A branching particle system as a model of semi pushed fronts
Abstract
We consider a system of particles performing a onedimensional dyadic branching Brownian motion with spacedependent branching rate, negative drift $\mu$ and killed upon reaching $0$, starting with $N$ particles. More precisely, particles branch at rate $\rho/2$ in the interval $[0,1]$, for some $\rho>1$, and at rate $1/2$ in $(1,+\infty)$. The drift $\mu(\rho)$ is chosen in such a way that, heuristically, the system is critical in some sense: the number of particles stays roughly constant before it eventually dies out. This particle system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population. Recent studies from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, semipushed and pushed fronts. Here, we rigorously verify and make precise this classification and focus on the semipushed regime. More precisely, we prove the existence of two critical values $1<\rho_1<\rho_2$ such that for all $\rho\in(\rho_1,\rho_2)$, there exists $\alpha(\rho)\in(1,2)$ such that the rescaled number of particles in the system converges to an $\alpha$stable continuousstate branching process on the time scale $N^{\alpha1}$ as $N$ goes to infinity. This complements previous results from Berestycki, Berestycki and Schweinsberg for the case $\rho=1$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2111.00096
 Bibcode:
 2021arXiv211100096T
 Keywords:

 Mathematics  Probability