next up previous
Next: Introducing herbivores into Daisyworld Up: Modelling the geosphere-biosphere feedback Previous: The Lovelock-Watson model of

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 [11, 5].

The ``climate'' here coincides with the temperature field T(x,y,t), which is governed by an elementary energy balance equation (see, e.g., [12]):
 equation47
where tex2html_wrap_inline515 denotes the diffusion constant and A(x,y,t) represents the spatiotemporal distribution of albedo. The latter reflects the prevailing vegetation pattern.

We consider an extended biosphere consisting of infinitely many different species, which may be conveniently classified by their specific albedos tex2html_wrap_inline519. 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. 5 then determine the coevolution of albedo and temperature field, respectively, in the plane. As the analytic solution of this intricate nonlinear 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 [13, 14]. 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_inline525, where tex2html_wrap_inline527, and the basic spatial units tex2html_wrap_inline529, tex2html_wrap_inline531 can be chosen arbitrarily. Time proceeds in discrete steps tex2html_wrap_inline533, where tex2html_wrap_inline535 and tex2html_wrap_inline537 is again an optional unit. Thus any systems variable F becomes a function tex2html_wrap_inline541.

The occurrence of vegetation in a particular cell tex2html_wrap_inline525 at time tex2html_wrap_inline545 can be indicated by a binary coverage map tex2html_wrap_inline547. The albedo dynamics is then determined by the following rules:

  1. tex2html_wrap_inline549, i.e. the cell is covered by vegetation.
     eqnarray62
  2. tex2html_wrap_inline555, i.e. the cell is uncovered.

    Then choose at random a next-neighbour cell tex2html_wrap_inline557 of tex2html_wrap_inline525 and make the following distinction:

    1. tex2html_wrap_inline561
       eqnarray85
    2. tex2html_wrap_inline563
       eqnarray96
Thus tex2html_wrap_inline503 and tex2html_wrap_inline571 denote again mortality and growth rate, respectively. The growth probability tex2html_wrap_inline571 depends only on the temperature of the uncovered cell at site tex2html_wrap_inline525, i.e., tex2html_wrap_inline577. The function f in (8) offers the opportunity to incorporate also more sophisticated biological effects: by choosing, for instance, f(A)=A+R, where R is a random number distribution with the properties tex2html_wrap_inline585, tex2html_wrap_inline587, it is possible to take mutations of the albedo into account. Here tex2html_wrap_inline589 can be interpreted as the mutation rate.


next up previous
Next: Introducing herbivores into Daisyworld Up: Modelling the geosphere-biosphere feedback Previous: The Lovelock-Watson model of

Werner von Bloh (Data & Computation)
Thu Jul 13 14:36:30 MEST 2000