1. WHAT IS A SYSTEM?
• An organized collection of interrelated physical components characterized by a boundary
and functional unity
• A collection of “communicating” materials and processes that together perform some set
of functions
• An interlocking complex of processes characterized by many reciprocal cause-effect
pathways
• Any collection of interacting objects or processes
• Any phenomenon, either structural or functional, having at least 2 separable components
and some interaction between these components
2. WHAT IS A MODEL?
• An abstraction or simplification of reality
• A description of the essential elements of a problem and their relationships
• A reconstruction of nature for the purpose of study (Levins 1968)
• “A model is a caricature of nature ...... the simplest version of nature can be perturbed
numerically with its responses being indications of the directions nature may take”
(Scavia, Lang and Kitchell 1988).
"Everything should be made as simple as possible, but not simpler."
--- Albert Einstein
"While intelligent people can often simplify the complex, a fool is more likely to
complicate the simple."
--- Swedish proverb
3. TYPES OF MODELS
1) Physical vs. Symbolic (or abstract) Models
o Physical: a scaled-down (or, less frequently, scaled-up) physical replicas of a real
system
o Symbolic: verbal, written language, or mathematical
2) Dynamic vs. Static Models
o Dynamic: with time varying variables
o Static: time invariant
3) Empirical (Correlative) vs. Mechanistic (Explanatory) Models
o Empirical: without internal dynamics; descriptive; prediction
o Mechanistic: with internal dynamics; explanatory; understanding/prediction
o Whether a model is empirical or mechanistic is often relative, depending on the level
of detail or the scale of observation.
4) Deterministic vs. Stochastic Models
o Deterministic: with no random variables; point estimation of parameters
o Stochastic: with one or more random variables; estimation of parameter variations
(e.g., mean, variance, distribution)
1
5) Simulation vs. Analytical Models
o Analytic models: pencil and paper; solvable in closed form analytically; relatively
complicated mathematics; greater mathematical power and usually higher generality
o Simulation models: use of computers; relatively simpler mathematics; usually less
generality and more realism
4. ORGANIZED VS. UNORGANIZED COMPLEXITIES AND MODELING APPROACHES:
1) Small-number system:
o Relatively few components, highly interrelated
o Organized simplicity
o Newtonian approach
2) Medium-number
o Intermediate number of components, closely interrelated
o Organized complexity
o Systems approach
3) Large-number systems
o Large number of components, loosely interrelated
o Unorganized complexity
o Statistical approach
v Rules of thumb for choosing methods of problem solving based on data availability and the
level of understanding on the system:
Amount of Data
Many
Many
Few
Few
Level of Understanding
Little
Good
Little
Good
Appropriate Method
Statistics
Physics
System Analysis & Simulation
System Analysis & Simulation
5. WHY MODELS?
Models can be and have been used for several purposes other than prediction:
• Prediction
• Understanding
• Generate and test hypotheses
• Synthesis
• Identify areas of ignorance
• Serve as management tools
2
SYSTEMS THINKING AND SYSTEM MODELING
Systems Thinking
•
Systems thinking is the art and science of making reliable inferences about system
behavior by developing an increasingly deeper understanding of underlying structure and
processes.
1) Structure-as-cause thinking (or the endogenous viewpoint):
§ Viewing the structure of a system as the cause of its behaviors, as opposed to
seeing the behaviors as being imposed upon the system by outside agents.
2) Closed-loop thinking:
§ Structural elements of the system are arranged in closed-loops (feedback
loops).
§ Emphasizing reciprocal causal relationships
3) Operational thinking:
§ What are closed-loops composed of? Primarily, stocks (state variables) and
flows (rate variables).
§ System components form feedback loops which ultimately determine the
behaviors of the system.
Systems Analysis
1) What Is Systems Analysis?
• A methodology originally developed by the military to deal with the complex logistical
problems during World War II.
• Systems analysis consists of the determination of important system components, systems
simulation, systems optimization, and systems measurement (Watt 1968).
• The application of scientific method to complex problems, which is distinguished by the
use of advanced mathematical and statistical techniques and by the use of computers
(Dale 1970)
2) Systems Models and Systems Diagrams
• Stocks (or state variables or accumulations) are the major concerns which change
through time (e.g., population size, biomass, etc.).
• Flows (or rate variables) represent flows of material or energy between state variables
and characterize the rate of change of these state variables as a result of specific
processes.
• Information links between variables are established by auxiliary variables (converters).
