Admissible speeds in spatially periodic bistable reaction-diffusion equations

TL;DR

This paper explores the range of admissible propagation speeds in spatially periodic bistable reaction-diffusion equations, revealing that these speeds can be highly asymmetrical depending on direction and equation parameters.

Contribution

It demonstrates that in bistable reaction-diffusion equations, the set of admissible wave speeds is large and can be tailored in multiple directions, unlike the symmetric case in Fisher-KPP.

Findings

Admissible speeds can be arbitrarily chosen in multiple directions.

In 1D, any pair of nonnegative or nonpositive wave speeds is possible.

Speed asymmetry influences multistable equation dynamics.

Abstract

Spatially periodic reaction-diffusion equations typically admit pulsating waves which describe the transition from one steady state to another. Due to the heterogeneity, in general such an equation is not invariant by rotation and therefore the speed of the pulsating wave may a priori depend on its direction. However, little is actually known in the literature about whether it truly does: surprisingly, it is even known in the one-dimensional monostable Fisher-KPP case that the speed is the same in the opposite directions despite the lack of symmetry. Here we investigate this issue in the bistable case and show that the set of admissible speeds is actually rather large, which means that the shape of propagation may indeed be asymmetrical. More precisely, we show in any spatial dimension that one can choose an arbitrary large number of directions , and find a spatially periodic bistable type equation to achieve any combination of speeds in those directions, provided those speeds have the same sign. In particular, in spatial dimension 1 and unlike the Fisher-KPP case, any pair of (either nonnegative or nonpositive) rightward and leftward wave speeds is admissible. We also show that these variations in the speeds of bistable pulsating waves lead to strongly asymmetrical situations in the multistable equations.