Spreading properties of the Fisher--KPP equation when the intrinsic growth rate is maximal in a moving patch of bounded size

TL;DR

This paper investigates how the spreading front of the Fisher--KPP equation behaves in a heterogeneous environment with a moving favorable patch, revealing complex phenomena and providing explicit formulas for spreading speed in specific cases.

Contribution

It offers a detailed analysis of spreading properties in a heterogeneous environment with a moving patch, including explicit formulas for spreading speed under certain conditions.

Findings

Complex front behaviors depend on the patch speed.

Nonlocal pulling and locking phenomena can occur.

Explicit formulas for spreading speed are derived for constant patch size and speed.

Abstract

This paper is concerned with spreading properties of space-time heterogeneous Fisher--KPP equations in one space dimension. We focus on the case of everywhere favorable environment with three different zones, a left half-line with slow or intermediate growth, a central patch with fast growth and a right half-line with slow or intermediate growth. The central patch moves at various speeds. The behavior of the front changes drastically depending on the speed of the central patch. Among other things, intriguing phenomena such as nonlocal pulling and locking may occur, which would make the behavior of the front further complicated. The problem we discuss here is closely related to questions in biomathematical modelling. By considering several special cases, we illustrate the remarkable diversity of possible behaviors. In particular, when the central patch has constant size and constant speed, we provide a complete set of explicit formulas for the spreading speed.