Numerical approximation of a coagulation-Fragmentation Model for Animal Group Size Statistics

TL;DR

This paper numerically investigates a coagulation-fragmentation model for animal group sizes, validating equilibrium approximations and analyzing the transition from discrete to continuous distributions.

Contribution

It introduces and compares three numerical methods to approximate the equilibrium distribution in a coagulation-fragmentation model, including a recursive algorithm, Newton method, and time-dependent analysis.

Findings

All three methods accurately approximate the equilibrium's asymptotic behavior.

The recursive algorithm effectively explores the transition from discrete to continuous distributions.

The time-dependent scheme demonstrates uniform convergence to equilibrium and estimates convergence rates.

Abstract

We study numerically a coagulation-fragmentation model derived by Niwa and further elaborated by Degond et al., where a unique equilibrium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al., a Newton method and the resolution of the time-dependent problem. All three schemes are validated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates.