

A PARABLE ABOUT A PLANETARY SYSTEM:
Watson and Lovelock's Daisy
World
INTRODUCTION
INTRODUCTIONIn the late 1960s and early 1970s, James Lovelock developed the Gaia Hypothesis. Traditional scientific thinking viewed Earth as a container with an exogenous set of abiotic conditions within which life appeared and then evolved. In Lovelock's Gaian view of the Earth System, the planet was described as a coupled system, with strong dynamic feedbacks between the coevolving biota and abiotic components  solid earth, atmosphere, ocean  working to produce an environment optimized for living things. Because of the language used and misunderstanding of Lovelock's thinking, the Gaia hypothesis was initially dismissed by most earth and life scientists as "new age nonscience". Today the model that life influences planetary processes (i.e., it has a substantial effect on abiotic processes) has become known as the weak Gaia hypothesis and is widely accepted by scientists. Lovelock's original Gaia hypothesis, that life has evolved to where in large part it controls planetary processes (i.e., life in essence created and continually tunes  selfregulates  the planetary environment it experiences), has become known as the strong Gaia hypothesis. It is still not widely accepted, though it has some prominent defenders, so the debate continues.
In defending his hypothesis in the scientific community, Lovelock developed in collaboration with Andrew Watson a simple numerical model as an elegant metaphor for a selfregulating planetary system  see Watson and Lovelock (1983). Termed "Daisy World", this model has received much analysis and been widely discussed in the scientific community. The Daisy World model illustrates one possible mechanism through which the biota might optimize its abiotic environment (simplified to be global temperature) by means by means of negative feedback. It is an heuristic model, that is, one that describes a mechanism by which such optimization might occur. After publication by Watson and Lovelock, Daisy World took on a life of its own, one independent of its relationship to the Gaia Hypothesis, as an example of a selfregulating system.
The set of coupled equations describing Daisy World can easily be solved via a realization of the model in STELLA. When exploring Daisy World with such a model, it quickly becomes apparent that even this very simple system exhibits a rich variety of complex and sometimes surprising behaviors.
Many variants of Watson and Lovelock's original Daisy World have been developed. It has also been used to illustrate a mechanism that can resolve the "Paradox of the Faint Young Sun".
HOW DAISY WORLD WORKSTeaching Note: Before reading this section, try running MonoWorld a few times using both black and white daisies. Some familiarity with the model and its output will increase comprehension of the material that follows.
Coming to understand how Daisy World works as a system is a good learning experience. It requires one to blend physical understanding about solar and terrestrial radiation, systems theory, and mathematical skills. The following discussion walks through the main points.
Imaginary planet illuminated by an aging main sequence star Transparent atmosphere, free from clouds and greenhouse gases Flat (more exactly, a segment of a sphere); no latitudinal, longitudinal or topographic effects No seasonality in climate Changes in surface temperature solely result of changing luminosity (energy from the star) and surface albedo Only two species in the biota:  Black daisies  dark in color, lower albedo than the soil surface  White daisies  light in color, higher albedo than the soil surface Daisy seeds  the same color as the mature plant  are initially scattered, randomly but uniformly, all across the planet. The seeds are so small they do not have to be taken into account when calculating planetary albedo and planetary temperature. In conditions either too cold or two hot for daisies to bloom, they remain present as dormant seeds awaiting more favorable conditions. Germination and the growth rate of both colors of daisy is a function of local surface temperature. Conditions suitable for growth of daisies over the entire surface of the planet N.B.: The above points describe
the original Daisy World as assembled by by Watson and Lovelock. Several
of these features can be adjusted to produce interesting variants of the
basic models. Watson and Lovelock do just a bit of this in their paper
and others have followed their lead, however, much of Daisy World needs
to be more fully explored.

Figure 1. Cartoon showing the energy flows to and
from Daisy World.
It is always high noon everywhere on Daisy World. As sketched by Watson and Lovelock, Daisy World is caricature of a planet. It really is a segment of a sphere. It is always facing the local star, which is everywhere directly overhead. The planetary surface is divided into bare surface (= "fertile ground", where daisies can grow if temperature conditions are right) and fields of daisies. (Some models allow for portions of the planet being unable to support daisies.) The substrate of Daisy World is a perfect insulator, so radiation enters and leaves the planet only from the starfacing side. (This is a good example of how far scientistmodelers will go in simplifying a system.)The planet supports only a very simple form of vegetation  daisies. In Watson and Lovelock's original incarnation, there were two species of daisy, black and white. However, more advanced Daisy Worlds can host multiple species of daisy that differ only in color; some also add herbivores to eat the daisies, and then predators to keep the herbivores in check.


It's not the color, its the albedo that counts. Although we will casually refer to different colors, each daisy specie and the bare planetary surface is more precisely defined in terms of its albedo (= fraction of incident solar radiation that is reflected). "Black" daisies have low albedo, between 0.0 and 0.5; they absorb most of the sunlight that they intercept. "White" daisies have a high albedo, between 0.5 and 1.0; they reflect most of the sunlight they intercept. The "bare ground" surface of the planet, i.e., the surface without vegetation, is characterized by an intermediate value of albedo, 0.5 (more or less "gray").
When you're hot, you're hot; when you're not, you're not (growing, that is). In the original model, all daisies species have the same temperaturedependent growth rate shown above. Daisies begin to grow at a minimum temperature of 5^{o}C, and grow faster as the temperature warms up. At the "optimum temperature" of 22.5^{o}C, daisies have their maximum growth rate (1.0). If the temperature warms still further, growth rates remain positive, but begin to decline in magnitude. Finally, there is a maximum temperature of 40^{o}C, above which no daisies grow. (The values cited here are those chosen by Watson and Lovelock. It is of interest to explore what happens when one or more daisy species have growth rate properties different from other species.) A essential point is that the temperature which is important from the daisy perspective is the local temperature of daisies of a particular color.
The local temperature of each particular kind of daisy (and for the bare ground) is determined by a balance between the insolation (= incoming solar radiation) from the star, the outgoing thermal (infrared or "IR") radiation from the surface (daisy or bare ground), and the transport of heat from warmer to cooler regions. In general, high albedo (light colored) daisies will be somewhat cooler than the planetary average temperature; low albedo (dark colored) daisies, somewhat warmer.If the interregional transport is weak, then the daisies are moreorless independent of one another and temperature contrasts between different daisy species (and with respect to bare ground) are very sharp. If this transport strong, then differences are smoothed out, and the contrasts between different areas are minor.
Seeds, seeds everywhere. We take that daisy seeds are initially scattered, randomly but uniformly, all across the planet. The seeds are taken to have the same color as the mature plant. The seeds are so small that we do not have to take them into account when calculating planetary albedo and planetary temperature. In conditions either too cold or two hot for daisies to bloom, they remain present as dormant seeds awaiting more favorable conditions.Teaching Note: Actually getting the seeds properly represented in the model turns out to require a few tricks.
Paradox of the Faint Young Sun. In the scenario laid out by Watson and Lovelock, environmental conditions on Daisy World are driven by the slowly warming star. The scenario opens in the far past, with the star's output, that is, its relative luminosity, L, at 0.75 the current value. (Relative luminosity is the ratio of the star's output to the presentday output of Earth's Sun, so currently L = 1.0.) The story ends in the distant future with the star at L = 1.25. In the early days of the life of this planetary system, conditions on Daisy World are cold and no mature daisies are present. Eventually, the star's luminosity increases to where the some of the seeds for black daisies (slightly warmer than their bare dirt surroundings and much warmer than their white counterparts) begin to germinate. Under these cold conditions, the black daisies have an initial advantage as they more readily absorb the insolation and become warmer than the bare soil. Mature black daisies spread across the planet, and the planetary temperature jumps upward.
At some later time, a combination of increased solar input and heating resulting from the presence of the black daisies raises the surface temperature to the point where white daisies begin to germinate. They gradually take hold in the increasing warmth (recall that white daisies reflect more energy, so they effectively cool the system; if they came on too strong too early, they would put themselves out of business). Competition for fertile ground begins. For a time, there are both black and white daisies present, but as the star continues to slowly warm, the species dominance shifts from the black daisies to the white ones. White daisies can survive warmer temperatures better than the black ones since they have a local temperature less than the planetary temperature. When heat increases to the point where black daisies are overheated, they die away, and the surface is covered by the highly reflective white daisies. Eventually the increasing heat is too much for even the cooling effect of the white daisies to hold down their local temperature; the white daisies die off quickly (crash!) and the planet becomes too hot for life.
