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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

Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

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.

Get the paper in its entirety as  15-CNA-017.pdf

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