Convergence and Wave Propagation for a System of Branching Rank-Based Interacting Brownian Particles
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arXiv
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| Format: | Preprint |
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2025
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| author | Demircigil, Mete Tomasevic, Milica |
| author_facet | Demircigil, Mete Tomasevic, Milica |
| contents | 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 $χ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08563 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence and Wave Propagation for a System of Branching Rank-Based Interacting Brownian Particles Demircigil, Mete Tomasevic, Milica Analysis of PDEs Probability 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 $χ$. |
| title | Convergence and Wave Propagation for a System of Branching Rank-Based Interacting Brownian Particles |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2505.08563 |