Transition fronts of monotone bistable reaction-diffusion systems around an obstacle

TL;DR

This paper investigates how planar traveling fronts in monotone bistable reaction-diffusion systems interact with obstacles, establishing conditions for their persistence and transition behavior, and applying results to ecological models.

Contribution

It introduces a method to analyze the interaction of traveling fronts with obstacles, proving existence, uniqueness, and convergence to transition fronts in exterior domains.

Findings

Existence and uniqueness of solutions with obstacle interaction

Convergence of solutions to stationary or traveling fronts

Geometric conditions ensuring complete propagation

Abstract

This paper is concerned with the interaction between a planar traveling front and a compact obstacle for monotone bistable reaction-diffusion systems in exterior domains. By constructing appropriate sub- and supersolutions, we first establish the existence, uniqueness and monotonicity of the entire solution emanating from a planar traveling front. In particular, we verify that regardless of the shape of the obstacle, the entire solution locally converges to a stationary solution as time tends to infinity. Under the complete propagation assumption, we further show that the entire solution recovers to the same planar traveling front as time tends to infinity after passing the obstacle, and it constitutes a transition front. In addition, we provide some geometric conditions on the obstacle to ensure that the complete propagation assumption is nonempty. Finally, we apply our theoretical results to the Lotka-Volterra competition-diffusion system.