The asymptotic behaviors of traveling waves for a single species model with delays on lattices were investigated.The existence of traveling wave solutions of the system was studied by using the monotone iterative technique with the help of the upper and lower solutions, whose constructed structures can ensure the exponential asymptotic behaviors of non-critical traveling wave solutions (the wave speed is greater than the critical speed).In the light of the asymptotic theory in the Ikehara theorem the exponential asymptotic behaviors of all non-critical traveling wave solutions were obtained and the algebraic exponential asymptotic behaviors of the critical traveling wave solutions (the wave velocity is equal to the critical velocity) were achieved.The results make up and improve the result on asymptotic behaviors of traveling wave solutions.