Spatial Dynamics For Partially Degenerate Reaction-Diffusion Systems In Periodic Environment

In recent years,many reaction-diffusion models with some zero diffusion coefficients（i.e.partially degenerate reaction-diffusion systems）have been introduced to accurately describe the interaction phenomena between species that differ in their mobility in epidemiology and population biology.Due to the lack of non-compactness with respect to compact open topology and lower regularity of solutions,some classical theories and methods（such as,classical Krein-Rutman’s theorem and the priori estimates of solutions to parabolic equa-tion）are not applicable.Therefore,it is exactly important and difficult to study the spatial dynamics of such systems.In addition,the mathematical models deriving from real biologi-cal phenomena should possess spatial-temporal heterogeneity since the nature environment is heterogeneous.Furthermore,the time delays that are prevalent in nature can not be ig-nored.However,limited by theory,methods,concepts and techniques,there are very few research results on the spatial dynamics of partially degenerate reaction-diffusion systems in heterogeneous environments.This paper is concerned with the spatial dynamics of partially degenerate（delayed）reaction-diffusion systems in heterogeneous environments.Some new concepts,methods and techniques are developed and introduced.Firstly,we establish the theory of principal eigenvalues for a large class of partially de-generate,linear and periodic parabolic cooperative systems with time-delay via generalized Krein-Rutman theorem.We also extend the earlier theory of basic reproduction ratio to abstract functional differential equations with time-delayed internal transition by using the variation of constants formulas of abstract delayed differential equations and the properties of principal eigenvalues of linear positive operators.Then we apply these theoretical results and present a threshold-type result in terms of basic reproduction ratio（R₀）for the dynamic behaviors of a blacklegged tick Ixodes scapularis population model.We show that the tick population persists if R₀>1 and the population becomes extinct if R₀<1.Secondly,we investigate the spreading speed and monostable traveling waves for partial-ly degenerate reaction-diffusion systems with time-periodic nonlinearity.In the monotone case,we prove the existence of periodic traveling fronts and the exponential stability of con-tinuous periodic traveling fronts.In the non-monotone case,we establish the existence of the minimal wave speed of periodic traveling waves and its coincidence with the spread-ing speed.More specifically,when the system is nondegenerate,the existence of periodic traveling waves is proved by using Schauder’s fixed point theorem and the regularity of an-alytic semigroup,while in the partially degenerate case,due to the lack of compactness and standard parabolic estimates,the existence result is obtained by appealing to the asymp-totic fixed point theorem with the help of some properties of the Kuratowski measure of noncompactness.Thirdly,we study the spreading speeds,monostable traveling waves and transition waves of a large class of time-space periodic and partially degenerate reaction-diffusion systems with time-delay.In the quasi-monotone case,based on the theory of principal eigenvalues for linear and partially degenerate systems with time delay,we establish the existence of spreading speeds and its coincidence with the minimal wave speed of time-space periodic traveling waves.In the non-quasi-monotone case,we introduce the definition of transition semi-waves and prove the existence and equality of the spreading speed and the minimal wave speed of transition semi-waves by constructing two auxiliary cooperative systems and establishing some new prior estimate.