Propagation Dynamics Of Two Species Reaction-Diffusion Systems In Time-Periodic Environments

Reaction-diffusion equations are widely used in population dynamics,epidemiology,ecology and other disciplines.Considering the background of the practical problems,an important issue of such equations is propagation dynamics.In population biology,the competitive systems and predator-prey systems are important models for simulating interactions between two or more species.In addition,the time-varying environment(such as the change of seasons,alternation of day with night)has a great influence on the dynamics of the population.Therefore,it is a factor that can not be neglected in the study of species evolution.Thus,it is of great practical value to study the propagation dynamics of time-periodic competition and predator-prey reaction-diffusion systems.In mathematics,the periodicity brings new difficulties to the study of these two kinds of systems.In particular,due to the non-monotonicity of predator-prey systems,many classical theories and methods are no longer applicable to time-periodic predator-prey systems.Therefore,there are substantial difficulties in the study of such problems.This thesis is mainly concerned with the propagation dynamics of time-periodic Lotka-Volterra(L-V for short)competitive reaction-diffusion systems and predator-prey diffusion systems,and some new techniques and methods are attempted to be introduced or developed.The main contents are as follows.Firstly,we study the long time behavior of bounded solutions to a time-periodic L-V reactiondiffusion system with strong competition.By constructing a suitable subsolution,it is proved that the solutions to the system with a class of compact support initial values converge to a pair of diverging periodic traveling fronts.Furthermore,a sufficient condition for solutions to spread is given.To the best of our knowledge,this may be the first time that the stability of a pair of diverging periodic traveling fronts for time-periodic reaction-diffusion systems is considered.Then,the spreading properties of solutions for the system with a class of non-compact support initial values are considered.More specifically,we prove that if the two species are initially absent from the right half-line > 0 and the slower one dominates the faster one on < 0,then the solutions approach a propagating terrace.Secondly,we study the time-periodic traveling waves for a periodic L-V competition system with nonlocal dispersal.Under strong competition assumption and among other things,the existence of periodic traveling fronts connecting two stable semi-trivial periodic solutions is proved by using the theory on monotone semiflows with weak compactness.It should be pointed out that the periodic traveling waves obtained may be discontinuous due to the presence of the nonlocal dispersal operators.Since the linearized system of L-V competitive system at the semi-trivial periodic solution is reducible,the standard strong comparison principle does not hold for such system.By constructing two auxiliary diffusion systems and establishing a new strong comparison principle,the strict monotonicity and global exponential stability of smooth periodic traveling waves are proved.The Liapunov stability of the periodic traveling fronts is also obtained.Finally,we study the propagation dynamics of a time-periodic predator-prey system with nonlocal dispersal.We prove the existence of the periodic traveling waves by appealing to the asymptotic fixed point theorem with the help of the Kuratowski measure of noncompactness.Then,by using comparison argument and constructing an auxiliary system,the nonexistence of periodic traveling waves is obtained.Lastly,we investigate the spreading properties of solutions starting from compactly supported initial conditions.Roughly speaking,we show that if predators disperse faster than the prey,then both species spread simultaneously;whereas if the prey disperses faster than predators,then there will appear two separate invasion fronts,one front occurs as the prey invades into open habitats,and the other front appears when predators catch up the prey.It is worth pointing out that due to the lack of compactness of solution operators for nonlocal diffusion problems,the lack of comparison principle for predator-prey systems and the occurrence of time periodicity,some new techniques need to be introduced.For example,the theory of persistence in dynamical systems is developed to study lower estimates for the spreading speed.