Long time behavior of random and nonautonomous Fisher-KPP equations. Part I. Stability of equilibria and spreading speeds

Abstract

In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$u_t=u_{xx}+a({\theta }_t\omega )u(1-u),x\in \mathbb{R}$$ where $\omega \in \Omega$, $(\Omega ,\mathcal{F},\mathbb{P})$ is a given probability space, ${\theta }_t$ is an ergodic metric dynamical system on $\Omega$, and $a(\omega )>0$ for every $\omega \in \Omega$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$u_t=u_{xx}+a_0(t)u(1-u),x\in \mathbb{R}$$ where $a_0(t)$ is a positive locally Hölder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on $a(\omega )$, we show that the constant solution $u=1$ of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval $[2\sqrt{\underline{a}},2\sqrt{\bar{a}}]$, where $\underline{a}$ and $\bar{a}$ are the least and the greatest means of $a(\cdot )$, respectively, and hence the spreading speed interval is linearly determinant. It is shown that the solution of (1) with the initial function which is bounded away from $0$ for $x\ll -1$ and is $0$ for $x\gg 1$ propagates at the speed $2\sqrt{\hat{a}}$, where $\hat{a}$ is the average of $a(\cdot )$. Under some assumption on $a_0(\cdot )$, we also show that the constant solution $u=1$ of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that $a(\omega )$ and $a_0(t)$ are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher-KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).

Cited by in corpus (2)

• Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on ${\mathbb{R}}^N$. III. Transition fronts
• Long time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts