. 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
iterations, the global average temperature approaches the optimum
growth temperature 
, 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.
Figure 1: Daisyworld in statistical equilibrium for S'=1: snapshot of typical
albedo distribution (right) and associated spatial temperature fluctuations around (left, in false colour representation).
The species spectrum B(A) is defined as follows:
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 
of our model biosphere and its variance
are then given by
where 
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.,
Figure 2: Species spectra after 
iterations for optimal constant insolation () 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 
-function only in the limit
of infinite lattice.
The mean albedo consistently turns out to equal the optimum albedo
, which is determined by
So we have
if the natural choice for 
is made.
From Fig. 2 we can also see that 
is a monotonically increasing
function of r. Without mutation (r=0) the spectrum actually collapses
into a -function at 
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 and r for
fixed S'=1. The saturation effect here can be explained easily: the uniform
spectrum 
, where each species has the same ecological weight
in Daisyworld, implies an upper limit for the variance, namely
The dashed horizontal line in Fig. 3 marks the value 
.
Figure 3: Root-mean-square deviation of species spectra as a function of mutation rate r.