A remarkable result emerges as the black and white daisies compete for fertile ground on the face of increasing luminosity. Surface temperature is initially increased by the absorptive black daisies, then stabilized for a long period by the competition between the decreasing concentration of heatabsorbing (warming) blacks and increasing concentrations of the heatreflecting (cooling) whites. The competition provides a surprisingly effective feedback control system. For a wide range of insolation, this keeps the global temperature within a narrow range around a central value  termed the set point  close to the optimum for daisies, black and white. In Daisy World, homeostasis  meaning self or internal regulation resisting change in response to a changing external environment  is achieved by this rather simple mechanism: white (black) daisies are fitter in hot (cold) climates as their comparatively high (low) albedo tends to reduce (increase) the local temperature, thus creating local cool (warm) spots that favor the growth of more daisies of the parent's color.In more abstract system terms: In the face of a changing changing external control parameter, a system variable is regulated, that is, held within some narrow bounds, by two effects that act in opposite directions on the variable and inhibit one another. In Daisy World, the external control parameter is the insolation from the aging star, the system variable is global planetary temperature, the "two effects" are the warming and cooling effects are due to black and white daisies, respectively. The two kinds of daisies inhibit one another by competing for the same space. The span of control of the system (also called the 'homeostatic plateau') is defined as that range of stellar luminosity for which the daisies provide thermal regulation.
The devil is in the (mathematical) details. In mathematical terms, Daisy World is a nonlinear system described by a set of (N + 1) coupled differential equations relating Planetary Albedo to Planetary Temperature. Here N = number of kinds of daisy (= different colors = different albedoes) in the model; N = 2 in the original model. Planetary Albedo is the areaweighted average of the albedoes of the different patches of daisies and of bare ground. In a loose way, this quantity provides the average "color" of the planet. Planetary Temperature is the areaweighted average of the temperatures of the different patches and of bare ground. It represents a balance between the insolation from the star and the outgoing IR from the planet.There are two types of equations: a radiative balance equation and "daisy equation(s)". The radiative balance equation relates Planetary Albedo and Planetary Temperature, with Luminosity as a parameter:
Planetary Temperature = f(Planetary Albedo, Luminosity)
A daisy equation describes how Planetary Albedo is determined by Planetary Temperature for a given kind of daisy:
Planetary Albedo = f(Planetary Temperature)
There is only one radiative balance equation but there is a daisy equation for each kind of daisy included in the model. These (N + 1) equations are coupled and must be solved simultaneously.
On first sight, this set of equations for Daisy World looks terribly complicated. However, for MonoWorld (a Daisy World with only one kind of daisy, and so is described by only two equations), solutions to the set of equations have simple graphical representations. On a plot showing Planetary Albedo as a function of Planetary Temperature, the radiation equation and the daisy equation each produce a curve. The point(s) of intersection of the two curves are the simultaneous solution(s) the equations. Each solution defines a state for the system. In some situations, multiple solutions occur, indicating the system has several possible states. Let us use this graphical technique to explore the behavior of MonoWorld for black daisies, and then again for white daisies.
MonoWorld with black daisies. We begin by plotting a family of curves showing the behavior of the radiative balance equation for increasing values of stellar luminosity L. For each L, a curve shows the range of Planetary Temperatures associated with the full range of Planetary Albedo (0.0 £ Planetary Albedo £ 1.0). Four such curves, labeled L1, L2, L3, and L4, are shown in Figure 4. In the scenario outlined previously, as time moves forward, L continuously increases and this curve sweeps to the right; the labeled curves should be considered four snapshots taken at particular times.Note that for a Planetary Albedo = 0.0, the planet is a perfect reflector so no energy is absorbed; the Planetary Temperature is absolute zero, independent of the value of L. For a Planet Albedo = 1.0, the planet is a perfect absorber (a "black body") so all incident energy is absorbed; the Planetary Temperature has its greatest possible value for given L.
