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].
Daisyworld Simulator 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.
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