Uniform dynamics for Fisher-KPP propagation driven by a line of fast diffusion under a singular limit

TL;DR

This paper explores how a nonlocal Fisher-KPP model with a line of fast diffusion converges to a local model with Robin boundary conditions, revealing the relationship between two approaches to modeling enhanced spreading.

Contribution

It demonstrates the strong convergence of the nonlocal model to the local model as integral terms approach Dirac masses, linking two different modeling frameworks for fast diffusion lines.

Findings

Nonlocal dynamics tend to the local model in a strong sense.

Enhanced spreading occurs in both models due to the line of fast diffusion.

The convergence bridges the gap between integral and boundary condition models.

Abstract

The purpose of this paper is to understand the links between a model introduced in 2012 by H. Berestycki, J.-M. Roquejofre and L. Rossi and a nonlocal model studied by the author in 2014. The general question is to investigate the influence of a line of fast diffusion on Fisher-KPP propagation. In the initial model, the exchanges are modeled by a Robin boundary condition, whereas in the nonlocal model the exchanges are described by integral terms. For both models was showed the existence of an enhanced spreading in the direction of the line. One way to retrieve the local model from the nonlocal one is to consider integral terms tending to Dirac masses. The question is then how the dynamics given by the nonlocal model resembles the local one. We show here that the nonlocal dynamics tends to the local one in a rather strong sense.