Entire solutions for a delayed nonlocal dispersal system with monostable nonlinearities

The paper is concerned with the existence and qualitative properties of entire solutions for a delayed nonlocal dispersal system with monostable nonlinearities. We first prove the existence of traveling wave fronts and a spatially independent solution of the system. Then we establish the comparison principles and some upper estimates for solutions of the system with quasimonotone nonlinearities. Mixing the traveling wave fronts with different wave speeds and the spatially independent solution, we derive the existence of entire solutions for the quasimonotone system. Moreover, we investigate various properties of the entire solutions. Of particular interest is the relationship between the entire solutions and the traveling wave fronts. Finally, by introducing two auxiliary quasimonotone systems, we improve our results to nonquasimonotone systems.