems approach to analyze subharmonic and chaotic solutions in predator-prey models with degenerate weights.

Findings

Existence and multiplicity of subharmonic solutions established.

Chaotic dynamics can arise in simple predator-prey models.

Chaotic behavior is absent in cooperative models due to maximum principle.

Abstract

This paper deals with the existence, multiplicity, minimal complexity and global structure of the subharmonic solutions to a class of planar Hamiltonian systems with periodic coefficients, being the classical predator-prey model of V. Volterra its most paradigmatic example. By means of a topological approach based on techniques from global bifurcation theory, the first part of the paper ascertains their nature, multiplicity and minimal complexity, as well as their global minimal structure, in terms of the configuration of the function coefficients in the setting of the model. The second part of the paper introduces a dynamical system approach based on the theory of topological horseshoes that permits to detect, besides subharmonic solutions, ``chaotic-type'' solutions. As a byproduct of our analysis, the simplest predator-prey prototype models in periodic environments can provoke chaotic dynamics. This cannot occur in cooperative and quasi-cooperative dynamics, as a consequence of the ordering imposed by the maximum principle.