*Remember to save your work along the way in your desktop folder and Google Drive or M+Box Account (which you can find at www.itcs.umich.edu/storage/box/). If you do not know how to do this ask your GSI.
Gaia and Daisyworld
Introduction
In the 1960s, James Lovelock was employed by NASA to help determine whether life existed on Mars. While Lovelock’s colleagues were inventing methods of detecting life on Mars, Lovelock argued based on the known composition of Mars’ atmosphere that it couldn’t contain life.
Lovelock noted that Venus’ and Mars’ atmospheric composition are similar and are close to chemical equilibrium. In other words, there is not net production or destruction of chemical components. In contrast, Earth’s atmospheric composition is not at chemical equilibrium. (Note in Table 1 below how different Earth’s chemical composition is from that of Venus and Mars.) The persistence of an atmosphere in chemical disequilibirum, and at a relatively constant composition for time periods longer than the reaction times of the gases, implies that the Earth’s atmosphere is somehow “regulated”  the regulating mechanism is life.
Table 1. Chemical composition of terrestrial planets.
Atmos. Composition  Venus 
Earth 
Mars 
CO_{2} 
96% 
0.04% 
95% 
N_{2} 
3.4% 
78% 
2.7% 
O_{2} 
0.0069% 
21% 
0.013% 
H_{2}O 
0.3% 
04% 
0.03% 
From this early observation of some regulation of Earth’s atmosphere, Lovelock and his colleagues formulated the Gaia hypothesis. What is Gaia? Gaia is a selfregulating system composed of the Earth’s climate and environment that is a consequence of an automatic, but not necessarily purposeful, goalseeking system. Gaia has also been described as a super organism, analogous to a beehive that has the goal of maintaining an environment suitable for its own existence.
According to Lovelock, Gaia is composed of at least four elements:
The Gaia hypothesis has rather astounding implications. It suggests that:
After first proposing the Gaia hypothesis, Lovelock encountered many critics. The main criticism was that Gaia is teleological. That is, it requires organisms to have a collective purpose. In addition, it was known that biological regulation is only half the story. The evolution of the atmosphere involved the coevolution of organic (biologic) and inorganic components.
Formulation of the Daisyworld model
In response to his critics, Lovelock invented the Daisyworld model to demonstrate that selfregulation could occur without a collective purpose. Daisyworld is a simple Earthlike world that is inhabited only by white daisies. (Yes, that’s right, the flower.) The temperature of Daisyworld is determined by the fraction of the planet covered by daisies and the fraction that is bare soil. Because of their white color, daisies have a high albedo (reflectivity). The dark, bare soil has a lower albedo. The planetary temperature of Daisyworld can be calculated if the fraction of daisies is known. In the rest of this section, we describe the details of the Daisyworld model. There is some algebra involved, but it looks more daunting than it actually is. You saw these concepts last week in class.
Figure 1. The white daisy, inhabitant of Daisyworld, in all of its glory.
To calculate the planetary temperature, radiative equilibrium is assumed (this equilibrium is achieved much more rapidly than the sun changes its luminosity). Radiative equilibrium assumes that the energy emitted by planetary longwave radiation is equal to the shortwave radiation received from the Sun. In the absence of radiative equilibrium, the planet will warm or cool until radiative equilibrium is established.
Energy emitted = Energy absorbed (equation 1)
The energy emitted is simply defined by the StefanBoltzmann Law, which states that the energy emitted from a body is proportional to the fourth power of the body’s temperature:
Energy Emitted = A * ε * σ * T^{4} (2)
In equation (2), A, ε, σ, and T are Daisyworld’s area, emissivity, the StefanBoltzmann constant, and temperature, respectively. The energy received by Earth (aka energy absorbed) is the energy received that is radiated from the Sun minus the portion that is reflected back to space:
Energy Absorbed = Energy Received – Energy Reflected (3)
The energy reflected is a function of how many daisies there are. More white daisies increase the planetary albedo (i.e., reflectance), thereby causing more sunlight to be reflected back to space and preventing the sunlight from warming the Earth.
Energy Reflected = Planetary Albedo * Energy Received (4)
The planetary albedo is calculated by multiplying the total fraction of Earth that is either daisies or soil by the albedo for each surface:
Planetary Albedo = f_{daisy} * α_{daisy} + f_{soil} * α_{soil} (5)
In equation 5, f is the fraction of Daisyworld that is either soil or daisies, and α is the albedo of these surfaces. If Daisyworld is completely covered with flowers, the fraction of daisies (f_{daisy}) will be 1 and the fraction of soil (f_{soil}) will be 0.
The above equations describe the relationship between the planet’s temperature and the fraction of daisies. In addition, an equation is needed to express the conditions for the growth of daisies. On Daisyworld, it is assumed that the planetary temperature determines whether and how fast daisies grow. Daisies can survive at temperatures between 5 and 40 ºC, but do best at temperatures near 22.5 ºC. This relationship is expressed through a complicatedlooking growth factor:
GF_{daisy} = 1 0.003265 *(22.5 – T_{daisy})^{2} (6)
When equation 6 is plotted it should look like this:
The temperature of the daisycovered areas, T_{daisy}, is written as:
T_{daisy} = FHA * ( α_{planet} – α_{daisy}) + T_{planet}
(7)
This equation, where FHA is a heat absorption factor, converts sunlight (energy) into temperature. Finally, it is necessary to convert the growth factor into an expression of the change in the area of daisies. The change will be represented as:
Area change = Area_{daisy} * ( Area_{soil} * growth factor – death rate) (8)
This expression consists of two elements. First, it gives the area lost through a constant death rate (Area_{daisy} * death rate). Second, it expresses the area gained through daisy growth (+Area_{daisy} * Area_{soil} * growth factor). This last term is written such that the optimal growth will occur when there is an equal area of daisies and soils. Growth will slow both with less daisy area (fewer flowers to germinate) and with more daisy area (no bare ground in which to grow).
Concept of Equilibrium States
Many systems, including the climate system, are thought to have equilibrium states. An equilibrium state is a state of no change. The classic example of equilibrium states is that of a ball on an uneven terrain (Fig. 2). In this example, the equilibrium states exist when the ball is either in a valley (E1) or on a hill (E2). There are two types of equilibrium states, stable and unstable. In an unstable equilibrium (E2), a small perturbation to the system (a small push on the ball) will cause the system to move to a new equilibrium (the valley in Fig. 2  note that this is NOT reversible; that is, if you give another small push on the ball it will not return to it's original state of being on the top of the hill where it started in the E2 position). In a stable equilibrium (E1), a small perturbation to the system will not alter the equilibrium state. Rather, the ball will return to its original position (E1). One of the most important questions facing us is whether our climate system is stable or unstable. If the climate system is unstable, a perturbation to the system (such as warming caused by carbon emissions) will send it into a very different state.
Figure 2. Classic example of equilibrium states. E1 represents stable equilibrium; E2 represents unstable equilibrium.
Daisyworld also has equilibrium states. They can be determined by comparing the effect of temperature on the daisy coverage (solid line) and the effect of daisy coverage on temperature (dotted line). The intersection of these curves at E1 and E2 represents the equilibrium states.
Figure 3. Relationship between daisy fraction and temperature. The lines intersect at equilibrium points (E1 and E2).
To determine whether the equilibrium states are stable or unstable consider the response of the system at E1 to a small negative temperature perturbation. Such a small perturbation will cause the daisy coverage to decline and the planetary albedo to decrease slightly. With a small reduction in albedo, the surface temperature will warm. As a result, the system eventually returns to its E1 state. Similarly, a small positive temperature perturbation will cause the daisy coverage to increase, increasing the planetary albedo, and reducing the temperature. E1 is a stable equilibrium.
Now consider a negative temperature perturbation at E2. In this case, the daisy coverage increases, the albedo increases, and the temperature is reduced even more! The system is moving away from the equilibrium position at E2. This equilibrium is unstable.
Assignment: Wreaking havoc on Daisyworld
Now that we understand the basics of Daisyworld, let’s mess with our model! Our goal here is to evaluate the Gaia hypothesis, and particularly to determine whether biological regulation of climate is possible or whether it requires divine intervention as the critics of Gaia have argued.
For the sake of time, we have already constructed the Daisyworld model for you, rightclick on this link and save it to your desktop and M+Box: WhiteDaisyWorld.STMX
Question 1. Answer all the following subquestions and include your graph from above: Is the system in equilibrium? How do you know? What is the equilibrium white daisy fraction? The soil fraction?
Now we’ll create a model for the evolution of Earth's climate. Our Sun has become more luminous since it formed.
Question 2. Answer these three subquestions: What is the effect of an increase in solar luminosity over time on the white daisy population? What is the mechanism that allows daisies to affect the planetary temperature? Explain how and why the daisy population changes. [Hint: think about stable and unstable equilibrium points discussed earlier]
Keep the solar luminosity function the same as from Question 2. The Optimal Growth Temperature (set at 22.5°C) is the preferred temperature for daisy growth. This factor is determined empirically, or assumed.
Question 3. Answer these questions and include your completed table: How does the optimal growth temperature affect the timing of equilibrium? Why? Copy the table below to your Word document, record your results for different alterations of the Optimal Growth Temperature then use the table to help you answer the question.
Optimal Growth Temperature 
Time of White Daisy Growth (myrs) 
Time of White Daisy Death (myrs) 









