Slowing Allee effect vs. accelerating heavy tails in monostable reaction diffusion equations

TL;DR

This paper investigates how heavy tails in initial data and a weak Allee effect in reaction-diffusion equations influence the speed of invasion, revealing conditions for acceleration or slowdown of spreading.

Contribution

It provides a precise analysis of the interplay between heavy tails and Allee effects, establishing exact criteria for acceleration in monostable reaction-diffusion equations.

Findings

Algebraic tails can cause acceleration despite Allee effects.

Exponential tails lighter than algebraic do not lead to acceleration with Allee effects.

The paper estimates the position of level sets during acceleration.

Abstract

We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between "no acceleration and acceleration". This implies in particular that, for tails exponentially unbounded but lighter than algebraic , acceleration never occurs in presence of an Allee effect. This is in sharp contrast with the KPP situation [19]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.