Tipping Points in Nature: The Daisyworld Model
The concept of a “tipping point,” as well as the graphs used to calculate such points, reminded me strongly of the Daisyworld model.
The model was first proposed by James Lovelock, who wanted to demonstrate that an ecosystem can self-regulate within a certain range of conditions. Daisyworld explores the growth of white and black daisies under certain conditions and perturbations. The basic premise is that white daisies have a higher albedo (reflectance of light) and therefore reflect more heat back into space, cooling down the environment. Since the darker daisies absorb more light (and therefore heat), they are better able to survive in these cooler conditions and become more abundant, thus leading to an increased temperature, at which the white daisies thrive better (and cool the planet, as mentioned above), leading to a cycle that results in a regulated temperature pattern on the planet.
The figure below shows the relationship between the two variables (temperature and daisies): the solid line shows how temperature affects white-daisy coverage; the relationship is a curve because there are certain temperatures at which white daisies thrive, and the further the surface temperature is from that ideal one, the fewer the daisies. The dotted line shows how daisy coverage affects temperature, which is linear: more daisies will reflect more light/heat, therefore reducing the surface temperature.
The points where the two lines intersect represent equilibria; if the environment were to remain at any of these points, the effect on daisies by temperature would balance out the effect of daisy coverage on temperature, and the system is at equilibrium. However, if a system at equilibrium were perturbed, what would happen?
If the system is initially at equilibrium 1 (marked “P1” in the figure above), a slight increase in temperature would result in a lower daisy coverage, which would then decrease the temperature again, which would in turn increase the white daisy coverage. This falls into the cycle mentioned above, and the temperature will ultimately be maintained. Therefore, P1 represents a stable equilibrium.
If, however, the system is initially at equilibrium 2 (marked “P2” in the figure), a slight increase in temperature would result in a drastic decrease in white daisy coverage, which would then increase the temperature even further. If the temperature were to be decreased from P2 instead, there would be a large increase in white daisies, which would then decrease the temperature further. In either case, the temperature moves away from the P2 equilibrium temperature, causing this point to be an unstable equilibrium.
Connection to course material:
The diagram above looks very similar to the model we have been using to study network effects in class. The curves, however, are slightly different in their meaning: here, the solid curve is simply an ecological observation, and the dotted line is a result of physics regarding reflectance and heat, whereas in class, the solid line here would be a product of reservation price and network effects and the dotted line would simply represent the price of interest.
However, the two systems are similar in that, given the two curves, the process for finding the equilibria is the same, as are the implications: both systems have a “tipping point” situation.
As with the models in class, the equilibria are found at the intersections of the two curves, and each system has one stable and one unstable equilibrium. Any perturbation from an unstable equilibrium point will move the state away from the unstable equilibrium: either to zero (which is also an equilibrium in both cases but does not present as involved a situation) or toward the other (stable) equilibrium.
The systems have further similarities: there is a tipping point that defines a range in which the system is able to self-regulate. This tipping point, in both situations, is found to be an unstable equilibrium point. In the market scenarios from class, this tipping point occurs before the stable equilibrium, suggesting that once a certain fraction of people are buying the product, the consumer base will be strong even with changes to pricing. In the Daisyworld model, however, the tipping point is at a higher temperature than the stable equilibrium; this means that under a certain temperature, the system can maintain a somewhat steady temperature, but crossing the tipping point has negative consequences, as temperature is no longer being regulated by the daisies, and any incoming energy will result in increasing temperatures. However, the general concept is the same in both: there exists a tipping point that allows for self-regulation of their respective systems.
Sources:
https://www.theguardian.com/environment/2022/jul/27/james-lovelock-obituary
https://globalchange.umich.edu/globalchange1/current/labs/Lab3_DaisyWorld/Intro_Stella.htm
https://atoc.colorado.edu/~fasullo/1060/gifs/daisy.gif