Coagulation-fragmentation model for animal group-size statistics

Pierre Degond , Imperial College London Jian-Guo Liu Duke University Robert L. Pego Center for Nonlinear Analysis, Carnegie Mellon University

Analysis of PDEs mathscidoc:1702.03009

Journal of Nonlinear Science, 1-46, 2017.1
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.
Detailed balance · Fish schools · Bernstein functions · Complete monotonicity · Fuss–Catalan sequences· Convergence to equilibrium
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  title={Coagulation-fragmentation model for animal group-size statistics},
  author={Pierre Degond, Jian-Guo Liu, and Robert L. Pego},
  booktitle={Journal of Nonlinear Science},
Pierre Degond, Jian-Guo Liu, and Robert L. Pego. Coagulation-fragmentation model for animal group-size statistics. 2017. In Journal of Nonlinear Science. pp.1-46.
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