Traveling Waves And Spreading Speeds Of Time Periodic Reaction Diffusion Equations

In general,the reaction-diffusion equation can be used to describe the growth of biological population and the spatial spread of epidemics.It is well known that the growth of popu-lation and the transmission of infectious disease in nature are largely influenced by time-varying environments.In particular,some regular variations,such as the replacement of day with night and seasonality,will lead to a periodic changes in population growth and disease spread.Therefore,it is interesting and valuable both in theory and practice to understand how seasonality influence the ecological dynamics.This thesis is mainly concerned with the spatial dynamics of two time-periodic reaction-diffusion equations.The main results in this thesis are as follows:For a time-periodic and delayed nonlocal reaction-diffusion model with age structure,under the quasi-monotone or non-quasi-monotone conditions,the uniqueness of the periodic trav-eling waves is first proved by the sliding plane technique.Then,in the quasi-monotone case,we prove the exponential stability of the non-critical periodic traveling waves by establish-ing two crucial comparison theorems for an initial value problem and an initial-boundary value problem associated with model.More precisely,using the method of principal eigen-value analysis of nonlocal linear operators,it can be showed that if the initial function is within a bounded distance from a non-critical periodic traveling waves with respect to a weighted maximum norm,then the solution will converge to the periodic traveling waves exponentially in time.The exponential convergent rate is also obtained.To describe the spatial propagation of a deterministic epidemic in multi-type of popula-tion,we propose a time-periodic and non-local multi-type SIS epidemic model.Firstly,the threshold dynamics of the spatially homogeneous system is analysed and the comparison theorem of corresponding initial value problem is also established.Since the established model is a cooperative system,the abstract theory of asymptotic speed of propagation of monotone periodic semiflow is applied to prove the existence of the spreading speed(8_*,and the nonexistence of traveling waves with wave speed 0<(8<(8_*.Then,we prove the existence and asymptotic behavior of the periodic traveling waves with speed(8>(8_*by the method of upper-lower solution.Finally,the existence of the critical periodic traveling waves is established by using a limiting argument.To overcome the difficulty of lack of compactness of solution maps of the nonlocal system with respect to compact open topolo-gy,we show that the solution sequence is pre-compact in_loc(R²,R⁸)).