Stability of travelling waves in a Wolbachia invasion

TL;DR

This paper investigates the spectral stability of travelling wave solutions in a reaction-diffusion model of Wolbachia spread, confirming their linear stability and discussing the uniqueness of the wavespeed.

Contribution

It provides the first spectral stability analysis of Wolbachia invasion travelling waves using Evans function methods.

Findings

Travelling wave solutions are linearly stable.

Essential spectrum lies in the left half plane.

Wavespeed is likely unique under realistic conditions.

Abstract

Numerous studies have examined the growth dynamics of Wolbachia within populations and the resultant rate of spatial spread. This spread is typically characterised as a travelling wave with bistable local growth dynamics due to a strong Allee effect generated from cytoplasmic incompatibility. While this rate of spread has been calculated from numerical solutions of reaction-diffusion models, none have examined the spectral stability of such travelling wave solutions. In this study we analyse the stability of a travelling wave solution generated by the reaction-diffusion model of Chan & Kim (2013) by computing the essential and point spectrum of the linearised operator arising in the model. The point spectrum is computed via an Evans function using the compound matrix method, whereby we find that it has no roots with positive real part. Moreover, the essential spectrum lies strictly in the left half plane. Thus, we find that the travelling wave solution found by Chan & Kim (2013) corresponding to competition between Wolbachia-infected and -uninfected mosquitoes is linearly stable. We employ a dimension counting argument to suggest that, under realistic conditions, the wavespeed corresponding to such a solution is unique.