Propagation Phenomena Of Integro-difference Equations And Bistable Reaction-diffusion Equations In Periodic Habitats

This dissertation is concerned with propagation phenomena of some evolution equations in periodic habitats. The main results consist of the following two parts.In the first part, we investigate the spatial spreading phenomena of some spatially periodic integro-difference equations for which the dispersal kernels are Radon probability measures on R. Firstly, we establish a general theory on the existence and characterization results of spreading speeds for noncompact evolution systems in periodic habitats, under the hypothesis that the linearized systems have principal eigenvalues. Secondly, we introduce the notion of uniform irreducibility for finite Radon measures on the circle. It is shown that, any generalized convolution operator generated by such a measure admits a principal eigenvalue. Finally, applying the above general theories, we prove the existence of spreading speeds for some spatially periodic integro-difference equations and integro-differential equations with uniformly irreducible dispersal kernels, and we also describe the effects of the spatial and temporal variations on the speeds.In the second part, we study the front propagation phenomena of spatially periodic reaction-diffusion equations with bistable nonlinearities. Firstly, we focus on the propagation solutions in the class of pulsating fronts. We find that the spatial period plays an important role in the existence of pulsating fronts. It is proved that, under various assumptions on the reaction terms, pulsating fronts exist when the spatial period is small or large. Furthermore, some properties of the set of the period for which there exist pulsating fronts with nonzero speed are established. We also characterize the sign of the front speeds and we show the global exponential stability of the pulsating fronts with nonzero speed. Secondly, we investigate the front propagation solutions in the larger class of transition fronts. It is shown that, under suitable assumptions, transition fronts reduce to pulsating fronts with nonzero wave speed. But we also prove the existence of new types of transition fronts which are not pulsating fronts.