Spatio-Temporal Propagation Of Nonlocal Dispersal Equations

Recently,since it can describe the large range diffusion,the nonlocal dispersal equation described by a convolution operator has aroused widespread concern and research,two of the most important research topic about the spatio-temporal propagation phenomenon of nonlocal dispersal equations are traveling wave solutions and entire solutions.At present,most existing results are based on the symmetry of the kernel function,however,in reality,the diffusion of population or individuals has directional selectivity,due to the influence of external environmental factors such as sun,wind,food and so on,this require us to use asymmetric kernel to simulate the diffusion process of population or individuals.In the first part,we study the traveling wave solutions and entire solutions of a nonlocal dispersal equation with asymmetric kernel.The second part will consider the traveling wave solutions and entire solutions of a nonlocal epidemic model with symmetric kernel.Firstly,we consider the entire solution of the nonlocal dispersal equation with asymmetric kernel and Fisher-KPP nonlinearity.Because the asymmetry of the kernel function affects the sign and size of the minimal wave speed,we first use analysis method to make a reasonable classification of the wave speeds,and then by studying the interactions of two traveling wave solutions and spatially homogeneous solutions,we construct different upper and lower solutions to obtain multi-types of entire solutions and their qualitative properties by the comparison principle and ArzelaAscoli theorem.We also obtain the uniqueness and the continuous dependence of the entire solutions which are established under the symmetric kernel function on the wave speeds and the translation parameters.Secondly,we discuss the exponential asymptotic behavior of the traveling wave solutions and the type of the entire solutions of the nonlocal dispersal equation with asymmetric kernel and bistable nonlinearity.In this case,by limit the range of the variable,we can consider the interactions of the monostable and bistable traveling waves,so as to obtain the merging front-like entire solutions.First of all,by using Jessen inequality,we classify the wave speeds,and use the bilateral Laplace transform and Ikehara theorem to obtain the asymptotic behavior of the traveling wave solutions,then by using upper and lower solution method and comparison principle,we establish the existence and qualitative properties of the entire solutions.In particular,we construct some examples of the kernel function and the nonlinear function to illustrate the results proposed in this chapter are meaningful and valuable.Thirdly,we study the exponential asymptotic behavior of the traveling wave solutions and the entire solution of the nonlocal dispersal equation with asymmetric kernel and ignition nonlinearity.Due to degenerative(i.e.f?(0)= 0)influence,we can not use the bilateral Laplace transform and Ikehara theorem to prove the asymptotic behavior of the traveling wave solution in-?.Therefore,we adopt a new method.By constructing appropriate barrier function and using comparison principle,we get the asymptotic behavior of traveling wave solution in-?,thus further obtain the existence and qualitative properties of the front-like entire solution.We also give some special examples of the kernel function and the nonlinear function in order to explain the theoretical significance of the results.Finally,we explore the entire solutions of a nonlocal epidemic model with symmetric kernel.We consider the type and qualitative properties of the entire solutions of the epidemic system with monostable and bistable nonlinearity,respectively,which shows that this kind of infectious disease has many modes of transmission from the beginning to survival or extinction.We first use the upper and lower solution method and the abstract theory of semigroups to obtain several types of entire solutions.But due to lack of the compactness of nonlocal operator and low regularity of solution,the entire solutions established here do not have enough smoothness with respect to the space variable,so we further obtain the smoothness of the entire solution on the space variable by using the ordinary differential theory and making some reasonable assumptions on the nonlinear function.