Transition between linear and exponential propagation in Fisher-KPP type reaction-diffusion equations

TL;DR

This paper investigates how the speed of invasion in Fisher-KPP equations with fractional Laplacian transitions from linear to exponential as the fractional order varies, revealing a phase change in propagation dynamics.

Contribution

It characterizes the transition in propagation speed from linear to exponential in fractional Fisher-KPP equations as the fractional order approaches 1.

Findings

Propagation is linear during a time of order -ln(1 - α).

After this period, the invasion speed becomes exponential.

The transition occurs smoothly as α approaches 1.

Abstract

We study the Fisher-KPP equation with a fractional laplacian of order {\alpha} \in (0, 1). We know that the stable state invades the unstable one at constant speed for {\alpha} = 1, and at an exponential in time velocity for {\alpha} \in (0, 1). The transition between these two different speeds is examined in this paper. We prove that during a time of the order - ln(1 - {\alpha}), the propagation is linear and then it is exponential.