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Equilibrium behaviour

  Let us first choose a fixed insolation well within the range where the biosphere is able to maintain optimal subsistence conditions. Let us assume, for instance, that S'=1, implying tex2html_wrap_inline1271. We vary, however, the mutation rate r in order to reveal the relations between mutation, biodiversity, and adaptive capacities.

The system is initialized by a random distribution of vegetation (albedo): then the rules of the cellular automaton are applied. After approximately tex2html_wrap_inline1275 iterations, the global average temperature approaches the optimum growth temperature tex2html_wrap_inline1277, i.e. the system has relaxed to a statistical equilibrium. Further iterations produce significant local fluctuations but do not modify the mean properties of the model planet. Fig. 1 depicts a typical equilibrium distribution of species (characterized by their albedo) and the associated two-dimensional temperature field.

  figure254
Figure 1: Daisyworld in statistical equilibrium for S'=1: snapshot of typical albedo distribution (right) and associated spatial temperature fluctuations around tex2html_wrap_inline1277 (left, in false colour representation).

The species spectrum B(A) is defined as follows:
equation259
Note that due to the finiteness of the lattice we have only finitely many ``daisies'' on our planet; therefore this and the following quantities are well defined. The mean albedo tex2html_wrap_inline1285 of our model biosphere and its variance tex2html_wrap_inline1287 are then given by
eqnarray263
where tex2html_wrap_inline1289 is the total ``biomass''.

The variation of the species spectrum as a function of the mutation rate r is depicted in Fig. 2 for version B of the automaton. Extensive computations corroborate the fact that B(A) can be approximated by a Gaussian distribution for r<0.1, i.e.,
equation276

  figure279
Figure 2: Species spectra after tex2html_wrap_inline1297 iterations for optimal constant insolation (tex2html_wrap_inline1271) and for different mutation rates r=0.0,0.005,0.01,0.02, and 0.1, respectively. The solid curves represent the appropriate Gaussian fits. Note that the curve for r=0 becomes a pure tex2html_wrap_inline1307-function only in the limit of infinite lattice.

The mean albedo tex2html_wrap_inline1285 consistently turns out to equal the optimum albedo tex2html_wrap_inline1311, which is determined by
equation286
So we have
equation290
if the natural choice for tex2html_wrap_inline1079 is made.

From Fig. 2 we can also see that tex2html_wrap_inline1315 is a monotonically increasing function of r. Without mutation (r=0) the spectrum actually collapses into a tex2html_wrap_inline1307-function at tex2html_wrap_inline1323 if we take the limit for an infinite lattice size, while for large r an almost uniform spectrum emerges.

Fig. 3 reproduces the quantitative relationship between tex2html_wrap_inline1315 and r for fixed S'=1. The saturation effect here can be explained easily: the uniform spectrum tex2html_wrap_inline1333, where each species has the same ecological weight in Daisyworld, implies an upper limit for the variance, namely
equation297
The dashed horizontal line in Fig. 3 marks the value tex2html_wrap_inline1335.

  figure303
Figure 3: Root-mean-square deviation tex2html_wrap_inline1315 of species spectra as a function of mutation rate r.


next up previous
Next: Homeostatic response to increasing Up: Results Previous: Results

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