Daisy World
an introduction to systems and
equilibria
Geological Example: Albedo
Feedback
Albedo 
reflected sunlight
received sunlight
• high albedo: most energy is reflected
back into space
• low albedo: most energy is absorbed and
heats the surface
Some Albedo Values
(from Table 2-1, Kump et al., 1999)
Sand:
Grass:
Forest:
Water (overhead sun):
Water (sun near horizon):
Fresh snow:
Thick cloud:
0.20 – 0.30
0.20 – 0.25
0.05 – 0.10
0.03 – 0.05
0.50 – 0.80
0.80 – 0.85
0.70 – 0.80
Image source: NASA Visible Earth, http://visibleearth.nasa.gov/view_detail.php?id=4701
Albedo Feedback cont.
Surface
Temperature
Snow
Coverage
Albedo
Albedo Feedback cont.
Surface
Temperature
Snow
Coverage
Albedo
A variation of the Albedo Feedback
• how do plants respond to climate ?
possible outcomes of feedback
loops
• Possible choices:
– Arrived at stable end state
– Never settles down in a stable end state but
oscillates
– Never reaches any stable end state
• What kind of feedback loops would
produce these options?
Equilibrium States
• If system is in equilibrium it will not change
unless it is disturbed
• two types: stable, unstable
Stable Equilibrium
• (within limits) system will return to old
equilibrium state
Unstable Equilibrium
• small perturbations will dramatically affect the
state of the system
What kind of feedback loops tend to produce
stable equilibria ?
Long-term temperature fluctuations
Daisy World
• Gaia hypothesis
• Earth a self-regulating system
• How can it do that without a brain
• Daisy World: local processes affect global
state of climate
Influence of Daisy Growth on
Surface Temperature
Daisy Coverage
negative or positive feedback ?
why ?
Avg. Surface Temp.
Daisy (Albedo) Feedback
Influence of Daisy Growth on
Surface Temperature
Influence of Daisy Growth on
Surface Temperature
• two modes:
– below optimum temperature → positive
feedback
– above optimum temperature → negative
feedback
Two Feedback Mechanisms
Some Mathematical Conventions
• y = f(x) means that y is a function of an
independent variable x
• mathematicians plot the independent
variable on the abscissa (horizontal axis)
and the dependent variable on the
ordinate (vertical axis)
• physicists and other scientists are not
always that rigid!
Two Feedback Mechanisms
How to Solve for Both:
Two Equilibrium Points
• Both conditions have to
be fulfilled
• Daisy coverage affects
surface temperature
• Surface temperature
affects daisy coverage
• under which conditions is
the system in
equilibrium?
How Does the System Reach
Equilibrium?
Are both Equilibria Created Equal?
• try out a few permutations yourself
• is the system always reaching an
equilibrium?
Two Types of Equilibria
External Forcing of the Daisy World
System
• Forcing: outside influence on the system
(e.g., insolation, chemical composition of
the climate system etc.)
• How does Daisy World respond to external
forcings ?
How will Insolation changes affect
the Daisy World ?
Effect of Luminosity on Temperature
How does Feedback Affect new
Equilibrium Temperature?
Temperature change without
Daisy feedback ?
(number of daisies is
constant)
Temperature change with
Daisy feedback ?
(Daisies will spread)
How to Quantify Importance of
Feedback?
Temperature change without
Daisy feedback ?
(number of daisies is
constant)
Temperature change with
Daisy feedback ?
(Daisies will spread)
How to Quantify Importance of
Feedback?
• compare ΔT0 and ΔTeq
• feedback factor:
f
temperature change with feedback

temperature change without feedback

Teq
T0
How will Stability of Equilibria be
Affected?
Stability of Equilibrium Positions
Effects of Daisy Feedback
• biological albedo feedback
• positive and negative feedback loop
• two equilibrium positions (one stable, one
unstable)
• resulting equilibrium is generally not
optimum growth temperature for daisies
• presence of daisies lowers planetary
temperatures
Effects of Daisy Feedback (cont.)
• Dampens effects of external forcings
• feedback factor f quantifies feedback
effect
• Stability of equilibria changes