Positive travelling fronts for reaction-diffusion systems with distributed delay

TL;DR

This paper establishes conditions for the existence of positive travelling wave solutions in reaction-diffusion systems with distributed delay, using functional analysis and perturbation methods, with applications to biological models like the delayed Fisher-KPP equation.

Contribution

It introduces a novel approach to prove positive travelling waves in delayed reaction-diffusion systems without requiring quasi-monotonicity of nonlinearities.

Findings

Existence of positive travelling waves under specific conditions.

Application of the method to delayed Fisher-KPP equation.

Positive wave profiles exhibit exponential decay at -infinity.

Abstract

We give sufficient conditions for the existence of positive travelling wave solutions for multi-dimensional autonomous reaction-diffusion systems with distributed delay. To prove the existence of travelling waves, we give an abstract formulation of the equation for the wave profiles in some suitable Banach spaces, and apply known results about the index of some associated Fredholm operators. After a Liapunov-Schmidt reduction, these waves are obtained via the Banach contraction principle, as perturbations of a positive heteroclinic solution for the associated system without diffusion, whose existence is proven under some requirements. By a careful analysis of the exponential decay of the travelling wave profiles at $-\infty$, their positiveness is deduced. The existence of positive travelling waves is important in terms of applications to biological models. Our method applies to systems of delayed reaction-diffusion equations whose nonlinearities are not required to satisfy a quasi-monotonicity condition. Applications are given, and include the delayed Fisher-KPP equation.