Transition waves for two species competition system in time heterogeneous media. (December 2018) - Bao, Xiongxiong
Abstract: This paper is concerned with the spreading speeds and transition waves of the following two species competition system in time heterogeneous media, u t = Δ u + u ( a 1 ( t ) − b 1 ( t ) u − c 1 ( t ) v ), v t = Δ v + v ( a 2 ( t ) − b 2 ( t ) u − c 2 ( t ) v ), x ∈ R N, t ∈ R, where a i ( ⋅ ), b i ( ⋅ ), c i ( ⋅ ) ( i = 1, 2 ) depend in a general way in t ∈ R . The notion of transition waves for such a system is first introduced in this paper. We first establish the lower bounds of spreading speed intervals and generalized spreading speed intervals. It then shows that, under certain conditions, there exists a pair of general transition waves ( u ( t, x ), v ( t, x ) ) connecting two semitrivial solutions of this system for a given class of speed c ( t ) with c ̲ > c ∗ and for any c ̲ < c ∗, there is no such transition waves.