Existence and nonexistence of monotone traveling waves for the delayed Fisher equation

In this paper a new approach based on a shooting method in a half line coupled with the technique of upper–lower solution pair is used to study the existence and nonexistence of monotone waves for one form of the delayed Fisher equation that does not have the quasimonotonicity property. A necessary and sufficient condition is provided. This new method can be extended to investigate many other nonlocal and non–monotone delayed reaction–diffusion equations.