Communications in Mathematical Sciences

Volume 18 (2020)

Number 1

A Markov jump process modelling animal group size statistics

Pages: 55 – 89



Pierre Degond (Department of Mathematics, Imperial College London, United Kingdom)

Maximilian Engel (Department of Mathematics, Technical University of Munich, Germany)

Jian-Guo Liu (Departments of Physics and of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Robert L. Pego (Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)


We translate a coagulation-fragmentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the 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., 224:451–457, 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(2):379–424, 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.


population dynamics, numerics, jump process, fish schools, self-consistent Markov process

2010 Mathematics Subject Classification

45J05, 60J75, 65C30, 65C35, 70F45, 92D50

P.D. acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under grant no. EP/M006883/1 and EP/P013651/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award no. WM130048 and by the National Science Foundation (NSF) under grant no. RNMS11-07444 (KI-Net). P.D. is on leave from CNRS, Institut de Mathématiques de Toulouse, France.

M.E. gratefully acknowledges support from the Department of Mathematics, Imperial College London through a Roth Scholarship and by the German Research Foundation (DFG) via grant SFB/TR109 Discretization in Geometry and Dynamics. This material is based upon work supported by the National Science Foundation under grants DMS 1812573 (JGL) and DMS 1812609 (RLP) and by the Center for Nonlinear Analysis (CNA) under National Science Foundation PIRE Grant no. OISE-0967140, and by the NSF Research Network Grant no. RNMS11-07444 (KI-Net).

Received 31 December 2018

Accepted 3 September 2019

Published 1 April 2020