Travelling wave fronts of Lotka-Volterra reaction-diffusion system in the weak competition case

This paper is concerned with spreading phenomena of the classical two-species Lotka-Volterra reaction-diffusion system in the weak competition case. More precisely, some new sufficient conditions on the linear or nonlinear speed selection of the minimal wave speed of travelling wave fronts, which connect one half-positive equilibrium and one positive equilibrium, have been given via constructing types of super-sub solutions. Moreover, these conditions for the linear or nonlinear determinacy are quite different from that of the minimal wave speeds of travelling wave fronts connecting other equilibria of Lotka-Volterra competition model. In addition, based on the weighted energy method, we give the global exponential stability of such solutions with large speed $c$ . Specially, when the competition rate exerted on one species converges to zero, then for any $c>c_0$ , where $c_0$ is the critical speed, the travelling wave front with the speed $c$ is globally exponentially stable.