The determinacy of wave speed sign for a reaction-diffusion system with nonlocal diffusion (Q6554695)

A bistable traveling wave is a special solution of a system of partial differential equations connecting two stable equilibrium points of its reaction system. The sign of its wave speed (called the bistable wave speed ) indicates the direction of its propagation, which means which of the two stable equilibrium points is more competitive, and in biology, which species ultimately survive and which species will die out over time. Hence, the determination of the speed sign of bistable travelling waves has been in a very active area of research.\N\NIn this paper, the authors study the sign of bistable wave speed of a two-component Lotka-Volterra competitive model with nonlocal diffusion. The upper and lower solution method is recently developed and currently the most effective tool for determining the sign of bistable traveling wave. However, for systems with nonlocal diffusion terms, it is extremely challenging to find upper and lower solutions. The authors of this paper develop a new idea for constructing upper and lower solutions to establish explicit conditions for obtaining positive or negative wave speed. The new technique not only overcomes the difficulties caused by the nonlocal dispersal term, but also does not require assumptions that the kernel functions \(J_1\) and \(J_2\) are asymmetric.