Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

Hongpeng Guo,Zhiming Guo

doi:10.3390/math9161999

Abstract

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ fixed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed c>c*, where c*>0 is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions.

Highlights

In contrast with integral difference equations, we found that there is no such problems for temporally discrete reaction diffusion equations like (8) (or (1))
In order to better understand the dynamical behaviors of (1) with non-local diffusion caused by immature individuals dispersion, we will study traveling wave solutions whenever the birth function is monotonic or non-monotonic, respectively
For readers’ convenience, we introduce some results on the following temporally discrete reaction-diffusion equation which will be used later, u(n + 1, x ) − u(n, x ) = D∆ x u(n + 1, x ) + f (u(n, x )), (15)

Summary

Introduction

We derived in [1] the following delayed temporally discrete non-local reaction-diffusion equation. In 2006, Lin and Li [20] studied following equation with delay: un ( x ) − un−1 ( x ) = d∆un ( x ) + f (un ( x ), un−τ ( x )), n ∈ N, x ∈ R They established the existence of traveling wavefronts and showed that (11) is a good approximation of its continuous time model in the sense of propagation. We note that in the existing research literature, researchers assumed that the non-standard discretizations preserve the dynamical consistency of the continuous-time reaction-diffusion equations, but they do not provide reasonable biological explanations for the modeling process. In order to better understand the dynamical behaviors of (1) with non-local diffusion caused by immature individuals dispersion, we will study traveling wave solutions whenever the birth function is monotonic or non-monotonic, respectively.

Traveling Wavefronts for the Monotone Case

Traveling Wave Solution for the Non-Monotone Case

Discussion