We want finally to demonstrate that the geometry of the remaining growth space is indeed of paramount importance for the ecological performance of our toy biosphere. To this end, we repeat our simulations for a trivial process of habitat reduction, namely progressive shrinking in time of the rectangular core growth area. The prescription for the process is as follows:
Let 
denote the size of the lattice, so the cells 
of
the system obey the inequalities
Then assume that
where 
, d(0)=L, and 
for 
.
That means that the habitable zone is a dwindling central square of approximate
area 
and is a decreasing function of time 
. The properties of the so-restricted system can be compared to those
for the above-described patchy one with identical total growth area, i.e. for
Figure 14: Variation of global mean temperature 
with progressive
trivial shrinkage of habitable area. As in Fig. 9, S' has been chosen to
generate 
. The tilted broken line indicates the linear
estimation as expressed in Eq. 39.
The evolution of the global mean temperature as a function of 
under an insolation that corresponds to 
is depicted in Fig. 14. Note
that, in contrast to the non-trivially fragmented system, the ``shrinking square
biosphere'' is not capable of planetary homeostatic control:
increases almost linearly with growing . This behaviour is approximated
by the formula
The markedly different adaptive capabilities induced by habitat geometry
can be explained in a straightforward way. In the first case, where the growth
area is reduced according to the percolation algorithm, we have approximately
non-coverable cells, and almost all of them are adjacent to cells
covered by vegetation. In the second case, where the planetary space decays into
a coverable central square and a non-coverable margin, only a very small number
of ``dead'' cells
are neighbour on a ``living'' cell. In other words: due to its intricate
patchiness, the surface-to-bulk ratio of a percolation cluster is very large
in comparison with the surface-to-bulk ratio of a simple square covering
the same area!
But the size of the surface is an indicator
for the total heat flow, which can be activated between the sterile and the
fertile zones of Daisyworld. We clearly find that the patchy and lacunary
distribution of living cells over the planet is sufficient to suppress ``hot spots''
via thermal relaxation.