# MathSciDoc: An Archive for Mathematician ∫

#### Analysis of PDEsmathscidoc:1702.03009

Journal of Nonlinear Science, 1-46, 2017.1
ABSTRACT We study coagulation-fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no $H$-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior crossing over from exponent $-\frac23$ for small size to $-\frac32$ for large size, with an exponential cut-off.
Detailed balance · Fish schools · Bernstein functions · Complete monotonicity · Fuss–Catalan sequences· Convergence to equilibrium
@inproceedings{pierre2017coagulation-fragmentation,
title={Coagulation-fragmentation model for animal group-size statistics},
author={Pierre Degond, Jian-Guo Liu, and Robert L. Pego},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170206155120685219195},
booktitle={Journal of Nonlinear Science},
pages={1-46},
year={2017},
}

Pierre Degond, Jian-Guo Liu, and Robert L. Pego. Coagulation-fragmentation model for animal group-size statistics. 2017. In Journal of Nonlinear Science. pp.1-46. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170206155120685219195.