2025/05/13 by Mete Demircigil, Demircigil, Mete, Milica Tomašević +1
Biochemistry, Genetics and Molecular Biology · Environmental Science · Mathematics · #Analysis of PDEs (math.AP) #Coagulation and Flocculation Studies #Diffusion and Search Dynamics #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2505.08563
openalex publication_date 2025/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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 ≥ 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 → ∞ 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.