The daisyworld model has a short and slightly controversial
history, being intimately connected with Gaia
Theory. Here we take the system on its merits and attempt in
clude all the possible effects reported thus far in the
literature.
Background
There have been numerous articles written on Daisyworld over
the years, so if the reader is interested in a general
background I would suggest reading something written by someone
more articulate than I. Typical examples may be found here, here
or here,
but this is far from exhaustive.
In summary the daisyworld parable consists of a model planet
that orbits some distance from an imaginary sun, but is
nonetheless similar to our own. We model this world in an
incredibly simplified way, firstly we assume that the living
things on this planet consists of daisies which appear in one of
two types – white or black. If a space on the planet is not
populated by on of these endemic species it will be bare. The
daisies (often called biota) will grow according to prescribed
(replicator) growth equations and both types distribute their
"seed" perfectly. The planetary dynamics are simplified by
assuming that the total amount of heat absorbed by this world is
only a function of the average colour or albedo (white has
albedo greater than one half, black has albedo less than one
half). White daisies reflect more of the suns energy back into
space, black daisies absorb more. The planetary temperature is
supposed to equilibrate fast so that it is in balance and that
the different patches of coloured daisies transfer heat between
each other. The model has no spatial component in its original
guise.
This model then exhibits the property of homoeostasis or
regulation as predicted by the Gaia hypothesis. More
specifically the relative proportions of the black and white
daisies adjust so that the overall planetary temperature is
maintained at the optimal for growth.
The simplifications in the model are extreme, and many
convincing arguments have levelled against it. To date however,
the model survives intact and criticisms against it have failed
to stick. Indeed most generalisations of model have continued to
preserve the central property of homoeostasis, despite even
increasing complexity. The model we introduce below is no
exception.
Daisyworld models
Non-spatial models
The model for daisyworld was introduced by Andrew Watson and
James Lovelock in 1983 and published in the journal Tellus B [1] (there is an
unusual story of how it wasn't published in nature). The model
is a system of differential equations in a series of variables
which describe the situation described above.
These eight equations can be solved in closed form, the
original solution for which was presented by Saunders [2]. This solution
demonstrates the homoeostatic property of the daisies. Despite
this success there are many critiques of the model, including
its failure to include evolution, the possibility of chaotic
behaviours and its pure simplicity. Here we shall concentrate on
the former the criticisms.
Mentioned in the paper by Saunders and later developed by
Robinson and Robertson [4] is the idea that the daisies
should, instead of altering there albedo, alter their optimal
temperature for growing, which is held fixed, at the same value,
for both daisy types. Would it not be more likely that instead
of voluntarily giving up space on an overly hot prototype world,
the black daisies would simply adapt to the hotter temperatures?
This effect was found to destabilised the regulation on
daisyworld. Lenton and Lovelock [5] later countered this by arguing that
there need to be some limits on this behaviour, life cannot
simply smoothly adjust to frozen cells for example – there
are necessarily physical bounds on adaptability. Introducing
this effect restabilises the world.
Spatial models
An important extension to the basic daisyworld model is the
explicit introduction of space, including a spatially dependent
temperature field. The model by von Bloh et. al. [3] is the first
example of this. Here the replication of daisies is now achieved
by stochastic growth and death rules, rather than differential
equations. Crucially the model thus developed displays even
better regulation than the simple daisyworld model.
A model with two kinds of evolution
The model we have developed is essentially a sum of the
component parts mentioned above. We have a fully spatial
temperature field with a thermal diffusion and heat capacity. In
addition our daisies evolve and mutate on the two-dimension
periodic surface of the "planet" with stochastic growth
rules. Importantly we allow the daisies to evolve both there
albedo and their preferred growth temperature
independently. This leads to number of effects, most striking of
which is the oscillatory nature of the resultant system. This
work is described in a recent publication submitted to the
Journal of Theoretical Biology [6].
The Applet
The applet below uses the same underlying code as our
simulation runs, but the size of the system, for convenience,
has been fixed at 100 sites per edge, so we have a potential
population of 10000 daisies. Our ideal temperature is fixed at
22.5 degrees Celsius and the bracketing width of both the growth
and bounding functions is set at 17.5 degrees Celsius. The other
parameters are:
Loop - number of complete iterations of the system between GUI updates.
Cycles - number of loops in a run.
Insolation - solar drive. 1 indicates it is driven at the ideal growth temperature.
Death rate - probability that a live site will die in a given time step.
T mutation - maximum amount which the growth temperature can change between generations.
A mutation - maximum amount which the albedo can change between generations.
The panels to the right indicate the different view-ports and
data one can monitor during a run. The panel at the top gives a
continuous readout of the albedo (white), average planetary
temperature (red) and the ideal temperature (green). Not that
changing the loop number will alter the scale of this
axis. Enjoy playing, or you can look at one of our walkthroughs
to look at some specific effects.
Walkthroughs
Click to bring up a separate box that gives some possible parameter combinations to try in the simulation
Relevant literature
^[1] Watson
A. J. and Lovelock J. E. (1983) Biological Homeostasis of
the global environment - the parable of daisyworld Tellus B
35 284 The original daisyworld model.
^[2]
Saunders P. (1994). Evolution without natural selection -
further implications of the daisyworld parable. Journal of
Theoretical Biology 166, 365. Exact analysis of
Daisyworld.
^[3] VonBloh
W. , Block A. , Schellnhuber H. J. (1997) Self-stabilization of the biosphere under global change: a
tutorial geophysiological approach. Tellus B 49
249. First Spatial Daisyworld.
^[4]
Robertson D. ,Robinson, J. (1998). Darwinian
Daisyworld. Journal of Theoretical Biology 195,
129. Allowing the evolution of optimal temperature.
^[5] Lenton
T. M. , Lovelock J. E. (2000). Daisyworld is Darwinian:
Constraints on adaptation are important for planetary
self-regulation. Journal of Theoretical Biology
206, 109. Placing a physically motivated bound on
the temperature adaption.
^[6] Wood A. J.,
Ackland G. J., Lenton T. M. (2006) Mutation of albedo and growth
response produces oscillations in a spatial Daisyworld.
Journal of Theoretical Biology, 242
