PIMS Postdoctoral Colloquium: KPP pulsating traveling fronts within large drift

Abstract: This talk is based on a joint work with St\'ephane Kirsch. Pulsating traveling fronts are solutions of heterogeneous reaction-advection-diffusion equations that model some population dynamics. Fixing a unitary direction $e$, it is a well-known fact that for nonlinearities of KPP type (after Kolmogorov, Petrovsky and Piskunov, f(u)=u(1-u) is a typical homogeneous KPP nonlinearity), there exists a minimal speed c* such that a pulsating traveling front with a speed $c$ in the direction of $e$ exists if and only if $c\geq c^*$. In a periodic heterogeneous framework we have the formula of Berestycki, Hamel and Nadirashvili (2005) for the minimal speed of propagation. This formula involves elliptic eigenvalue problems whose coefficients are expressed in terms of the geometry of the domain, the direction of propagation, and the coefficients of reaction, diffusion and advection of our equation. In this talk, I will describe the asymptotic behaviors of the minimal speed of propagation within either a large drift, a mixture of large drift and small reaction, or a mixture of large drift and large diffusion. These ``large drift limits'' are expressed as maxima of certain variational quantities over the family of ``first integrals'' of the advection field. I will give more details about the limit and a necessary and sufficient condition for which the limit is equal to zero in the 2-d case.