Reaction-diffusion Equations with Infinite Delay

We have developed several results on the existence and asymptotic behavior of nlild solutions to rextiondiffusion systems that have infinite delirys in the nonlinear reaction terms. We find that the semiflow generated by a cooperative and irreducible reaction-tliffilsio system with infinite delay is not compact but set-condcrrsing, ancl not strongly order-preserving but quasi strongly ordc~r-1)rest:rviiig. These set-condenseness and quasi strong ordt!r-prosc:rvi~~g properties allow us to use a ~nodificirt,ion, recent.1~ givol~ by Freedman, Miller and one of the ilutliors of tllis l,iqier, of the wellknown monotone dynan~ical systenl tl~c:ory due to Dancer, Hess, Hirsch, Matano, Smith, Tllicme, 1301iCik and TakaE to obtain some results about convergence and stability of solutions. Examples of Loth-Volterra co~npetition-diffusion models with distributed delay are given to illustrate the obtained results.