In DaisyWorld, as constructed by Ginger Booth in the Java-based Lab DaisyWorld, daisies find 22.5 degrees Celsius just perfect. At that temperature, their birthrate is a maximal 1.0. Their birthrate is zero at 5 and 40 degrees Celsius. Their death rate is constant. Determining the temperature is more complex. Different color daisies, and the bare ground, have different albedos, the amount of incoming light reflected back into space. An albedo of 1.0 is a perfect reflector and 0.0 is a perfect absorber. Insulation (0 to 1.0) is the degree to which each color daisy's area maintains its own local temperature rather than equilibrating to the global temperature.  In DaisyWorld, solar input varies from 0.6 to 2.0 times the base solar luminosity. Like Earth's sun, the sun of DaisyWorld is getting brighter. See Booth for more detailed background discussion.

Because black daisies absorb more radiation (they reflect less light and have a lower albedo), they are warmer than white daisies and can survive better in a colder environment. The white daisies reflect more radiation (have a higher albedo) and can therefore survive with a greater amount of solar radiation (they live better in warmer conditions.)

There are therefore four (4) possible scenarios:

1. All black daisies with a lower solar input;
2. Both black and white daisies with increased solar input;
3. All white daisies with even a higher input; and
4. No daisies when the temperature is outside their range of tolerance (5-40 degrees Celsius).

The Daisyworld model is valuable because it illustrates a feedback mechanism in a system, i.e., the reflection and absorption of radiation. A feedback mechanism exists when one process interacts with a second process, which in turn effects the original process. For example, black daisies growing at a certain level of solar luminosity absorb heat, increasing the temperature and making it easier for white daisies to grow.  As the population of white daisies increases, more sunlight is reflected, cooling the planet, making it easier for black daisies to grow.  This process continues until a level of equilibrium between white and black daisies is established.

Daisyworld provides an illustration of climate and biological feedback. While GCMs (General Circulation Models) do not usually include biological feedback loops, this model does illustrate some of the feedback mechanisms in GCMs. Of course, GCMs are much more complex.

It is important to realize that models such as Daisyworld and GCMs are based on mathematics.  While the feedback mechanism in this model is relatively simple, the math is much more complicated, as will be explained below.

Mathematics involved in DaisyWorld (as taken from the Booth)

Looking at the formulas used to run the DaisyWorld model gives some idea of the complexity of models and the importance of mathematics in the sciences. An explanation of the formulas is below. (Most of this information is taken from the DaisyWorld site. Refer to "Under the Hood" for the algorithm used and other information.)

Formulas
Planetary Albedo
Planetary Temperature
Local Temperature
Birthrate
Area Change

Symbols
	Albedo 0 perfect light absorber 1 perfect light reflector
A	Area, percent of planet surface population measure for daisies
subscript p	Planetary
subscript color, barren	For daisy/barren color
T	Temperature, Kelvin
L	Luminosity multiplier here, 0.6 2.0 (dimensionless)
	DaisyWorld solar input at L = 1.0 Watts per square meter planet surface energy flux density
	Earth solar input at L = 1.0
	Stefan-Boltzmann constant fundamental constant of physics
R	Temperature insulation, local to planet 0 perfect conduction 1 perfect insulation
	Birthrate
	Deathrate

1. The planetary albedo is how shiny the planet is, how much incoming light it reflects back out into space. Planetary albedo is the weighted average of the albedos of each color making up the surface area. Multiplying each albedo with its percent surface area, and summing the result, is the weighted average. The capital sigma (S) means the sum of what ever is to the right:
(equation 1)
2. This is a mathematical description of an energy balance of a planet with no atmosphere: solar (shortwave) radiation coming in; infrared (longwave radiation) going out. The planetary temperature is calculated using the Stefan-Boltzmann Law. This Law states that outgoing long-wave radiation is proportional to the 4th power of the temperature. The physics is lucidly explained online in Wong, Appendix A. Basically, the formula says that the temperature depends on the energy input multiplied by the percent that is not being reflected back out into space.
3. The local temperature is calculated similarly to the planetary temperature. It varies from the planetary temperature by an amount set by the local albedo's difference from the planetary albedo, and the insulation factor. Again note the Stefan-Boltzmann Law: the longwave radiation is proportional to the 4th power of the temperature.
4. Each species of daisy's birthrate falls off by the local temperature's deviation from the daisy's ideal temperature, squared. That's a downward facing parabola with a maximum birthrate of 1.0 at ideal temperature, falling off to 0 at min and max temperature. The birthrate is 0 below minimum and 0 above maximum temperature.
5. At each step, the area of each color daisy may grow or shrink. The current area times the deathrate gives the death of daisies, or loss of area. The current area, times the birthrate, times the barren area available for colonization, gives the birth of daisies, or increase in area.

The differential equation can be approximated by a finite difference formula which is then easily solveable by the student. The formula is:

where <symbol for tau - T> is the time index and (<symbol for delta> _t <delta t>) is the time step or increment.