Gaucherel/Online Resource 1 p. 1

Online Resource 1: Daisyworld Modeling

Appendix to the article:

Ecosystem Complexity Through the Lens of Logical Depth:

Capturing Ecosystem Individuality

by

Cédric Gaucherel

French Institute of Pondicherry (IFP-CNRS)

Pondicherry, India

e-mail:

published (2014) in:

Biological Theory: Integrating Development, Evolution, and Cognition

Appendix: Daisyworld Modeling

The Original Daisyworld Modified

Daisyworld is a flat cloudless planet with negligible atmospheric greenhouse effect. Its entire biota consists of two species of daisies (or individuals) that differ only in color, i.e., their characteristic traits. The original equations are retained where possible (Watson and Lovelock 1983). Those governing the growth of biota are:

,,(1)

where t is the dimensionless time, aw and ab are the normalized areas covered by light (called “white”) and dark (“black”) daisies, respectively, ag the normalized area of fertile ground not covered by daisies. The entirely planet is potentially fertile, i.e., there is no habitat restriction (Von Bloh et al. 1997). pkill is the death rate per dimensionless unit time, here equal to 0.3 for numerical calculations.  is the growth rate of (either white or black) daisies per unit time, allowing daisies to spread over the planet. In the original model, growth rates follow a parabolic function of the temperature Ti = w,b of the concerned daisy i. We modified the original model into a similar Gaussian function for a more realistic shape and an easier control of its dependence to the environment (the local temperature):

(2)

With T0 = 22.5 C° playing the role of the central trait value of the previous model,  = 17.5 C°. Notice thatw,b can be neglected for temperatures below 4.85 C° (278 K) and above 39.85 C° (313 K) as in the original Daisyworld.

In steady-state conditions, and the temperature of daisies ranges between 7.75 C° and 36.95 C°, since the growth rates must be at least equal to the death rate. It is finally possible to define the mean albedo and the effective planetary temperature linked to the insolation or solar luminosity S:

,(3)

Insolation is playing the role of the environmental pressure, which drives the evolution of the whole planet (see Eq. 5). The difference between NS and NA evolution rules (and therefore systems) is related to the way the central temperature T0 (equation 2) is defined: it is equal to a global Te temperature for NS, and a local mean neighbor temperaturefor NA evolution. In the case of LA evolution rule, each daisy temperature is directly computed on the basis of a local albedo Aw,b defined as the mean neighbor albedo, instead of a growth rate based on random mutation of the parent albedo of the daisy concerned. We will vary the insolation S in a way that helps us to understand the response of the three evolution rules and their corresponding ecosystems.

The Spatially Mutating Daisyworld

Early versions of the model neglected local interactions between daisies as well as mutations. This allows for focusing on the effect of local interactions on the ability for adaptive system behavior. We have here followed the lead of others to construct a spatial Daisyworld, using the same numerical values when possible (Lenton and van Oijen 2002). We consider a square grid with eight neighbors to each cell. The probability of seeding an empty cell (poff = 0.001) is used to enable populations to establish, but once they are established it has little impact. For an empty cell with nb black neighbors each with growth rate b and nw white neighbors each with growth rate w, the following reproduction (called colonization principle, in the initial model) principles apply. The probability that black will colonize when only they are present is:

(4)

The probability that white will colonize when only they are present is:

The probability that the cell remains empty is:

The probability that black will colonize when both daisies are present is:

The probability that white will colonize when both daisies are present is:

The spatial Daisyworld model has less smooth dynamics than the original zero-dimensional model, in that it can exhibit bifurcations and limit cycling as shown in details by Lenton (Lenton and van Oijen 2002). Note that the system is stabilized by at least three effects: if one daisy type reaches an uninhabitable temperature, the other may still be able to spread because of their different local temperature; in the mid-range of forcing, a combination of both daisy types generates a global temperature closer to optimal; and the presence of one daisy type reduces the area available for the other to expand. We concentrated here on the spatial effects. Life may develop the ability to drive an environmental variable into an uninhabitable or barely habitable state, if the environmental effect is strong and local interactions that encourage growth outweigh negative feedback on growth from the environment. In the initial growth phase, clusters form and cells at the edge of clusters are better able to spread than those within clusters.

This first version of the model does not include natural selection, in that all the daisies are identical and are always equally fit in terms of growth rate. Following Lenton, we used a Daisyworld version with random mutation of albedo to the model, generalizing the previous reproduction equations to deal with any albedo type (Lenton and van Oijen 2002). Mutations are assumed to always occur in the process of reproduction of an empty cell. Albedo can mutate within a parameterized width of the parent albedo a (here, , by 10-5 steps). Limits to mutations are set between Ab (0.15) and Aw (0.65). As mentioned in the literature, with mutation and natural selection within the model system, its behavior could be said to be “adaptive” in that it evolves to counteract forcing. Increasing amax drives the average planetary temperature more rapidly towards the optimum for daisy growth. We developed on this basis a Daisyworld model with two competing evolution rules.

Simulated Daisyworld Dynamics

NS evolution, or the original Lenton model, corresponded to the population dynamics already described in the literature (Fig. 1). The dark types (close to albedo Ab) flourished initially, when alone, and still had a significant population later, before insolation (S) is such that there is no need for regulation. Then, paler mutants of these dark albedos (still less than Ag) arose and dominated at low insolation. Progressively paler albedo mutants (up to Aw) were favored in sequence when insolation increased background temperatures above the optimum for daisy growth. The combination of different albedo types stabilized the planetary temperature evolution at ~39 °C between ~1 and ~1.3 insolation, a finding confirmed by the cited literature. Differing from Lenton’s model, the spatial albedo mutation did not extend the range of temperature regulation beyond that in the original model, because we chose that no paler types than the original “white” ones, nor darker types than the original “black” ones, could arise.

We observed the emergence of two distinct behaviors of dark (< Ag) and pale (> Ag) daisy populations, both arising quickly (after S ~0.6). This was caused by the spatial cohesion favoring the presence of pale daisies even when dark daisies dominated (in a non-spatial case), because they were geographically separated. Hence, both groups of albedo competed up to S ~1, with the main consequence that the dark daisies did not succeed in warming the global temperature to stabilize it (the pale daisies cool it simultaneously). Temperature stabilization was only possible after S ~1, when pale daisies dominated and when the high insolation imposed a cooling on the global temperature. This has also been observed in the literature, although it is not easy to detect this effect with multiple mutant range curves.