On this same graph, we plot using a dashed line the curve for the daisy equation. This shows the relationship between Planetary Temperature and Planetary Albedo for black daisies. This curve has three segments: an initial horizontal line, where the Planetary Albedo is equal to the albedo of bare ground (no daisies are active  too cold); a curved segment representing the range of Planetary Temperatures where black daisies are present and the Planetary Albedo is the sum of contributions from bare ground and black daisies; and a final segment where the Planetary Albedo is again equal to the albedo of bare ground (no daisies are active  too hot).When we consider the two curves together, the path denoted by red dots shows the evolution of the state of the system (as represented by (Planetary Albedo, Planetary Temperature) in the face of increasing L. The labeled points tell the story:
A  No daisies have germinated, so the planet's surface is barren; the Planetary Albedo is equal to the albedo of bare soil.It is instructive to replot the above evolution on a graph showing the evolution of Planetary Temperature with time.B1, B2  The luminosity of the star has increased, shifting the curve for the radiation equation to the right (this is indicated by the line of fine black dots passing through B1 and B2). At B2, this curve for the first time is tangential to the the dashed curve. At this point, the system has two possible states: B1, a stable state (bare ground only, no daisies), and B2, an unstable state (daisies present). We discuss this (in)stability in a following section. The tendency of the system is to remain at B1.
As Luminosity increases and moves to the right, solution B1 also moves to the right. However, state B2 splits into two, one unstable and one stable. With increasing L, the unstable state moves upward from B2 along the daisy curve while the stable state moves downward, along the dashed curve. In this region, the there are three solutions to the coupled equations.
C1, C2  The increasing luminosity eventually drives the system to C1, the rightmost point on the initial segment. A further small increase in L causes an abrupt, fundamental shift in the system as it jumps from C1, a bare, daisyfree surface, to C2, in which the surface is largely covered by daisies. This jump is a sudden transition from one state to another; the occurrence of such abrupt transitions is a characteristic of nonlinear systems. Note that the system never visits any of the solutions along the curve C1B2C2.
E, F  Further increases in luminosity slowly drive the system to the right across the middle segment. This is the region in which the system is stabilized. F is the rightmost point in the portion of the curved segment where daisies are active.
G  No daisies are active as it's too hot for them. Once again, the Planetary Albedo is equal to the albedo of bare soil and the path of solutions is a horizontal line.
Figure 5. The evolution of temperature on MonoWorld with black daisies (path shown by
red dots) as a function of increasing luminosity. Labeled points correspond to points labeled
similarly in Figure 4. Black daisies are active over the portion of the curve C1C2F.In the plot above, the solid line shows the trace of Planetary Temperature if there were no daisies ever present  Planetary Temperature continually increases as L increases, and Planetary Albedo is fixed at the value for bare ground. The red dots show the impact of the black daisies. The labeled points correspond to the points with the same labels in Figure 4. Note in particular, the abrupt transition from C1 (no daisies) to C2 (daisies covering much of the planet).
Finally, let us close this discussion by examining a plot of the actual relationship between Planetary Temperature and Planetary Albedo for black daisies on MonoWorld. This curve was generated using a variant of the MonoWorld model provided below.
This curve is not as symmetric as the curve in the schematic considered above. It also lacks the sharp corners of the sketch. Trying sketching a plot of Planetary Temperature versus time using this (black) daisy equation curve.
Figure 6. Curve showing Planetary
Albedo as a function of Planetary
Temperature for black daisies.
Compare with curve shown with
dashed line in Figure 4.
MonoWorld with white daisies. Let us consider the alternative case for MonoWorld, where only white daisies are present. The schematic diagram showing the intersections of the radiative equation with the daisy equation then takes the form shown below.
Once again we have a family of curves, labeled L1, L2, L3, and L4, which provide snapshots of the radiative equation for a sequence of increasing times. As with the black daisy equation, the white daisy equation has three segments. The evolution of the system through time is shown by the path marked by red dots. The labeled points again tell the story:A  No daisies have germinated, so the planet's surface is barren; the Planetary Albedo is equal to the albedo of bare soil.The corresponding plot of Planetary Temperature as a function of time is shown below. Note that the beginning at B, the continually growing patches of white daisies cool Daisy World extensively, so that the increase in Planetary Temperature from B to F1 is about onehalf of the increase from B to F2. However, when the crash in white daisies does occur, the jump from F1 to F2 represents a dramatic temperature increase.B  The luminosity of the star has increased, shifting the curve for the radiation equation to the right. Any further shift raises the temperature of the white daisy seeds to the point where they begin to grow. (Note that this can get a little complicated. If the Planetary Temperature is just a little warmer than the temperature required to activate the daisies, the appearance of the daisies will cool the planet below what is needed to sustain the daisies. We should thus expect that the sharp corner shown at B is in reality somewhat more complex in shape.)
C  As L increases, the common solution (= state of the system) ascends the curved segment; more and more white daisies appear, greatly limiting the increase in Planetary Temperature.
D1, D2  When L has increased to a value where the solution of the radiative equation passes through D1 and D2 (noted the thin dotted line), the set of governing equations transitions to having two solutions. For slightly larger L, the solution at D2 splits so that there are three solutions, e.g., E1, E2, and E3. As before, one solution (= state of the system) is unstable, in this case the solution that lies along the curve between D2 and F1, e.g., E2. Solutions along the curve from D1 to F1, such as E1, are stable, as are solutions such as E3 in the horizontal segment running from D2 to F2.
L eventually increases to the point where the curve for the radiation equation passes through F1 and F2. The system jumps from F1 (white daisies covering the plant) to F2 (bare ground). The "ecosystem" of white daisies reaches its greatest extent across the face of Daisy World, and then crashes abruptly. (Note that for increasing L, the system never visits any solutions along the curve F1E2D2E2F2.
F2, G  The increasing luminosity drives the system further to the right. No daisies are active as it's too hot for them. Once again, the Planetary Albedo is equal to the albedo of bare soil and the path of solutions is a horizontal line.
Finally, let us close this discussion by examining a plot of the actual relationship between Planetary Temperature and Planetary Albedo for white daisies on MonoWorld. This curve was generated using a variant of the MonoWorld model provided here.This curve lacks the symmetry of the curve in the schematic considered above. In particular, note the abrupt rise on the left half of the curve. Trying sketching a plot of Planetary Temperature versus time using this (white) daisy equation curve.
Figure 9. Curve showing Planetary
Albedo as a function of Planetary
Temperature for white daisies.
Compare with curve shown with
dashed line in Figure 7.
Yeek! Cob webs  stability on Daisy World.
Let us consider what is meant by stability in the MonoWorld system. In the usual sense, a state of a system can be characterized in one of three ways:Stable = when disturbed from an initial equilibrium state, the system return to that state, that is, the disturbance dies away.To illustrate (in)stability, we will use a cobweb diagram. This is a graphical tool that allows the user to estimate (in)stabiilty under many conditions.Neutrally Stable = when disturbed from an initial equilibrium state, the system remains in the state to which it was disturbed, that is, the disturbance do not die away but neither does it become any larger. The "disturbed state" is a new equilibrium state.
Unstable = when disturbed from an initial equilibrium state, the system does not return to that state, that is, the disturbance grows. The state of the system evolves over time to a new equilibrium state some distance from the original one.
Consider the situation for black daisies. The cartoon in Figure 10 is an enlargement of the plot of Planetary Albedo versus Planetary Temperature considered Figure 4:
Consider the point B2. This is an equilibrium state for MonoWorld with black daisies. But is it stable or unstable? To answer this question, let us assume first that a small cooling occurs. This disturbance is shown by the short horizontal line  a drop in temperature  drawn to the left from B2 and labeled with a minus () sign. How then does the MonoWorld/black daisy system respond to this disturbance?We use a "cob web" diagram to estimate how the system will respond to such a disturbance. One constructs such a diagram in an interactive fashion. For the disturbance, determine the subsequent change in Planetary Albedo by drawing a vertical line intersecting the daisy equation curve. The point of intersection gives a value for the albedo resulting from the disturbance (cooling) in temperature. Note 1) the albedo is reduced (which makes sense since we expect a cooling to reduce the area covered by daisies) and 2) the point on the daisy equation curve is not an equilibrium point as the radiative equation does not pass through this point for the given value of L. So we determine the temperature that results from the change in albedo by drawing a horizontal line to intersection the radiative equation curve. We see 3) that a further cooling occurs and 4) again the result is not an equilibrium point. We continue this zigzagging process until the iterative leads to another equilibrium point. As shown this process leads to B1. Clearly the initial negative disturbance leads to a move away from B2.
