Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions

TL;DR

This paper investigates how nonlinear diffusion, initial data tails, and Allee effects interact to influence the speed of biological invasions, identifying conditions for acceleration and providing precise invasion front estimates.

Contribution

It analyzes the combined impact of nonlinear diffusion, initial tail heaviness, and Allee effects on invasion speed, distinguishing scenarios with and without acceleration.

Findings

Heavy tails can accelerate invasion fronts.

Weak Allee effects can slow down spreading.

Clear criteria for when acceleration occurs.

Abstract

We focus on the spreading properties of solutions of monostable equations with non-linear diffusion. We consider both the porous medium diffusion and the fast diffusion regimes. Initial data may have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity may involve a weak Allee effect, which tends to slow down the process. We study the balance between these three effects (nonlin-ear diffusion, initial tail, KPP nonlinearity/Allee effect), revealing the separation between "no acceleration" and "acceleration". In most of the cases where acceleration occurs, we also give an accurate estimate of the position of the level sets.