Propagation Dynamics For Several Types Of Reaction-Diffusion Models In Periodic Media

There is no absolutely ideal homogeneous media in nature.Due to spatial heterogeneity,habitat fragmentation,specific areas,obstacles,and diurnal seasonal variations,it is only natural to consider diffusion and propagation phenomena in heterogeneous media.Concerning a simple but fundamental approximation for heterogeneous media,this thesis is devoted to the mathematical study of spatial propagation in periodic media for some classical reaction-diffusion models arising in population and epidemiological dynamics.By periodic media we mean that the reaction-diffusion equations and systems under consideration are equipped with coefficients or nonlinearities which are periodic with respect to the space variables.Our goal is threefold.In the first part,we study the wave propagation phenomena for reaction-diffusion equations of KPP type in multi-dimensional periodic media.In such a periodic setting,the notion of planar travelling waves has been extended to the notion of pulsating travelling waves.The profile of pulsating travelling waves constructed in the classical literature satisfies a degenerate elliptic equation,which can be largely solved by a vanishing viscosity method,namely an elliptic regularization procedure in this context.We construct a new profile function of pulsating waves under a moving frame dependent on an orthogonal transformation.It satisfies a non-degenerate parabolic equation with periodicity conditions(in both space and time)for a dense set of directions.Hence the difficulties caused by the elliptic degeneracy are successfully overcome for a dense subset of propagation directions,while the case of a general direction of propagation can be obtained by limiting approximation arguments.Based on such a framework,we revisit the existence of pulsating travelling fronts for the Fisher-KPP equation posed in multidimensional periodic media.Our methodology somewhat refines and simplifies some earlier proofs.In fact,we have also succeeded in devising another approach to discuss the existence of pulsating waves for multidimensional reaction-diffusion equations.In the second part,we study the well-known Kermack-McKendrick SIR epidemic model with Fickian diffusion posed in multi-dimensional periodic media,which is a nonmonotone reaction-diffusion system.Using the methodology developed in the first part coupled with some properties of the periodic principal eigenvalues associated with the elliptic equations linearized at disease-free equilibrium,we prove the existence and nonexistence of pulsating travelling waves for this model.Since there is no a priori maximum norm estimate for the entire solutions of this kind of systems,we develop new techniques to prove the uniform boundedness of two sequences of the solution derived from a finite domain approximation procedure and a rational approximation one,which ensures that the limit function during both limiting arguments,namely the pulsating travelling wave propagating along any given direction,is globally bounded.Under some stronger assumptions,the asymptotic behaviour of pulsating waves propagating in each rational direction is also investgated.Our analysis yields that the travelling epidemic wave still transitions from one disease-free equilibrium to another and behaves a pulse-like shape for the infections,but the profile of the front is periodic with respect to time in the moving frame along each rational direction.In the third part,we investigate the propagation dynamics for an epidemic model describing the spatial spread of rabies between two populations in a periodic habitat.A classical and interesting trick allows us to transform this model into a cooperative parabolic system.Then we use the abstract theory for monotone semiflows coupled with some auxiliary Cauchy problems to study the spreading properties of this system.Coming back to the original epidemic model,we conclude that the existence and nonexistence of pulsating travelling waves is entirely determined by the basic reproduction number,and the minimal wave speed is linearly determined and given by a variational formula involving an eigenvalue problem for a linear cooperative and irreducible elliptic system of equations with periodic coefficients.Furthermore,for some initially compactly supported infected population,the solution of such a model has an asymptotic spreading speed in large time along both directions of the left and the right,which is exactly coincident with the minimal wave speed.