Entire Solutions For Nonlocal Dispersal Equations (Systems) In Different Media

Nonlocal dispersal equations are an important kind of evolution equations,which have wide application background in many disciplines such as population ecology,material science,neural networks,epidemiology,physics and so on,and have aroused great interest of experts and scholars in various disciplines.Compared with the classical local dispersal operator,the nonlocal dispersal operator has obvi-ous advantages in describing the spatial distribution mechanism of the organisms.Therefore,many mathematicians and biologists pay close attention to the propaga-tion dynamics of nonlocal dispersal equations,such as asymptotic speeds of spread,traveling wave solutions and entire solutions.In particular,the study on entire solu-tions is crucial to the understanding of two important aspects of infinite-dimensional dynamical systems:transient dynamics and global attractors.In this paper,we are mainly concerned with entire solutions of nonlocal dispersal Lotka-Volterra compe-tition systems in homogeneous and periodic media and nonlocal dispersal equations with bistable nonlinearity in spatially periodic media.Firstly,we are concerned with entire solutions of the Lotka-Volterra competi-tion system with nonlocal dispersals.By studying the eigenvalue problem of Lotka-Volterra competition system linearized at the traveling wave solution consisting of a single species,we construct a pair of super-and sub-solutions,and then establish the existence of entire solutions originating from the traveling wave solution consist-ing of a single species.Meanwhile,we give a detailed description on the long-time behavior of such entire solutions as→∞.The main results include both the weak and strong competition cases and we present some simulations to illustrate analytical results.Further,since the strong competition systems have more com-plex dynamical behaviors than other competition cases.Thus,we consider entire solutions originating from multiple traveling fronts for Lotka-Volterra strong com-petition system with nonlocal dispersals.By constructing appropriate super-and sub-solutions,and with the aid of corresponding comparison principle,we establish the existence and related qualitative properties of entire solutions originating from three and four traveling wave solutions.Meanwhile,we derive the non-existence of entire solutions originating from more than seven traveling fronts by introducing the definition of non-extendable(terminated)sequence.Secondly,we study the new types of entire solutions originating from pulsating fronts of nonlocal dispersal equations with bistable nonlinearity in spatially periodic media.In order to prove the existence of these entire solutions.We first establish the existence of pulsating fronts with small periods by the implicit function theorem,and then give the asymptotic behavior near infinity of the pulsating fronts.Further,we investigate the existence and qualitative properties of new types of entire solutions by the sub-and super-solutions method and the comparison principle.Finally,we investigate the new types of entire solutions which are different from pulsating fronts of Lotka-Volterra competition systems with asymmetric dis-persal in spatially periodic habitats.The asymmetry of kernel function leads to a great influence on the profile of pulsating fronts and the sign of wave speeds,which makes the properties of the entire solution more complex and diverse.We first give a relationship between the critical speeds c~*(ξ)and c~*(-ξ),corresponding to the minimal speeds of two pulsating fronts propagating in the direction of ξ and -ξ,respectively.Then,the exponential behavior of pulsating fronts as they approach their limiting states is obtained.Finally,by considering the interactions of two dif-ferent pulsating fronts coming from two opposite/same directions and applying the super-/subsolutions techniques as well as the comparison principle,we establish the existence and various qualitative properties of some different types of entire solu-tions.Moreover,we give some numerical simulations to describe intuitively these entire solutions.