First, we test the decay of self-stabilizing power with increasing patchiness in the 2D Daisyworld without animals.
Fig. 2 summarizes our findings regarding the relation between global mean
temperature 
and the percolation parameter 
. It can be observed 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: Convergence of numerical results for p--relationship
for increasing lattice size ranging from 
(wiggliest line) to
(smoothest line).
S has been fixed to a value that
generates a geophysical planetary temperature 
.
The numerical results are robust. As a matter of fact, Fig. 2 shows for a series of extensive
calculations with
increasing lattice dimensions that finite-size effects can be neglected.
It actually turns out that the above-mentioned threshold value
for patchiness has universal character. 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 connectivity!
For detailed results concerning, for example, the corresponding change in the species spectrum see again von Bloh[5].