Changing the parameters may lead to a small time delay before the figures adjust to the new values.

The fraction of daisies responds to the equations of population growth and the evolution of area fractions of white (${\mathit{a}}_{\mathit{w}}$) and black (${\mathit{a}}_{\mathit{b}}$) daisies is given by the following differential equations:

where ${\mathit{a}}_{\mathit{g}}$ is the area fraction of bare ground, $\mathit{\beta }\left(\mathit{T}\right)$ is the birth rate for a given temperature $\mathit{T}$ and $\mathit{\gamma }$ is the death rate. As a consequence, ${\mathit{a}}_{\mathit{g}}\equiv \mathit{p}-{\mathit{a}}_{\mathit{b}}-{\mathit{a}}_{\mathit{w}}$, with $\mathit{p}$ refering to the proportion of fertile ground in the system (we take $\mathit{p}=1$). $\mathit{t}$ is a unitless time as we only look for steady state solutions (see below). Here, $\mathit{\gamma }$ is kept fixed and has the same value for both black and white daisies ($\mathit{\gamma }=0.3$). ${\mathit{T}}_{\mathit{w}}$ and ${\mathit{T}}_{\mathit{b}}$ are local temperatures felt by each daisies species. $\mathit{\beta }\left(\mathit{T}\right)$ is defined as follows:

where =295.5 K (22.5°C) is the optimal temperature. The parabolic width $\mathit{k}$ is chosen so that the daisies grow for temperature between 278 K and 313 K (5 °C and 40 °C), i.e. $\mathit{k}={17.5}^{-2}$ K${}^{-2}$.

Fixed albedos are prescribed for the white daisies (${\mathit{\alpha }}_{\mathit{w}}$), for the black daisies (${\mathit{\alpha }}_{\mathit{b}}$) and for the bare ground (${\mathit{\alpha }}_{\mathit{g}}$). The planetary albedo ${\mathit{\alpha }}_{\mathit{p}}$ is therefore given by:

where ${\mathit{\alpha }}_{\mathit{g}}=\frac{1}{2}$ by convention and .

The local temperatures ${\mathit{T}}_{\mathit{w}}$ and ${\mathit{T}}_{\mathit{b}}$ and the bare ground temperature ${\mathit{T}}_{\mathit{g}}$ are obtained through a simple heat balance including a heat transfert between the different regions. Local temperatures are obtained as:

where $\mathit{T}$ is the planetary temperature and $\mathit{q}=2.06\times {10}^9$ K${}^4$ is a heat transfert coefficient. The planetary temperature is derived from the global heat balance of Daisyworld:

where ${\mathit{S}}_{\mathit{o}}\mathit{L}$ is the average solar energy flux incident on the Daisyworld. ${\mathit{S}}_{\mathit{o}}=917$ Wm${}^{-2}$. $\mathit{L}$ is an adjustable parameter representing the luminosity of the star and $\mathit{\sigma }$ is the Stefan-Boltzmann constant ($\mathit{\sigma }=5.67\times {10}^{-8}$Wm${}^{-2}$K${}^{-4}$).

The evolution of the model variables is shown for increasing and decreasing values of luminosity. The model equations are integrated using the finite difference method. The methodology applied here is the one introduced by Watson and Lovelock (1983). For a given value of $\mathit{L}$, the model equations are integrated until a steady state is reached. The value of $\mathit{L}$ is then incremented and the initial conditions for ${\mathit{a}}_{\mathit{w}}$ and ${\mathit{a}}_{\mathit{b}}$ of the new simulation are set to the steady state value of the previous simulation, or 0.01 if these equal 0. The same procedure is applied to decreasing values of $\mathit{L}$. The system variables display an hysteresis, i.e. their behaviour for increasing luminosity is not the same as the one for decreasing luminosity.

In order to answer the questions in the following quiz, you can modify several parameters of the daisyworld model and identify their effect on the state variables. After answering each question, please check it using the box on the left before going to the next question.

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