If we repeat the process, beginning again from B2, but this time with a positive temperature disturbance, that is, a slight warming, then we get a quite different result. An initial positive disturbance leads to a zigzag return to the initial state B2.
Thus B2 is unstable with respect to negative perturbations and stable to positive ones.
Note: A cob web diagram provides only an estimate of how the system will likely behave to disturbances. It should not be considered a definitive tool. In reality, the system's response is likely to be more complicated than that implied by the simple iteration used to construct the diagram.Figure 11 is a second carton showing an enlargement of another part of Figure 4:
We can see that when the unstable point shown as B2 in Figure 10 split into two solutions (= system states), one is unstable and the other is stable.Teaching Note: Given the arguments on stability made above for system states in MonoWorld with black daisies, repeat the exercise for MonoWorld with white daisies. Draw the appropriate cartoons and the appropriate cob web diagrams. Which solutions (= system states) are stable? ... which are unstable?
DAISY WORLD MODELSI provide here a set of Daisy World models for exploration and further development. These are really works in progress, with many notes tacked here and there on the models. In some respects, NineWorld is the most polished.
In developing these models originally, I had two goals in mind:
 Understanding how Daisy World behaves as the number of kinds of daisy is increased. Seeing how well Daisy World responds to a cyclic input, such as would come from a variable star.
MonoWorld  Start by exploring the effects of black, gray, and white daisies, each operating on their own; hysterisis; maximum area covered by daisies. A picture of MonoWorld with black daisies on gray bare surface.
DuoWorld  The original Daisy World; repeat the experiments you did for MonoWorld, but now with black daisies and white daisies present at the same time. A picture of DuoWorld, with black and white daisies on gray bare surface.TriWorld  Three daisies; explore the impact of kudzu following Keeling (1991).FiveWorld  Five daisies
SevenWorld  Seven daisiesNineWorld  Nine daisiesAs a first step, use these Daisy World models to investigate the effects of one color of daisy working alone, then two, then three, and so on. Is Daisy World stable? How is the span of control impacted by add of daisies of intermediate albedoes?Teaching Note: Note that in each of the models, daisies spread across the planet but never fill all spaces completely, even when all the planet is covered with fertile ground. This is because at each time step, a fraction of all daisies, specified by the Death Rate Coefficient, die. If one images the planet surface to be cut up into "daisy positions", then at any instant some of these are empty because their occupant just died. Note also that that the average lifetime of a daisy is 1/(Death Rate Coefficient) in units of time steps.
SUGGESTIONS FOR EXPLORING AND EXTENDING DAISY WORLD MODELSBasic questions:
* Assess the impact of limiting the amount of fertile ground.More difficult challenges:* Explain how Daisy World offers a possible mechanism to resolve the "Paradox of the Faint Young Sun" through albedo feedback. (The actual resolution for this apparent paradox for Earth uses a different mechanism.)
* Add simple clouds to Daisy World (see Watson and Lovelock's original paper).
* Add simple evolution to Daisy World: What happens if 1) the colors of the daisies change over time; 2) if the Death Rate Coefficient changes over time.
* Explore Kudzu World. Start with TriWorld. Let "kudzu" be the plant with an albedo the same as the planetary surface. Normal kudzu competes with the daisies for space while not contributing anything toward regulating the temperature. However, suppose a mutation produces "Killer Kudzu" that has a growth rate 3 times that of daisies. What is the impact on Daisy World? Try this for FiveWorld, SevenWorld, and NineWorld. See Keeling (1991).
* An alternative evolution produces "Killer Kudzu 2" which has the same set point and maximum for the growth rate curve, but has a wider range of temperature for growth than the standard black and white daisies. What is the impact on Daisy World?