3
v Symbols used in systems modeling (Forrester diagram)
OR
Level
Auxiliary
Rate
Table Function
Exogenous Variable
Information Link
Constant
Material Flow
Source OR Sink
Four Basic Phases in Systems Analysis (or Systems Modeling)
Phase 1: Conceptual model formulation
• Define the problem
• Specify model objectives
• Delineate system boundaries
• Construct causal diagrams
• Understand feedback loop structure
Phase 2: Quantitative model specification
• Select general quantitative structure for the model
• Choose basic time step for simulations
• Identify functional forms of model equations
• Estimate parameters of model equations
• Code model equations for the computer
• Execute baseline simulation
• Present model equations
4
Phase 3: Model evaluation
• Components of model evaluation
• Model verification
• Model validation
Phase 4: Model application
• Develop and execute experimental design for simulations
• Analyze and interpret simulation results
• Examine additional types of management policies or environmental situations
• Communicate simulation results
* NOTE: The four phases are interactive, meaning that one often has to go back and forth
among the four phases during model development.
System Dynamics and STELLA
1) System Dynamics (SD)
• Founded by Forrester and his associates at MIT in 1950s (Forrester, 1961, 1968)
• Holds that the macro behavior of a system is primarily determined by its internal micro
structure
• Emphasizes the connections among the various parts that constitute a system and applies
feedback principles to model and analyze dynamic problems
• Holds that the core of system structure is composed of feedback loops (positive and
negative) that integrate the three fundamental constituents: state, rate, and information.
• DYNAMO – the computer language born with System Dynamics.
• Based on general systems theory, incorporates cybernetics and information theory, and
has become a unique, powerful simulation modeling methodology
2) STELLA
•
•
•
•
•
STELLA (acronym for Structural Thinking Experiential Learning Laboratory with
Animation)
Icon-oriented and self-debugging systems simulation package
First released in August 1985 by High Performance Systems, Hanover, New Hampshire
Designed to facilitate System Dynamics modeling and make it available for even those
lacking computer experience and mathematical expertise
Awarded as “the best published piece of work in the [SD] field between 1984 and 1989”
(Forrester Award Committee)
5
Material Flow
Source
Level_1
Input Rate
Sink
Output Rate
Auxiliary
~
Table Function
Outflow (solid arrow )
Constant
Information Link
Ghost Variables
Level_2
Biflow
Input Rate
Uniflow
Net Rate Output Rate
Symbols used for structural diagrams in STELLA
6
GENERIC MODULES - BUILDING BLOCKS FOR SYSTEMS MODELS
Quote of the Day
“To do science is to search for repeated patterns,
not simply to accumulate facts..."
- Robert H. MacArthur (1972)
I. WHY BUILDING BLOCKS?
Building blocks, or simulation modules are simple models that represent some basic
system structures and dynamics. These modules are very important for understanding many
fundamental processes common in biological, physical, and socioeconomic phenomena. One
certainly needs to understand them well before attempting to deal with complex feedback
systems. In the same time, the model building blocks demonstrate how these commonly
occurring basic processes are represented in systems simulation, and often become convenient
and effective for constructing complex systems models.
Linear Growth in STELLA
X
II. BUILDING BLOCKS FOR SYSTEMS MODELS
(I) FIRST-ORDER SYSTEMS
dX dt
o First-order: One state variable (or 1 stock)
o Linear: No non-linear combinations of the state variable
of any sort in the algebraic equation of rates.
X(t) = X(t - dt) + (dX_dt) * dt
INIT X = 0
INFLOWS:
dX_dt = 2
1. Linear Change
o One state variable.
o One rate variable.
o The rate of change is a constant, not dependent on the state variable. Thus, there is no
feedback loop.
1: X
(1) Linear Growth
o The rate of change is positive, and thus
represented as an inflow.
o Examples:
o Increase in water level in a tank with a
constant inflow of water into the tank.
o Plant biomass sometimes is found to
increase linearly with certain soil
nutrients.
2: dX dt
1:
2:
30.00
3.00
1:
2:
15.00
2.00
1
2
2
2
2
1
1
1:
2:
0.00
1.00
1
0.00
3.00
Graph 1 (Linear Growth)
7
6.00
Time
9.00
12.00
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(2) Linear Decline
o The rate of change is negative, and thus
represented as an outflow.
o Examples:
Ø Decrease in the amount of materials left in
a stock when the materials are taken out of
the stock in the same amount per unit of
time.
Ø Decrease in crop yield with declining
annual precipitation sometimes exhibits
linear pattern.
Linear Decline in STELLA
X
dX dt
X(t) = X(t - dt) + (- dX_dt) * dt
INIT X = 40
OUTFLOWS:
dX_dt = 2
(3) Linear Growth and Decline
o Combination of the linear growth and decline module.
o One state variable.
o Either two uniflows (one inflow and
one outflow) or one biflow (compare
the structural diagrams below).
o The uniflow version gives more
details, whereas the biflow version is
more concise. The choice of the two
forms is dependent on the degree of
details is needed.
o Linear growth occurs when the total
rate of change (inflow minus outflow) is positive, and linear decline does otherwise.
1: X
1:
2:
2: dX dt
45.00
3.00
1
1
1:
30.00
2:
2.00
2
2
2
2
1
1
1:
2:
15.00
1.00
0.00
3.00
6.00
Graph 1 (Linear Growth)
Linear Growth and Decline in STELLA
(1) Two-uniflow model:
(2) One-biflow model:
X
dX dt in
Y
dX dt out
dY dt
X(t) = X(t - dt) +
Y(t) = Y(t - dt) +
(dX_dt_in- dX_dt_out) * dt
(dY_dt) * dt
INIT X = 0
INIT Y = 40
INFLOWS:
INFLOWS:
dX_dt_in = 2
dY_dt = -1
OUTFLOWS:
dX_dt_out = 1
8
Time
9.00
12.00
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2. Exponential Growth and Decay
Exponential Growth in STELLA
N
(1) Exponential Growth
• Also known as compounding process
• One state variable.
• One rate variable.
dN dt
• One auxiliary variable.
• The rate of change is equal to a constant proportion of
r
the state variable.
N(t) = N(t - dt) + (dN_dt) * dt
• Because the rate variable depends on the state
INIT N = 2
variable itself, they form a positive feedback loop, in
INFLOWS:
which the state variable and the rate of change
dN_dt = r*N
reinforce each other, generating an accelerating, runr = 0.1
away behavior.
• The rate of change is always positive (i.e., the state variable increases monotonically), and is
represented it as an inflow.
• Time Constant Tc: The reciprocal of the constant r in
the exponential model, Tc = 1/r, with the dimension of
units of time.
o For exponential growth, Tc is the time required
for the state variable to become e (= 2.7183)
times of its current value. This can be seen
from the following simple manipulations:
N = N0 e rt .
When t is equal to Tc, we have
N Tc = N 0 e
1
Tc
Tc
= N 0e
In general, if t = nTc,
NnTc = N0 e
1
nTc
Tc
= N0 e n
o Note: For continuous
systems, the time step, ∆t, in
simulation must be smaller
than the smallest time
constant in the model.
1:
2:
1: N
300.00
30.00
1:
2:
150.00
15.00
2: dN dt
2
1
1:
2:
0.00
0.00
1 2
0.00
1
12.50
2
1
25.00
Graph 1: Page 1 (Exponenti… Time
9
2
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50.00
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•
Doubling Time Td: The time required for the state variable in the exponential model to
double in value.
To calculate the doubling time, let N = 2N0 and let t = Td.
Thus, from N = N0 e rt , we have:
2N0 = N 0 erT d
ln 2 = rTd
0.69
Td =
r
or Td = 0.69 Tc
o Therefore, the doubling time for exponential growth is a
constant. It is conversely proportional to the rate of
change: doubling time = 70 / percent natural growth rate.
This is where “the rule of 70” in population demography
comes from.
• Exponential growth is a common type of dynamics that
exists in all different disciplines. Any phenomena
described by words such as “snowball effect”, “vicious
circles”, “virtuous circles”, and “bandwagon effect” can be
represented as a positive feedback loop structure.
Examples include: Savings increase in a bank account due
to interest income; Population growth when resources are
not limiting; Drug addiction; Arms race; etc.
• Super-exponential (or supra-exponential) growth:
o In some positive feedback
systems, doubling time
1: Super N
2: N
1:
100.00
2:
decreases, rather than remain
constant, as the state variable
increases. This means that the
rate of change is a nonlinear
1:
2
51.00
2:
function of the state variable,
2
e.g.:
1
2
dN
1 2
= (rN)N = rN2
1:
2:
2.00
dt
0.00
5.00
10.00
15.00
20.00
Graph 1: Page 1 (Superexponen… Time
1:36 PM 4/8/97
o The dynamics generated by this
nonlinear system, with a rate of
change faster than a fixed proportion of the state variable, is called superexponential
growth.
o The superexponential growth has the same feedback structure as the exponential.
10
Exponential Decay in STELLA
N
(2) Exponential Decay
• Also known as draining process
• 1 state variable; 1 rate variable.
