Propagation Dynamics Of Reaction-Advection-Diffusion Systems In Heterogeneous Media

Propagation phenomena for reaction-diffusion equations(systems)arising from population dynamics,epidemiology and many other fields have been the purpose of very active research in the past few decades,which have attracted rather con-siderable interest especially in heterogeneous media as it not only creates more patterns of propagation but also brings some other interesting mathematical ob-servations.On the other hand,the interaction of individuals posed in advective environments,where individuals are exposed to unidirectional flow or biased dis-persal,makes the situation more complicated and may cause some very different propagation phenomena.As standard models,periodic reaction-diffusion monotone systems have been frequently utilized to the study of individual movements in het-erogeneous media.We study in this thesis the propagation phenomena of periodic monotone reaction-diffusion systems with advection terms,especially on periodic traveling waves,spreading speeds and pulsating front-like entire solutions.Firstly,we study the propagation phenomena for a bistable Lotka-Volterra com-petition system with advection in a periodic habitat.Under certain assumptions,the system is bistable between two periodic semitrivial steady state solutions.We first establish the existence of the pulsating traveling front connecting two periodic semi-trivial steady state solutions by using an abstract monotone semiflow theory.Furthermore,we confirm that the pulsating traveling front is globally asymptotically stable for wave-like initial values.We finally show that such pulsating traveling fron-t is unique(up to translation).The methods involve the sub-and supersolutions,spreading speeds of monostable systems,and the dynamics approach.Secondly,we study some pulsating front-like entire solutions for a bistable Lotka-Volterra competition system with advection in a periodic habitat.We first study the decay properties of bistable pulsating fronts near the infinity.Then we construct a pair of appropriate sub-and super solutions using both leftward and rightward pulsating fronts.By the comparison principle we establish some pulsat-ing front-like entire solutions,and investigate some other qualitative properties of the entire solution such as stability,uniqueness and continuously dependence on parameters.Some of these entire solutions behave as the two pulsating fronts ap-proaching each other from both sides of the x-axis,which turn out to be unique and Liapunov stable 2-dimensional manifolds of solutions.The others behave as two pulsating fronts propagating from one side of the x-axis,the faster one invades the slower one as t?+?.Thirdly,we study the asymptotic behavior of traveling waves and entire solu-tions for a time periodic bistable competition-diffusion system.It has been proved that the system admits periodic traveling waves(X(t,x-ct),Y(t,x-ct))connecting two periodic semitrivial solutions under certain conditions.We first investigate the asymptotic behavior of periodic bistable traveling waves at infinity using a dynami-cal approach combined with the two-sided Laplace transform method,and it shows that the fronts decay with an exponential rate or a power of | x-ct| times an expo-nential rate.With these asymptotic properties and using(X(t,x-ct),Y(t,x-ct))and its reflection(X(t,-x-ct),Y(t,-x-ct)),we establish the existence and vari-ous qualitative properties of these entire solutions.In particular,the entire solutions exhibit a 'T-jumping' monotonicity with respect to t variable.Finally,we study the propagation phenomena of cooperative reaction-diffusion systems with advection in multi-dimensional periodic habitats.We first establish an theory concerning traveling waves and spreading speeds of monotone semi-flows posed in RN.Then we obtain the existence of spreading speeds and pulsating traveling waves for the system under appropriate assumptions.We also obtain a set of sufficient conditions for the linear determinacy of the spreading speed.Then we study the asymptotic behavior of pulsating traveling waves near infinity with noncritical and critical speeds respectively.We further construct some pulsating front-like entire solutions based on the interaction of rightward and leftward pul-sating traveling waves.These spreading properties of the reaction-diffusion system then exhibit some invasion and spreading ways for species or epidemics in multi-dimensional heterogeneous habitats.We finally consider a two-species competition model in a periodic habitat and study the propagation phenomena of the system.