Super-linear spreading in local and non-local cane toads equations

TL;DR

This paper investigates super-linear invasion dynamics in nonlocal and local cane toads models, demonstrating that populations spread at a rate proportional to t^{3/2} when diffusivity is unbounded, advancing mathematical understanding of biological invasions.

Contribution

The paper establishes super-linear spreading rates in nonlocal and local cane toads equations with unbounded diffusivity, providing new mathematical insights into invasion speeds.

Findings

Population spreads as t^{3/2} in nonlocal model

Sharp spreading rate determined for local model

Unbounded diffusivity leads to super-linear invasion dynamics

Abstract

In this paper, we show super-linear propagation in a nonlocal reaction-diffusion-mutation equation modeling the invasion of cane toads in Australia that has attracted attention recently from the mathematical point of view. The population of toads is structured by a phenotypical trait that governs the spatial diffusion. In this paper, we are concerned with the case when the diffusivity can take unbounded values, and we prove that the population spreads as $t^{3/2}$. We also get the sharp rate of spreading in a related local model.