Stability of traveling waves of the nonlocal Fisher–KPP equation

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 ϵ-1 (ϵ > 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 ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ-1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

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Non-cooperative Fisher–KPP systems: Asymptotic behavior of traveling waves

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202518500288 ◽

2018 ◽

Vol 28 (06) ◽

pp. 1067-1104 ◽

Cited By ~ 9

Author(s):

Léo Girardin

Keyword(s):

Traveling Waves ◽

Traveling Wave ◽

Traveling Wave Solutions ◽

Reaction Diffusion ◽

Positive Cone ◽

Precise Description ◽

Kpp Equation ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Null State

This paper is concerned with non-cooperative parabolic reaction–diffusion systems which share structural similarities with the scalar Fisher–KPP equation. In a previous paper, we established that these systems admit traveling wave solutions whose profiles connect the null state to a compact subset of the positive cone. The main object of this paper is the investigation of a more precise description of these profiles. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and superlinear competition.

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Doubly nonlocal Fisher–KPP equation: Speeds and uniqueness of traveling waves

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2019.02.010 ◽

2019 ◽

Vol 475 (1) ◽

pp. 94-122 ◽

Cited By ~ 3

Author(s):

Dmitri Finkelshtein ◽

Yuri Kondratiev ◽

Pasha Tkachov

Keyword(s):

Traveling Waves ◽

Kpp Equation

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Stochastic traveling waves of a stochastic Fisher–KPP equation and bifurcations for asymptotic behaviors

Stochastics and Dynamics ◽

10.1142/s021949371950028x ◽

2019 ◽

Vol 19 (04) ◽

pp. 1950028

Author(s):

Zhenzhen Wang ◽

Zhehao Huang ◽

Zhengrong Liu

Keyword(s):

Traveling Waves ◽

Traveling Wave ◽

Wave Speed ◽

Threshold Value ◽

Environmental Capacity ◽

Kpp Equation ◽

Asymptotic Behaviors ◽

Transition Front ◽

Stochastic Transition ◽

Explicit Relation

In this paper, traveling wave for a Fisher–KPP equation with stochastic advection and stochastic environmental capacity is investigated. Some conditions are imposed on the reaction rate and noise intensities such that the stochastic transition front exists. Following the results on stochastic transition front, the existence of stochastic traveling waves for the equation is established. Explicit relation between the wave speed and noise attributes including noise intensities and correlation is shown, which can realize the noise effects. It is found that noises reduce the wave speed. In addition, the positive correlation of noises may complement this reduction in a way. But the negative correlation of noises will further aggravate this reduction. There exists a threshold value on the noise correlation making the traveling wave wandering. If the correlation is larger than this threshold value, the wave travels with a forward tendency. Otherwise, the wave travels with a backward tendency. Bifurcations for asymptotic behaviors of the equation induced by the noise intensities and correlation are presented.

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