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An ODE Model of the Motion of Pelagic Fish Bjorn Birnir

Summary: An ODE Model of the Motion of Pelagic Fish
Bjšorn Birnir
Center for Complex and Nonlinear Science
Department of Mathematics
University of California, Santa Barbara
January 15, 2007
A system of ordinary differential equations (ODEs) is derived from a discrete system of
Vicsek, CzirŽok et al. [35], describing the motion of a school of fish. Classes of linear and sta-
tionary solutions of the ODEs are found and their stability explored using equivariant bifurca-
tion theory. The existence of periodic and toroidal solutions is also proven under deterministic
perturbations and structurally stable heteroclinic connections are found. Applications of the
model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in
the North Atlantic, are discussed and simulation of the ODEs presented.
1 Introduction
The internal dynamics of a school of fish and the migration of the school as a whole present a
fascinating problem with many applications. The individual fish tend to adjust their direction and
speed to the direction and speed of the school as a whole, see Partridge [28], but the internal
structure of the school can be very complicated. We discuss the biology that the model is based on


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics