dynamic equilibrium and system archetypes...dynamic equilibrium and system archetypes todd bendor...
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Dynamic Equilibrium and System Archetypes
Todd BenDor Associate Professor Department of City and Regional Planning [email protected] 919-962-4760 Course Website: http://todd.bendor.org/datamatters
mailto:[email protected]://todd.bendor.org/datamatters
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What is equilibrium?
• Static equilibrium – No change – Rates of change are zero – System at a stand still
• Dynamic equilibrium – Inflow=outflow – System changing, but rates are constant
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Where is the equilibrium?
Water in Silver Lake
big creek flow
little creek flow
net evaporation
silver creek flow
~ surface area
net evaporation rate
=40 KAF/yr
=80 KAF/yr
=60 KAF/yr
= ? Kacres
= 4 feet/yr
= ? KAF/yr
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Can do the same thing here… Fill in 6 question marks (Carbon in a Forest)
carbon flowing into the system
carbon in leaves carbon in branches
carbon in stems
carbon in roots
flow to leaves
flow to branches
flows to stems
flow to roots
carbon in the litter
leave to litter flowbranch to litter flow
carbon in the humus
litter to humus flow
root to humus flow
carbon stored as charcoalhumus to charcoal flow
humus exit flow
humus exit rate
humus to charcoal transfer rateroot to humus transfer rate
litter to humus transfer rate
stem to litter flow
branch tanser rate
stem transfer rate
leaf transfer rate
root fraction
leaf fraction
branch fraction
stem fraction
litter exit flow litter exit rate
30 GT/yr
9 GT/yr
6 GT/yr
6 GT/yr9 GT/yr
9 GT/yr
?
60 GT
24 GT
9 GT
160 GT
18 GT
.3.3
.2.2
.05/yr
.1/yr1.0/yr
.5/yr12 GT/yr
.5/yr
12 GT/yr
?
?
??
?
0.18 GT/yr
.01/yr
.99/yr
17.82 GT/yr
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The Circulatory System
pulmonary storage = 432 ml
left heart storage= 168 ml
arterial storage =624 ml
storage in thearterioles andcapillaries =
336 ml
storage in theveins, venules
and thevenous
sinuses =3,072 ml
right heart storage= 168 ml
arterial flowto heart
venus flowto lungs
cardiac output =80 ml/sec
venousreturn flow
flow to arterioles
capilary outflow
lung time
arterialtime
capilarytime
right hearttime
venoustime
left hearttime
168 ml/2.1 sec =
80 ml/sec
What are each of the flows? What is the arterial time? What is the capillary time?
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Stability and Equilibrium
• Dynamic and static equilibrium – Tell us nothing about stability
• What does stability mean? – ‘Resilience’ to exogenous shocks – Does system collapse? – Does system return to previous state? – Does system achieve a new state?
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System Archetypes
• Typical behaviors seen in many systems • Basic behaviors are common
– Growth (exponential, linear) – Decline (exponential, linear)
• Combined – behaviors become archetypes – Important archetype – S-shaped growth – Will talk about another later
• Oscillations
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Exponential Growth and Decay
• “Nothing is as powerful as an exponential whose time as come.”
• Rule of 70: “…in order to estimate the number of years for a variable to double, take the number 70 and divide it by the growth rate of the variable.”
• You open a savings account on the day your child is born. The bank guarantees that the balance in the account will grow exponentially at 10%/yr forever.
• Your goal is to have $1 million in the account by the time your child reaches 70 years of age, and you plan no further deposits. How much do you deposit in the account?
http://donellameadows.org/archives/nothing-is-so-powerful-as-an-exponential-whose-time-has-come/
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S-Shaped Growth
empty areaf lowered area
+
total area
growth
decay rate
growth rate
decay
What affects growth and decay?
(TA = flowered area + empty area)
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‘Density dependent’ change
empty areaf lowered area
+
total areagrowth
f raction occupied
decay rate
actual growth rate~growth rate multiplier
intrinsic growth rate
decay
0.00.20.40.60.81.0
0.0 0.2 0.4 0.6 0.8 1.0
What would you expect to happen in this model?
Now what affects growth and decay?
Any analogies in your field?
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Seeking an equilibrium…
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Years
1:
1:
1:
2:
2:
2:
3:
3:
3:
0
500
1000
0
200
400
1: f lowered area 2: growth 3: decay
1
1
1
1 1
2
22
2 2
3
3
3 3 3
empty areaflowered area
+
total area
growth
fraction occupied
decay rate
actual growth rate~
growth rate multiplier
intrinsic growth rate
decay200 acres 800 acres
1,000 acres 1.0/yr
0.2
0.80.2/yr
0.2/y
160 acres/yr
160 acres/yr
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We can experiment the effect of density on growth rates
0.00.20.40.60.81.0
0.0 0.2 0.4 0.6 0.8 1.0
comparativ e graph
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Years
1:
1:
1:
0
500
1000
f lowered area: 1 - 2 - 3 -
1
1
1 1
2
2
2 2
3
3
3 3
Alternate Growth Rate Multipliers
Whi
ch G
R M
ultip
lier c
orre
spon
ds w
ith w
hich
gra
ph?
