Non-trivial, non-negative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

TL;DR

This paper establishes conditions for the existence of non-trivial, non-negative periodic solutions in a system of singular-degenerate parabolic equations with nonlocal terms, modeling biological species interactions.

Contribution

It introduces a novel application of Leray-Schauder degree theory to complex singular-degenerate systems with nonlocal interactions, ensuring coexistence conditions.

Findings

Existence of periodic solutions under specific conditions

Conditions for coexistence of two biological species

Application of topological degree theory to singular systems

Abstract

We study the existence of non-trivial, non-negative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray-Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence of the two species.