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Introducing spatial dependence, competition, and mutation into Daisyworld

  In general, the stable coexistence of many different species in Daisyworld can be brought about either by temporal fluctuations or by extending the spatial dimensionality. In this paper the second approach is used and our planet will be represented by a 2D plane with coordinates x and y [16].

The ``climate'' here coincides with the temperature field T(x,y,t), which is governed by an elementary energy balance equation (see, e.g., Henderson-Sellers and McGuffie 1990):
 equation60
where tex2html_wrap_inline1121 denotes the diffusion constant and A(x,y,t) represents the spatiotemporal distribution of albedo. The latter reflects the prevailing vegetation pattern. The homogeneous solutions of Eq. 6 are equivalent to the solutions of Eq. 1 of the original LWM.

We consider an extended biosphere consisting of infinitely many different species, which may be conveniently classified by their specific albedos tex2html_wrap_inline1125. So the variable A serves a twofold purpose, namely (i) to label the ``daisies'' stored in the genetic pool, and (ii) to express their radiative properties. As a consequence, the vegetation dynamics within our model can be directly represented by the albedo dynamics.

To achieve this we have to translate the vegetation growth rules, which can be set up in the spirit of the LWM, into albedo modification rules. Their dependence on T and Eq. 6 then determine the coevolution of albedo and temperature field in the plane. As the analytic solution of this intricate non-linear dynamics is unfeasible, we will have to resort to numerical computation schemes based on discretization of the system. It is therefore reasonable to employ the CA approach from the outset [17, 18]. One major advantage of this approach is the fact that consistent albedo modification rules can be written down immediately.

The CA is constructed as follows: the plane is replaced by a quadratic lattice tex2html_wrap_inline1131, where tex2html_wrap_inline1133 and the basic spatial units tex2html_wrap_inline1135, tex2html_wrap_inline1137 can be chosen arbitrarily. Time proceeds in discrete steps tex2html_wrap_inline1139, where tex2html_wrap_inline1141 and tex2html_wrap_inline1143 is again an optional unit. Thus any systems variable F becomes a function tex2html_wrap_inline1147.

The occurrence of vegetation in a particular cell tex2html_wrap_inline1131 at time tex2html_wrap_inline1151 can be indicated by a binary coverage function tex2html_wrap_inline1153. The albedo dynamics is then determined by the following rules:

  1. tex2html_wrap_inline1155 , i.e. the cell is covered by vegetation.
     eqnarray77
  2. tex2html_wrap_inline1161, i.e. the cell is uncovered.

    Then choose at random a next-neighbour cell tex2html_wrap_inline1163 of tex2html_wrap_inline1131 and make the following distinction:

    1. tex2html_wrap_inline1167
       eqnarray100
    2. tex2html_wrap_inline1169
       eqnarray111
Thus tex2html_wrap_inline1105 and tex2html_wrap_inline1177 denote again mortality and growth rate per each time step tex2html_wrap_inline1151, respectively. Regarding the functional dependence of the growth probability on the spatial distribution of temperature, two obvious choices can be made:
(A) tex2html_wrap_inline1177 depends only on the temperature of the uncovered cell at point tex2html_wrap_inline1131, i.e.,
 equation133
(B) tex2html_wrap_inline1177 is determined by the temperature of the next-neighbour cell tex2html_wrap_inline1163, i.e.,
 equation138

So in the first version, (A), the growth rate depends on the state of the area that will be covered by the vegetation in the next time step, while in the second version, (B), the temperature of the vegetation patch which initiates the growth determines the growth probability. It turns out that this innocent-looking local distinction induces rather different behaviour even at the systems scale (see below).

The function f in (9) offers the opportunity to incorporate also more sophisticated biological effects: by choosing, for instance,
 equation144
where R is a random number distribution with the properties
equation147
it is possible to take mutations of the albedo into account. Here tex2html_wrap_inline1193 can be interpreted as the mutation rate.

The CA is completed by solving the temperature evolution equation (6) by an explicit finite-difference scheme on the square lattice with the same resolution as employed for the growth dynamics. The discretization step must be chosen in the way that the stability of the explicit scheme is guaranteed and no bifurcation (as, e.g., in the logistic map) takes place. A list of the chosen parameter settings is shown in Table 1. These values are used unless stated otherwise.

 
Parameter Value
tex2html_wrap_inline1085 tex2html_wrap_inline1197
tex2html_wrap_inline1121 500
C 2500
tex2html_wrap_inline1079 0.5
tex2html_wrap_inline1105 0.02
tex2html_wrap_inline1143 1
tex2html_wrap_inline1135 1
tex2html_wrap_inline1137 1
Table 1: List of the values assigned to the adjustable parameters in all computer simulations referred to in this paper.

 

Comparison with the original model Our extended geophysiological model for biosphere-geosphere interactions contains all the dynamics of the zero-dimensional LWM as a special subprocess. To demonstrate this we have to consider the true evolution of the vegetation density N. Within the discrete CA formalism, N is defined as follows:
equation163
where tex2html_wrap_inline1217 denotes an averaging over a statistical ensemble. Thus tex2html_wrap_inline1219 represents the probability of encountering vegetation of any type in cell tex2html_wrap_inline1131 at time tex2html_wrap_inline1151. The dynamics of N is implicitly determined by the rules summarized in Eqs. 7 to 9.

To simplify the calculations we restrict the vegetation to one species in two spatial dimensions without considering mutations (f(A)=A). We have
 eqnarray167
The next step is to take the continuous limit by making a Taylor expansion in tex2html_wrap_inline1135, tex2html_wrap_inline1137, and tex2html_wrap_inline1143 up to second order in space and first order in time, which results in the following expressions:
eqnarray172
Substituting the approximations in Eq. 15 and performing some transformations, the following PDE can be obtained for growth version A, if we neglect the terms of higher order:
equation189
where tex2html_wrap_inline1235, tex2html_wrap_inline1237, tex2html_wrap_inline1239, and tex2html_wrap_inline1241. In order to have a non-vanishing diffusion for low values of tex2html_wrap_inline1135, tex2html_wrap_inline1137, tex2html_wrap_inline1177 must be enlarged. This can be compensated by a lower value of tex2html_wrap_inline1143, which ensures also the stability of the discretization scheme.

For growth version B the density N is determined by
 eqnarray201
The resulting PDE for N is
equation206
Let us emphasize here that for homogeneous solutions, i.e., tex2html_wrap_inline1255, the equations (15) and (18), respectively, are identical to the zero-dimensional LWM for one species (see Eq. 3).

If we include mutations of the albedo according to (12) for the homogeneous solution N(A;t) we get:
 equation218
Taylor expansion in A yields the following expression for tex2html_wrap_inline1261:
 equation225
Inserting (21) into (20) and taking the continuous limit in t, the following equation, valid for small mutation rates r, is obtained:
 equation236
This analysis can be expanded to all aspects of our full CA to demonstrate that the discretization preserves the right physics and biology.


next up previous
Next: Results Up: Self-stabilization of the biosphere Previous: The original Daisyworld model

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