Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case

This paper deals with entire solutions for a general nonlocal dispersal monostable equation with spatio-temporal delay, i.e., solutions that are defined in the whole space and for all time t∈R. We first derive a particular model for a single species and show how such systems arise from population biology. Then we construct some new types of entire solutions other than traveling wave solutions and equilibrium solutions of the equation under consideration with quasi-monotone and non-quasi-monotone nonlinearities. Various qualitative properties of the entire solutions are also investigated. In particular, the relationship between the entire solutions and the traveling wave fronts which they originated is considered. Our main arguments are based on the comparison principle, the method of super- and sub-solutions, and the construction of auxiliary control systems.