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A model for speed adaptation of individuals and existence of weak solutions

Published online by Cambridge University Press:  01 July 2003

F. LUTSCHER
F. LUTSCHER
Affiliation:
Department of Mathematics, University of Tübingen, 72076 Tübingen, Germany email: flutscher@math.ualberta.ca Present address: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada.
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Abstract

When individuals move together in large groups, as seen in schools of fish, they adapt their speed and direction to that of their neighbours. We present and analyse a model for the speed adaptation process in the case in which all individuals move in the same or in two opposite directions. The model consists of a hyperbolic conservation law for the density of individuals coupled to a parabolic or elliptic equation for speed. A detailed linear analysis reveals several mechanisms for the appearance of instabilities of the homogeneous steady state, which trigger the formation of schools, herds, flocks, etc. Long-term existence of weak solutions is shown using the vanishing viscosity approach.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 14 , Issue 3 , June 2003 , pp. 291 - 311
Copyright
2003 Cambridge University Press

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