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

Abstract

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.