Numerical approximation of a coagulation-fragmentation model for animal group size statistics

Pierre Degond; Maximilian Engel; Pierre Degond; Maximilian Engel

doi:10.3934/nhm.2017009

Networks and Heterogeneous Media

2017, Volume 12, Issue 2: 217-243. doi: 10.3934/nhm.2017009

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Numerical approximation of a coagulation-fragmentation model for animal group size statistics

Pierre Degond ^,,
Maximilian Engel

Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK

Received: 01 April 2016 Revised: 01 January 2017
Primary: 45J05, 49M15, 82B99, 92D50

We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] 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. [12], 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.
- Coagulation-fragmentation equation,
- numerics,
- convergence to equilibrium,
- fish schools,
- Newton method,
- Euler scheme
Citation: Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics[J]. Networks and Heterogeneous Media, 2017, 12(2): 217-243. doi: 10.3934/nhm.2017009

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Abstract

We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] 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. [12], 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.

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