Models for Mutual Attraction and Aggregation of Motile Individuals

Abstract

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

$${\partial _t}u + {\partial _X}w = 0$$

((1))

. Modelling dispersion by Ficks law would result in

$$w = - {\mu _0}(u) \cdot {\partial _x}u,\;{\mu _0}(u) \geqslant 0$$

((2))

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.