A Simple Model

First let's consider a simple model. In this model, we will consider one type of daisy that initially covers 1% of the Earth (at \(t = 0\)). The rate of change of the amount of daisies will depend several factors:

• \(F\) = fraction of land covered by daisies
• \(G\) = The natural growth rate of daisies
• \(D\) = The natural death rate of daisies

A More Complicated Model...

Here are some parameters we will be using

• \(P\) = power (\(W\))
• \(S\) = surface area (\(m^2\))
• \(\sigma\) = Stefan-Boltzmann constant = \(5.67\times 10^{-8} J m^{-2} s^{-1} K^{-4}\)
• \(T\) = temperature (we want to work in Celsius and will convert when necessary)
• \(L\) = solar luminosity factor
• \(X\) = solar flux constant = 917 \(W m^{-2}\)
• \(A\) = albedo
• \(D\) = death rate
• \(G\) = growth factor
• \(F\) = fraction of land
• \(H\) = heat absorption constant = 20 \(^{\circ}C\)

The Sun is an important factor in this model. The Sun has a solar flux of 917 \(W m^{-2}\). This is a baseline amount of power that the Sun emits to Daisy World.

Like any star, the Sun gets hotter and hotter as it ages. The solar luminosity of the sun starts at 0.6 at time \(t=0\), and increases linearly to 1.8 at time \(t=200\). This represents the proportion of the solar flux that the Sun actually emits

Daisy World would be in big trouble if it absorbed all of the energy received by the Sun. Albedo is a measure of how much energy is reflected back into space. We will use \(A_P\) to denote the albedo of Daisy World. For example, if \(A_P = 0.5\), this means that 50% of the incoming solar power is reflected.

Because our time scale is on the scale of millions of years, we will assume that the planet is in radiative equilibrium. This means that the amount of energy that Daisy World absorbs is equal to the amount it emits. The amount of energy absorbed by Daisy World (per unit area) is given by \[X\times L\times(1-A_{P})\] This also represents the amount of energy emitted. We can now use the Stefan-Boltzmann Law to estimate average global temperature, \(T_P\). We will assume Daisy World is an ideal blackbody with emissivity = 1. \[\frac{P}{S}=\sigma T^4\] Plugging in known values gives us \[X\times L\times(1-A_P)=\sigma T_P^4\] We can use this equation to solve for \(T_P\)

Now we can introduce the daisies! Daisies have the special ability to affect temperature because of their albedo. Uncovered Daisy World area has an albedo of \(A_U=0.5\). We will start by assuming that black daisies have an albedo of \(A_B=0.25\) and white daisies have an albedo of \(A_W=0.75\). The albedo of the planet can be calculated as follows: \[A_{P} = F_UA_{U} + F_BA_{B} + F_WA_{W} \] Daisy World starts off empty because daisies can only survive in temperatures between 5 and 40 degrees Celsius. This means that \(F_B=F_W=0\) and \(F_U=1\), so \(A_P = A_U\).

Here we make an important distinction. There is a difference between the average global temperature \(T_P\) and the local temperature \(T_{B}\) or \(T_W\) for a given color of daisies. Daisies can only survive when their local temperature is between 5 and 40 Celsius. At the start of the simulation, \(T_P=T_B=T_W\) because there are no daisies. Once daisies begin to grow however, these temperatures will differ. The local temperature for a species is given by the following equation: \[T_{B/W} = H \times (A_P-A_{B/W}) + T_P\] Here, \(H\) is a constant that simply controls how much the local temperature differs from the average global temperature.

The growth factor helps determine the growth rate of daisies. It is 0 when outside this temperature range. When inside the range, it follows the given quadratic function: \[G_{B/W} = 1-0.003265\times(22.5-T_{B/W})^2\] The overall rate of change of a given amount of daisy is given by the following equation: \[\frac{dF_{B/W}}{dt} = F_{B/W}\times(F_U\times G_{B/W} - D)+0.001\] The "+ 0.001" causes daisies to spontaneously sprout once the local temperature is within the acceptable range.

We've also added an interesting [PLAGUE] option...

During a plague, the death rate is bumped to 0.9. After the plague ends, it returns to normal

And that's about it! Explore how this system works using the graphs below

Legend
Black: Black Daisies
White: White Daisies
Brown: Uncovered Area
Orange: Empty Planet Temperature
Purple: Daisy World Temperature