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S-Shaped Growth in a Sales Company: 1st cut at a model
size of salesforce
new hires
exit rate
hiringfraction
widgetsales
widgetprice
annual revenuefractionto sales
salesdepartment
budget
budgeted sizeof sales force
departureseffectiveness
averageannual salary
Start with 50 people. This model would
double every 2.7 years.
There would be 10 doublings in 27 years. That’s up to 50,000
people.
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2nd Version of Sales Model
size of salesforce
new hires
exitrate
hiringfraction
effectiveness
widgetsales
widget price
annualrevenues fraction to
sales
salesdepartment
budgetbudgeted sizeof sales force
averageannual salary
lookup foreffectivenessdepartures
SalesGrowthLoophiringcontrol
saturation
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2nd Version of Sales Model
size of w idgets totalsales per day w idgetsforce per person per day
0 2.0 0200 2.0 400400 2.0 800600 1.8 1080800 1.6 1280
1000 0.8 8001200 0.4 480
800
700
600
500
400
300
200
100
00 4 8 12 16 20
Time (Year)
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Punchline: Systems are Analogies - Flowers and Sales Company
• Both show S-shaped growth for the same reasons • What links the two together?
• Sales company - diminishing marginal returns for the sales force effectiveness
• Flowers - density dependent growth rate multiplier – Diminishing marginal growth of flowers in smaller and smaller
available (empty) space
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Example from World War I era: Influenza Epidemic
Total Deaths: over 20 million, more than deaths in WW I
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Example from World War I era: Influenza Epidemic
Total Deaths: over 20 million, more than deaths in WW I
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Dynamics of Epidemics Demonstrate S-Shaped Growth
Suscepible PopulationInf ected Poplulation Recov ered Population
?
inf ections recov eries
Af f ected Population
duration of inf ection
What determines the rate of infection?
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What causes infections?
Susceptible Population Inf ected Poplulation Recov ered Populationinf ections recov eries
contacts per day per inf ected person
total contacts per dayinf ectiv ity + Af f ected Population
duration of inf ection
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Day s
1:
1:
1:
2:
2:
2:
3:
3:
3:
0
5000
100001: Susceptible Population 2: Inf ected Poplulation 3: Recov ered Population
11
1 122
2
23
3
3
3
Which of the archetypes does the red line follow?
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Contagion and ‘Depletion’ in the Same Model
Susceptible Population Inf ected Poplulation Recov ered Populationinf ections recov eries
+
Total Population
Fr Susceptible
f r of contacts that are with a susceptible person
contacts per day per inf ected person
total contacts per day
dangerous contacts per day
inf ectiv ityduration of inf ection
+
Af f ected Population
Contagion
Depletion
Causal Loop Diagrams Come Later – the loops will be clearer
Do contact rates change as people get sick? How does likelihood of contact between infected and susceptible people change?
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Epidemic runs its course in 20 days
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Day s
1:
1:
1:
2:
2:
2:
3:
3:
3:
0
5000
10000
1: Susceptible Population 2: Inf ected Poplulation 3: Recov ered Population
11
1
1
2
2
2
23
3
3
3
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Affected population shows S-shaped growth
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Infected Population: Sensitivity to ‘Infectivity’
Run 1: 50% Run 2: 25% Run 3: 12.5%
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Infected Population: Sensitivity to Infection Duration
Run 1: 4 days Run 2: 2 days Run 3: 1 day
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Policy Analysis: Contact Avoidance
Susceptible Population Infected Poplulation Recovered Populationinfections recoveries
+
Total Population
Fr Susceptible
fr of contacts that are with a susceptible person
~contacts per day
per infected persontotal contacts per day
dangerous contacts per day
infectivityduration of infection
+
Affected Population
Contact Avoidance Depletion
Contagion
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Infected population when we test contact avoidance
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Day s
1:
1:
1:
0
2000
4000
Inf ected Poplulation: 1 - 2 - 3 -
1
1
1
1
2
2
2
2
3
3 3
3
Run 1: Reference case (‘Base Case’): no avoidance Run 2: when infected reaches 2,000, cut contacts in half Run 3: when infected reaches 1,000, cut contacts in half
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Total Affected Population
Run 1: reference case: no avoidance Run 2: when infected reaches 2,000, cut contacts in half Run 3: when infected reaches 1,000, cut contacts in half
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Systems as learning analogies
• What do the flowers model, the sales model, and the epidemic model have in common?