In a previous post I wrote about chemotaxis, the motion of organisms in response to the concentration of a chemical substance. Here I want to discuss aspects of modelling this phenomenon on a population level. I consider the motion of E. coli in agar with chemotaxis in response to the concentration of aspartate. A presentation of mathematical models for this related to the Keller-Segel model can be found in Betterton and Brenner (Phys. Rev. E64, 61904). The phenomena of interest are the formation and propagation of bands of high concentration of bacteria and formation of localized aggregates. In the latter case this translates in the mathematical models into the occurrence of a singularity in the solution. In these models the phenomena taken into account are motion of the bacteria due to a chemical gradient, production of chemical substances (e.g. aspartate) and consumption of substances (e.g. succinate). The experimental observations and the first attempts to explain them are due to Adler (Science 166, 1588) and Budrene and Berg (Nature 376, 49).
I now want to focus on the situation where a band of bacteria breaks up into localized concentrations which then (in the mathematical models) proceed to singularity formation. Concentrating on a region close to a point of the band it seems reasonable to look at the collapse of an almost uniform cylinder of bacteria. This leads to the problem of formation of singularities in solutions of the Keller-Segel system in three space dimensions which are almost cylindrically symmetric. The case of exact cylindrical symmetry reduces to the Keller-Segel system in two space dimensions. From the point of view of scaling this is a critical problem and the asymptotics of collapse is subtle. (In three space dimensions there are self-similar solutions describing collapse to a point.) This asymptotic behaviour was analysed rigorously by Herrero and Velázquez (Math. Ann. 306, 583). On the basis of numerical work Betterton and Brenner suggested that the solutions of Herrero and Velázquez (HV) were unstable and that they (Betterton and Brenner) had found the generic behaviour. Velázquez (SIAM J. Appl. Math. 62, 1581) showed that the blow-up behaviour which Herrero and he found is stable in the sense of formal expansions. Later sophisticated numerical work by Budd et al. (J. Comp. Phys. 202, 463) led the authors to the conclusion that the HV behaviour is generic. A central idea of the numerical method is to use an adaptive mesh in such a way that the mesh points follow any scaling behaviour present in a solution. This gives a way of avoiding rescaling the mesh according to an expectation of the scaling behaviour and thus perhaps ‘finding the scaling you expect’.
There is a region close to the centre where the density of bacteria can be approximated by the result of applying a time-dependent rescaling to a the corresponding function coming from a static solution of the system. The scaling is the natural scaling of the system and generates a one-parameter family of time-independent solutions starting from any one of them. At least superficially all of this bears a close resemblance to what happens in the formation of singularities in critical wave maps, as described in a previous post. Does this have a deeper meaning?