On Positive Solutions of Some System of Reaction-Diffusion Equations with Nonlocal Initial Conditions

TL;DR

This paper investigates positive solutions of a coupled reaction-diffusion system with nonlocal initial conditions, relevant to predator-prey models, using bifurcation methods to analyze coexistence states based on fertility parameters.

Contribution

It introduces a global bifurcation approach to characterize the structure of positive solutions in a reaction-diffusion system with nonlocal initial conditions, highlighting coexistence states.

Findings

Existence of co-existence steady-states

Structure of positive solutions with respect to fertility parameters

Application to age-structured predator-prey models

Abstract

The paper focuses on positive solutions to a coupled system of parabolic equations with nonlocal initial conditions. Such equations arise as steady-state equations in an age-structured predator-prey model with diffusion. By using global bifurcation techniques, we describe the structure of the set of positive solutions with respect to two parameters measuring the intensities of the fertility of the species. In particular, we establish co-existence steady-states, i.e. solutions which are nonnegative and nontrivial in both components.