Microscopic Weak Selection Principle for the Logistic Branching Brownian Motion with selection

In this work, we present the logistic branching Brownian motion with selection (Log-BBM), a modification of the N-BBM defined by Groisman et. al (2020), in which birth and competition events are decoupled to allow for a variable population size that follows the branching process with logistic growth defined by Lambert (2005). We study the representation of the Log-BBM as a microscopic model for a FKPP-type equation. In the large population limit, the renormalised empirical measure of the Log-BBM converges weakly to a probability measure whose density solves a nonlocal version of the FKPP equation, while its cumulative distribution function solves the classical FKPP equation. We also show that this model exhibits behaviour that is characteristic of the Brunet-Derrida family of systems, in particular the so-called weak selection principle. Indeed, we show that, in the Log-BBM, the particles select the minimal propagation speed, in contrast to the FKPP equation, where the selected front speed depends on the initial condition.