| This paper is concerned with a class of nonlocal diffusion systems of three species with delays,where(Ji*ui)(x,t)=∫R Ji(x-y)ui(y,t)dy(i=1,2,3),Ji denote the diffusive kernel func-tions of i specie respectively,and ui(x,t)represent the density of i specie.The parameters di,ri>0,0<aij<1(i,j=1,2,3 and i≠j),and the delays τij≥0(i,j=1,2,3).Mor-eover,aij,(i,j)={(1,2),(2,1)},denote the inter-specific cooperation rates;aij,(i,j)={(1,3),(3,1),(2,3),(3,2)},denote the inter-specific competition rates.This paper divides into four chapters to discuss the existence,the asymptotic behavior,and the monotonicity and uniqueness of traveling wave solutions of the above system when the parameters meet certain specific conditions.In Chapter 1,we introduce the research background of nonlocal diffusive competition-cooperation systems of three species with delays,and the existing research results.In Chapter 2,we make some assumptions about the parameters in the system.Then it is proved by using the upper and lower solutions that,when c≥c*,the system has traveling wave solutions connecting different equilibria.While the non-existence of the traveling wave solutions when 0<c<c*,is proved by the contradiction method.In Chapter 3,we discuss the asymptotic behavior of the nondecreasing solutions at the negative infinity when τii=0 or τii satisfy 0<τii<min i≠j{τij}(i=1,2,3).Then,forτij=0 and Ji(x)=δ(x)-δ"(x)(i,j=1,2,3),where δ is the Dirac delta function,we obtain the asymptotic behavior of the nondecreasing solutions at the positive infinity by the stable manifold theorem,and then explore the monotonicity and uniqueness of traveling wave solutions. |
