Diffusive KPP Equations with Free Boundaries in Time Almost Periodic Environments: II. Spreading Speeds and Semi-Wave

TL;DR

This paper studies the spreading speeds of diffusive KPP equations with free boundaries in time almost periodic environments, establishing the existence of semi-wave solutions and unique spreading speeds in such ecological models.

Contribution

It introduces the concept of semi-wave solutions for these equations and proves the existence of a unique spreading speed, advancing understanding of species invasion dynamics.

Findings

Existence of a unique semi-wave solution.

Proof of a unique spreading speed.

Application to ecological spreading models.

Abstract

In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In the first part of the series, we showed that a spreading-vanishing dichotomy occurs for such free boundary problems (see [16]). In this second part of the series, we investigate the spreading speeds of such free boundary problems in the case that the spreading occurs. We first prove the existence of a unique time almost periodic semi-wave solution associated to such a free boundary problem. Using the semi-wave solution, we then prove that the free boundary problem has a unique spreading speed.