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

TL;DR

This paper analyzes a rank-based branching Brownian particle system modeling cell movement, establishing its limit behavior as the number of branching particles grows, and numerically exploring wave propagation and front types based on key parameters.

Contribution

It introduces a new model of rank-based branching Brownian particles with discontinuous coefficients and studies its limit behavior and wave propagation characteristics.

Findings

Limit behavior of the population as K approaches infinity.

Threshold behavior in propagation speed depending on parameter χ.

Categorization of traveling fronts as pushed or pulled.

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 $\chi$ > 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 $\chi$ 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 $\chi$.