# A Markov jump process modelling animal group size statistics

## TL;DR

This paper models animal group size dynamics using a Markov jump process derived from coagulation-fragmentation equations, enabling numerical approximation of equilibrium states and analysis of statistical properties, with connections to stochastic differential equations.

## Contribution

It formalizes a coagulation-fragmentation model as a Markov jump process and develops a numerical scheme for equilibrium approximation, extending previous models with new analytical and simulation insights.

## Key findings

- Validated numerical scheme matches analytical equilibrium results.
- Analyzed population and size distributions for complex rates.
- Established connection between jump processes and stochastic differential equations.

## Abstract

We translate a coagulation-framentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the \blue{nonlinear} evolution equation with a Markov jump process of a type introduced in classic work of H.~McKean. In particular this formalizes a model suggested by H.-S. Niwa [J.~Theo.~Biol.~224 (2003)] with simple coagulation and fragmentation rates. Based on the jump process, we develop a numerical scheme that allows us to approximate the equilibrium for the Niwa model, validated by comparison to analytical results by Degond et al. [J.~Nonlinear Sci.~27 (2017)], and study the population and size distributions for more complicated rates. Furthermore, the simulations are used to describe statistical properties of the underlying jump process. We additionally discuss the relation of the jump process to models expressed in stochastic differential equations and demonstrate that such a connection is justified in the case of nearest-neighbour interactions, as opposed to global interactions as in the Niwa model.