* More generally, how sensitive is Daisy World to the details of the growth rate curve (shape, minimum and maximum temperatures, optimum temperature, maximum growth rate value)?
* Use MonoWorld with black and then white daisies, then DuoWorld, TriWorld, ..., and NineWorld in the normal mode to estimate the impact of having progressively more daisies on the temperature regulation. Is having more kinds of daisy "good"? ... or "bad"?
* What happens on Daisy World if, instead of a warming star, we have a cooling star? Is Daisy World reversible (i.e., can you go home again)? Hint: Make Daisy World's star a variable that periodically warms and cools. Look up what is meant by the phenomenon termed hysterisis.
* Recovery Of Daisy World from Major Disturbances  Periodically drop asteroids onto Daisy World and assess the system's "recovery time". This requires adding a deterministic disturbance process to Daisy World that periodically eliminates some proportion of the daisies and reverts that area to bare ground. The disturbance process should have two parameters: Frequency and Severity. Frequency is used to set the number of years between disturbance events, while Severity sets the size of the area to be disturbed. Does the addition of disturbance expand or shrink the span of control? Now add stochasticity to the model by drawing the interval to next disturbance and the area to be disturbed from a statistical distribution. The parameter values should now specify the means of these distributions. Reanalyse the system behavior using the stochastic version of the model and compare results to those found using the deterministic model. We have assumed that all daisies had the same susceptibility to disturbance. How would the results change if the black daisies were more likely to be killed by a disturbance? What if the white daisies were more susceptible? Add a differential mortality rate to the disturbance process and report on how this changes the results. Is Daisy World "robust" and/or resilient?* Herbivores and Carnivores on Daisy World: Add a more complex ecosystem to Daisy World by providing "grazers" which eat the daisies (perhaps preferentially by color) and "predators" to eat the grazers. Is an enhanced Daisy World, either with more colors of daisies or with grazers and predators in addition to daisies, more stable in some sense than the basic model?
* Add A Greenhouse Effect To Daisy World:
* Reflective Permafrost on Daisy World: Daisy World has been criticized because it emphasizes too strongly the stabilizing tendency of negative feedback. A possible destabilizing feedback loop involves the loss of reflective properties of high latitude permafrost with the onset of a warmer climate (never mind that Daisy World does not have high latitudes as we know them). Add reflective permafrost to Daisy World. Begin by assigning a separate stock to the area covered by permafrost. Assume that the permafrost albedo is 0.85, a typical value for fresh snow. Initialize the area of permafrost 0.1 Include the permafrost area in the calculation of the planet's average albedo. Next, assume that the permafrost expands or shrinks depending on its local temperature. (When the local temperature falls below zero degrees, the permafrost expands. When it rises above zero degrees, it shrinks.) Assume that the temperature in the vicinity of the permafrost is 25 degrees below the planet's average temperature. Should the average temperature rise above 25 degrees, the permafrost temperature will rise above freezing. This will trigger shrinkage in the permafrost and a reduction in its reflective power. Daisy World would then become warmer, causing further shrinkage in the permafrost. Use the model to study the the span of control. Compare the span of control with the findings from the same arrangement but no permafrost. Does the addition of permafrost expand or shrink the span of control?
* Methane Trapped in the Permafrost: Expand the preceding permafrost model to include methane trapped in the permafrost. Assign a stock to methane in the permafrost and a separate stock to methane in the atmosphere. Assume that amount M of methane is trapped in the permafrost region at its initial size. Set the initial value of atmospheric methane at zero. Assume that methane enters the atmosphere only with shrinkage in the permafrost. Assume that atmospheric methane acts as a greenhouse gas to increase the fraction of stellar luminosity absorbed on Daisy World. If there is no atmospheric methane, the planet's average albedo is the same as the previous model. If a fraction = 0.1 of the stored methane ends up in the atmosphere, the planet's average albedo would be lowered by 8% from the values calculated in the previous exercise. If the entire amount M escapes to the atmosphere, the albedo would be lowered by 80%. Build this new model and compare its results with Watson and Lovelock with the solar luminosity constant at 1.0. Use the model to study the span of control. Compare the span of control with the findings from the previous exercise. Does the addition of trapped methane expand or shrink the span of control?