Theory

The dimensionless model

Watson & Lovelock’s 1983 Daisy World[1] was deterministic: a planet held in thermal equilibrium at every instant. We keep the daisies and break that constraint. The star flickers, the temperature lags, and the model becomes a four-degree-of-freedom stochastic differential equation for the black- and white-daisy fractions \(f_B, f_W\) (the biotic agent) coupled to the planetary temperature \(T\) and stellar luminosity \(L\) (the environment). In the dimensionless form of Appendix B of Sowinski, Ghoshal & Frank (2025) - temperature in units of the optimal temperature \(T_{opt}\), luminosity in units of \(L_{opt}\) - we write

\[\begin{split}\frac{df_B}{dt} &= w(T_B-1)\,(f - f_B - f_W)\,f_B - \frac{\gamma_D}{\gamma_G} f_B, \\ \frac{df_W}{dt} &= w(T_W-1)\,(f - f_B - f_W)\,f_W - \frac{\gamma_D}{\gamma_G} f_W, \\ \frac{dT}{dt} &= \frac{1}{\gamma_G \tau_E}\!\left[\left(1 - \frac{\delta A_B f_B + \delta A_W f_W}{1-A_G}\right)L - T^4\right], \\ \frac{dL}{dt} &= \frac{1}{\gamma_G \tau_S}(1 + \lambda - L) + \sqrt{\frac{2}{\gamma_G \tau_S}}\,\delta\,(1+\lambda)\,\eta,\end{split}\]

with the smooth growth window

\[w(x) = e^{-\alpha x^4}, \qquad \alpha = 8\left(\frac{T_{opt}}{\Delta T}\right)^4,\]

and the scaled daisy temperatures

\[T_\alpha^4 = T^4 + Q\,\delta A_B f_B + Q\,\delta A_W f_W - Q\,\delta A_\alpha, \qquad \delta A_\alpha = A_\alpha - A_G.\]

The luminosity is the only noisy dof, an Ornstein–Uhlenbeck process[2] with mean \(1+\lambda\), so the driving Wiener process is scalar. That is what lets ExoDaisyWorld use the strong-order-1 stochastic Runge–Kutta scheme of A. Roberts (2012)[3]; the step is a line-for-line port of updateExoDaisyWorld.m.

Rein control

Because \(A_B < A_G < A_W\), black daisies warm their surroundings and white daisies cool them. As the star brightens the mix shifts from black to white, adjusting the planetary albedo \(A = A_G + \sum_\alpha (A_\alpha - A_G) f_\alpha\) so that the surface temperature stays near \(T_{opt}\) across a wide band of luminosities. This is Gaian rein control.

The information architecture

We partition the system into agent \(a = (f_B, f_W)\) and environment \(e = (T, L)\), in the spirit of Kolchinsky & Wolpert’s semantic-information program[4], and read the coevolution off information measures estimated from the joint distribution \(p_{AE}\) (Section 4):

• Viability \(V = \mathbb{E}^A[(f_B+f_W)/f]\) (Eq. 9), the expected occupied fraction of the habitable area.

• Mutual information \(I(A{:}E)\), the agent–environment correlation.

• Correlation change \(\Delta I_{\varnothing\to A} = I(e_1{:}e_2) - I_0(e_1{:}e_2)\) (Eq. 12), how much the biome strengthens or weakens the intrinsic temperature–luminosity correlation.

• Cooperation \(C(a_1{:}a_2\|E)\) (Eq. 14), the interaction information among the two species and the environment; negative values signal a synergy mediated by the environment.

We estimate every entropy by histogramming, with the square-root binning rule \(N_{bins} = 1 + \lceil\sqrt{n}\rceil\) of Appendix B.3.

This page compresses Exo-Daisy World: Revisiting Gaia Theory through an Informational Architecture Perspective (Sowinski, Ghoshal & Frank, Planet. Sci. J. 6, 176, 2025)[5], where the derivations live.

References

[1]

Andrew J Watson and James E Lovelock. Biological homeostasis of the global environment: the parable of Daisyworld. Tellus B: Chemical and Physical Meteorology, 35(4):284–289, 1983.

[2]

George E Uhlenbeck and Leonard S Ornstein. On the theory of the Brownian motion. Physical Review, 36(5):823–841, 1930.

[3]

Anthony J Roberts. Modify the improved Euler scheme to integrate stochastic differential equations. arXiv preprint arXiv:1210.0933, 2012.

[4]

Artemy Kolchinsky and David H Wolpert. Semantic information, autonomous agency and non-equilibrium statistical physics. Interface Focus, 8(6):20180041, 2018.

[5]

Damian R Sowinski, Gourab Ghoshal, and Adam Frank. Exo-daisy world: revisiting Gaia theory through an informational architecture perspective. The Planetary Science Journal, 6(7):176, 2025.