Propagation Dynamics Of Nonlocal Dispersal Problems In Heterogeneous Media

A lot of nonlocal dispersal equations have been derived from population dy-namics,epidemiology,material science and many other fields,which aroused rather considerable interest in recent years.Compared with the classical random diffusion mode,the nonlocal dispersal operator owns a prominent advantage in terms of de-scribing the spatial distribution mechanism of the organisms.At present,among the most important dynamical issues about reaction-diffusion equations are propa-gation dynamics such as the asymptotic speeds of spread,traveling waves and entire solutions.In addition,numerous mathematical models originating from practical problems possess spatial-temporal heterogeneity owing to the complexity of the un-derlying environment.However,due to the limitations of theoretical methods and conceptual techniques,the current research results on the propagation dynamics of the nonlocal dispersal problems in heterogeneous media are still very limited.In this dissertation,we continue to study this subject by choosing some typical nonlo-cal dispersal equations(systems)which have important theoretical significance and application value.Firstly,we study some new types of entire solutions for nonlocal dispersal equa-tions with monostable nonlinearity in spatially periodic habitat of RN.By estab-lishing the existence of a globally spatially periodic solution connecting zero solution and positive periodic steady state,and considering the interactions of it and two different pulsating waves coming from two opposite directions,respectively,we con-struct some different types of entire solutions.Also,the qualitative properties of these entire solutions are investigated.In particular,for a class of special heteroge-neous reaction,we further establish the uniqueness and continuous dependence of such an entire solution on parameters such as wave speeds and the shifted variables.Moreover,we obtain some higher dimensional entire solutions by studying the in-teractions of finite number of pulsating fronts and a global spatial periodic solution.Meanwhile,we find the condition that certain pulsating front can highlight when time goes infinity.Secondly,we study the propagation dynamics for a time periodic nonlocal dis-persal model with stage structure.In the case where the birth rate function is monotone,we establish the existence of the spreading speed and its coincidence with the minimal wave speed for monotone periodic traveling waves by appealing to the theory developed for monotone semiflows.In the case where the birth rate function is non-monotone,we first obtain the spreading property by the squeeze technique combined with some known results for the monotone case,which implies the nonexistence of subcritical waves,and then investigate the existence of super-critical periodic traveling waves by using the asymptotic fixed point theorem due to the lack of parabolic estimates and compactness.Also,we apply the general results to a time periodic Nicholson's blowflies model for its spatial dynamics.Thirdly,we study the spatial dynamics of a nonlocal dispersal population mod-el in a shifting environment where the favorable region is shrinking.By the method of super-and subsolutions and comparison arguments,we show that the species becomes extinct in the habitat if the speed of the shifting habitat edge c>c*(?),while the species persists and spreads along the shifting habitat at an asymptotic speed c*(?)if c<c*(?),where c*(?)is determined by diffusion ability(disper-sal kernel J and diffusion rate d)and the maximum linearized growth rate r(?).Moreover,we demonstrate that for any given speed of the shifting habitat edge,the model system admits a nondecreasing traveling wave connecting 0 and r(?)with the wave speed at which the habitat is shifting,which indicates that the extinction wave phenomenon does happen in such a shifting environment.Then the unique-ness of the extinction wave with the same forced speed is obtained by the sliding technique.Finally,we study the global dynamics and spreading speeds of a partially de-generate nonlocal dispersal cooperative system in periodic habitats.We first obtain the existence of principal eigenvalue for a periodic eigenvalue problem with partial-ly degenerate nonlocal dispersal and then consider the coexistence and extinction dynamics by using the part metric method.Further,we investigate the existence of the spreading speed interval and also prove that this interval is indeed a singleton by combining the upper-lower linear systems which dominate the original system.Meanwhile,the linear determinacy and computational formulae of the spreading speed are given.At last,we have a simple discussion on the pulsating waves and new types of entire solutions for such a degenerate system.