Now let’s add a little ‘manmade’ cataclysmic variability to the system by including a plague in the Daisyworld model. (Make sure the optimal growth temperature is 22.5°). This work will allow you to answer Question 4.
Run the model and graph the White Daisy Fraction and the planetary temperature. Think about how the daisies respond to this plague, and think about how the plague affects the planetary temperature.
Now we’ll explore the effects of plague timing and plague magnitude. In the table below we have provided three scenarios for you to observe. In the steps above you have already completed the first scenario. To complete the subsequent scenarios, reset all death rates to 0.3, prior to adding a new plague.
Question 4: Answer the following questions: Does each plague respond similarly in magnitude and planetary temperature change? Why or why not? Explain your observations in terms of equilibrium concepts. Include your table from below with all values filled in.
Plague Death Rate  Year of Plague (myrs) 
Duration of Plague (myrs)  White Daisy Response  Temperature Response 
60 
103 
20 

90 
103 
20 

60 
155 
20 

Question 5. Let’s think about what we have learned from the Daisyworld model. We’ve seen (in problem 2) that the daisies can control the planetary temperature. Answer the following questions:
Lab 3 Assignment: By the beginning of next week's lab, on your section's Canvas site, please turn in the following in a single Word document. Also turn in your finished Stella file.
Don't forget to also turn in your Stella file (.stmx) of the final model.
References
Bice, D., 2001, Exploring the Dynamics of Earth Systems: Models Using Stella, http://specialpapers.gsapubs.org/content/413/171.full
Lovelock, J., 1988, The Ages of Gaia, W.W. Norton & Company, New York, 255 pp.