Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations

Zhenzhen Wang; Tianshou Zhou

doi:10.3934/dcdsb.2020323

Article Contents

2021, Volume 26, Issue 9: 5023-5045. Doi: 10.3934/dcdsb.2020323

This issue Previous Article A comparative study of atomistic-based stress evaluation Next Article Stochastic modelling and analysis of harvesting model: Application to "summer fishing moratorium" by intermittent control

Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations

Zhenzhen Wang^1, and
Tianshou Zhou^2, ,

1.
School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China
2.
Guangdong Province Key Laboratory of Computational Science, School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

^* Corresponding author: Tianshou Zhou
^* Corresponding author: Tianshou Zhou

Received: May 31, 2020

Revised: July 31, 2020

Early access: November 2020

Published: September 2021

This work was partially supported by the National Natural Science foundation of P.R. China through Grant grants 11931019, 11775314 and 11701115

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Abstract

Fisher-KPP equations are an important class of mathematical models with practical background. Previous studies analyzed the asymptotic behaviors of the front and back of the wavefront and proved the existence of stochastic traveling waves, by imposing decrease constraints on the growth function. For the Fisher-KPP equation with a stochastically fluctuated growth rate, we find that if the decrease restrictions are removed, the same results still hold. Moreover, we show that with increasing the noise intensity, the original equation with Fisher-KPP nonlinearity evolves into first the one with degenerated Fisher-KPP nonlinearity and then the one with Nagumo nonlinearity. For the Fisher-KPP equation subjected to the environmental noise, the established asymptotic behavior of the front of the wavefront still holds even if the decrease constraint on the growth function is ruled out. If this constraint is removed, however, the established asymptotic behavior of the back of the wavefront will no longer hold, implying that the decrease constraint on the growth function is a sufficient and necessary condition to ensure the asymptotic behavior of the back of the wavefront. In both cases of noise, the systems can allow stochastic traveling waves.

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Mathematics Subject Classification: 60H15, 60H30.

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