1. Daisyworld

2. What is a System?
Definition: A system is a group of different components that interact with each other
Example: The climate system includes the atmosphere, oceans, polar caps, clouds, vegetation…and lots of other things

3. How do we study systems? Identify the components
Determine the nature of the
interactions between components

4. Systems Notation
= system component
= positive coupling
= negative coupling

5. Positive Coupling Atmospheric CO2 Greenhouse effect
An increase in atmospheric CO2 causes
a corresponding increase in the greenhouse
effect, and thus in Earth’s surface temperature
Conversely, a decrease in atmospheric CO2
causes a decrease in the greenhouse effect

6. Negative Coupling An increase in Earth’s albedo causes a
(reflectivity)
Earth’s
surface
temperature
An increase in Earth’s albedo causes a
corresponding decrease in the Earth’s surface
temperature by reflecting more sunlight back to
space
Or, a decrease in albedo causes an increase in
surface temperature

7. Conditions under which the system will remain indefinitely
Equilibrium State:
Conditions under which the system will remain indefinitely
--If left unperturbed

8. An Unstable Equilibrium State

9. An Unstable Equilibrium State
Perturbation

10. When pushed by a perturbation, an unstable equilibrium state shifts to a new, stable state.

11. A Stable Equilibrium State

12. A Stable Equilibrium State
Perturbation

13. When pushed by a perturbation, a stable equilibrium state, returns to (or near) the original state.

14. Daisy World

15. Gaia hypothesis
Earth as a single living superorganism (James Lovelock)
Gaia - a new look at life on Earth, Oxford University Press, 1979.

16. Lovelock’s Questions
James Lovelock: NASA atmospheric chemist analyzing distant Martian atmosphere.
Why has temp of earth’s surface remained in narrow range for last 3.6 billion years when heat of sun has increased by 25%?

17. Lovelock’s Questions Why has oxygen remained near 21%?
Martian atmosphere in chemical equilibrium, whereas Earth’s atmosphere in unnatural low-entropy state.

18. Our Earth is a Unique Planet in the Solar System
Runaway greenhouse ::
No water cycle to remove carbon from atmosphere
Earth
Harbor of Life
Loss of carbon ::
No lithosphere motion on Mars to release carbon
Earth is unique in our solar system in its capacity to sustain highly diversified life
from Guy Brasseur (NCAR)

19. Lovelock´s answers
Earth can’t be understood without considering role of life
Abiotic factors
(physical, geological and chemical)
determine biological possibilities
Biotic factors feed back to control abiotic factors
Increased Planetary
Temperature
Increased Planetary Albedo
Sparser Vegetation, More Desertification
Reduced Temperature

20. Gaia Hypothesis
Organisms have a significant influence on their environment
Species of organisms that affect environment in a way to optimize their fitness leave more of the same – compare with natural selection.
Life and environment evolve as a single system – not only the species evolve, but the environment that favors the dominant species is sustained

21. Daisy world
White daisies
Black daisies
Available fertile land

22. About Daisyworld…
Daisyworld: a mythical planet with dark soil, white daisies, and a sun shining on it.
The dark soil have low albedo – they absorb solar energy, warming the planet.
The white daisies have high albedo – they reflect solar energy, cooling the planet.
There’s a simple flash animation of daisyworld concept out there too.

23. The number of daisies affects temperature
The number of daisies influences temperature of Daisyworld.
More white daisies means a cooler planet.

24. Temperature affects the number of daisies
At 25° C many daisies cover the planet.
Daisies can’t survive below 5° C or above 40° C.

25. White Daisy Response to Increasing Solar Luminosity
Relative solar luminosity

26. Daisies can live between a min.T & a max. T
daisy
coverage T
daisy
coverage T
Daisy coverage
min.
max.
optimum

27. Effects of daisy coverage on T
Intersection of 2 curves means the 2 effects are balanced => equilibrium points P1 & P2. T
daisy
coverage T
daisy
coverage P1
Effects of daisy coverage on T P2 T
Daisy coverage
Effects of T on
daisy coverage
ENSC 425/625 Chapter 3UNBC

28. Feedback loops P1 P2 T ENSC 425/625 Chapter 3UNBC
Daisy coverage
Effects of T on
daisy coverage P1
Effects of daisy coverage on T P2
ENSC 425/625 Chapter 3UNBC

29. Perturb daisy coverage at P1 => sys. returns to P1 (stable equil. pt.)
A large perturb.
=> daisies all die
from extreme T

30. Large incr. in daisy cover => very low T =>
decr. in daisy cov. => very high T => lifeless. P1 T
Daisy coverage P2

31. From P2, incr. daisy cov. => decr. T =>
further incr. in daisy cov. => converge to P1 P1 T
Daisy coverage P2 T
daisy
coverage
unstable
equilib. pt.
ENSC 425/625 Chapter 3UNBC

33. Gradual incr. in solar luminosity
For any particular value
of daisy cov., T incr. T
Daisy coverage P1 P2
The effect of T on
Daisy unchanged
P1
P2
Teq
To
Tf
ENSC 425/625 Chapter 3UNBC

34. The key variables
Later, we’ll see that we also need T, the “effective temperature”, but that isn’t obvious until we get a bit further on in the modelling.

