期刊
EUROPEAN JOURNAL OF APPLIED MATHEMATICS
卷 14, 期 -, 页码 291-311出版社
CAMBRIDGE UNIV PRESS
DOI: 10.1017/S0956792502005041
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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.
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