Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions

We consider the nonlocal KPP-Fisher equation ut(t,x)=uxx(t,x)+u(t,x)(1−(K⁎u)(t,x))ut(t,x)=uxx(t,x)+u(t,x)(1−(K⁎u)(t,x)) which describes the evolution of population density u(t,x)u(t,x) with respect to time t and location x  . The non-locality is expressed in terms of the convolution of u(t,⋅)u(t,⋅) with kernel K(⋅)≥0K(⋅)≥0, ∫RK(s)ds=1∫RK(s)ds=1. The restrictions K(s)K(s), s≥0s≥0, and K(s)K(s), s≤0s≤0, are responsible for interactions of an individual with his left and right neighbors, respectively. We show that these two parts of K play quite different roles as for the existence and uniqueness of traveling fronts to the KPP-Fisher equation. In particular, if the left interaction is dominant, the uniqueness of fronts can be proved, while the dominance of the right interaction can induce the co-existence of monotone and oscillating fronts. We also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on K.