Monotone traveling wavefronts of the KPP-Fisher delayed equation

In the early 2000's, Gourley (2000), Wu et al. (2001), Ashwin et al. (2002) initiated the study of the positive wavefronts in the delayed Kolmogorov–Petrovskii–Piskunov–Fisher equationequation(⁎)ut(t,x)=Δu(t,x)+u(t,x)(1−u(t−h,x)),u⩾0,x∈Rm. Since then, this model has become one of the most popular objects in the studies of traveling waves for the monostable delayed reaction–diffusion equations. In this paper, we give a complete solution to the problem of existence and uniqueness of monotone waves in Eq. (⁎). We show that each monotone traveling wave can be found via an iteration procedure. The proposed approach is based on the use of special monotone integral operators (which are different from the usual Wu–Zou operator) and appropriate upper and lower solutions associated to them. The analysis of the asymptotic expansions of the eventual traveling fronts at infinity is another key ingredient of our approach.