Dynamic Equilibrium and System Archetypes

Todd BenDor Associate Professor Department of City and Regional Planning bendor@unc.edu 919-962-4760 Course Website: http://todd.bendor.org/datamatters

mailto:bendor@unc.eduhttp://todd.bendor.org/datamatters

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

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

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

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?

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?

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

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/

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)

‘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?

Seeking an equilibrium…

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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

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?

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.

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

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)

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

Example from World War I era: Influenza Epidemic

Total Deaths: over 20 million, 
more than deaths in WW I

Example from World War I era: Influenza Epidemic

Total Deaths: over 20 million, 
more than deaths in WW I

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?

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:

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2:

2:

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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?

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?

Epidemic runs its course in 20 days

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1: Susceptible Population 2: Inf ected Poplulation 3: Recov ered Population

11

1

1

2

2

2

23

3

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3

Affected population shows S-shaped growth

Infected Population: Sensitivity to ‘Infectivity’

Run 1: 50% Run 2: 25% Run 3: 12.5%

Infected Population: Sensitivity to Infection Duration

Run 1: 4 days Run 2: 2 days Run 3: 1 day

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

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

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

Systems as learning analogies

• What do the flowers model, the sales model, 
and the epidemic model have in common?