Lattice Brownian bees with cooperative reproduction: steady states, collapse, and spreading
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916061426745344 |
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| author | Vilk, Ohad Meerson, Baruch |
| author_facet | Vilk, Ohad Meerson, Baruch |
| contents | We extend the ``Brownian bees'' model of Berestycki et al. (2021, 2022) to cooperative reproduction, $kA\to(k{+}1)A$, 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 $N\to\infty$, 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 $k\le 2$ and instability for $k\ge 4$. 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 $k\ge 4$, the unstable steady state separates regimes of a finite-time collapse and a 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29958 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lattice Brownian bees with cooperative reproduction: steady states, collapse, and spreading Vilk, Ohad Meerson, Baruch Populations and Evolution Statistical Mechanics Probability Pattern Formation and Solitons We extend the ``Brownian bees'' model of Berestycki et al. (2021, 2022) to cooperative reproduction, $kA\to(k{+}1)A$, 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 $N\to\infty$, 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 $k\le 2$ and instability for $k\ge 4$. 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 $k\ge 4$, the unstable steady state separates regimes of a finite-time collapse and a 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. |
| title | Lattice Brownian bees with cooperative reproduction: steady states, collapse, and spreading |
| topic | Populations and Evolution Statistical Mechanics Probability Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2605.29958 |