Spreading of two competing species governed by a free boundary model in a shifting environment

We investigate the long-time spreading behavior of two competing species in a shifting environment. The evolution of the population densities of the species is governed by a one space dimension diffusive Lotka-Volterra competition system with two different free boundaries, describing a dynamical process of two competitors invading into the new habitat in the same direction. It is assumed that the unfavourable region of the environment moves into the otherwise favourable homogeneous environment with a given speed c>0 in the spreading direction of the species. By close examination of certain simple cases, we show that such a shifting environment could reverse the fates of the species. A complete classification of the long-time dynamical behavior of the system is obtained for such cases.