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The Daisyworld Model

by on January 13, 2012

DaisyWorld with black and white daisies. Notice how the daisies control the planet temperature close to the daisy ideal temperature (pink line, bottom plot). The planet even cools as luminosity increases. Without daisies, the planet's temperature simply rises with luminosity (brown line, bottom plot).

Daisyworld, a computer simulation, is a hypothetical world orbiting a star whose radiant energy is slowly increasing. It is meant to mimic important elements of the Earth-Sun system, and was introduced by James Lovelock and Andrew Watson in a paper published in 1983 to illustrate the plausibility of the Gaia hypothesis. In the original 1983 version, Daisyworld is seeded with two varieties of daisy as its only life forms: black daisies and white daisies. White petaled daisies reflect light, while black petaled daisies absorb light. The simulation tracks the two daisy populations and the surface temperature of Daisyworld as the sun’s rays grow more powerful. The surface temperature of Daisyworld remains almost constant over a broad range of solar output. Source http://en.wikipedia.org/wiki/Daisyworld

Geophysiology: a new look at earth science (1986)

What properties does this close coupling of life and its environment confer on the whole system? Does it explain the homeostasis and homeorhesis that is observed? The difficulty is that the diagram (Fig. 1, top) is much too simple; in reality, the biota and the environment are vastly complex entitles, interconnected in multiple ways, and there is hardly a single aspect of their interaction that can confidently be described by a mathematical equation. It occurred to me that a drastic simplification was needed, namely, reduction of the environment to a single variable, temperature, and of the biota to a single species, daisies. I first described Daisyworld in 1982, and I am indebted to my colleague Andrew Watson for the clear graphic way of expressing it illustrated in Fig. 2.

Figure 2. The effect of daisy cover on the mean temperature of Daisyworld, curve (A) and the effect of temperature on the growth of daisies, curve (B). In this example, the daisies are taken to be lighter in color than the bare planetary surface so that increasing daisy cover reflects more sunlight and so lowers the mean temperature. The left hand intersection of curves A and B is a stable equilibrium between daisies and temperature. The dashed curve (Al) is drawn for a lower solar luminosity than with curve (A). If the daisy cover did not respond to temperature, the difference in planetary temperature for the two solar luminosities would be (dT), about 16°C. If, as is more normal, the daisies responded to the cooler sun by covering less of the planet, the temperature difference would be (dTl), about 2°C. This simple responsive coupling between life and its environment is the basis of geophysiological regulation.

Daisyworld model

Daisyworld is a cloudless planet with a negligible atmospheric greenhouse that bears life only in the form of daisies. To start with, let us assume that the daisies are white. Because they are lighter than the ground in which they grow, they tend to increase the albedo of their locality, and, as a consequence, are cooler than a comparable area of bare ground. Where the daisies cover a substantial proportion of the planetary surface they will influence the mean surface temperature of the planet. The variation will be as illustrated by the curve (A) in Fig. 2. The parallel dotted line shows how the curve (A) might shift if there were a change in some external variable that influenced the planetary temperature. An example of such a variable is the output of radiant energy from Daisyworld’s sun.

Like most plant life, the daisies grow best over a restricted range of temperatures. The growth rate peaks near 20°C and falls to zero below 5°C and above 40°C. As a function of temperature, the steady-state population of daisies will be as in the curve (B) of Fig. 2. Curves (A) and (B) relate the temperature to the Daisyworld population at steady state, and the steady state of the whole system must be specified by the point of intersection of these two curves. In the example, it can be seen that there are two possible steady-state solutions. It turns out that the solution where the derivatives of the two curves have opposite signs is unconditionally stable, whereas the other solution is unstable. If the system is initialized at some arbitrary point, it will normally settle down at the stable solution.

What happens to this stable solution when some change of the external environment alters the planetary temperature? Suppose, for example, that the sun warms up as our sun is said to be doing. If the daisy population is artificially held constant, the planetary temperature will simply follow the change of heat output of the sun; there will be a much larger temperature change than if we allow the daisy population to grow to its new natural steady state. In this new steady state, the daisy population has changed so as to oppose the effect of a change in solar output.

Very few assumptions are made in this model. It is not necessary to invoke foresight or purpose on the part of the daisies. It is merely assumed that the growth of daisies can affect the mean planetary temperature and vice versa. Note that the mechanism works equally well whatever direction the effect is. Black daisies would have done as well; as long as the Daisyworld albedo is different from that of the bare ground, some thermostasis will result. The assumption that growth is restricted to a narrow range of temperatures is crucial to the working of the mechanism, but all main-stream life is observed to be limited within this same narrow range; indeed, the peaked growth curve (B) is common to other variables besides temperature, for example, pH and the abundance of nutrients.

