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

Rachidi B. Salako, Wenxian Shen|arXiv (Cornell University)|Jun 4, 2018

Mathematical and Theoretical Epidemiology and Ecology Models37 references3 citations

TL;DR

This paper investigates the long-time behavior of nonnegative solutions to random and nonautonomous Fisher-KPP equations without requiring boundedness or uniform positivity of the growth rate. It establishes asymptotic stability of the positive equilibrium $ u=1 $, derives a deterministic spreading speed interval $[2\sqrt{\underline{a}}, 2\sqrt{\bar{a}}]$, and proves linear determinacy of spreading speeds, even when $ a(\omega) $ or $ a_0(t) $ is unbounded or vanishes in parts of the domain.

ABSTRACT

In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(θ_tω)u(1-u),\quad x\in\mathbb{R} $$ where $ω\inΩ$, $(Ω, \mathcal{F},\mathbb{P})$ is a given probability space, $θ_t$ is an ergodic metric dynamical system on $Ω$, and $a(ω)>0$ for every $ω\inΩ$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),\quad x\in\mathbb{R}$$ 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 $a(ω)$, 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 $a(ω)$ 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).

Motivation & Objective

• To analyze the long-time dynamics of nonautonomous and random Fisher-KPP equations with unbounded or vanishing growth rates.
• To establish asymptotic stability of the positive equilibrium $ u=1 $ under general conditions on the time- and space-dependent growth rate.
• To determine the spreading speed interval for both random and nonautonomous Fisher-KPP equations without assuming $ \inf a(\omega) > 0 $ or $ \sup a_0(t) < \infty $.
• To extend classical results on spreading speeds and take-over properties to more general, less regular growth rate functions.
• To lay the foundation for studying transition fronts in the second part of the series.

Proposed method

• Utilizes ergodic metric dynamical systems to model random time-dependent coefficients $ a(\theta_t\omega) $ in the random Fisher-KPP equation.
• Applies the subadditive ergodic theorem to analyze the long-term behavior and derive spreading speeds in the random case.
• Employs comparison principles and auxiliary solutions with constant coefficients to bound solutions of the nonautonomous equation.
• Constructs explicit supersolutions using exponential functions and time-dependent potentials to control solution decay and propagation.
• Uses the method of moving frames and traveling wave-type estimates to analyze spreading speeds in nonautonomous settings.
• Establishes linear determinacy of spreading speeds by comparing with constant-coefficient Fisher-KPP equations with effective growth rates $ \underline{a} $ and $ \bar{a} $.

Experimental results

Research questions

• RQ1 Under what conditions is the positive equilibrium $ u=1 $ asymptotically stable for the random Fisher-KPP equation with unbounded or vanishing $ a(\omega) $?
• RQ2 What is the spreading speed interval for the random Fisher-KPP equation when $ a(\omega) $ is not uniformly bounded away from zero or infinity?
• RQ3 How does the spreading speed of the nonautonomous Fisher-KPP equation depend on the time-averaged growth rate $ \hat{a} $ when $ a_0(t) $ is not bounded?
• RQ4 Is the spreading speed linearly determinate in the absence of uniform lower and upper bounds on $ a(\omega) $ or $ a_0(t) $?
• RQ5 Can the take-over property (i.e., propagation at speed $ 2\sqrt{\hat{a}} $) be preserved when the growth rate is irregular or unbounded?

Key findings

• The constant solution $ u=1 $ is asymptotically stable with respect to strictly positive initial perturbations in the random Fisher-KPP equation, even when $ a(\omega) $ is not bounded away from zero.
• The random Fisher-KPP equation admits a deterministic spreading speed interval $[2\sqrt{\underline{a}}, 2\sqrt{\bar{a}}]$, where $ \underline{a} $ and $ \bar{a} $ are the essential infimum and supremum of $ a(\omega) $, respectively.
• The solution propagates at speed $ 2\sqrt{\hat{a}} $, where $ \hat{a} $ is the mean of $ a(\cdot) $, for initial data that are 1 for $ x \ll -1 $ and 0 for $ x \gg 1 $.
• For the nonautonomous Fisher-KPP equation, the positive equilibrium $ u=1 $ is asymptotically stable under mild conditions on $ a_0(t) $, even when $ \inf a_0(t) = 0 $ or $ \sup a_0(t) = \infty $.
• The spreading speed interval for the nonautonomous case is bounded and depends on the time-averaged growth rates $ \underline{a}_0 $ and $ \overline{a}_0 $, with linear determinacy preserved.
• An explicit example of $ a_0(t) $ is constructed with $ \inf a_0(t) = 0 $, $ \sup a_0(t) = \infty $, yet satisfying $ \underline{a}_0 = 1 $, $ \overline{a}_0 = 2 $, demonstrating the applicability of the theory to highly irregular growth rates.