Super-linear spreading in local bistable cane toads equations

TL;DR

This paper investigates how an Allee effect influences the invasion speed of cane toads in Australia, revealing that acceleration persists even with bistable non-linearities, contrasting previous findings.

Contribution

It demonstrates that super-linear spreading occurs in a bistable reaction-diffusion model with unbounded diffusivity, challenging prior assumptions about the effects of bistability on invasion speed.

Findings

Super-linear spreading persists with bistable non-linearity.

Acceleration occurs despite the Allee effect killing small populations.

Contrasts with previous models where bistability prevented acceleration.

Abstract

In this paper, we study the influence of an Allee effect on the spreading rate in a local reaction-diffusion-mutation equation modelling the invasion of cane toads in Australia. We are, in particular, concerned with the case when the diffusivity can take unbounded values. We show that the acceleration feature that arises in this model with a Fisher-KPP, or monostable, non-linearity still occurs when this non-linearity is instead bistable, despite the fact that this kills the small populations. This is in stark contrast to the work of Alfaro, Gui-Huan, and Mellet-Roquejoffre-Sire in related models, where the change to a bistable non-linearity prevents acceleration.