Pushing Up The Daisies J. G. Dyke & I. R. Harvey Evolutionary and Adaptive Systems Group University of Sussex [email protected] Abstract consider it as a model of homeostasis employed by both When components of an interacting dynamical system (such real and artificial systems. Therefore Daisyworld-type as organs within an organism, or daisies within the homeostasis need not be limited to daisies, or global Daisyworld model) have a limited range of viability to regulation, but can be seen operating within diverse areas changes in some essential variable, intuition suggests that such as a simple agent that performs phototaxis (Dyke & increasing any individual range of viability will also Harvey 2005), or the regulation of blood glucose in increase viability in the context of the whole system. We humans (Saunders et al 2000). show circumstances in which the reverse is true. The results presented and discussed here may apply to other formulations of the Daisyworld model. Preliminary results indicate that the results and analysis presented here Introduction do not apply specifically to the original Watson and An organism may be defined as an open physico-chemical Lovelock model, but the fact that they do apply to the even system existing in steady state with its external simpler variant model presented here suggests that these environment. It is this ability to maintain a steady state possibilities could exist with many other similar versions. within a constantly changing environment that contributes towards the success of both real and artificial living Daisyworld systems. Homeostasis in this context describes the Using realistic constants and modelling assumptions, a mechanisms and processes that organisms employ in order number of equations sketch out the Daisyworld planet. The to minimise the impact of external perturbations on their model consists of a star that emits short wave energy which internal states. Increased robustness to external is received on the surface of a grey planet that is home to perturbations can be achieved in two main ways: 1, two daisy species: a black species and a white species. increase in the efficacy of homeostatic mechanisms to Albedo is a measure of the reflectivity of an object. In minimise the impact of external perturbations on internal Daisyworld the black daisies have a low albedo (0.25), the variables; 2, increase in the range of values that the internal grey planet intermediate albedo (0.5) and the white daisies variables can take without endangering the viability of the a high albedo (0.75). The white daisies having the highest organism. In this paper we present results from a radically albedo in the model will reflect more of the short wave simplified version of the Daisyworld model in which the energy from the star and so have a lower temperature than second approach is adopted. We investigate the effects of either the grey planet or the black daisies. The same increasing the viability range, and detail circumstances in applies, but in reverse to the black daisies. The black and which, counter-intuitively, doing this actually decreases the white daisies share a viability range of temperature. They systems robustness to external perturbations. are only able to grow when the local ambient temperature The ‘parable’ of Daisyworld (Lovelock 1983, Watson & is within 5–40 degrees Celsius. Within this range growth Lovelock 1983) is a simple mathematical model designed rates of the daisies vary, with optimum growth being to demonstrate global homeostasis in which despite large achieved when the temperature is 22.5 degrees. changes in heat input from the sun, the ambient planetary Simulations begin when the luminosity is low (brightness temperature is regulated by the biota to within the range of the star is dim) and the ambient temperature of the most favourable for life. The original results have been planet is below 5 degrees. As luminosity increases the replicated many times and many extensions and ambient temperature reaches 5 degrees and black daisies developments of the original zero-dimensional model have begin to grow. The increase in the number of black daisies been produced. In this paper we develop a radically initiates a feedback loop that increases their temperature simplified ‘two box’ Daisyworld model introduced in and a population explosion of black daisies; the greater the (Harvey 2004) and recently extended as a two dimensional coverage of black daisies, the lower the albedo and so the cellular automata version (Williams & Noble 2005). warmer the planet which increases the growth rate of the Lovelock’s motivation for the original Daisyworld model black daisies. This feedback loop is regulated by the was to borrow ideas of homeostasis at the organism level parabolic growth rate of the daisies. As the ambient and then apply them to the operation of a hypothetical temperature increases past 22.5 degrees, the daisy growth planetary homeostatic system. Another