ASSESSING THE KELLER-SEGEL MODEL: HOW HAS IT FARED?

Nine years ago, Lee Segel and I (1970) introduced an equation to describe the macroscopic motion of motile, chemotactic organisms, namely: $$ {b_t} = \nabla \cdot \left( {\mu (s)\nabla b} \right) - \nabla \cdot \left( {\chi (s)b\nabla s} \right) $$ where b(x,t) is the density of organisms, s(x,t) of substrate, μ(s) represents the motility and x(s) the chemotactic sensitivity of the organisms. Wishing to apply this equation to the migrating bands of chemotactic bacteria which had been studied by Adler (1966), we added a second equation to describe the diffusion and consumption of substrate (Keller and Segel, 1971): $$ {s_t} = D{\nabla^2}s - k(s)b $$ and looked for a solution to these equations which would mimic the behavior which Adler had observed. Accordingly, we sought one dimensional traveling waves, reducing the equation to a pair of ordinary differential equations: $$ cs' = k(s)b - Ds'',\;and\;cb' = \left( {\chi (s)bs'} \right)' - \left( {\mu (s)b'} \right)' $$ where the prime denotes differentiation with respect to ξ = x - ct, the wave variable. To solve these, a number of important assumptions were made regarding the form of the functions μ(s), x(s), and k(s).