# American Institute of Mathematical Sciences

September  2021, 26(9): 5023-5045. doi: 10.3934/dcdsb.2020323

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

 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

Received  June 2020 Revised  August 2020 Published  September 2021 Early access  November 2020

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

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.

Citation: Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5023-5045. doi: 10.3934/dcdsb.2020323
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