A branching particle system as a model of semi pushed fronts

Abstract

We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift −µ and killed upon reaching 0, starting with N particles. More precisely, particles branch at rate ρ/2 in the interval [0, 1], for some ρ > 1, and at rate 1/2 in (1, +∞). The drift µ(ρ) 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, semi-pushed and pushed fronts. Here, we rigorously verify and make precise this classification and focus on the semi-pushed regime. More precisely, we prove the existence of two critical values 1 < ρ 1 < ρ 2 such that for all ρ ∈ (ρ 1 , ρ 2 ), there exists α(ρ) ∈ (1, 2) such that the rescaled number of particles in the system converges to an α-stable continuous-state branching process on the time scale N α−1 as N goes to infinity. This complements previous results from Berestycki, Berestycki and Schweinsberg for the case ρ = 1. Contents

Relative vs absolute fitness in a population genetics model. How stronger selection may promote genetic diversity

“…Intricate phase transitions in front propagations have been observed in previous works [5,33,42,20]. One interesting example is a population whose large scale behavior is described by the noisy F-KPP equation with Allee effect [4,5]…”

“…log(1), where for the last equality we have again used Lemma C.2 which implies E w −1 k = k. Thus, taking expectations in(42) and plugging in(43) we obtain, for every > 0, log(χ − ) + log log (N ) + o (1) ≤ E log N k=1 w I k ; A N, , B N, ≤ log(χ + ) + log log (N ) + o (1) . This, together with (41), implies log(1 − χ ) ≤ lim N →∞ log χ log(N ) − E log…”

Genealogies in bistable waves

“…Before discussing the definition of our model, we mention recent rigorous results of Tourniaire [33] on a related model. She studies a model that mimics a population expanding according to a travelling wave, and her model also exhibits fully pushed, semi-pushed and pulled regimes.…”