The Climate Change Problem is very fashionable today. A huge amount of publications mask some simple facts that we can hardly conceive:
 How does the biosphere machine operate?
 Is the Earth biosphere unique, or do there exist other virtual biospheres?
A few words about the history of this problem. It
was initially formulated (at a qualitative level, of course) by Vernadsky (1927) in the
form of an idea about the interdependence between vegetation and climate. Then Kostitzin (1935) realized this idea in the first mathematical
model for the coevolution of atmosphere (climate) and biota. It is interesting that he has obtained the "epoques glaciaires" as selfoscillations
in this model. Recently Watson and Lovelock (1983) further developed Vernadsky's idea. They considered a causal loop between the surface
temperature and two types of vegetation (differing in albedo). A competition
between them on a temperature ecological niche generates different spatial
vegetation and temperature patterns ("Daisy World"). In 1993 Schellnhuber and von Bloh realized this model, using cellular automata, in the form
of a 2dimension structure. This approach showed an important
role of
fluctuations in forming spatial patterns (von Bloh et al., 1997).
Svirezhev (1994) formulated a concept of "virtual biospheres".
According to this concept the contemporary Earth biosphere is one of many
possible (virtual) biospheres corresponding to different equilibria
of a nonlinear dynamical "climate + biosphere" system (Svirezhev and von Bloh, 1999).
In the course of the
planetary history and own evolution, this system passed through several
bifurcation points in which some random factors (small perturbations)
determined solution branches actually followed by the system.
A driving force in this evolution could be the
evolution of the "Earth green cover", which had, in turn, several bifurcation
points, for instance, the appearance of terrestrial vegetation, or
the change from the coniferous forest to the deciduous forest.
Note that this concept contradicts Vernadsky's "ergodicity axiom",
according to which the contemporary Earth biosphere is unique,
irrespective of the initial and previous states.
What kind of bifurcations are admissible in this system?
We can observe a change of planet's "status"
from a "cold desert" to either a "cold green planet", or a "hot
green planet" (the first bifurcation).
Then either a "wet hot" planet covered with a
tropical rain forest, or a "dry hot" planet (savannah)
develops from a "green hot"
planet as a result of the second bifurcation.
Analogically, either a "wet cold" (a temperate forest) or a "dry cold" (steppe)
planet arises from a "green cold" planet (see Fig. 1).
Fig. 1: Bifurcation diagram for the "climate+biosphere" system.
A socalled minimal model describing the climate biosphere mechanisms of a
hypothetical zerodimensional point planet is presented in
(Svirezhev and von Bloh, 1996). Unlike other
attempts of modelling the global vegetation (e.g., the Osnabrück biosphere model
(Esser, 1991) or the Frankfurt biosphere model
(Lüdeke et al., 1995)) the system can be fully understood through analytical as well as numerical inspections. This model consists of two coupled differential equations and is based on two hypotheses:
Hypothesis 1:
The albedo α depends only on the vegetation density N, so that α=α(N)
is a monotonous decreasing function of N.
and
Hypothesis 2:
The growth function β depends only on temperature T, i.e. β=β(T); it is an unimodular
function of T.
The dynamics of coupled climatevegetation can be described by two coupled differential equations:
dT(t)/dt=S(1α(N))4σT^{4}
dN(t)/dt=β(T)NγN^{2}.
The evolution of climate is determined by the balance between ingoing (S(1α))and outgoing radiation (4σT^{4}), while the evolution of the vegetation is determined by the balance between growth (&beta(T)N) and decay (γN^{2}).
Stationary solutions can be determined by setting the righthand side of the equations to zero:
dT(t)/dt=0
dN(t)/dt=0.
Analytical calculations for this coupled
climatebiosphere system indicate that up to two different stable equilibria are possible. One is the "dead" planet without any vegetation (N=0) and the other is the living planet (N>0) (see Fig. 2).
Fig. 2: Zeros of the two differential equations (I) and (II) in the climatevegetation domain {T,N} for different insolations. The intersections of the two curves indicate stationary points and are marked with ac.
Further on a carbon cycle was added to the system of two differential equations (Svirezhev and von Bloh, 1997) increasing the number of equations to three. Because the total amount of carbon A is conserved the set of equations can be reduced by one. The system has two causal loops: (1) vegetation → albedo → temperature → vegetation, (2) vegetation ↔
atmosphere carbon → temperature → vegetation.
In this case the system can have up to five different equilibria: three of them are
stable and two others are unstable. The projection of its phase portrait onto the
climate vegetation plane is shown in Fig. 3.
Two bifurcation parameters can be identified, A and the maximum biological productivity P_{max}. Depending on these two parameters the "living" planet bifurcates into the "hot green" planet and the "cold green" planet.
Fig. 3: Phase portrait of the climatebiosphere system with carbon cycle. The different colors indicate the basins of attraction of the three stable equilibria
"cold desert", "cold green planet", and "hot green planet".
