Coagulation-fragmentation model for animal group-size statistics

TL;DR

This paper analyzes coagulation-fragmentation equations inspired by fisheries science, revealing universal equilibrium profiles with power-law behaviors and exponential cut-offs in animal group-size distributions.

Contribution

It provides a comprehensive description of equilibrium profiles and large-time behavior for a class of coagulation-fragmentation equations without detailed balance.

Findings

Equilibrium profiles exhibit power-law behavior with exponents -2/3 and -3/2.

Universal scaling profile determines all equilibria in the large-population limit.

Profiles show a crossover from small to large size with an exponential cut-off.

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.