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Erschienen in:
1985 | OriginalPaper | Buchkapitel
Models for Mutual Attraction and Aggregation of Motile Individuals
verfasst von : Wolfgang Alt
Erschienen in: Mathematics in Biology and Medicine
Verlag: Springer Berlin Heidelberg
Enthalten in: Professional Book Archive
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Let u = u(t,x) be the density distribution of individuals over x ∈ ℝ and w = w(t,x) their mean flux. Then without birth and death the simple conservation law holds (1)$${\partial _t}u + {\partial _X}w = 0$$. Modelling dispersion by Ficks law would result in (2)$$w = - {\mu _0}(u) \cdot {\partial _x}u,\;{\mu _0}(u) \geqslant 0$$ and (1) would be the usual diffusion equation. In contrast, modelling aggregation would require the opposite sign of μ0 leading to an ill-posed problem for (1) in general.