Within our 2D model the disposable area for vegetation growth is the full square, i.e., a simply connected domain. In the real world, however, the area available for biospheric adaption to Global Change forces is highly fragmented by civilisatory activities: urban settlements, infrastructures, agriculture, tourism, etc.. The implications of habitat fragmentation on biodiversity is at present a much-debated issue.
Our toy planet constitutes an ideal theatre for investigating this and related questions in some depth; we specifically ask how the species spectrum and the resulting homeostatic properties of the biosphere depend on landscape heterogeneity. The latter is simulated here in a well-defined way: we employ the percolation model from solid state physics [19] in order to simulate successive non-trivial reduction of growth space.
The percolation model on a square lattice is formulated in the following way: for a given probability 
, each site will be randomly occupied with
probability p. As a consequence, it will remain empty with probability 1-p.
A connected group of occupied sites is called a ``cluster''. The size of the clusters clearly grows with increasing p. ``Percolation'' is said to set in when
the largest cluster extends from one end of the system to the other (``spanning cluster''). In the limit of infinitely large lattices there exists a sharp threshold value 
for percolation. The spanning cluster associated with
this phase transition is a multiple-connected fractal object with a power-law
hole-size distribution. Fig. 9 gives an example of such a critical configuration
which allows to traverse the entire lattice via next-neighbour steps.
Figure 9: Patch-work of occupied sites in the standard percolation model
at criticality 
. The fractal spanning cluster is marked
by the darker shade. Lattice size is 
.
Therefore, we have to distinguish between three qualitatively different regimes determined by the occupation probability:
: the collection of occupied sites does not form any
spanning cluster, but the collection of unoccupied sites represents a connected
''void space''.
: neither the occupied nor the void sites form a
connected structure.
: the collection of occupied sites does form a connected
structure, but the void space is now disconnected .
We introduce civilisatory land-use into our extended Daisyworld by gradually
diminishing the potential growth area in the following way: choose a (small)
generating probability 
. In every time step n all cells
within the finite lattice are considered one by one and excluded from the growth space
with probability . At time 
, the probability that any specific site
has been ``civilized'' is therefore given by
Note that
is then the statistical fraction of habitable area after n time steps.
Our fragmentation scheme is independent of the actual status of the cell under consideration. Furthermore, all physical properties, such as diffusive heat transport remain unaffected. We now present some computer simulation results, which shed light on the systems behaviour of ``anthropomorphic Daisyworld''.
First, we test the decay of self-stabilizing power with increasing patchiness
parameter 
, i.e., for growing n. For fixed S' and 
we perform 
time steps, to destroy almost (
) all growth sites.
Fig. 10 reproduces our findings regarding the relation between global mean
temperature 
and the percolation parameter p. We observe that even
the fragmented biosphere is able to stabilize the planetary temperature near
the optimal value, unless p exceeds a value of approximately 0.4.
Figure 10: Dependence of global mean temperature on the fragmentation parameter p. S' corresponds to 
for the temperature 
of the ``dead'' planet. The broken vertical line at 
indicates the disconnection threshold for the habitable space.
Our numerical results are robust. A series of extensive calculations with
increasing lattice dimensions shows that finite-size effects can be neglected:
the homeostatic response of the biosphere to fragmentation results in a well-defined
p--curve for any fixed S' (see Fig. 11).
Figure: Convergence of numerical results for p--relationship
for increasing lattice size, (a) 
, (b) 
, (c)
. S' has been fixed to a value generating a geophysical planetary
temperature 
.
As a matter of fact it turns out that the above-mentioned threshold value
for patchiness has universal character, i.e., the behaviour depends neither on the system size nor the parameter settings. In particular, the strength of the
insolation, which represents an external driving force, does not affect the
threshold value.
This is demonstrated in Fig. 12, where
the self-organized mean temperature is plotted as a two-dimensional
surface over the control space spanned by the driving-force variables 
(i.e. S') and p. The adaptive power of Daisyworld clearly breaks down
when p approaches the value
The explanation for this phenomenon is simple but illuminating: for 
the growth space has lost its connectivity and is broken up into many isolated
domains. Our toy model hence provides us with clear-cut evidence that the
ecological performance of a system directly depends on its topology!
Figure 12: Bioplanetary temperature as a function of insolation
(as represented by ) and fragmentation (as represented by p).