Emergent traveling waves in spatially extended system with finite memory of transport

Abstract

Standard diffusion equation is formalized on the basis of Brownian movement of dispersing species in absence of any persistence in the transport motion of individuals. As a result, this description accounts for instantaneous spreading of dispersing species over an arbitrarily large distance from their original location predicting infinite velocities. This feature appears to be unrealistic, particularly while considering invasion dynamics in biological systems. Therefore, a better description demands the consideration of dispersal with inertia. Here, we investigate the spatiotemporal dynamics of a reaction-transport system including Cattaneo's modification of flux and a source term with cubic nonlinearity, reminiscent of population dynamics or flame propagation models. We show that the spatially extended system admits a traveling wave solution. The presence of a small finite relaxation time of the diffusive flux modifies the speed of the traveling wave, specifically, resulting decay in speed with an increase of relaxation time of the flux. Our investigation reveals a complex interplay of reaction parameters and the finite memory of transport. The observed traveling wave solution reveals that there are conditions inbuilt in the choice of the reaction rate parameters, which governs the evolution of the wave solution. Our analytical predictions are qualitatively in good agreement with the numerically simulated results.