The Daisyworld System

Abstract

In this bachelor thesis, we study a version of the daisyworld model without spatial dynamics. The daisyworld model was made to study how the strength of solar luminosity affects the balance of plant life in an ecosystem. Since differently colored plants have a different albedo, they reflect different amounts of sunlight, which can influence the surface temperature of the planet. To simulate this, we observe two types of flowers in the daisyworld system: black daisies which reflect a low amount of sunlight and white daisies which reflect a high amount of sunlight. The system consists of three differential equations, corresponding to growth of white respectively black daisies, and the change of surface temperature.

We study three variations of this model with increasing complexity, where most of the analysis is done on the final variation. In variation 1, we assume both species to have the same constant growth rate. In variation 2, we instead let the two species have the same, temperature dependent growth rate. In variation 3, through addition of a heat diffusion term, we let the two daisy species have different growth rates.

The analysis of the system is done through the following steps: First we find the equilibrium points, we do this by reducing the system to two dimensions by letting the temperature be a parameter. We find the two dimensional equilibria for a wide ranges of temperatures and use mathematical continuation to construct the equilibrium curves. Then, we determine which points are equilibria of the full system. Using Matcont, the stability of these equilibrium points can be determined.

Of the three differential equations, the time equation has constants of significantly smaller magnitude than those of the other two equations. As a consequence, for most initial conditions, the daisy population changes rapidly, while the surface temperature of daisyworld changes slowly. To still be able to determine the behavior of orbits in the daisyworld system, we apply the theory of fast-slow systems. We approximate the development of orbits into the fast dynamics and slow dynamics. For initial daisy population and temperature, we first keep the temperature constant and let the daisy population develop towards a stable equilibrium, then we let the slow temperature dynamics play out, following the equilibrium curves we previously found. Through this analysis, we find that for a solar luminosity L equal to the earth's sun, set at L =1, the planet will eventually only be covered by white daisies, or will not be covered by daisies at all. Which case occurs depends on the initial temperature of the planet.

When we repeat this analysis for different values of solar luminosity, we find that for L<0.44 and L>1.14, there is either too little or too much incoming solar energy, leading to the daisy coverages tending to 0 for all initial conditions. For 0.44<L<0.65, the planet will only be covered by white daisies as time progresses. For 0.65<L<1, the planet cycles between either daisy species covering most of daisyworld. For 1<L<1.14 the white daisies will cover most of the planet as time progresses.

Through this analysis, we can conclude that the black daisies are more capable of survival under low amounts of solar luminosity, which is consequence of their ability to absorb more of the solar radiation, while the white daisies are better suited for higher amounts of solar luminosity, considering they reflect a higher fraction of the solar radiation.