Entire solutions of time periodic bistable Lotka–Volterra competition-diffusion systems in $${\mathbb {R}}^N$$

Abstract

This paper is concerned with entire solutions of a two species time periodic bistable Lotka–Volterra competition-diffusion systems in $${\mathbb {R}}^N$$ . Here an entire solution refers to a solution that is defined for all time and in the whole space. It is known that “annihilating-front” type entire solutions have been obtained recently in $${\mathbb {R}}$$ . In the present paper, we first prove that there is a new type of entire solution which behaves as three time periodic moving planar traveling fronts as time goes to $$-\infty $$ and as a time periodic V-shaped traveling front as time goes to $$+\infty $$ in $${\mathbb {R}}^N$$ . Furthermore, we show that the propagating speed of such entire solutions coincides with the unique speed of the time periodic planar front, regardless of the shape of the level sets of the fronts.