Daisyworld

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

How do we study systems?

• Identify the components

• Determine the nature of the interactions between components

Systems Notation

= system component

= positive coupling

= negative coupling

Positive Coupling

AtmosphericCO2

Greenhouseeffect

• 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

Negative Coupling

Earth’s albedo(reflectivity)

Earth’ssurface

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

Equilibrium State:

Conditions under which the system will remain indefinitely

--If left unperturbed

An Unstable Equilibrium State

An Unstable Equilibrium State

Perturbation

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

A Stable Equilibrium State

A Stable Equilibrium State

Perturbation

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

Daisy World

Gaia hypothesis

Earth as a single living superorganism (James Lovelock)

Gaia - a new look at life on Earth, Oxford University Press, 1979.

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%?

Lovelock’s Questions

Why has oxygen remained near 21%?Martian atmosphere in chemical equilibrium, whereas

Earth’s atmosphere in unnatural low-entropy state.

Lovelock’s Questions

Runaway greenhouse ::

No water cycle to remove carbon from atmosphereEarth is unique in our solar system in its capacity to sustain highly diversified life

Our Earth is a Unique Planet in the Solar System

Loss of carbon ::

No lithosphere motion on Mars to release carbon

Earth

Harbor of Life

from Guy Brasseur (NCAR)

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

Sparser Vegetation, More Desertification

Increased Planetary

Albedo

Reduced Temperature

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

White daisies

Black daisies

Available fertile land

Daisy world

About Daisyworld…About Daisyworld…

Daisyworld: a mythical planet Daisyworld: a mythical planet with dark soil, white daisies, with dark soil, white daisies, and a sun shining on it.and a sun shining on it. The dark soil have low albedo – they The dark soil have low albedo – they

absorb solar energy, warming the absorb solar energy, warming the planet.planet.

The white daisies have high albedo – The white daisies have high albedo – they reflect solar energy, cooling the they reflect solar energy, cooling the planet.planet.

The number of daisies affects temperature

The number of daisies The number of daisies influences temperature influences temperature of Daisyworld. of Daisyworld.

More white daisies means More white daisies means a cooler planet.a cooler planet.

Temperature affects the number of daisiesTemperature affects the number of daisies

At 25° C many daisies At 25° C many daisies cover the planet.cover the planet.

Daisies canDaisies can’’t survive t survive below 5° C or above below 5° C or above 40° C. 40° C.

White Daisy Response to Increasing Solar Luminosity

Relative solar luminosity

26

Daisies can live between a min.T & a max. T

T

Dai

sy c

over

age

min. max.

optimum

T daisycoverage

T daisycoverage

ENSC 425/625 Chapter 3UNBC

27

Intersection of 2 curves means the 2 effects are balanced => equilibrium points P1 & P2.

T

Dai

sy c

over

age

Effects of T ondaisy coverage

P1

Effects of daisy coverage on T

P2

T daisycoverage

T daisycoverage

Feedback loops

ENSC 425/625 Chapter 3UNBC

28

P1

Effects of daisy coverage on T

P2T

Da

isy

cove

rag

e

Effects of T ondaisy coverage

Perturb daisy coverage at P1 => sys. returns to P1 (stable equil. pt.)

29

T

Dai

sy c

over

age P1

P2

A large perturb. => daisies all die from extreme T

Large incr. in daisy cover => very low T => decr. in daisy cov. => very high T => lifeless.

P1

T

Dai

sy c

over

age

P2

ENSC 425/625 Chapter 3UNBC

31

From P2, incr. daisy cov. => decr. T => further incr. in daisy cov. => converge to P1

P1

T

Dai

sy c

over

age

P2

unstable equilib. pt.

T daisycoverage

ENSC 425/625 Chapter 3UNBC

33

Gradual incr. in solar luminosity

T

Dai

sy c

over

age

P1

P2Teq

To Tf

P1

P2

The effect of T on Daisy unchanged

For any particular value of daisy cov., T incr.

The key variables

b: Fraction of planet covered in black daisies

w: Fraction covered in white daisies

Tb: Temperature where the black daisies areTw: Temperature where the white daisies are

L: Solar luminosity

An equation for the black daisies

dαb/dt = αb ( 1 – αb – αw) β(Tb) - γαb

= αb (αg β(Tb) – γ)

(T) is a function that is zero at 5C, rises to a maximum ofone at 22.5C and then falls to zero again at 40C

A simple and convenient choice is ( )( . )

.T

T

1

22 5

17 5

2

2

An equation for the white daisies

We use a similar equation for the white daisies:

We don’t have to use the same and but itkeeps things simple. We can use different oneslater if we want to.

dαw/dt = αw (αg β(Tw) – γ)

Heat Flow

Because different regions of Daisyworld are at differenttemperatures, there will be heat flow. We include this in the model using the equations

Tb4 = T4 + q(A-Ab) Tw

4=T4 + q(A-Aw)

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 dothe temperature differences. Don’t worry about the 4th powers; they’re only there to makethe calculations easier and don’t make any real difference.

The Daisyworld Equations

db/dt = b(g

(Tb) - ) d

w/dt=w(g

(Tw) - ) (T+273)4 = SL(1-A) A = gAg + bAb + wAw

Tb4 = T4 + q(A-Ab) Tw

4 = T4 + q(A-Aw)

No daisies

TSL

2273

1 4

/

Black daisies only

0 11

b b b bb

T a T( ( )( ) ) ( )

( )( . )

.T

Tb

b

122 5

17 5

2

2

Tbb

b

22 5 17 5

1

1. .

Gaia Hypothesis

• Proposed by James Lovelock • 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)

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

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

4)1(

SB

pp

LST

bbbbununp aaa

Daisyworld Model (3)

• Area of daisies is modified according to the following equations

pspHAs

ss

sunss

TFT

Tg

deathrategaadt

da

)(

)5.22()540(

41

001.0)(

22

Daisyworld Model (4)

• 2D CA rules:• 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

Example of Daisy Crowding

• Spreading-threshold = 6

=> Start new patch of daisies

=> Don’t start new patch

Parameter Settings

• Two different temperature models• 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

Spatial Daisyworld vs. Mathematical Daisyworld

(Mathematical Model)

Area Occupied by Daisies

(Spatial Model)

Spatial Daisyworld vs. Mathematical Daisyworld (2)

(Mathematical Model)

Temperature of Daisyworld

(Spatial Model)

Effects of Solar Luminosity on Daisyworld0.80.7 0.9 1.0

1.1 1.2 1.3 1.4

The Effects of Death Rate on Daisyworld

death-rate = 0.1

death-rate = 0.3

death-rate = 0.5

Daisyworld with Four Species of Daisies

Area covered by daisies Temperature of Daisyworld

Effects of Solar Luminosity on Daisyworld with Four Species

0.80.7 0.9 1.0

1.1 1.2 1.3 1.4