Understanding Tipping Points in Climate and Sustainability
Waterloo Women in Math Aug. 2014 Mary Lou Zeeman Bowdoin College
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Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support
Components of Sustainability Hum
an Fa
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Peter March
Math Gives Sustainability Coherence Hum
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Mathematical Sciences
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Peter March
Bringing Coherence to Sustainability Hum
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Geography
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Literature “Flight Behavior” Barbara Kingsolver
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ctors
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Needs Interdisciplinary Courage
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Intergovernmental Panel on Climate Change
Google IPCC 5th Assessment Report WG1 -‐ Science Basis “Warming of the climate system is unequivocal, and since the 1950s, many of the observed changes are unprecedented over decades to millennia.”
Observed Change: Carbon Dioxide
Mauna Loa South Pole
IPCC AR5
Observed Change: Temperature
Twelfth Session of Working Group I
Approved Summary for Policymakers
Figure SPM.1 [FIGURE SUBJECT TO FINAL COPYEDIT]
Global average surface temperature change, 1850-‐2012
Annual averages
Decadal averages
1850
2000
IPCC AR5
Observed Change: Ocean AcidificaEon
CO2
pH
IPCC AR5
Climate Change and IPCC IPCC
5th Assessment Report, WG1 – Science Basis “Human influence on the climate system is clear.”
Climate Change and IPCC IPCC
5th Assessment Report, WG1 – Science Basis “Human influence on the climate system is clear.” But climate predicEon is sEll difficult… Why? • Complex System • Chaos • Uncertainty • Tipping Points
Climate Change and IPCC IPCC
5th Assessment Report, WG1 – Science Basis “Human influence on the climate system is clear.” But climate predicEon is sEll difficult… Why? • Complex System • Chaos • Uncertainty • Tipping Points
Climate Change and IPCC IPCC
5th Assessment Report, WG1 – Science Basis “Human influence on the climate system is clear.” But climate predicEon is sEll difficult… Why? • Complex System • Chaos • Uncertainty • Tipping Points
Similar Challenges Across Sustainability Hum
an Fa
ctors
Math
Natural Resources
But climate predicEon is sEll difficult… Why? • Complex System • Chaos • Uncertainty • Tipping Points
Our Plan today • • • • •
Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support
Research Theme: Model Hierarchies Insight Simple Dynamical
PredicEve ???
Complex ComputaEonal
What’s in between?
Research Theme: Model Hierarchies Insight Simple Dynamical
I’ll be talking here
PredicEve ???
Complex ComputaEonal
But you can be thinking here
When do insights persist thru’ to complex version?
Example: Earth’s Energy Balance
Kiehl, J. T. and Trenberth, K. E., 1997
Math and Climate
Hans Kaper & Hans Engler SIAM
John Marshall & Alan Plumb
Ray Pierrehumbert Ka-‐Kit Tung
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy In
Balances
Energy Out
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation). 17
17
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy In
Balances
Energy Out
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave Depends o n A lbedo radiation). radiation). > 0.8 for ice and snow 17 17 < 0.2 for blue ocean
Albedo = FracEon reflected
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy In
Balances
Energy Out
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave Depends o n A lbedo radiation). radiation). > 0.8 for ice and snow 17 17 < 0.2 for blue ocean
Energy In, extreme cases Temp very low => ice covered earth => Energy in is low
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy In
Balances
Energy Out
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave Depends o n A lbedo radiation). radiation). > 0.8 for ice and snow 17 17 < 0.2 for blue ocean
Energy In, extreme cases Temp very low => ice covered earth => Energy in is low Temp very high => ice free earth => Energy in is high
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy
Energy In
Balances
Energy Out
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation).
