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 nonnegative solutions to the following random Fisher–KPP equation, 1 $$\begin{aligned} u_t =u_{xx}+a(\theta _t\omega )u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$ 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 nonnegative solutions to the following nonautonomous Fisher–KPP equation, 2 $$\begin{aligned} u_t=u_{xx}+a_0(t)u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$ where $$a_0(t)$$ is a positive locally Holder 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 determinate. It is shown that the solution of (1) with a nonnegative 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 mean 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).