Lattice Brownian bees with cooperative reproduction: steady states, collapse, and spreading
Abstract
We extend the ``Brownian bees'' model of Berestycki et al.\ (2021, 2022) to cooperative reproduction, kA→(k+1)A, of a population of N symmetric random walkers with removal, at each birth event, of the particle farthest from the origin. Working in the limit N→∞, we formulate a hydrodynamic free-boundary problem for this model. Using this formalism, we determine steady state population densities for all~k and prove their linear stability for k≤ 2 and instability for k≥ 4. In the marginal case k=3, there is a whole continuous family of steady states at a single critical ratio of the reproduction and diffusion rates. Above criticality the population undergoes an asymptotically self-similar finite-time collapse to the origin. Below the criticality the population spreads diffusively, but the reproduction remains quantitatively relevant. For k≥ 4, the unstable steady state separates regimes of a finite-time collapse and diffusive spreading. Here the collapse dynamics is asymptotically self-similar, and the population density exhibits a scale separation requiring a matched-asymptotic description. Our analytical predictions are confirmed by numerical solutions of the hydrodynamic free-boundary problem and by Monte Carlo simulations of the original microscopic model.
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BibTeX
@misc{vilk2026lattice,
title = {{Lattice Brownian bees with cooperative reproduction: steady states, collapse, and spreading}},
author = {Ohad Vilk and Baruch Meerson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.29958},
eprint = {2605.29958},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.29958},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lattice Brownian bees with cooperative reproduction: steady states, collapse, and spreading
AU - Ohad Vilk
AU - Baruch Meerson
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2605.29958
UR - https://arxiv.org/abs/2605.29958
AB - We extend the ``Brownian bees'' model of Berestycki et al.\ (2021, 2022) to cooperative reproduction, kA→(k+1)A, of a population of N symmetric random walkers with removal, at each birth event, of the particle farthest from the origin. Working in the limit N→∞, we formulate a hydrodynamic free-boundary problem for this model. Using this formalism, we determine steady state population densities for all~k and prove their linear stability for k≤ 2 and instability for k≥ 4. In the marginal case k=3, there is a whole continuous family of steady states at a single critical ratio of the reproduction and diffusion rates. Above criticality the population undergoes an asymptotically self-similar finite-time collapse to the origin. Below the criticality the population spreads diffusively, but the reproduction remains quantitatively relevant. For k≥ 4, the unstable steady state separates regimes of a finite-time collapse and diffusive spreading. Here the collapse dynamics is asymptotically self-similar, and the population density exhibits a scale separation requiring a matched-asymptotic description. Our analytical predictions are confirmed by numerical solutions of the hydrodynamic free-boundary problem and by Monte Carlo simulations of the original microscopic model.
ER -