This talk is inspired by recent findings concerning the noisy F-FPP equation with Allee effect
$$ u_t =\frac{1}{2} u_{xx} + u(1 − u)(1 + Au) + \sqrt{\frac{u(1 − u)}{N}}\eta $$
where $\eta$ represents space-time white noise, $N$ is a large demographic parameter and $A$ encodes an Allee effect in the population. Empirical observations and theoretical considerations point to the presence of a fascinating phase transition encompassing a pulled, a semi-pushed, and a fully pushed regime. From a biological standpoint, this phase transition has some important consequences on the genetic diversity in expanding populations.
To elucidate this phenomenon, I will introduce a category of branching Brownian motions that exhibit varying branching rates. This particular model was recently proposed by Tourniaire (22) and can be seen as a modification of the well-known Berestycki, Berestycki, Schweinsberg model (13). I will demonstrate the existence of a phase transition analogous to the one observed in the noisy F-KPP equation. The proof will rely on a general methodology, involving the computation of "moments" of a generalized branching process through spinal decompositions (as seen in Foutel–Rodier, Schertzer 22).
