Hyperbolic traveling waves driven by growth

Abstract

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ${\epsilon }^{-1}$ ($ε\textgreater{}0$), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter $\epsilon$: for small $\epsilon$ the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large $\epsilon$ the traveling front with minimal speed is discontinuous and travels at the maximal speed ${\epsilon }^{-1}$. The traveling fronts with minimal speed are linearly stable in weighted $L^2$ spaces. We also prove local nonlinear stability of the traveling front with minimal speed when $\epsilon$ is smaller than the transition parameter.

24 pages

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• A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources
• Probabilistic approach to a proliferation and migration dichotomy in the tumor cell invasion
• On a nonlinear flux--limited equation arising in the transport of morphogens

Cited by in corpus (4)

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