Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries

TL;DR

This paper investigates the spreading speed of a nonlocal Fisher-KPP equation with free boundaries, establishing conditions for linear spreading and solving the semi-wave problem to determine exact spreading speeds.

Contribution

It provides a complete characterization of spreading speeds for the nonlocal Fisher-KPP equation, including conditions for linear and accelerating spreading, and solves the semi-wave problem.

Findings

Spreading grows linearly under certain kernel conditions.

Accelerating spreading occurs when kernel conditions are violated.

Complete solution to the semi-wave problem for spreading speed.

Abstract

In Cao, Du, Li and Li [8], a nonlocal diffusion model with free boundaries extending the local diffusion model of Du and Lin [12] was introduced and studied. For Fisher-KPP type nonlinearities, its long-time dynamical behaviour is shown to follow a spreading-vanishing dichotomy. However, when spreading happens, the question of spreading speed was left open in [8]. In this paper we obtain a rather complete answer to this question. We find a condition on the kernel function such that spreading grows linearly in time exactly when this condition holds, which is achieved by completely solving the associated semi-wave problem that determines this linear speed; when the kernel function violates this condition, we show that accelerating spreading happens.