Spreading Speeds And Traveling Waves Of Nonlocal Dispersal DS-I-A Models

This paper studies the spatial propagation dynamics of the following DS-I-A model with non-local diffusion(?)in which all the parameters are nonnegative.The susceptible is divided into different groups according to their characteristics.From the viewpoint of monotonic dynamic systems,this system does not satisfy the comparison principle when?i(i=1,2,…,n)are different.We formulate the propagation dynamics of this system by traveling wave solutions and asymptotic spreading in the correspond-ing initial value problems,which models the transition process from disease free steady state to epidemic state.We first consider the minimal wave speed of traveling wave solutions.Without the monotonicity,we show the existence of nontrivial traveling wave solutions if the wave speed is not less than a given threshold,which is finished by generalized upper-lower solutions and Schauder's fixed point theorem.Then the asymptotic behavior is obtained by the theory of asymptotic spreading.When the wave speed is less than the threshold,we confirm the nonexistence of traveling wave solutions,which implies that the minimal wave speed equals to the threshold.We then study the asymptotic spreading of initial value problem.When the initial condition of the infected admits nonempty compact support.By constructing a pair of generalized upper-lower solutions,the upper bounds of spreading speed is obtained,which is not larger than the minimal wave speed.Moreover,the lower bounds of spreading speed is given by showing proper auxiliary equations,which indicates the invasion speed of the infected is not less than the minimal wave speed.Thus,the minimal wave speed of traveling wave solutions equals to the spreading speed in this model.