next up previous
Next: Fragmentation and biodiversity Up: Self-stabilization of the biosphere Previous: Hysteresis of Life

The impacts of fragmentation

 

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 tex2html_wrap_inline1467, 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 tex2html_wrap_inline1475 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.

  figure402
Figure 9: Patch-work of occupied sites in the standard percolation model at criticality tex2html_wrap_inline1477. The fractal spanning cluster is marked by the darker shade. Lattice size is tex2html_wrap_inline1479.

Therefore, we have to distinguish between three qualitatively different regimes determined by the occupation probability:

  1. tex2html_wrap_inline1481: the collection of occupied sites does not form any spanning cluster, but the collection of unoccupied sites represents a connected ''void space''.
  2. tex2html_wrap_inline1483: neither the occupied nor the void sites form a connected structure.
  3. tex2html_wrap_inline1485: 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 tex2html_wrap_inline1487. In every time step n all cells within the finite lattice are considered one by one and excluded from the growth space with probability tex2html_wrap_inline1487. At time tex2html_wrap_inline1151, the probability that any specific site has been ``civilized'' is therefore given by
equation409
Note that
equation411
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 tex2html_wrap_inline1497, i.e., for growing n. For fixed S' and tex2html_wrap_inline1503 we perform tex2html_wrap_inline1505 time steps, to destroy almost (tex2html_wrap_inline1507) all growth sites. Fig. 10 reproduces our findings regarding the relation between global mean temperature tex2html_wrap_inline1345 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.

  figure416
Figure 10: Dependence of global mean temperature tex2html_wrap_inline1345 on the fragmentation parameter p. S' corresponds to tex2html_wrap_inline1523 for the temperature tex2html_wrap_inline1525 of the ``dead'' planet. The broken vertical line at tex2html_wrap_inline1527 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-tex2html_wrap_inline1345-curve for any fixed S' (see Fig. 11).

  figure423
Figure: Convergence of numerical results for p-tex2html_wrap_inline1345-relationship for increasing lattice size, (a) tex2html_wrap_inline1267, (b) tex2html_wrap_inline1541, (c) tex2html_wrap_inline1543. S' has been fixed to a value generating a geophysical planetary temperature tex2html_wrap_inline1547.

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 tex2html_wrap_inline1345 is plotted as a two-dimensional surface over the control space spanned by the driving-force variables tex2html_wrap_inline1081 (i.e. S') and p. The adaptive power of Daisyworld clearly breaks down when p approaches the value
equation430
The explanation for this phenomenon is simple but illuminating: for tex2html_wrap_inline1559 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!

  figure434
Figure 12: Bioplanetary temperature tex2html_wrap_inline1345 as a function of insolation (as represented by tex2html_wrap_inline1081) and fragmentation (as represented by p).




next up previous
Next: Fragmentation and biodiversity Up: Self-stabilization of the biosphere Previous: Hysteresis of Life

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