Traveling Waves For Predator-prey System With Nonlocal Diffusion

This thesis is mainly studied the traveling wave solution of the ratio-dependent predator-prey model with non-local diffusion terms,and the traveling wave solution of the predator-prey model with non-local diffusion term and time delay considering different functional responses.The first chapter is mainly reviewed the background and current status of the research problem,and introduced the main work done in this article and the significance of the research.In Chapter 2,we will study the following ratio-dependent predator-prey model with non-local diffusion(?)where di>0(i=1,2)represent the diffusion rate of the prey and predator,re-spectively,0<?<a,a>0,r>0,J*u(x,t)=(?)J(y)u(x-y,t)dy,J*v(x,t)=(?)(y)v(x-y,t)dy.Firstly,we will construct a pair of upper and lower solutions when the wave speed c is greater than the minimal wave speed c*.Then by applying Schauder's fixed point theorem with the suitable upper and lower solutions,we can obtain the existence of traveling waves when the wave speed c>c*.Moreover,in order to prove the limit behavior of the traveling waves at infinity,we construct a sequence that converges to the coexistence state.Finally,by using the comparison principle,we can obtain the nonexistence of the traveling waves when the wave speed 0<c<c*.In Chapter 3,we will discuss the following non-local predator-prey model with time delay(?)where d>0,r>0,??(0,1),J*u(x,t)=(?)(y)u(x-y,t)d,7 J*v(x,t)=(?)(y)v(x-y,t)dy and K*v(x,t)=(?)K(y,s)v(x-y,t-s)dyds.We will mainly study the existence of the traveling waves for the nonlocal predator-prey model with spatio-temporal delay.We will establish the existence of the traveling waves for c>c*by applying Schauder's fixed point theorem and squeeze method.Here,the compactness of the supported set of dispersal kernel is needed when passing to an unbounded domain in the proof.Finally,by using the comparison principle,we can obtain the nonexistence of the traveling waves for 0<c<c*.