Convergence and Wave Propagation for a System of Branching Rank-Based Interacting Brownian Particles

TL;DR

This work analyzes a rank-based branching diffusion on $\mathbb{R}$ with a Go or Grow mechanism: the $K$ rightmost particles grow (branch) at unit rate, while the others drift with intensity $\chi>0$, all undergoing Brownian motion. By weighting individuals with $1/K$ and studying the empirical distribution, the authors derive a limit PDE for the cumulative distribution and prove tightness and uniqueness results for the limit, connecting the microscopic stochastic dynamics to a macroscopic parabolic limit in weak form. Numerical experiments reveal linear spreading and a transition between pushed and pulled traveling waves governed by $\chi$, with a pushmi-pullyu case at $\chi=1$; they also develop an ancestral-lineage methodology showing how the backward-in-time distribution of ancestors converges to a diffusion with drift dependent on position within the wave, distinguishing regimes. A conjectured large-$K$ ancestral limit yields a one-dimensional diffusion with drift $\beta(z)=\sigma^*-\chi\mathbf{1}_{\{z<0\}}+2\frac{\partial_z u^{\sigma^*}}{u^{\sigma^*}}$, linking microscopic lineages to macroscopic wave dynamics. Overall, the paper advances the understanding of how rank-based branching and localized drift shape traveling waves and provides a rigorous continuum limit together with insightful numerical illustrations.

Abstract

In this work we study a branching particle system of diffusion processes on the real line interacting through their rank in the system. Namely, each particle follows an independent Brownian motion, but only K $\ge$ 1 particles on the far right are allowed to branch with constant rate, whilst the remaining particles have an additional positive drift of intensity $χ$ > 0. This is the so called Go or Grow hypothesis, which serves as an elementary hypothesis to model cells in a capillary tube moving upwards a chemical gradient. Despite the discontinuous character of the coefficients for the movement of particles and their demographic events, we first obtain the limit behavior of the population as K $\rightarrow$ $\infty$ by weighting the individuals by 1/K. Then, on the microscopic level when K is fixed, we investigate numerically the speed of propagation of the particles and recover a threshold behavior according to the parameter $χ$ consistent with the already known behavior of the limit. Finally, by studying numerically the ancestral lineages we categorize the traveling fronts as pushed or pulled according to the critical parameter $χ$.

Figures (9)

• Figure 1: Cartoon representation of the model. The first $K$ cells are in the Grow regime and branch with rate 1, whilst the trailing cells are in the Go regime and drift with a drift coefficient $\chi$. Moreover, all cells undergo Brownian motion.
• Figure 2: The average velocity of the $K$-th particle over the interval $[100,200]$, i.e.$\frac{\xi^K_{200}-\xi^K_{100}}{100}$, where $\xi^K_t=H_K(\mu^K_t)$ is the position of the $K$-th particle at time $t$, serves as an estimator of the expansion speed of the population. As $K$ increases, in average this estimator gets closer and closer to the deterministic traveling wave speed $\sigma^*=\left\{\chi+\frac{1}{\chi}\text{ if }\chi>12\text{ if }\chi\leq 1 \right.$ (see Figure \ref{['fig:speed']} for more details).
• Figure 3: Graphical representation of the particle trajectories in (a) the stationary frame $(t,x)$, and (b) the moving frame $(t,z)=(t,x-\xi^K_t)$, where $\xi^K_t$ is the position of the $K$-th particle at time $t$. The red curve represents the position of the $K$-th particle and in (a) we observe a linear evolution of its position. Four more trajectories have been arbitrarily highlighted for illustration purposes and represent the ancestral lineage that evolves backwards in time, i.e. that should be read from right to left (see Figure \ref{['fig:trajectories']} for more details).
• Figure 4: Graphical representation of the ancestral lineage distribution. (a): The pushed case with $\chi=2$. (b): The pulled case with $\chi=\frac{1}{2}$. The top figure represents the histogram of $\mu_T^K$ in the frame $z=x-\xi^K_T$ for $T=200$, $K=4096$. The distribution of $\mu_T^K$ is qualitatively close to the deterministic traveling wave $u^{\sigma^*}$ (see also Figure \ref{['fig:histogram']}) The two red bars stake out the positions of the particles selected, whose ancestral lineages are tracked, which are all the particles whose position is in $[-20,10]$. The bottom figure represents the position of the ancestral lineages in the frame $z=x-\xi^K_t$ at time $t=100$, i.e.$s=100$, of the selected particles (which are between red bars in the top figure). The minimum of the cyan and the magenta curves represents the predicted equilibrium of Equation (\ref{['introSDE']}) in the case $\chi>1$. (see Figure \ref{['fig:backward-distribution']} for more details).
• Figure 5: Histogram of $\mu_T^K$ for $\chi=2,K=4096,T=200$ and initial data satisfying $(\text{ID})$. The width of a single bin in the histogram is $dx=0.1$. The red curve represents $y=C\left\{ e^{-\chi (x-\xi^K_T))}\text{ if } x>\xi^K_T1\text{ if } x\leq \xi^K_T \right.$, where we recall that $\xi^K_T=H_K(\mu_T^K)$ denotes the position of the $K$-th particle at time $T$ and $C$ has been chosen a priori with $C:=K\chi dx$, which is consistent with the discretization of the histogram.

Theorems & Definitions (21)

• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Proposition 2.5
• Theorem 2.6
• Conjecture 2.7
• Conjecture 2.8
• Lemma 3.1
• proof