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Angels and Mortals

by Iain Weaver (Submitted: 02/01/2010)

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WHAT IS IT?

This is a replication of Solomon's model to demonstrate a system where a description of the means is insufficient to predict it's evolution, as presented in "The importance of being discrete: Life always wins on the surface".

HOW IT WORKS

The landscape is initially populated by a uniform cloud of mortals with density nB. These are treated as continuous, and this cloud diffuses at a rate given by DB.

Next we add a distribution of angels with density nA. These are treated as discrete, and diffuse at a rate given DA.

Each tick, a fraction of mortals on each patch die, given by exp(-mu).

Is there is an angel occupying the patch, the mortals multiply by a factor given by exp(lambda).

We might naively predict that the population will evolve according to the differential equation
dnB(t)/dt = (lambda * nA - mu) * nB(t)
whose solution is
nB(t) = nB(0) * e ^ (t * (lambda * nA - mu))
which predicts an exponential decay to extinction for lambda * nA < mu.

HOW TO USE IT

Setup sets each patches mortals to nB and scatters world-width*world-width*nA across the patches.

Each tick, angels and mortals diffuse, and then the mortals at each patch decay or multiply according to the rules above.

The patches are colored acording to the log of mortals there.

'Concentration' shows the density of mortals predicted by the above equation, compared to the true value determined from the model.

THINGS TO NOTICE

For the following parameters (used in Solomon's paper):
nB=10 nA=0.05
DB=10 DA=1
mu=6
lambda=10
Initially, the population appears to follow the predicted course. However, very soon the discrete angels allow clusters of mortals to form in what Solomon describes as a genuinely adaptive system.

Where the equations predict extinction, the model shows that population can flourish.

THINGS TO TRY

Under what conditions does the model behave more, or less like the equations predict?

Increasing the diffusion rates, particularly for angels means the continuum approximation used to construct the equations becomes more accurate.

Conversely, reducing diffusion rates allow the mortals to benefit from the less mobile angels, particularly if they themselves diffuse from these sites more slowly.

CREDITS AND REFERENCES

Nadav M. Shnerb, Yoram Louzoun, Eldad Bettelheim, and Sorin Solomon - "The importance of being discrete: Life always wins on the surface"

Any suggestions or questions? e-mail: isw3@le.ac.uk

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