motivation is to rate decreases. At steady state the ambient temperature, take the principles of Daisyworld-type homeostasis and growth rate and constant death rate are at equilibrium. As luminosity continues to rise, coverage in black daisies Two Box Daisyworld decreases and white daisies begin to grow. This initiates a feedback loop that decreases their temperature; the greater In order to better understand the effects of changing the the white daisies, the higher the albedo and so the cooler viability range, a simplified ‘two box’ variation of the daisies which reduces the growth rate of the daisies. Daisyworld first discussed in (Harvey 2004) will be used. Again, this feedback loop is regulated by the parabolic Whilst Watson and Lovelock’s Daisyworld was intended growth rate of the daisies. Increasing luminosity results in a as a simple parable of a complex planetary system, it still progressive increase in white daisies (and decrease in black contains number of features that make analysis a non- daisies) until the maximum coverage of white daisies is trivial exercise. The use of non-linear equations that reached. Any further increase in luminosity takes the employ the Stephan-Boltzmann constant for the calculation ambient temperature past the point where growth rates of heat, space competition between the two different daisy balance death rates and so the coverage of white daisies types and the heat transport mechanism produce a number decreases. This initiates a rapid collapse of white daisies of complex relationships (see (Saunders 1994) for an similar in nature to the population explosion of the black analytical treatment). The two box model seeks to reduce daisies. The differential coverage of white and black Daisyworld to the bare minimum in order to highlight the daisies results in a system that effectively regulates principal interactions and relationships at work. Rather ambient planetary temperature to within the optimum range than both black and white daisies competing for space on for daisy growth. Whereas the temperature of a bare the same planet, two separate daisy boxes, or daisy beds lifeless planet would increase in an approximately linear are seeded with either black or white daisies. These two fashion with increases in luminosity, when black and white separate beds are coupled in the form of heat conductance. daisies are present, ambient temperature remains very close Heat will flow from the black, warm daisy bed to the white to the optimum of 22.5 degrees over a wide range of solar cool daisy bed in proportion to a conducting parameter. forcings (Fig. 1). Reflected Star Reflected 0.8 Energy Energy 0.6 Incoming Energy 0.4 0.2 White Black Fractional Coverage 0 Heat Flow 0 2000 4000 6000 8000 10000 12000 14000 Black Daisy Bed White Daisy Bed 80 60 Fig 2. The Two Box Daisyworld which can be regarded as 40 analogous to a simplified two box energy balance climate model. 20 Only black daisies are seeded in the black daisy bed and only 0 Planet white daisies in the white daisy bed. Black daisies, being darker Degrees Celsius -20 than the grey daisy bed earth will absorb more energy from the 0 2000 4000 6000 8000 10000 12000 14000 Time (Million Years) star. White daisies, being lighter, absorb less energy. Hence the Fig 1. Time is plotted on the horizontal axis on the top and black daisies are warmer and the white daisies are cooler than the grey bare earth. The two beds are coupled via a heat conducting bottom plot. Daisy numbers are plotted on the vertical axis on the medium which allows heat to flow from the black to white daisy top plot. The temperature of the planet is plotted on the vertical bed. axis on the bottom plot. As luminosity increases over time, the temperature on the planet increases initially linearly but then rapidly increases and then stabilises around the optimum growth The following table lists the differences between the temperature. Watson & Lovelock and Two box Daisyworld models: Watson & Lovelock Two Box Organisation of Paper Variable birth rate and constant Single variable used to find The following sections detail a radically simplified two box death rate used to find total coverage of daisies, no Daisyworld model and present results obtained when the numbers of daisies separate birth and death rates Parabolic growth function Piece-wise linear coverage viability ranges of the daisies are increased. Contrary to function initial intuitions, there are certain circumstances, in this Space competition between Black and white daisies simplified model, where increasing the viability range of black and white daisies occupy separate daisy beds the daisies (making them able to survive over a greater Temperatures calculated as Simplified ‘cut-down’ physics range of ambient temperatures) actually decreases the proportional to fourth power and employed to calculate employ Stephan-Boltzmann temperatures in dimensionless range of solar forcing over which they are able to survive.