In a recent paper, Watson & Lovelock (1983), this model is discussed in depth, and we emphasize there that the exercise was conducted not because we thought that daisies or any other colored plants regulate the earth’s temperature by this mechanism, but because it is easily understood as a model of the close coupling between the biota and the environment. The daisy populations were modeled by differential equations borrowed directly from theoretical ecology, Carter and Prince (1981).

Daisyworld models have a novel and wholly unexpected property. Their mathematical solution is not limited, as are similar simple ecological models, to two species only. Indeed, the number of species that can be accommodated appears to be limited only by the speed and size of the computer used and by the patience of the user. The inclusion of feedback from the environment appears to stabilize the system of differential equations used to model the growth and competition of the species. Theoretical ecology models have nearly always ignored the environment of their imaginary species, just as geophysical and geochemical models have tended to ignore the biota. Daisyworld models are admittedly primitive and, as yet, limited to a few species and a single environmental variable. But they are models of an active system where the biota and the environment are closely coupled, and they do share with the real world the same strong tendency to homeostasis and stability.

The power of the Daisyworld models is perhaps best illustrated by the imaginary world depicted in Fig. 3. It illustrates the time history of a planet where thermostasis is maintained during the progressive increase of luminosity of its sun and in spite of repeated disasters that destroy a substantial proportion of the daisies. In addition, the world is populated also by rabbits that eat the daisies and by foxes that cull the rabbits. The health of a self-regulating system is measured by its capacity to resist perturbation and by the rapidity and smoothness of the return to normal. The system illustrated in Fig. 3 (top) was perturbed four times during the course of its evolution by the abrupt but temporary deletion of 40 percent of the plant population.

These four perturbations were effectively resisted and the system rapidly recovered its former stable state. Figure 3 (bottom) illustrates the variation of the population of the species of this imaginary world. Between the perturbations, both the environment and the populations are seen to be stable. At the disturbances, the changes in the population and in the temperature take place in synchrony. The model is not concerned with the cause of the perturbations, and these could have been either internally or externally generated. This response resembles that described by S. J. Gould in his hypothesis of the punctuated evolution of the species. Gaia models are limited neither to daisies nor to the regulation of temperature by albedo change. Other environmental variables, such as the pH of the soil or sea or the abundance of oxygen and other elements, can plausibly be shown to keep within a narrow range by the same homeostatic processes that were illustrated in Fig. 2.

The regulation of the climate as a consequence of an evolutionary feedback system, involving atmospheric CO2 and the weathering of the rocks by the biota, has already been described by Lovelock and Whitfield (1982) and Lovelock and Watson (1982). The model is based on that of Walker et al. (1981), who assumed that when life started the climate was warm enough, in spite of a cooler sun, on account of a much higher concentration of CO2 in the air. It was thought to make up between 10 and 30 percent of the atmosphere. As the sun evolves and increases its flux of radiation, the temperature is kept constant by a progressive decrease of CO2. The process of CO2removal is the weathering of calcium silicate rocks. The Gaian variant of Walker’s model assumes that the biota are actively engaged in the process of weathering and that the rate of this process is directly related to the biomass at any time. If conditions are too cold, the rate of weathering declines, and as a consequence of the constant input of CO2 by degassing from the earth’s interior, the CO2 partial pressure rises.

Models of this kind about CO2 and climate could add to the current interest in this most important environmental concern. They are based on an active feedback-control system, and with such systems, it is possible to predict instability and oscillation enhanced by positive feedback; these instabilities are most probable when the system nears the limits of its capacity to regulate. It is interesting to compare this prediction with the observations of the correlation between CO2 and climate that characterized the last glaciation, particularly the sudden and apparently synchronous rise of both CO2 and temperature at the termination of the glaciation. The exact sequence of these events is still uncertain but few doubt that the end was sudden and that both CO2 and temperature rose substantially on a global scale. These rapid changes some 12,000 years ago cannot be explained by geochemical or geophysical theory alone. They suggest a sudden change of biomass; most probably the sudden death of a proportion of the marine phytoplankton, an event that would reduce the rate of pumping of CO2 from the air. The geophysiological prediction of oscillatory instability near the limits of regulation fits with these observations. It is well known that glaciations are in synchrony with variations of solar illumination consequent upon the earth’s orbital position and inclination, the Milankovich effect. This alone cannot account for the rapid reversal of the glaciation. But the Milankovich effect could be the trigger that synchronizes an otherwise free-running geophysiological oscillator.

