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A branching particle system as a model of semipushed fronts

Julie Tourniaire

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 $(1,+\infty)$. 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, 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 $1<ρ_1<ρ_2$ such that for all $ρ\in(ρ_1,ρ_2)$, there exists $α(ρ)\in(1,2)$ 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 $ρ=1$.

A branching particle system as a model of semipushed fronts

Abstract

We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift and killed upon reaching , starting with particles. More precisely, particles branch at rate in the interval , for some , and at rate in . The drift 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 such that for all , there exists such that the rescaled number of particles in the system converges to an -stable continuous-state branching process on the time scale as goes to infinity. This complements previous results from Berestycki, Berestycki and Schweinsberg for the case .

Paper Structure

This paper contains 34 sections, 48 theorems, 452 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Noisy FKPP-type equations and semipushed fronts
  3. Continuous-state branching processes
  4. The model: assumptions and main result
  5. Comparison with results on fluctuating fronts
  6. Overview of the proof
  7. Perturbation of the Laplacian on the half-line
  8. Related models
  9. Biological motivations: the Allee effects
  10. Structure of the article
  11. Some notation
  12. BBM in an interval: the density of particles
  13. Spectral analysis
  14. Heat kernel estimates
  15. The Green function
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\rho_1:=1+\frac{\pi^2}{4}$. There exists $\rho_2>\rho_1$ such that for all $\rho\in(\rho_1,\rho_2)$, there exists $\mu(\rho)>1$ and $\alpha=\alpha(\rho)\in(1,2)$ such that, if we consider the BBM with branching rate def:r and drift $-\mu(\rho)$, the process $(\bar{N}_{N^{\alpha-1}t})_{t>0}$ con

Figures (4)

  • Figure 1: The expansion velocity as a function of cooperativity. Figure (a): in the particle system. Graph of $\mu$ as a function of $\rho$ (see Equations \ref{['defmu1']} and \ref{['def:murho']}). The transition between the pulled and the pushed regimes occurs at $\rho_1=1+\frac{\pi^2}{4}\approx 3.47$. Figure (b): in the PDE \ref{['limit:PDEh']}. Graph of $v$ as a function of $B$ (see Equation \ref{['eq:speedPDE']}) for $r_{0}=\frac{1}{2}$. The transition between the pulled and the pushed regimes occurs at $B=2$. Note that $\mu$ and $v$ have the same asymptotic behaviour when $\rho$ and $B$ tend to $+\infty$.
  • Figure 2: The expansion velocity as a function of cooperativity. Figure (a): in the particle system. Graph of $\mu$ as a function of $\rho$ (see \ref{['defmu1']} and \ref{['def:murho']}). The weakly pushed regime corresponds to $\mu\in(1,\mu_c)$. The transition between the weakly pushed and fully pushed regime occurs at $\rho=\rho_2$ (see \ref{['hwp']}). Figure (b): in the PDE. Graph of $v$ as a function of $B$ (see Equation \ref{['eq:speedPDE']}) for $r_0=\frac{1}{2}$. In the noisy FKPP equation, the transition between weakly pushed and fully pushed waves occurs when $v=\mu_c$ (see \ref{['defalphat']}), which corresponds to $B=4$.
  • Figure 3: Location of the eigenvalues of the SLP \ref{['eq:defT']} for $\rho=4$ and different values of $L$. The blue line represents the LHS of \ref{['square']}. The red line corresponds to the RHS of \ref{['square']}. The eigenvalues are located at the intersections of the blue and red solid lines. Note that the negative eigenvalues tend to a continuous spectrum as $L\to\infty$. For $\rho=9$, we have $K=1$.
  • Figure 4: Location of the eigenvalues of the SLP \ref{['eq:defT']} for $\rho=30$ and different values of $L$. For $\rho=30$, we have $K=2$.

Theorems & Definitions (91)

  • Theorem 1.1: informal version
  • Theorem 1.2
  • Conjecture 1
  • Conjecture 2
  • Lemma 1.3: Many-to-one lemma
  • Lemma 1.4: Many-to-two lemma, see Ikeda:1969tj, Theorem 4.15
  • Proposition 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 81 more