Traveling Wave Solutions For Two Classes Of Diffusion Systems With Nonlocal Dispersal And Delay

As a kind of steady state solutions of reaction-diffusion equations,traveling wave solutions have space translation invariance,and thus many propagation phenomena in nature can be described by them,such as the spread of infectious diseases,the growth of the population,the migration and intrusion of species,and so on.For example,in the epidemic models,traveling waves show that the transmission of infection source in space.If the speed of infection source can always be controlled within the minimum propagation speed that the infectious disease occurs exactly,then infectious diseases will not be transmitted.In addition,by the effect of delay and spatial non-locality,the changes of dynamic behavior of equations are led to appear,such as time delay can reduce the minimal wave speed,non-local diffusion can also accelerate the minimal wave speed,and so on.Therefore,it is significant on theory and practice on the investigation of the existence of traveling waves for non-local diffusion equations with delay and the impact on traveling waves derived from delay and nonlocal dispersal.Based on the above fact,we are not only concerned with the existence of traveling waves for a delayed SIR model with nonlocal dispersal and nonlinear incidence and the existence of monostable traveling waves of a non-local diffusion system with delay and without quasi-monotonicity,but the impact on traveling waves derived from delay and nonlocal dispersal.The main results in this paper are included as follows.· The existence of traveling wave solutions for a delayed SIR model with nonlocal dispersal and nonlinear incidence is established.First of all,in a limited area,by upper and lower solutions and Schauder fixed point theorem,we establish the existence of the solutions in a bounded area and thus get the existence of traveling wave solutions of system in the whole space by using a prior estimate combining with the limit theory.Secondly,by two-sided Laplace transform,we establish the non-existence of traveling wave solutions of the system.Finally,we obtain the results that the spread speed c is influenced by the dispersal rate of the infected individuals and the delay ?.· The existence of monostable traveling wave solutions of a non-local diffusion system with delay and without quasi-monotonicity is established.Firstly,by using the results of wavefront solutions under the quasi-monotone conditions,we construct a quasimonotone comparison system and wave profile set in an appropriate Banach space.Secondly,we prove that the fixed point of the operator in wave profile set is exactly the traveling wave solution of the system without quasi-monotonicity by Schauder fixed point theorem.