The intention of geophysiological modelling is the analysis of the feedback
mechanisms acting between geo and biosphere (Lovelock, 1989, Lovelock, 1991). It is one approach to Earth System Analysis (Schellnhuber and Wenzel, 1998). The planet Earth can be understood as a "superorganism" which itself regulates to a state allowing life on Earth. (Gaia hypothesis, see, e.g., Lovelock and Margulis, 1974, Lenton, 1998). First attempts were done by Vernadsky (1927) on a qualitative level in the form of an idea about the interdependence between vegetation and climate (see, e.g., Vernadsky, 1998). Later on Kostitzin (1935) realized this idea in the first mathematical model for the coevolution of atmosphere and biota.
Socalled
"minimal models" of the Earth are described in (Svirezhev and von Bloh, 1996, 1997). According to the "virtual biospheres" concept (Svirezhev and von Bloh, 1999) the evolution of the Earth system can be understood as a sequence of bifurcations.
An example of geophysiological regulation behaviour can be found in the Earth system: Silicaterock weathering as a process to regulate the surface temperature.
Due to silicate rock weathering carbon dioxide is taken out of the atmosphere. At geologic time scales there is an equilibrium between this sink of carbon and the sources by volcanic emissions. An increase in temperature accelerates the chemical processes of weathering reducing the amount of carbon in the atmosphere. Due to the lower greenhouse effect the temperature decreases; we have a negative feedback regulating the surface temperature.
Based on the work of Lovelock and Whitfield (1982)
Caldeira and Kasting (1992) have developed a model
to estimate the life span of the biosphere taken into account the above
mentioned feedback mechanism. In Fig. 1 the global
surface temperature is plotted under increase of the solar luminosity for the
whole history of the Earth. In contrast to the curve for fixed carbon dioxide
content of the atmosphere the temperature is always above the freezing point
of water. Such a regulation mechanism can be an explanation for the "faint
young Sun" paradox (Sagan and Mullen, 1972), where we
have at least for 3.8 Gyr in the past always liquid water on the surface of
Earth. The role of the biosphere is to increase the weathering rate.
The Caldeira/Kasting model assumes constant volcanic activity and continental
area during the Earth evolution, Franck et al. (2000) extend this model by adding a geodynamic description of the geospheric processes. A short description of the model used can be found here.
Such a model can be used to determine the habitability of Earthlike planets in the solar system and for extrasolar planetary systems.
Fig. 1: Evolution of the global surface temperature T for the Caldeira/Kasting
model (solid curve) and a model with fixed (present) atmospheric CO_{2} (dashed curve). S denotes the corresponding solar luminosity in solar units.
The basic characteristics of such feedback models can be illustrated using the
"Daisyworld" scenario (Watson and Lovelock, 1983):
In this model a planet with surface albedo A_{0} can be covered
by two types of vegetation: one species ("white daisies") with an albedo
α_{1}> α_{0} and the other
species with albedo α_{2} < α_{0} ("black daisies") covering fraction N_{1} and N_{2} of the planetary surface normalized to one. Then the dynamic of the two daisies can be described by the following differential equations:
dN_{1}(t)/dt=N_{1}(&beta(T_{1})xγ)
dN_{2}(t)/dt=N_{2}(&beta(T_{2})xγ)
x=1N_{1}N_{2},
where β(T) is the growth rate and γ the death rate of daisies.
The global temperature is determined by the balance between ingoing solar radiation and outgoing blackbody radiation
T=(S(1&alpha)/4σ)^{1/4},
where α=α_{1}N_{1}+&alpha_{2}N_{2}+&alpha_{0}x is the global albedo.
Due to their different albedos the
local temperatures T_{1,2} differ from each other.
T_{1}^{4}=T^{4}+q(α&alpha_{1})
T_{2}^{4}=T^{4}+q(α&alpha_{2}),
where the factor q depends on the heat exchange.
The growth function β depends only
on the local temperature T_{1,2}. Because the global albedo of the planet is changed
by the growth process, the global temperature is changed altering the
growth rates.
One interesting aspect is the behaviour of the coupled vegetationclimate
model on external perturbations, e.g., an increase of the external solar luminosity.
In Fig. 2 the solar luminosity was successively increased. It must be pointed
out that the global temperature of the populated planet  in contrast to
the unpopulated one  is remarkable constant in a wide range of solar
luminosity: the planet itself regulates the optimum state of life (homeostasis).
The original Daisyworld model has been analyzed by other authors in great detail, informations can be found in Isakari and Somerville (1989) and Saunders (1994).
Fig. 2: Coevolution of climate and vegetation at increasing
(red) and decreasing (blue) insolation. T_{0} is the global temperature of the corresponding planet without vegetation.
The conceptual simple zerodimensional Daisyworld model
allows in an equilibrium state due to theoretical considerations only the
coexistence of at most two different species. This restriction can be avoided
if we introduce spatial dependency into the model. The local temperatures
T_{1}, T_{2} and albedos α_{1}, α_{2} are
replaced by 2dimensional fields T(x,y) and α(x,y).
The temporal and spatial evolution of the temperature is determined by a partial differential equation based on an energybalance model (EBM) (see, e.g., HendersonSellers and McGuffie, 1983). The evolution of the albedo
connected directly with the vegetation is modelled by a cellular automaton (CA)
approach (for a description of the Cellular automaton concept see, e.g., Wolfram, 1983, Goles, 1994). CA models have been successfully applied to a variety of problems. An application of such a CA approach to the modelling of calcite formation in microbial mats is presented in (Kropp et al., 1996). Fig. 3 describes the nondeterministic rules for the update of the lattice.
Fig. 3: Update rules for the cellular automaton for an empty cell(righthand side) and a covered cell (lefthand side). The state of the cell in the following time step depends only on the state of the cell itself and their next neighbours.
The model was implemented on a RS/6000 SP parallel computer (IBM).
Because of only nextneighbour interactions a high efficiency of the parallelization can be expected. The parallelization was performed using the GeoPar library based on the messagepassing paradigm. Some documentation and links about parallel programming can
be found here.
A typical state of the temperature and albedo distribution is
plotted in Fig. 4.
Fig. 4: Typical configuration for the albedo distribution (righthand side) and
the temperature distribution (lefthand side).
The 2dimensional model has a similar selfregulation behaviour as the original model. Due to the cellular automaton approach the model can be simply extended to
incorporate several biological effects like
 mutations
 seeding
 habitat fragmentation
into the model. Only the set of rules defining the CA must be modified. One
expects that adding mutations, i.e., slight random changes of the albedo in
the growth process, will decrease the regulation capability. But after doing the
simulation one gets a counterintuitive result: with mutations the solar luminosity can be significantly increased
until the vegetation breaks down (see Fig. 5).
Fig. 5: Global temperature T vs. increasing insolation S with (b) and
without (a) mutation of the albedo.
An animation is available as a MPEG video(11 MByte). It illustrates the temperature and albedo evolution under increasing insolation. After the vegetation is wiped out the insolation is decreased:
It can be clearly seen that the CA version of the Daisyworld exhibits hysteresis, i.e. the behaviour of the model depends on its history. The breakdown of the vegetation is a firstorder phase transition. This model has also been used by Ackland et al. (2003) in order to describe the formation of deserts in Daisyiworld.
In the real world the growth area is not simply connected, but is fragmented
due to anthropogenic influences by, e.g., human settlements and infrastructure.
It is possible to model such fragmentations with our 2dimensional CA and
to analyze the effects on the regulation capability.
The fragmentation is modelled by a percolation model (Stauffer and Aharony, 1992). In each time step
a cell can be marked as "infertile" with a given probability p independent of
its state. Configurations for different concentrations p of infertile
cells are shown in Fig. 6:
Fig. 6: Fragmentation of landscape for different concentrations of infertile cells p. Some
connected clusters are marked (red color).
For a critical p an infinite cluster of connected cells is formed. At 1p the
growth area splits up into disconnected areas. The critical concentration
p_{c} can be numerically determined to p_{c}=0.537...
The simulations of the 2dimensional Daisyworld model with fragmentation are done for different constant solar luminosities S. The regulation capability breaks down at p=0.407 independent of
the initial solar luminosity S. More details can be found in (von Bloh et al., 1997).
Fig. 7: Global average temperature as a function of the fragmentation p and the temperature T_{0} of the uncovered planet.
In order to reflect the fundamental ecosystems dynamics on trophic interactions as well, we extend the 2D model even further and add herbivores to our planet (see also, e.g., Harding and Lovelock, 1996). These
vegetarian "lattice animals" move on the grid as random walkers. The state of the herbivore can be changed by four different events:
 Death