35. An equation for the black daisies
( 1 – αb – αw)
β(Tb)
- γαb
dαb/dt =
= αb (αg β(Tb) – γ)
b(T) is a function that is zero at 5C, rises to a maximum of
one at 22.5C and then falls to zero again at 40C
A simple and convenient choice is

36. An equation for the white daisies
We use a similar equation for the white daisies:
dαw/dt = αw (αg β(Tw) – γ)
Another reason for using a different growth function and death rate later on is to check that the result doesn’t depend on using the same for both. But we have only two equations for four unknowns, so we have to think about what else is going on that we haven’t included so far.
We don’t have to use the same b(T) and g but it
keeps things simple. We can use different ones
later if we want to.

37. Heat Flow Because different regions of Daisyworld are at different
temperatures, there will be heat flow. We include this in
the model using the equations
Note that if q=0 the whole planet is at the same temperature,
i.e., the heat flow is very rapid indeed. As q increases, so do
the temperature differences.
Don’t worry about the 4th powers; they’re only there to make
the calculations easier and don’t make any real difference.
T is now properly defined. Note that if q=0, then heat flow is so rapid that the whole planet is at the same temperature.

38. The Daisyworld Equations

39. No daisies
On a dead planet, as the solar luminosity steadily increases, so does the temperature.

40. Black daisies only
This is not a full derivation because we haven’t got L into it yet, but you can already see that the nonlinearity of the equations means the value of the local temperature may not be uniquely determined by L.

41. Gaia Hypothesis Proposed by James Lovelock Definition of Gaia:
Developed in 1960s
First published in 1975
Definition of Gaia:
a complex entity involving the Earth's biosphere, atmosphere, oceans, and soil; the totality constituting a feedback or cybernetic system which seeks an optimal physical and chemical environment for life on this planet. (Lovelock)

42. Daisyworld Model
Daisyworld is a hypothetical planet orbiting a sun that increases in intensity
The planet is inhabited by 2 species
Black daisies
White daisies
Original Daisyworld model consisted of a system of differential equations
This project uses these equations to build a 2D cellular automata representation of Daisyworld

43. Daisyworld Model (2)
Temperature of Daisyworld is based on the assumption that the planet is in radiative equilibrium (i.e. energy emitted = energy absorbed)
Albedo of the planet is computed based on the albedos of each type of daisy and the area covered by them
-SB= 5.669e-8W/degK^4
-S= 917W/m^2

44. Daisyworld Model (3)
Area of daisies is modified according to the following equations

45. Daisyworld Model (4) 2D CA rules: If da/dt > 0 If da/dt <= 0
If neighbors with no daisies < spreading threshold
Bare neighbors grow daisy with probability: p = c*da/dt
Else if neighbors with no daisies >= spreading threshold
Start new patch of daisies
If da/dt <= 0
Daisies die with probability p = -da/dt

46. Example of Daisy Crowding
Spreading-threshold = 6
=> Start new patch of daisies
=> Don’t start new patch

47. Parameter Settings Two different temperature models Death-rate: 0.3
Automatic linear increase of solar luminosity
Manual adjustment of solar luminosity
Death-rate: 0.3
Albedo of white daisies: 0.75
Albedo of black daisies: 0.25
Albedo of bare land: 0.50
Spreading threshold: 8
Optimal daisy growth temperature: 22.5 C

48. Spatial Daisyworld vs. Mathematical Daisyworld
Area Occupied by Daisies
(Mathematical Model)
(Spatial Model)

49. Spatial Daisyworld vs. Mathematical Daisyworld (2)
Temperature of Daisyworld
(Mathematical Model)
(Spatial Model)

50. Effects of Solar Luminosity on Daisyworld
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4

51. The Effects of Death Rate on Daisyworld

52. Daisyworld with Four Species of Daisies
Area covered by daisies
Temperature of Daisyworld

53. Effects of Solar Luminosity on Daisyworld with Four Species
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4