Snowball
Ice free 17
17
Energy In, extreme cases Temp very low => Energy in is low Temp very high => Energy in is high
Global average temperature, T
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy
Energy In
Energy Out
Balances
Energy radiated out:
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation). 17
Global average temperature, T
17
ε σT4
Blackbody Stefan-‐ Boltzmann RadiaEon constant
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy
Energy In
Balances
Energy Out
Energy radiated out:
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation). 17
Global average temperature, T
17
ε σT4
Atmospheric greenhouse gases, 0<ε<1
Blackbody RadiaEon
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy Out
Balances
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation). 17
Global average temperature, T
Energy
Energy
Energy In
εσT4
17
Global average temperature, T
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance
Energy
dT = C(energy in -‐ energy out) dt energy out, εσT4
energy in
Global average temperature, T
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -‐ energy out) dt Energy
energy out, εσT4
a What happens when T=a?
energy in
T
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -‐ energy out) dt Energy
energy out, εσT4
a dT = 0 dt So T is in equilibrium At a:
energy in
T
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -‐ energy out) dt Energy
energy out, εσT4 energy in
a
b
c
dT At a, b and c: = 0 dt So T is in equilibrium
T
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -‐ energy out) dt Energy
energy out, εσT4 energy in
a
b
c
What if T is between b & c?
T
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -‐ energy out) dt Energy
energy out, εσT4 energy in
a
b Between b & c: So T is increasing
c dT > 0 dt
T
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -‐ energy out) dt Energy
energy out, εσT4 energy in
a
b
c
T
What if T is above c?
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -‐ energy out) dt Energy
energy out, εσT4 energy in
a
b
c
T
dT < 0 dt So T is decreasing Above c:
Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -‐ energy out) dt Energy
energy out, εσT4 energy in
a
b
c
T
Example: Earth’s Energy Balance dT = C(energy in -‐ energy out) dt
Energy
energy out, εσT4 energy in
a
b
c
T
a
b
c
T
Example: Earth’s Energy Balance dT = C(energy in -‐ energy out) dt
Phase line for T
Energy
energy out, εσT4 energy in
a
c
b
b
c
T
a
Example: Earth’s Energy Balance dT = C(energy in -‐ energy out) dt
Phase line for T
Energy
energy out, εσT4 energy in
a
Earth with c Ice caps b
b
c
T
Snowball a Earth
AlternaEve Stable States Examples • Earth’s energy balance
Phase line Temp. Earth with c Ice caps
b
Snowball Earth a
AlternaEve Stable States Examples • Earth’s energy balance Geological evidence: 600 million years earth was a snowball. QuesEon: How did we ever get out of that “stable” state?
Phase line Temp. Earth with c Ice caps
b
Snowball Earth a
AlternaEve Stable States Examples • Earth’s energy balance Geological evidence: 600 million years earth was a snowball. QuesEon: How did we ever get out of that “stable” state? First: Let’s see some more examples
Phase line Temp. Earth with c Ice caps
b
Snowball Earth a
AlternaEve Stable States in Shallow Lakes Clear Lake: • Lots of vegetaEon • Biodiversity among plankton, fish and birds • Sediment anchored. Turbid Lake: • Algae dominate, block light. • Lisle or no vegetaEon. • Loss of biodiversity • Waves and bosom feeders sEr up sediment. Ecology of Shallow Lakes, Marten Scheffer, Springer, 2005
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake
Phase line VegetaEon Clear c High vegetaEon
b
Sudden dramaEc change can, and does, happen.
Turbid a Low vegetaEon
Example: Florida Everglades
Sawgrass
Casails
AlternaEve Stable States in Florida Everglades • Sawgrass and casails compete • Casails are taking over
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades
Phase line Angle
Sawgrass c
b
Sudden dramaEc change can, and does, happen.
Casails a
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs
Phase line Angle
Coral dominated c
b
Sudden dramaEc change can, and does, happen.
Algal dominated a
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries
Phase line Biomass
Fish stock c
b
Sudden dramaEc change can, and does, happen.
No fish a
Somatotrophe Growth hormone cell in the pituitary.
AlternaEve stable states • Quiescent • Tonic firing • Tips back and forth
nt
!
hA, of of ihe o-
oaded from jn.physiology.org on July 3, 2008
fel ll he
BursEng in Excitable Cells
* Detail
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron
Phase line Membrane voltage Tonic firing c
b
Sudden dramaEc change can, and does, happen.
Quiescent a
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron • Neural network
Phase line Overall network acEvity High acEvity c
b
Sudden dramaEc change can, and does, happen.
Low acEvity a
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron, Neural network • Kayak
Phase line Angle
Upright c
b
Sudden dramaEc change can, and does, happen.
