Traveling Wave Solutions In A Delayed Competitive Model

Reaction diffusion equations with time delays are a kind of important mathematical models,of which the dynamics has been paid much attention.In the study of delayed reaction diffusion equations,an important topic is the traveling wave solutions and many results have been established.However,for many coupled models,some important questions remain open and deserve further investigation.The thesis studies the traveling wave solutions of a delayed competitive system under two significant cases.Firstly,when the interspecific competition is strong,the corresponding functional differential equation has two stable and two unstable steady states,which is the socalled bistable case.Under the condition,a traveling wave solution connecting two stable steady states is a bistable wave.Changing the system to a cooperative system by variable transformation and using the traveling wave theory of abstract monotone semiflows,we establish the existence as well as strict monotonicity of bistable traveling wave solutions.With the squeezing technique based on comparison principle,the asymptotic stability of traveling wave solutions is proved,which further implies the uniqueness of wave speed and wave profiles in the sense of phase shift.Secondly,when the interspecific competition is weak,the system may have a unique positive steady state that is asymptotically stable.Due to the time delay,this system may be non-monotonic.When the domain is bounded,we obtain the persistence and stability of any positive mild solutions by constructing proper contracting rectangles.When the domain is whole space,the minimal wave speed of traveling wave solutions is given by presenting the existence or nonexistence of nontrivial traveling wave solutions with any wave speeds.Our main recipe is the generalized upper and lower solutions as well as the asymptotic spreading.In particular,when the system is monotone,the convergence of traveling wave solutions is also established.