Traveling Wave Solutions Of Time-periodic Reaction-diffusion Systems With Delay

From Physics,Chemistry to Biology,reaction-diffusion equations have been widely used to describe various natural phenomena such as population growth and the spread of infectious diseases.On the one hand,because many biological phenomena in nature change periodi-cally with seasonal changes,the spread of disease and population growth are closely related to seasonal factors.It is well known that time delay is often unavoidable.Therefore,tak-ing into account the combined effects of both time periodicity and time delay,it is more reasonable to describe some biological phenomena by using a time periodic and delayed reaction-diffusion model.However,there are few results on such models at present.In this thesis,we consider traveling wave solutions of time-periodic reaction-diffusion systems with delay.For a time-periodic competition system with delay,the existence and asymptotic behavior of periodic traveling wave solutions are mainly studied.Firstly,by the classical monotone itera-tion technique coupled with the method of upper-lower solutions,we establish the existence of periodic traveling wave solutions with wave speed(8>(8^*connecting two semi-trivial periodic solutions of the corresponding kinetic system.Then,it is shown that the periodic traveling wave solution is monotone with respect tovia comparison arguments.We fur-ther obtain the asymptotic behavior of periodic traveling waves by utilizing the monotony.Finally,the existence of periodic traveling wave solution for(8=(8^*is proved.For a delayed SIR epidemic model with nonlinear incidence and time periodicity,we main-ly investigate the existence and asymptotic behavior of periodic traveling wave solutions at minus/plus infinity.We first determine the basic reproduction number₀via the next generation method.Then,combining the method of super-and sub-solutions and compar-ison arguments,we consider a fixed point problem and establish the existence of periodic traveling wave solutions on a finite interval by fixed point theorem.We further prove the existence of periodic traveling wave solutions connecting two disease-free steady state by limiting arguments.Finally,it is proved that the traveling wave solutions satisfy the asymp-totic boundary condition.