Upside down a
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron, Neural network • Kayak • Stress axis
Phase line Behavior
Fight c
b
Sudden dramaEc change can, and does, happen.
Flight a
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron, Neural network • Kayak • Stress axis • Mood
Sudden dramaEc change can, and does, happen.
Phase line Mood
Manic c
b
Depressed a
Research Theme: Model Hierarchies Insight Simple Dynamical
I’ll be talking here
PredicEve ???
Complex ComputaEonal
But you can be thinking here
When do insights persist thru’ to complex version? Sudden dramaEc change can, and does, happen.
Our Plan today • • • • •
Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support
AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries Policy/management quesEons: • Neuron, Neural network Are we at risk of a regime shiv? • Kayak Is oaur state ‘resilient’ enough? • Stress xis • Mood How do we avoid a regime shiv? How do we maintain enough resilience?
How do we induce a regime shiv? How do we reduce resilience?
Phase line Mood
Manic c
b
Depressed a
Resilience… of what? to what? Resilience: How much disturbance the system can withstand.
Phase line
c
b
a
Resilience… of what? to what? Resilience: How much disturbance the system can withstand.
Phase line
Suppose the system is at state c c
b
a
Resilience… of what? to what? Resilience: How much disturbance the system can withstand.
Phase line
Suppose the system is at state c c Disturb the system to here What happens? b
a
Resilience… of what? to what? Resilience: How much disturbance the system can withstand.
Phase line
Suppose the system is at state c c The system returns to state c. It “recovers”
Disturb the system to here What happens? b
a
Resilience… of what? to what? Resilience: How much disturbance the system can withstand.
Phase line
Suppose the system is at state c c
Now disturb the system to here What happens?
b
a
Resilience… of what? to what? Resilience: How much disturbance the system can withstand.
Phase line
Suppose the system is at state c c
Now disturb the system to here What happens?
b
Regime shiv to state a. a
System does not recover it’s old state
Resilience… of what? to what? Resilience: How much disturbance the system can withstand.
Phase line
Suppose the system is at state c c
Now disturb the system to here What happens?
b
b acts as a threshold Regime shiv to state a.
a
System does not recover it’s old state
Example: Shallow Lake Resilience: How much disturbance the system can withstand.
VegetaEon
c
Clear High vegetaEon,
b
a
Turbid Low vegetaEon,
Example: Shallow Lake Resilience: How much disturbance the system can withstand.
VegetaEon
Suppose the lake is clear and vegetated c
Clear High vegetaEon,
b
a
Turbid Low vegetaEon,
Example: Shallow Lake Resilience: How much disturbance the system can withstand.
VegetaEon
Suppose the lake is clear and vegetated c
If too much algae grow (e.g. from high nutrient loading)
Clear High vegetaEon,
b
a
Turbid Low vegetaEon,
Example: Shallow Lake Resilience: How much disturbance the system can withstand.
VegetaEon
Suppose the lake is clear and vegetated c
If too much algae grow (e.g. from high nutrient loading) Lake shivs to turbid state
Clear High vegetaEon,
b
a
Turbid Low vegetaEon,
Example: Shallow Lake Resilience: How much disturbance the system can withstand.
VegetaEon
c
Clear High vegetaEon,
b
Suppose the lake is turbid a
Turbid Low vegetaEon,
Example: Shallow Lake Resilience: How much disturbance the system can withstand.
VegetaEon
c
If sediment disturbance is reduced (e.g. by removing some bosom feeder fish)
Clear High vegetaEon,
b
Suppose the lake is turbid a
Turbid Low vegetaEon,
Example: Shallow Lake Resilience: How much disturbance the system can withstand.
VegetaEon
Lake self-‐recovers to clear state c If sediment disturbance is reduced (e.g. by removing some bosom feeder fish)
Clear High vegetaEon,
b
Suppose the lake is turbid a
Turbid Low vegetaEon,
QuanEfying Resilience Resilience: How much disturbance the system can withstand. Resilience: Radius of basin of asracEon of the stable state
c
Basin of asracEon of stable state c
b
a
Basin of asracEon of stable state a
QuanEfying Resilience Resilience: How much disturbance the system can withstand. Resilience: Radius of basin of asracEon of the stable state
c
QuesEons: What’s the distribuEon of disturbance/noise/shocks?
