Positive solutions of some parabolic system with cross-diffusion and nonlocal initial conditions

TL;DR

This paper investigates positive solutions of a coupled nonlinear parabolic system with cross-diffusion and nonlocal initial conditions, using bifurcation theory and regularity results, relevant to age-structured predator-prey models.

Contribution

It introduces a novel application of global bifurcation methods to a cross-diffusion parabolic system with nonlocal initial conditions, advancing the understanding of such models.

Findings

Global bifurcation of positive solutions established

Conditions for existence of positive solutions derived

Application to age-structured predator-prey systems demonstrated

Abstract

The paper is concerned with a system consisting of two coupled nonlinear parabolic equations with a cross-diffusion term, where the solutions at positive times define the initial states. The equations arise as steady state equations of an age-structured predator-prey system with spatial dispersion. Based on unilateral global bifurcation methods for Fredholm operators and on maximal regularity for parabolic equations, global bifurcation of positive solutions is derived.