There are four attractive domains corresponding to the following equilibria:
 "cold desert" when the planet has no vegetation and the planetary
temperature is less than 3°C.
 "cold green planet" with a rich vegetation and relatively low temperature,
approximately equal to the current Earth's temperature, 15°C.
 "hot green planet" with a poor vegetation and relatively high temperature,
higher than 30°C that is typical for Earth's hot deserts.
 "hot desert" is a lifeless planet, its temperature is higher than the upper
boundary of the tolerable interval (>60°C).
Let us consider the evolution of the system topology when the total amount of carbon
slowly decreases. An animation is available as a MPEG video(3 MByte). Fig. 4 plots the temperatures of the different equilibria as a
function of A, where the black lines indicate trajectories of the evolution for
either
decreasing or increasing carbon. The colorshaded areas are the socalled basins of attraction,
i.e. the set of initial conditions, starting in which trajectories come to the same stable
equilibrium. The area denoted as the "cold desert" is the solution without vegetation on
the planet. The equilibrium temperature is below the lower tolerable
temperature, while the "hot desert" marks a solution with a
temperature, which is too hot for the vegetation. The "hot green planet" and "cold
green planet" are the nontrivial solutions with low and high vegetation densities,
respectively.
Fig. 4: Bifurcation diagram of the "climate + biosphere" system. Black lines
indicate trajectories of the evolution for either decreasing or increasing carbon. Colorshaded areas denote the basis of attraction of the stable equilibria.
Two hysteresis loops can be identified (Svirezhev et al., 2003). Starting with the "hot desert" the planet
evolves in the following way: hot desert → hot green planet →
cold green planet → cold desert.
The transition "hot green planet → cold green planet" is accompanied by
almost an explosive increase in vegetation biomass. Starting from the cold
desert, however, a different path is realized: cold desert → hot green
planet → hot desert.
This means that a current state of the coupled climatebiosphere system
depends on its history. The change from one to another equilibrium is done in a
noncontinuous way. The arrows in the diagram indicate noncontinuous transitions. The
rise of the hot green planet from the hot desert only is a continuous process.
 The evolution of the Earth system can be understood as a sequence of
bifurcations.
 The number of equilibria increases with increasing complexity of the model.
 Small changes in temperature or vegetation could force a drastic change in the climatevegetation system; a new equilibrium could be realized.
This research was part of the POEM core project at the Potsdam Institute for Climate Impact Research.
Esser, G., 1991. Osnabrück Biosphere Modelling and Modern Ecology.. In: G. Esser, D. Overdieck (Eds.). Modern Ecology, Basic and Applied Aspects. Elsevier, Amsterdam, 773804.
Kostitzin, V. A., 1935. L' evolution de 'l atmosphere: Circulation organique, epoques glaciaries. Hermann, Paris.
Lüdeke, M. K. B., S. Dönges, R. D. Otto et al., 1995. Responses in NPP and carbon stores
of the northern biomes to a CO_{2}induced climatic change as evaluated by the Frankfurt biosphere model (FBM). Tellus 47B, 191205.
Svirezhev, Y. M., 1994. Simple model of
interaction between climate and vegetation: virtual biospheres, IIASA Seminar, Laxenburg.
Svirezhev, Y. M., and W. von Bloh, 1996.
A minimal model of interaction between climate and vegetation: qualitative approach, Ecol. Mod. 92, 8999 (abstract, full text in a HTMLdocument).
Svirezhev, Y. M., and W. von Bloh, 1997.
Climate, vegetation, and global carbon cycle: the simplest zerodimensional model , Ecol. Mod. 101, 7995 (abstract, full text in a HTMLdocument).
Y. M. Svirezhev and W. von Bloh
, 1999.
The climate change problem and dynamical systems: virtual biospheres concept. In: G.
Leitmann, F. E. Udwadia, A. V. Kryazhimski (eds.), Dynamics and control, Gordon and Breach,
161173 (abstract).
Svirezhev, Y. M., A. Block and W. von Bloh, 2003. "Active planetary cover" concept and longterm evolution of planetary climate. In: W.E. Krumbein, D.M. Paterson and G.A. Zavarzin (eds.), Fossil and Recent Biofilms, A Natural History of Life on Earth, Kluwer Academic Publishers, Dordrecht/ Boston/ London, 415427 (abstract).
Vernadsky, V., 1998. The Biosphere, Complete Annotated Edition. Springer, New York,
250 p..
von Bloh, W., A. Block, and H.J. Schellnhuber, 1997. Self stabilization
of
the biosphere under global change: a tutorial geophysiological approach. Tellus 49B, 249262 (abstract, full text as a HTMLdocument).
Watson, A. J., and J. E. Lovelock, 1983. Biological homeostasis of the global
environment: the parable of Daisyworld . Tellus 35B, 286289.
