Publication 15-CNA-017
Coagulation-Fragmentation Model for Animal Group-Size Statistics
Pierre Degond
Department of Mathematics
Imperial College London
London SW7 2AZ
Jian-Guo Liu
Departments of Physics and Mathematics
Duke University
Durham, NC 27708
jliu@phy.duke.edu
Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@cmu.edu
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.
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