The herbivore deceases with a certain probability γ_{h}.
 Ingestion

If the herbivore survive and if the cell is covered by a daisy then the animal consumes the plant.
 Reproduction

The herbivore in question may give birth to a "lattice child". The probability of such an event depends on the local temperature and the number of daisies "eaten" by the herbivore in a multiplicative manner.
 Move

The herbivore may also walk to a randomly chosen next neighbour cell if the cell is not already occupied by another herbivore.
Fig. 8 shows some results for increasing fragmentation at different mortality rates γ_{h} for the herbivores. In von Bloh et al. (1999) the Daisyworld model including herbivores is described in detail.
(a)
(b)
Fig. 8: (a) Dependence of global mean temperature T on fragmentation parameter p for distinct herbivore mortality rates γ_{h}. (b) Dependence of herbivore concentration on p for the same mortality rates as in (a).
Starting from a very simple geophysiological model the developed 2D model
shows some remarkable effects:
 Local interactions of
competing species lead to global effects: Self organized regulation behaviour of
the global
temperature under external perturbations.
 The system undergoes a first order phase transition:
After the regulation capability is exhausted, an irreversible breakdown
occurs immediately.
 By splitting the growth area
into disconnected regions via, e.g., successive fragmentation the regulation
behaviour also breaks down.
This research was part of the POEM core project at the Potsdam Institute for Climate Impact Research.
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