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

This article 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. Without the complete propagation assumption, we prove that the entire solution still propagates in the form of the planar traveling front far behind the obstacle for large time. Finally, we apply our theoretical results to the Lotka-Volterra competition-diffusion system and the buffered bistable system.