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We consider the nonlocal KPP-Fisher equation $u_t(t,x) = u_{xx}(t,x) +u(t,x)(1-(K *u)(t,x))$ which describes the evolution of population density$u(t,x)$ with respect to time $t$ and location $x$. The non-locality isexpressed in terms of the convolution of $u(t, \cdot)$ with kernel $K(\cdot)\geq 0,$ $\int_{\mathbb{R}} K(s)ds =1$. The restrictions $K(s), s \geq 0,$ and$K(s), s \leq 0,$ are responsible for interactions of an individual with hisleft and right neighbors, respectively. We show that these two parts of $K$play quite different roles as for the existence and uniqueness of travelingfronts to the KPP-Fisher equation. In particular, if the left interaction isdominant, the uniqueness of fronts can be proved, while the dominance of theright interaction can induce the co-existence of monotone and oscillatingfronts. We also present a short proof of the existence of traveling waveswithout assuming various technical restrictions usually imposed on $K$.

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