Turing instability: cannibals and missionaries

An island is supposed to be populated by a population of cannibals and missionaries.

The missionaries are all celibate and thus depend on recruitment from the external world to maintain the population as its members gradually die.

Cannibals also die, but can also reproduce, so that the population naturally increases. However when two missionaries meet a cannibal, the cannibal is converted to missionary status. (If this seems a rather imperialist island it might be worth pointing out that under a commoner interpretation the cannibals are the growth promoters and the missionaries are the poison).

This tension between production and transformation means that a balance is reached when both populations are mixed together. If this balance is disturbed by a small amount of noise, the tension will act to restore the balance: the system is stable.

Now we imagine that the two populations, instead of mixing completely together, are spread out in a thin ring around the rather narrow beach of the island.

Now individuals react (that is, reproduce or convert) only with their immediate neighbours, but they also move around at random in a diffusive way.

Moreover the members of the two populations move at different speeds: the missionaries have bicycles and move faster.

This is enough to destabilize the system.

For if there is at any point a small excess of cannibals, say, then this will be followed by excess ‘production’ of more cannibals, and then of more missionaries (since they have more targets for conversion).

Without the spatial dimension, the extra production of missionaries would in turn reduce the cannibal excess and the system would return to balance.

But because the missionary excess is transported away more quickly, a pattern develops in which there is a near excess of cannibals and a far excess of missionaries.

Moreover the distance between these zones of relative excess is determined by the interaction between the reaction and the diffusion: a lenght scale, which is what is required for the emergence of pattern from non-pattern, has emerged from the dynamics.

The key to making this idea work is the missionaries’ bicycles: more technically that the inhibitor morphogen has a higher coefficient of diffusivity.

 

Further reading:

A. M. Turing (1952). “The Chemical Basis of Morphogenesis”. Philosophical Transactions of the Royal Society of London, volume B 237, pages 37-72.

A. M. Turing (1992). “The morphogen theory of phyllotaxis”. In Saunders (1992).

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