Nonlinear dynamics of emergent traveling waves in a reaction-Cattaneo system

TL;DR

This paper investigates how finite relaxation time in a reaction-Cattaneo system influences the emergence and speed of traveling waves, revealing that increased relaxation slows wave propagation, supported by analytical and numerical analysis.

Contribution

It introduces the effect of finite flux relaxation time on traveling wave dynamics in a reaction-Cattaneo system, extending previous linear stability analysis with nonlinear and numerical insights.

Findings

Traveling wave speed decreases as flux relaxation time increases.

Nonlinear effects lead to growth of perturbations into traveling waves.

Analytical predictions are confirmed by numerical simulations.

Abstract

Standard diffusion equation is based on Brownian motion of the dispersing species without considering persistence in the movement of the individuals. This description allows for the instantaneous spreading of the transported species over an arbitrarily large distances from their original location predicting infinite velocities. This feature is unrealistic particularly while considering biological invasion dynamics and a better description needs the consideration of dispersal with inertia. We here examine the behavior of non-infinitesimal perturbation on the steady state of an one-dimensional reaction-Cattaneo system with a cubic polynomial source term describing population dynamics or flame propagation models. It has been shown analytically that while linear analysis predicts stability of the homogeneous state, consideration of nonlinear contribution leads to a growth of spatiotemporal perturbation as a traveling wave. We show that the presence of a small finite relaxation time of the diffusive flux modifies the speed of the traveling wave. Specifically, we find that the wave speed decays with an increase of a finite relaxation time of flux. Our analytical predictions are well corroborated with the numerical results.