Global dynamics for a class of reaction-diffusion equations with distributed delay and Neumann condition

TL;DR

This paper studies the global behavior of certain reaction-diffusion equations with distributed delay and Neumann boundary conditions, establishing conditions for stability and applying results to biological models.

Contribution

It introduces new stability criteria for reaction-diffusion equations with distributed delay, using sub and super-solution methods, and applies these to biological models like Nicholson blowflies.

Findings

Established conditions for global attractivity of the positive steady state.

Derived exponential stability criteria for the models.

Obtained new stability results for Nicholson blowflies and Mackey-Glass equations.

Abstract

In this paper, we investigate a class of non-monotone reaction-diffusion equations with distributed delay and a homogenous boundary Neumann condition, which have a positive steady state. The main concern is the global attractivity of the unique positive steady state. To achieve this, we use an argument of a sub and super-solution combined with fluctuation method. We also give a condition for which the exponential stability of the positive steady state is reached. As an example, we apply our results to diffusive Nicholson blowflies and diffusive Mackey-Glass equation with distributed delay. We point out that we obtain some new results on exponential stability of the positive steady state for these cited models.