Review - STELLA Model Elements
• Reservoirs – These are the default stock type
– Think of a reservoir as an undifferentiated pile of
stuff (many instances of the same stuff)
– Reservoirs passively accumulate inflows minus
outflows (they are simply containers)
– Any units which flow into a Reservoir lose their
individual identity - Reservoirs mix together all
units into an undifferentiated mass as they
accumulate
• Flows – Function is to fill and drain stocks
– To bend a flow pipe, depress the shift key and
change the direction of mouse movement as you
drag the flow. Each time you depress the shift key,
a 90 degree bend will be put in the flow pipe
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Review - STELLA Model Elements
• Flows Cont.
– To draw an inflow to a stock, make sure that your cursor
makes contact with the stock before you release the mouse
button. The stock will turn gray on contact to let you know
that it will receive the flow. If you release the mouse button
prematurely, a cloud will appear at the destination end of the
flow pipe
– To replace a cloud with a stock, select the stock with the
Hand tool. Drag the stock over the cloud. When the cursor
(the tip of the index finger on the hand) is directly atop of the
cloud, the cloud will turn gray. Release the mouse button,
and the flow will be connected to the stock (the cloud will
disappear)
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Review - STELLA Model Elements
• Converters – These serve a utilitarian role in
the software
– They hold values for constants, define external
inputs to the model, calculate algebraic
relationships, and serve as repositories for
graphical functions
– In general, they convert inputs into outputs,
hence the name "converter"
• Connectors – These connect elements
– There are two types of connectors available in
STELLA
– Action connectors are shown as solid, directed
wires
– Information connectors (which we most likely
will not be using) are signified by a dashed wire
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Rules for Building Systems Models
1. Make systems diagrams as simple as possible
2. Relationships between elements should be
defined mathematically
3. If a mathematical expression is not available,
define relationships using graphs
4. Observe the conservation law and maintain
consistency in units
5. Reservoir values can only be changed by
inflows and outflows
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Model Structures and Behavior Patterns
• The systems modeler believes that the behavior of the
system is a function of the system itself
• Translating that idea to the model realm, this means
that certain structures of elements should produce
certain types of behavior patterns
• We are going to look at five common behavior
patterns and their associated structures:
Linear
Growth
or Decay
Exponential
Growth
or Decay
Logistic
Growth
Overshoot
and
Collapse
Oscillation
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Linear Growth or Decay - Example
• Consider the following example of a system:
– An oil reserve contains 10,000,000 barrels of oil
– The oil is consumed at a rate of 10,000 barrels per day
• What would the system diagram look like here?
•What entity changes with
time here?
•What is/are the process(es)
that cause that change?
•What determines the rate of
change?
•Because it is constant, no
converter is required
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Linear Growth or Decay - Example
• What is the difference equation for this system?
Oil Reserves tomorrow = Oil Reserves Today – 10,000 barrels
or more generally:
Oil Reserves in t days = Present Oil Reserves – 10,000 * t days
and shown mathematically:
R(t+∆t) = R(t) – (10,000 * ∆t)
• How will this system behave?
–
–
–
–
Reserve begins with 107 barrels
Decrease 10,000 barrels per day
After 1000 days, reserve is empty
NOTE: Once empty, eqn. not valid
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Linear Growth or Decay – System
Features, Diagrams, and Equations
• For a reservoir to exhibit linear growth or decay, the
sum of all inflows minus the sum of all outflows to the
reservoir must be constant
– A positive constant indicates growth
– A negative constant indicates decay
– If the constant is zero, the reservoir content remains constant
• System can have any number of
inflows and outflow
• Not all flows need be constant
• It is necessary for the difference
between the sums to be constant
Generic Linear System
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Linear Growth or Decay – System
Features, Diagrams, and Equations
• For our generic example, the difference equation is:
R(t+∆t) = R(t) + {(Inflow 1 + Inflow 2 + Inflow 3) – (Outflow 1 + Outflow 2)} * ∆t
or in general:
R(t+∆t) = R(t) + {(Inflow 1 + … + Inflow n) – (Outflow 1 + … + Outflow n)} * ∆t
• This can be rearranged as:
n
R(t+∆t) - R(t) =
n
Outflow
Inflow Σ
Σ
i=1
j=1
j
i
* ∆t
and divided by ∆t to give:
n
R(t+∆t) - R(t) =
∆t
n
Outflow
Inflow Σ
Σ
i=1
j=1
i
j
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Linear Growth or Decay – System
Features, Diagrams, and Equations
• We are now set up to find the instantaneous rate of
change of the reservoir with respect to time by taking
the derivative of the expression:
lim
∆t Æ 0
R(t+∆t) - R(t)
∆t
=
dR(t)
dt
n
=
n
Outflow
Inflow Σ
Σ
i=1
j=1
i
j
= constant = k
• In a linear system, the value k
is the slope of the line
• We have positive values of k
for growth and negative
values for decay
• What would k be in the Oil
Reserve example?