Basin of asracEon of stable state c
b
a
Basin of asracEon of stable state a
QuanEfying Resilience Resilience: How much disturbance the system can withstand. Resilience: Radius of basin of asracEon of the stable state
c
QuesEons: What’s the distribuEon of disturbance/noise/shocks?
Basin of asracEon of stable state c
b
How does disturbance accumulaEon interact with dynamics? a
Basin of asracEon of stable state a
QuanEfying Resilience Resilience: How much disturbance the system can withstand. Resilience: Radius of basin of asracEon of the stable state
QuesEon: Is the basin ‘large’ enough to contain accumulated disturbance?
c
QuesEons: What’s the distribuEon of disturbance/noise/shocks?
Basin of asracEon of stable state c
b
How does disturbance accumulaEon interact with dynamics? a
Basin of asracEon of stable state a
Research Theme: Model Hierarchies Insight Simple Dynamical
I’ll be talking here
PredicEve ???
Complex ComputaEonal
But you can be thinking here
Size and strength of basins of asracEon relaEve to Noise characterisEcs & accumulaEon behavior?
Our Plan today • • • • •
Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support
Channeling Christopher Zeeman Google: Sir Christopher Zeeman Christmas Lectures
EC Zeeman’s Framework
Discrete ConEnuous
BEHAVIOR
Discrete
THINGS
Dice Symmetries DISCRETE BOX
ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons
Finite Probability Finite Groups
BifurcaEon Theory Catastrophe Theory
Planets PopulaEons
Waves ElasEcity
Ordinary DifferenEal Eq ParEal DifferenEal Eq
EC Zeeman’s Framework
Discrete ConEnuous
BEHAVIOR
Discrete
THINGS
Dice Symmetries DISCRETE BOX
ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons
Finite Probability Finite Groups
BifurcaEon Theory Catastrophe Theory
Planets PopulaEons
Waves ElasEcity
TIME BOX Ordinary DifferenEal Eq ParEal DifferenEal Eq
EC Zeeman’s Framework
Discrete ConEnuous
BEHAVIOR
Discrete
THINGS
Dice Symmetries DISCRETE BOX
ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons
Finite Probability Finite Groups
BifurcaEon Theory Catastrophe Theory
Planets PopulaEons
Waves ElasEcity
TIME BOX CONTINUOUS BOX Ordinary DifferenEal Eq ParEal DifferenEal Eq
EC Zeeman’s Framework
Discrete ConEnuous
BEHAVIOR
Discrete
THINGS
Dice Symmetries DISCRETE BOX
ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons
Finite Probability Finite Groups
BifurcaEon Theory Catastrophe Theory
Planets PopulaEons
Waves ElasEcity
TIME BOX CONTINUOUS BOX Ordinary DifferenEal Eq ParEal DifferenEal Eq
EC Zeeman’s Framework
Discrete ConEnuous
BEHAVIOR
Discrete
THINGS
Finite Probability Finite Groups
ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons PANDORA’S BOX BifurcaEon Theory Catastrophe Theory
Planets PopulaEons
Waves ElasEcity
Dice Symmetries DISCRETE BOX
TIME BOX CONTINUOUS BOX Ordinary DifferenEal Eq ParEal DifferenEal Eq
Tipping point as fold bifurcaEon
Behavior
1-‐d slowly varying parameter
Research Challenge: generalize thru’ model hierarchies
Example: Earth’s Energy Balance
To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system. This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.
Energy Out
Balances
Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation). 17
Energy
Energy
Energy In
17
εσT4
ε represents
greenhouse gases
Global average temperature, T
Global average temperature, T
Example: Earth’s Energy Balance dT = C(energy In -‐ energy out) dt
Phase line for T
Energy
energy out, εσT4 energy in
a
c
b
b
c
Let’s increase ε (lower greenhouse gases)
T
a
Example: Earth’s Energy Balance dT = C(energy In -‐ energy out) dt
Phase line for T
Energy
energy out, εσT4 energy in
a
c b
b
c
Increase ε (lower greenhouse gases)
T
a
Example: Earth’s Energy Balance dT = C(energy In -‐ energy out) dt
Phase line for T
Energy
energy out, εσT4 energy in
a
c
Increase ε (lower greenhouse gases)
T
a
Example: Earth’s Energy Balance dT = C(energy In -‐ energy out) dt
Phase line for T
Energy
energy out, εσT4
a
Increase ε (lower greenhouse gases)
energy in
T
Snowball Earth a
Example: Earth’s Energy Balance dT = C(energy In -‐ energy out) dt
Phase line for T
Energy
energy out, εσT4 energy in
a
c
b
b
c
T
Now let’s decrease ε (increase greenhouse gases)
a
Example: Earth’s Energy Balance dT = C(energy In -‐ energy out) dt
Phase line for T
Energy
energy out, εσT4 energy in
a
c
b
b
c
Decrease ε (increase greenhouse gases)
a
T
Example: Earth’s Energy Balance
Energy
dT = C(energy In -‐ energy out) dt
Phase line for T Earth with ice caps c or no ice
energy out, εσT4
energy in
c
Decrease ε (increase greenhouse gases)
T
Global average temp, T
Stack up all the phase lines Stable States
Low
High
Greenhouse gas conc.
Example: Earth’s Energy Balance T
Snowball Earth Low
Greenhouse gas conc.
Example: Earth’s Energy Balance T
Earth with ice caps or no ice
High
Greenhouse gas conc.
Example: Earth’s Energy Balance T
Medium
Greenhouse gas conc.
Remember this quesEon? Phase line
Geological evidence: 600 million years earth was a snowball. QuesEon: How did we ever get out of that stable state?
Temp. Earth with c Ice caps
b
Snowball Earth a
Remember this quesEon? Phase line
Geological evidence: 600 million years earth was a snowball. QuesEon: How did we ever get out of that stable state? Watch what happens to the resilience (basin of asracEon)
Temp. Earth with c Ice caps
b
Snowball Earth a
Change Environment T
Snowball Low
Medium
High
Greenhouse gas conc.
Gradually increase greenhouse gas concentraEon (volcanoes)
Change Environment T
Snowball Low
Medium
High
Greenhouse gas conc.
Gradually increase greenhouse gas concentraEon (volcanoes)
Change Environment T
Snowball Low
Medium
High
Greenhouse gas conc.
Gradually increase greenhouse gas concentraEon (volcanoes)
Change Environment T
Snowball Low
Medium
High
Greenhouse gas conc.
Gradually increase greenhouse gas concentraEon (volcanoes)
Change Environment T
Earth with ice caps or no ice
Low
Medium
High
Greenhouse gas conc.
Gradually increase greenhouse gas concentraEon (volcanoes)
Change Environment T
Earth with ice caps or no ice
Low
Medium
High
Greenhouse gas conc.
Now decrease greenhouse gas concentraEon (rocks and oceans)
Irreversibility as hysteresis T
Earth with ice caps or no ice
Low
Medium
High
Greenhouse gas conc.
PaleoClimate Record, 70M years Temperature
Million years ago
Now
Zachos
70 mill yrs ago
Irreversibility as hysteresis T
Earth with ice caps or no ice Last 70 mill yrs
Low
Medium
High
Greenhouse gas conc.
PaleoClimate Record, 70M years Temperature
Million years ago
Now
Zachos
Lots of interesEng abrupt events & Epping points within here, too -‐ Requires more detailed models 70 mill yrs ago
Revisit Our Examples Examples • Earth’s energy balance • Shallow lake
Phase line VegetaEon Clear c High vegetaEon
b
Turbid a Low vegetaEon
Nutrient loading in shallow lake
Turbidity
Clear
Turbid High
Medium
Low
Nutrient Loading
Gradually Changing Environment Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs
Nutrient run-‐off from agriculture shrinks basin of asracEon
Then shock triggers regime change
Phase line
Clear Sawgrass c Coral dominated
b
Turbid Casails a Algal dominated
Our Plan today • • • • •
Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support
Can Critical Transitions be predicted ?
Even if we do not understand the system ? Marten Scheffer
Research Theme: Model Hierarchies Insight Simple Dynamical
Spot a pasern here
PredicEve ???
