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Stability of travelling waves in a Wolbachia invasion. (English) Zbl 1403.35302
Summary: 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 M. H. T. Chan and P. S. Kim [Bull. Math. Biol. 75, No. 9, 1501–1523 (2013; Zbl 1311.92173)] 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 [loc. cit.] 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.
MSC:
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35C07 Traveling wave solutions
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
62M15 Inference from stochastic processes and spectral analysis
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
92B05 General biology and biomathematics
92D30 Epidemiology
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