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Comparison between patchy and trivial reduction of growth space

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 tex2html_wrap_inline1593 denote the size of the lattice, so the cells tex2html_wrap_inline1595 of the system obey the inequalities
equation453
Then assume that
equation455
where tex2html_wrap_inline1597, d(0)=L, and tex2html_wrap_inline1601 for tex2html_wrap_inline1603. That means that the habitable zone is a dwindling central square of approximate area tex2html_wrap_inline1605 and is a decreasing function of time tex2html_wrap_inline1151. 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
equation459

  figure463
Figure 14: Variation of global mean temperature tex2html_wrap_inline1345 with progressive trivial shrinkage of habitable area. As in Fig. 9, S' has been chosen to generate tex2html_wrap_inline1613. The tilted broken line indicates the linear estimation as expressed in Eq. 39.

The evolution of the global mean temperature tex2html_wrap_inline1345 as a function of tex2html_wrap_inline1443 under an insolation that corresponds to tex2html_wrap_inline1619 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: tex2html_wrap_inline1345 increases almost linearly with growing tex2html_wrap_inline1443. This behaviour is approximated by the formula
 equation474

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 tex2html_wrap_inline1625 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
equation479
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.


next up previous
Next: Conclusion Up: The impacts of fragmentation Previous: Fragmentation and biodiversity

Werner von Bloh (Data & Computation)
Thu Jul 13 13:46:37 MEST 2000