Complex ComputaEonal
Test it here…
Resilience in Changing Environment Resilience Ability of system to withstand perturbaEon Size/strength of basin of asracEon of stable state
Equilibrium states of system
Slowly changing environment
Resilience in Changing Environment Resilience Ability of system to withstand perturbaEon Size/strength of basin of asracEon of stable state Resilience shrinks to zero as we approach Epping point Exploit that structure! Equilibrium states of system
Slowly changing environment
Early Warning Signs? Scheffer et al, Nature 2009 & Science 2012 How does system respond to stochasEc perturb.? 1.5
1
0.5
0
−0.5
−1
−1.5
−2 0
50
100
150
200
250
300
Early Warning Signs? Scheffer et al, Nature 2009 & Science 2012 How does system respond to stochasEc perturb.? 2
1.5
1.5
1 1
0.5 0.5
0
0 −0.5
−0.5
−1
−1 −1.5
−1.5
−2 −2.5
0
50
100
150
200
250
300
−2 0
50
100
150
200
250
300
Early Warning Signs? How does system respond to stochasEc perturb.? CriEcal slowing: as system approaches bifurcaEon, rate of asracEon to equilibrium approaches 0 2 1.5 1
PerturbaEons die down more slowly
0.5
0
• Increased system ‘memory’ • Increased autocorrelaEon • Increased variance
−0.5 −1 −1.5 −2 −2.5
0
50
100
150
200
250
300
−0.6 −1.6 −1.65
−0.7
−1.7
−0.8
−1.75 −1.8
−0.9
−1.85 −1.9
−1
−1.95
−1.1 −2 −2.05
−1.2
−2.1 45
50
55
60
65
70
75
225
230
235
240
245
250
255
PaleoClimate Record, 70M years Temperature
Million years ago
Now
Zachos
Lots of interesEng abrupt events & Epping points within here, too
70 mill yrs ago
Slowing down precedes ancient shifts shifts Critical slowing down announced 8 climate abrupt climate
Dakos et al PNAS 2008
Research Theme: Model Hierarchies Insight Simple Dynamical
Spot a pasern here
PredicEve ???
Complex ComputaEonal
Test it here… Shallow lakes Financial markets Power grid Psychology
Research Theme: Model Hierarchies Insight Simple Dynamical
Spot a pasern here
PredicEve ???
Complex ComputaEonal
Test it here… InteresEng how oven it works. Inspiring new research quesEons.
Policy Decision Context We have some control over our gradually changing environment.
E.g. Land Use, Water, Fisheries, Urban planning, TransportaEon…
Equilibrium states of system
control of environment
Policy Decision Context We have some control over our gradually changing environment.
E.g. Land Use, Water, Fisheries, Urban planning, TransportaEon…
Policy goal: Manage for resilience Modeling goal: quanEfy effect of policy opEons on resilience Equilibrium states of system
control of environment
Policy Decision Context
What if decision maker has two dimensions of control opEons? Equilibrium states of system
1-‐d control of environment
Channeling EC Zeeman again
ECZ and his catastrophe machine
Channeling EC Zeeman again Generic picture for systems with minimizing dynamics
2-‐d control of environment
Channeling EC Zeeman again Generic picture for systems with minimizing dynamics
No
r 2-‐d control of o t c a f rmal environment
Channeling EC Zeeman again Generic picture for systems with minimizing dynamics
No
r 2-‐d control of o t c a f rmal environment
Channeling EC Zeeman again Generic picture for systems with minimizing dynamics Splizng Factors:
Strength of Feedback
Network ConnecEvity
Depth of lake
Level of homogenizaEon
No
r o t c a f rmal
Channeling EC Zeeman again Generic picture for systems with minimizing dynamics Splizng Factors:
Strength of Feedback
Network ConnecEvity
Depth of lake
Level of homogenizaEon
Could offer insight into more policy opEons…
Channeling Christopher Zeeman Google: Sir Christopher Zeeman Christmas Lectures
THANK YOU!
Google: MPE 2013 MCRN Math Sir Christopher Zeeman, Christmas Lectures Thanks to: many friends & colleagues
MCRN