Traveling Wave Solutions Of A Three Species Competitive-Diffusion System

In recent years,a large number of mathematical models have been proposed to study and solve problems in biology.Many models of population dynamics can be summarized as reaction-diffusion systems,among which Lotka-Volterra competition-diffusion system is one of the classic models to describe inter-specific competition.At present,the dynamics of interactions between competing species have attracted many researcher's attention.In bistable competition-diffusion models,the sign of wave speed for the traveling wave solutions can be used to predict the result of competition among species.In monostable competition-diffusion models,it is necessary to consider the minimum wave speed of species diffusion into the unstable state to understand the invasion of species.Therefore,it has great biological significance to study the sign of wave speed for traveling wave solutions and the range of minimum wave speed of reaction-diffusion systems.Nowadays,many results have been obtained about the wave speed of traveling wave solutions in the competition-diffusion systems of two species.In this paper,we will further study the above two properties of the traveling wave solution speed of the competition-diffusion systems of three species.The first part of this paper mainly study the sign of wave speed for the bistable traveling wave solutions in the three species Lotka-Volterra competition-diffusion system.Firstly,we transform the competitive system to the cooperative system,and obtain the stability of the traveling wave solutions of the cooperative system by the degree theoretic approach.Then,based on the stability results,the value range of the traveling wave solution speed of the system and the sufficient conditions for judging the sign of the traveling wave solution speed of the system are obtained by using the comparison principle.Finally,the parameter range to judge the sign of wave speed for the traveling wave solutions of the system is obtained by constructing appropriate upper and lower solutions.The second part of this paper mainly study the selection mechanism of minimum wave speed of the monostable traveling wave solutions for the three species Lotka-Volterra competitiondiffusion system.Firstly,the monotone iteration technique combined with the upper and lower solutions method is used to prove the existence of the traveling wave solution.Then,based on the comparison principle,we obtain sufficient conditions for the linear and nonlinear selection of the minimum wave speed of the system.Finally,by constructing novel upper and lower solutions,the parameter range for judging the minimum wave speed selection mechanism of the system is obtained.