daisy-math

The math behind the daisyworld model

by Ralph H. Abraham, abraham@vismath.org

This is simple account of the mathematical analysis behind the daisyworld model, as originally published in Andrew J. Watson and James E. Lovelock, "Biological homeostasis of the global environment: the parable of Daisyworld", Tellus (1983), 35B, 284-289. We will refer to this article as "WL". The science behind the model is discussed in WL and elsewhere, see the bibliography.

As indicated in the title of WL, the heart of the model is a point attractor of a dynamical scheme. In this case, the main control parameter is

L, the solar luminosity.

A number of constants appear in the model, such as,

A_G, the albedo of bare ground,
A_B, the albedo of black daisies,
A_W, the albedo of white daisies.

These are fixed at 0.5, 0.25, and 0.75, respectively.

The state variables are:

alpha_G, relative area of bare fertile ground,
alpha_B, relative area covered by black daisies,
alpha_W, relative area covered by white daisies,
T_G, average temperature over the bare ground,
T_B, average temperature over the black daisies,
T_W, average temperature over the white daisies.

The sum of the three areas is assumed to be P, a constant, usually taken to be one. The temperatures are assumed to reach equilibrium rapidly, on the slow scale of time in which the daisy areas change. There values are given as functions of L and the three albedos, in the fourth order equations (4) and (6), again in a linear approximation in equation (7). Here a parameter q' is introduced, which indicates the effect of mixing of temperatures over different areas due to conduction of heat. In the simulations, q' = 20. The average albedo, A, is given by equation (5) of WL,

A = alpha_GA_G + alpha_BA_B + alpha_WA_W

Thus we have a two-dimensional dynamical system, given in equation (1) of WL, for the rates of change of alpha_B and alpha_W,

alpha_W' = alpha_W(x beta - gamma)
alpha_B' = alpha_B(x beta - gamma)

where x = alpha_G, gamma is the death rate of all daisies, taken as 0.3 in the simulations, and beta is a quadratic function of the local temperature, equation (3) of WL,

beta(T) = max {0, 1 - 0.003265 (22.5 - T)^2}

Now we look for the critical points. Assuming that both daisy areas are positive (a zero value means the game is over) we find the conditions for a critical point, as given in equations (14) of WL,

T*_B = 22.5 + (q'/2)+(A_W - A_B)
T*_W = 22.5 - (q'/2)+(A_W - A_B)

which are constants independent of L, a surprising and hopeful result. From these equilibrium conditions, we find beta*, and from (1) we have (from the vanishing of the right hand sides) x beta* = gamma, so we may calculate the sum of the two daisy areas.
But to find them individually, it is necessary to proceed with numerical integration. The results of these simulations occupy the bulk of tje WL paper.
Revised 06 May 2000 by Ralph Abraham.