The patterns we have discussed above are complex, but fairly regular patterns formed by spots and rings. The bacterium Bacillus subtilis, when inoculated onto a agar medium which has little nutrient, can exhibit quite different fractal-like patterns not unlike those found in diffusion-limited aggregation (see, for example, Matsuyama and Matsushita 1993). When the agar is semi-solid, however, the bacterial colonies formed by Bacillus subtilis are dense-branching patterns enclosed by a smooth envelope. The stiffness of the medium affects the patterns formed. Shigesada and her colleagues (Kawasaki et al. 1997) have studied this particular bacterium and constructed a relatively simple reaction diffusion model which captures many of the pattern characteristics found experimentally: they compare the results with experiments. Here we briefly describe their model and show some of their results. Although their model is a reaction diffusion one it is original and fundamentally different to those reaction diffusion systems we have studied up to now. It highlights, once again, the richness of pattern formation by such relatively simple systems.

They propose a model consisting of a conservation equation for the bacterial cells and the nutrient given by d n dt

9 b knb

d t 1 + y n where n and b are the concentration of the nutrient and bacterial cell densities respectively. Here the function knb/(1 + yn), where k and y are constants, is the consumption rate of the nutrient by the bacteria and 6 (knb/(1 + yn)) is the growth rate of the cells with 6 the conversion rate factor. Dn and Db are the diffusion coefficients of the nutrient and cells respectively. We now motivate the form given for Db.

The reasoning behind the form Db = a nb is based on the work of Ohgiwara et al. (1992) who observed the detailed movement of the bacteria and found that the cells did not move much in the inner region of the expanding colony where the level of nutrient was low but that they moved vigorously at the periphery of the colony where the nutrient level is much higher. They also noted that at the outermost front of the colony, where the cell density is quite low, the cells were again fairly inactive. Kawasaki et al. (1997) then argued that the bacteria are immobile where either the nutrient n or the bacteria density b are small. They modelled these effects by taking the bacterial diffusion as proportional to nb with the proportionality factor a. It was also observed that although each cell moves in a typical random way some of them exhibit stochastic fluctuations. They quantified this by setting a = 1 + A where the parameter A is a measure of the stochastic fluctuation from the usual random diffusion.

Kawasaki et al. (1997) studied the pattern formation potential of these model equations in two dimensions subject to initial conditions n(x, 0) = n0, b(x, 0) = b0(x), (5.101)

where n0 is the concentration of the initial uniformly distributed nutrient and b0(x) is the initial inoculum of bacteria. Since the nutrient concentration in the experiments is relatively low the saturation effect, accounted for by the y n term, is negligible so the consumption of nutrient can be taken as approximately knb, the functional form we use below.

We nondimensionalise the equations by setting

"• = (D)"2n b' = (m)"2b r' = (tf y (5102)

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