Spatial Dynamics Of Time–periodic And Non–monotone Reaction–diffusion System

As an important research object in chemistry, ecology and epidemiology, reactiondiffusion equations have attracted more and more attention and studies. It is well known that many nonlinear reaction-diffusion systems modeling the growth of population and the interaction of multi-species, such as predator and prey, the disease transmission among the susceptible and infective, etc, are non-monotone. Due to the lack of the comparison principle and monotonic properties of such evolution systems, the study of spatial dynamics of such model systems is very difficult. In addition, in the investigation of the growth of population and the transmission of infectious disease, the periodic factors, such as alternation of day with night and seasonality, etc, can not be ignore. Therefore, it is of important significance to study the non-autonomous reaction-diffusion equations. This thesis is devoted to the spatial dynamics of some non-autonomous and non-monotone reaction-diffusion models. The main contents are as follows:Firstly, we study the spreading speeds and periodic traveling waves for a time periodic reaction-diffusion population model with stage structure and non-monotone birth function. Since the birth function is non-monotone, the standard monotonic method is invalid. Additionally, the time periodicity makes the strategy for the traveling waves for autonomous reaction-diffusion equations lose efficacy. Consequently,we try our best to develop a new tactic to prove the asymptotic speeds of spread and periodic traveling waves. Through sandwiching the given birth function in between two appropriate nondecreasing functions to obtain two control equations, we use the comparison arguments combined with the related results on the asymptotic speeds of spread for monotone equation to get the existence of spreading speeds for the original equation. Then, by constructing an appropriate nonlinear and non-monotone operator on a closed and convex set, we apply Schauder’s fixed point theorem to obtain the existence of periodic traveling waves. Here the construction of the nonlinear operator is very different from the autonomous non-monotone system.On the basis of the obtained results on the spreading speed, we further prove the asymptotic boundary conditions and the non-existence for periodic traveling waves,respectively.Secondly, we discuss the periodic traveling waves for a time periodic reactiondiffusion SIR model with standard incidence. The basic idea to prove the existence of periodic traveling waves is similar to the last model. However, it is quite difficult to verify the asymptotic boundary conditions of the periodic traveling waves. To this end, we mainly apply Landau type inequalities, Harnack inequality for cooperative parabolic system and comparison arguments for scalar equation. Finally, by using the properties of spreading speed for the scalar periodic reaction-diffusion equation combined with the comparison arguments, we prove the nonexistence of the periodic traveling waves for two cases.Thirdly, we explore the dynamics of a periodic reaction-diffusion SIR model with latency. Taking the seasonality, diffusion and latent period into consideration,we derive a time-periodic nonlocal and delayed reaction-diffusion system. In contrast to the autonomous differential equations with delay, the stability for linear periodic equations with delay is not coincident with the associated periodic equations without delay. This leads to some difficulties to develop the theory of the basic reproduction number ?0for periodic population models with delay. We introduce ?0for the model system via a next generation operator and obtain the relationship between?0and the spectral radius of Poincar′e map of the associated linear equation. In terms of ?0, we establish the threshold dynamics by comparison arguments and persistence theory.Finally, we consider a nonlocal periodic reaction-diffusion SIR model with infection period. In contrast to the well-known reaction-diffusion epidemic models where the delayed term is positive, in the current model system, the term with nonlocal delay is negative and the initial value satisfies a nonlinear constraint condition,which result in some new mathematical difficulties in investigating the dynamics.We introduce the basic reproduction number ?0via a next generation operator and establish the sufficient conditions for the disease extinction and persistence by means an integral equation but not a differential equation, which is very different from general approach to do those for many epidemic models.