Figure 3. (top) The evolution of the temperature on an imaginary Daisyworld populated by daisies. The color of Daisyworld changes in response to temperature over the range 5-40°C. It is also populated by rabbits that graze upon the daisies and by foxes that hunt the rabbits. In addition, at intervals Daisyworld is perturbed by catastrophes that cause the death of 40 percent of the daisy population. It is assumed that Daisyworld is warmed by a star that increases its luminosity linearly with time. When the planetary temperature exceeds 5°C, the daisies grow rapidly and are dark colored in response to the initial low temperature. The mean temperature rises by positive feedback until the operating temperature for homeostasis is reached. Homeostasis is maintained and the four perturbations resisted. The capacity of the system to restore homeostasis after a disturbance is seen to decline as the increasing solar output carries the system nearer to its limit. The dashed line illustrates the planetary temperature expected in the absence of life. (bottom) The populations of daisies (A), rabbits (R) and foxes (F) during the evolution of Daisyworld.

The Gaia Hypothesis

In some ways the ecosystem of, for example, a forest in the humid tropics is like a human colony in Antarctica or on the moon. It is only self-supporting to a limited extent, and its continued existence depends upon the transport of nutrients and other essentials from the world. At the same time ecosystems and colonies try to minimize their losses by conserving water, heat, and nutrients; to this extent they are self regulating. The tropical rain forest, likewise, keeps wet by modifying its environment so as to favor rainfall. Traditional ecology has tended to consider ecosystems in isolation. Geophysiology reminds us that all ecosystems are interconnected. By analogy, in an animal, the liver has some capacity for the regulation of its internal environment, and its liver cells can be grown in the isolation of tissue culture. But neither the animal nor its liver can live alone; they depend upon their interconnection.

We do not know if there are vital ecosystems on the earth, although it would be difficult to imagine life continuing without the anoxic ecosystems of the sediments. The forests of the humid tropics do not add significantly to the world’s oxygen budget nor to the exchange of essential elements through the atmosphere. Their intensive biosynthesis is recycled within their boundaries. Where they may be significant on a global scale is in their effect on climate through evapotranspiration and the effect of their presence on the regional albedo. The transfer of nutrients and the products of weathering down tropical rivers are obviously part of their interconnection and may have a global significance.

If evapotranspiration or the movements of materials down tropical rivers to the oceans are vital to the present homeostasis, then the replacement of forests with agricultural or grassland surrogates would not only deny those regions to their surviving inhabitants but also might threaten the rest of the system as well. We do not yet know whether the tropical forest systems are vital to the present planetary ecology. They might be like the temperate forests that seem to be expendable without serious harm; temperate forests have suffered extensive destruction during glaciations as well as during the recent expansion of agriculture.

Relevance to Earth

Because Daisyworld is so simplistic, having for example, no atmosphere, no animals, only one species of plant life, and only the most basic population growth and death models, it should not be directly compared to Earth. This was stated very clearly by the original authors. Even so, it provided a number of useful predictions of how Earth’s biosphere may respond to, for example, human interference. Later adaptations of Daisyworld (discussed below), which added many layers of complexity, still showed the same basic trends of the original model.

One prediction of the simulation is that the biosphere works to regulate the climate, making it habitable over a wide range of solar luminosity. Many examples of these regulatory systems have been found on Earth. Source http://en.wikipedia.org/wiki/Daisyworld

from Vulnerability of Permafrost Carbon to Climate Change: Implications for the Global Carbon Cycle (2008)

Plant Albedo

Apart from changes in the C cycle, changes in ecosystem energy balance can directly affect regional climate, and in the case of permafrost thaw may be inextricably linked with the C cycle changes already discussed. Changes in albedo brought about by changes in plant species composition, length of the snow season, lake area, or fire frequency can have positive or negative effects on climate warming. Increases in shrub cover in graminoid-dominated tundra ecosystems result in greater absorption of solar radiation in summer and winter, leading to local warming in the summertime (Chapin et al. 2005).

Similar patterns can be expected as the treeline moves north. Changes in ecosystem albedo can also have a cooling effect. Increased fire frequency in boreal forests alters the proportion of forest dominated by broadleaf deciduous trees. These early-successional species reflect more solar radiation in summer than do the needle-leaved evergreen species they replace, and expose high-albedo snow on the ground in winter.

These long-term albedo effects can offset increased warming from both the transfer of C to the atmosphere from fire and the short-term decrease in albedo immediately following fire, and may actually cool the climate (Randerson et al. 2006). Lake or wetland expansion may serve to regionally warm or cool the climate, depending on the type of vegetation replaced.  Source 2008 Schuur (Bioscience) Related Ecosystem science perspectives on boreal forests and climate change mitigation

Lovelock explains Daisyworld computer model (at around minute 5:30)

Related http://www.gingerbooth.com/flash/daisyball/index.htmlGaia Hypothesis

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