Daisyworld was formulated by Watson and Lovelock to demonstrate mathematically how climate regulation by the biota could work. In their paper they describe the close-coupling beteen their much simplified biota and climate as follows: 'The growth rate of the daisies depends on only one environmental variable, temperature, which the daisies in turn modify because they absorb different amounts of radiation. Regardless of the details of the interaction, the effect of the daisies is to stabilize the temperature.' Subsequent work has shown that homeostasis is indeed a robust property of many types of Daisymodels. The present paper examines which of the assumptions underlying Daisyworld lead to self-regulation as well as to other properties of the system. Although Daisyworld was not devised as a realistic model of the Earth, its properties are meant to be important for the Earth. By studying different aspects of the steady state solutions of Daisyworld, we attempt to answer the question of how this model is relevant to the Earth system. All results are obtained analytically. Numerical integrations are only used to confirm analytical results.
The different surface types in Daisyworld are characterised by the area which they occupy and by their local temperature, given their local albedo. In equilibrium the local temperatures do not depend on the solar luminosity. This implies that the equilibrium expansion rate of the daisies and the equilibrium area of bare ground also do not depend on the solar luminosity. As the area of bare ground fixes the total area covered by the daisies, only the difference in daisy coverage depends on the solar luminosity.
The equilibrium effective temperature can be determined without actually solving for the difference in daisy coverage. The fixed local temperatures result in a direct relation between the effective temperature and the effective albedo: an increase in the effective albedo with respect to a pre-defined fixed value goes with a decrease in the effective temperature with respect to the fixed optimal temperature. Substitution of this relation into the heat budget results in a transparant expression for the equilibrium effective temperature in terms of model parameters. It is clear from this expression that the equilibrium effective temperature only weakly depends on incoming solar radiation. This homeostatic property depends crucially on the pre-hypothesized relation between the equilibrium effective temperature and albedo.
The thermostat of Daisyworld is thus found to have a fixed set point: the optimal temperature for plant growth. Temperature variations around this fixed value are small for small q (an inverse measure of the diffusive heat transport in the system). At the same time, the size of the domain over which the self-regulating solution exists is found to be large for small q. This indicates that a system which can redistribute heat in an efficient manner (q small) is best fit to regulate its temperature. The size of the domain is also found to increase with the difference in albedo between the black and white daisies. This clearly suggests that the biota have greater regulatory power over climate, when they are capable of altering the climate more profoundly. These conditions for optimal temperature regulation seem to be fairly realistic.
The appearance of a set point is somewhat surprising. This global property of the stationary state is closely linked to the fixed local temperatures (of covered as well as uncovered ground). The property of fixed local temperatures translates into a prescribed reference global temperature through the temperature-albedo relations. It is difficult to see a parallel in the real climate for this aspect of Daisyworld. On the one hand, it is not clear by which mechanism the biota could achieve local temperatures which depend on biological parameters rather than on the incoming solar radiation. This would imply a mechanism of heat redistribution which maintains perfect local homeostasis. On the other hand, the temperature-albedo relations suggest that such a mechanism would strongly connect the perfectly self-regulating surface types to each other. Again, it is not clear by which mechanism such strong links could be maintained.
The present analysis has shown that the regulating effect of the biota in Daisyworld is based on the introduction of a fixed reference temperature into the global equilibrium heat budget. It is an interesting property of the model that the close-coupling between the biota and climate, as described by Watson and Lovelock, only comes into play when the system is perturbed away from the equilibrium state. A stability analysis shows that the dependence of the expansion rate on the (perturbed) local temperatures, which are directly related to the (perturbed) effective temperature, is crucial for the equilibrium state to be attracting: in response to an increase in temperature away from its equilibrium value the daisies increase the effective albedo. This constitutes a negative feedback, which drives the system back toward the equilibrium state. However, the 'daisy-feedback' does not determine the long-term evolution of the equilibrium in response to changes in the solar luminosity.
Daisyworld allows two different species to exist, which gives rise to three different surface types with their own local properties. Linking the different surface types to each other and to the global climate is a non-trivial task. The heat exchange between the different surface types should probably depend on the assumed distribution of ecosystems in space. At one end of the spectrum this might be in coherent regions, well-separated from each other. At the other end, individual black and white daisies might be close neighbours influencing each other directly. The formulation of heat transport seems to be the weakest point of the model. At the same time, the chosen solution is crucial for the homeostatic property of Daisyworld.
Daisyworld is a highly idealised model of climate regulation by the biota. Although albedo variatons over the range considered in Daisyworld are unlikely in the real world, it is not unrealistic to assume that even smaller albedo variatons can have an impact on climate. This could be through direct changes in the surface albedo and, in addition, through feedbacks related to cloud formation, snow budgets, etc. However, the question of whether the biota introduce a 'stabilising' reference temperature into the system has not yet been answered.
Back to homepage of Nanne Weber