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Linear Growth or Decay – Feedbacks
and Steady State Conditions
• The generic linear system
contains no loops, therefore it
cannot have any feedbacks
• This system changes as a
constant rate and has linear
Generic Linear System
chain of cause and effect
• For a system to be in a steady state, the rate of change
of the contents of the reservoir must be equal to zero
• We have shown that the rate of change here is constant
• In the Oil Reserve example, this is true until we run out
of oil on day 1000, then the system is in steady state
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth
or Decay - Examples
• Consider the following example of a system:
– A pair of white mice escape from their cage
– They mate and have offspring, which mature and then do the
same, generation after generation …
• What would the system diagram look like here?
• This model is going to be able more complex than the
linear model, so we will attempt to construct the model
using a series of steps:
Step 1: Identify the reservoir(s)
– Since we are tracking the number of mice:
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth
or Decay - Examples
Step 2: Identify the process(es) that will change the
contents of the reservoir(s) over time:
– We obviously need to have mice being born, and mice dying
of old age:
Step 3: Identify the converter(s) that determine the
rates of inflow and outflow:
– We will need a Birth Rate and Death Rate:
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth
or Decay - Examples
Step 4: Define relationships between system elements
with connectors:
• Just on the basis of the model structure, we should be
able to guess that this will produce different behavior
from our linear model
• The key is how we define the Birth and Death Rates
• Birth Rate = 1.1 births/capita/month
• Death Rate = 0.08 deaths/capita/month
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth
or Decay - Examples
• What is the difference equation for this system?
W(t+∆t) = W(t) + ({Birth - Death} * ∆t)
• Using Birth Rate = 1.1 and Death Rate = 0.08, we can
find the expressions for the Birth and Death processes:
Birth = Birth Rate * W(t) = 1.1 * W(t)
Death = Death Rate * W(t) = 0.08 * W(t)
• Substituting those back into the equation:
W(t+∆t) = W(t) + ({1.1 * W(t) – 0.08 * W(t)} * ∆t)
= W(t) + (1.02 * W(t) * ∆t)
.
• In each time step (selected to be a month for this
model), the number of mice will more than double
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth
or Decay - Examples
• Such a system will exhibit the following behavior:
• Initially, the increasing
number of mice is not
particularly noticeable
• As time goes on, the mice
population grows quite
rapidly
• We can also come up with a system that will exhibit the
opposite behavior:
– A hanging bucket of water has a hole in the bottom
– The rate at which the water drips out is a function of the
amount of water remaining in the bucket
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth
or Decay - Examples
• What would the system diagram look like here?
• Note: We don’t need to
have inflow and outflow
processes necessarily to
produce this particular sort
of behavior
• Exponential growth (or decay) occurs if and only if the
reservoir increases (or decreases) at a rate that is
proportional to its size
– If the reservoir increases in size, this is exponential growth
– If the reservoir decreases in size, this is exponential decay
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth or Decay – System
Features, Diagrams, and Equations
• A system that has exponential behavior generally takes
the following form:
• Unlike a linear system, its possible to trace loops in this
diagram, thus these systems do have feedback, which
helps explain why their behavior runs “out of control”
• To see why we refer to this behavior as exponential, we
need to have a look at the difference equation for one
of these systems …
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth or Decay – System
Features, Diagrams, and Equations
• For our generic example, the difference equation is:
R(t+∆t) = R(t) + {[Inflow Rate * R(t)] – [Outflow Rate * R(t)]} * ∆t
• This can be rearranged (by subtracting R(t)) as:
R(t+∆t) - R(t) = {[Inflow Rate – Outflow Rate] * R(t)} * ∆t
• Then divide by ∆t:
R(t+∆t) - R(t) = {[Inflow Rate – Outflow Rate] * R(t)}
∆t
• Finally, take the derivative with respect to time:
lim
∆t Æ 0
R(t+∆t) - R(t)
∆t
=
dR(t)
dt
= kR(t), where k = [Inflow Rate – Outflow Rate]
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth or Decay – System
Features, Diagrams, and Equations
• The solution to the rate equation is:
R(t) = R0ekt
where:
R0 is the value of R(t) at t = 0
k = [Inflow Rate – Outflow Rate]
• So, we call it exponential growth because k is the rate
constant, which appears in the equation as an exponent
–
–
–
–
k is the net growth or decay rate in the system
If k > 0, the system will exhibit exponential growth
If k < 0, the system will exhibit exponential decay
The ratio of change in the reservoir over one unit of time is ek,
i.e. in one time step, the reservoir changes from R(t) to ek * R(t)
– The larger | k | is the more rapid the growth or decay
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth or Decay – System
Features, Diagrams, and Equations
| k | in Exponential Growth
| k | in Exponential Decay
•While the rate of change in the reservoir is slow initially
in an exponential growth situation, and slow eventually in
a exponential decay situation, it never quite reaches
zero, but instead the reservoir value asymptotes at some
value, and R(t) will asymptotically approach a steady
state value of R = 0
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005