• The rate of change is equal to a constant
dN dt
proportion of the state variable, but in contrast
with exponential growth it is negative.
r
• The rate variable depends on the state variable
N(t) = N(t - dt) + (- dN_dt) * dt
itself, and form a feedback loop between them.
INIT N = 100
Because an increase in the value of the state
OUTFLOWS:
variable increases the rate of change, but an
dN_dt = r*N
increase in the rate decreases the value of the
r = 0.1
state variable, the feedback is negative (or goalseeking).
• Time constant Tc: The reciprocal of the constant r in the exponential model, Tc = 1/r, with
the dimension of units of time.
o For exponential decay, Tc is the time required for the state variable to become e -1 (=
0.368 ) times of its current value, or the time required for 63% of the contents in the
stock to vanish. This is also called the relaxation time, which is sometimes used as a
measure of the speed with which a system is absorbing disturbances.
o Mathematically, relaxation time can be derived as follows:
N = N0 e −rt ,
1:
2:
For a time period from t=0 to t=Tc,
−
NTc = N 0 e
1
Tc
Tc
NnTc = N0 e
•
2: dN dt
1
2
−1
= e N0 = 0.368 N0 ,
1:
2:
In general, when t = nTc, we have
−
1: N
100.00
10.00
1
nTc
Tc
−n
50.00
5.00
1
n
= e N0 = 0.368 N 0
2
1:
2:
0.00
0.00
1
2
Half-life Th: The time required for the
state variable to reduce its current value
by a half. It is an analog of the
doubling time in exponential growth. It can be calculated as follows:
1
N0 = N 0 e− rTh ,
2
1
ln = −rTh , or ln 2 = rTh , thus,
2
0.69
Th =
or Th = 0.69 Tc
r
o Therefore, the half-life time for exponential decay is a constant (about 70% of the
time constant), independent of the value of the state variable.
0.00
12.50
25.00
Graph 1: Page 1 (Exponentia… Time
11
1
2
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Exponential Growth/Decline in STELLA
•
N
Examples:
o Draining water from a tank
o Population-death rate process
o Redioactive decay process
(3) Exponential Growth and Decay Combined
o Simply a combination of exponential
growth and exponential decay.
o The behavior of the first-order linear
system exhibits three different patterns:
1) exponential growth when growth
constant is larger than decay
constant;
2) exponential decay when growth
constant is smaller than decay
constant; and
N(t) = N(t -­‐ dt) + (inflow -­‐ outflow) * dt INIT N = 150 INFLOWS: inflow = growth_constant*N OUTFLOWS: outflow = decay_constant*N decay_constant = 0.1 growth_constant = 0.1 Exponential Collapse in STELLA
3. Exponential Collapse
• Also known as accelerated decay, indicating
that the rate of change gets faster as the level
goes down.
• A simple exponential collapse model may
consist of 1 state variable, 1 outflow, and 2
constants.
• A positive feedback: increasing level -->
increasing rate of change --> increasing
level --> ...
• A simple mathematical model for
exponential collapse:
1:
dN
= −r( M − N)
dt
where N (< M) is the state variable, and
r and M are two constants.
1:
Examples:
o Population shrinking when smaller
than MVP (minimum viable
population).
o Change in the interactive force
decay constant
growth constant
3) remaining unchanged when the two constants are equal. •
outflow
inflow
level
outflow
M
collapse constant
level(t) = level(t - dt) + (- outflow) * dt
INIT level = 99
OUTFLOWS:
outflow = collapse_constant*(M-level)
collapse_constant = 0.15
M = 100
1: N
300.00
2: N
3: N
2
3
150.00
3
3
3
1
2
1
1:
0.00
1
0.00
1
12.50
25.00
Graph 1: Page 1 (Exponenti… Time
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2
2
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50.00
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between molecules when water is heated up and boils.
o Decline of species diversity with increasing habitat fragmentation.
o Some other threshold phenomena in physical and biological processes may exhibit
behavior that resembles exponential collapse.
4. Exponential Growth and Collapse Combined
• A simple exponential collapse consists of 1 state variable, 1 bi-directional flow, and two
constants.
• This simple model can be mathematically expressed as:
dN
= r(N − M)
dt
Exponential Growth/Collapse in STELLA
N
dN dt
r
M
level(t) = level(t - dt) + (- outflow)*dt
INIT level = 99
OUTFLOWS:
outflow = collapse_constant*(M-level)
collapse_constant = 0.15
M = 100
•
The simple structure exhibits 3 different
behavioral patterns:
o Exponential growth when N > M;
o Exponential collapse when N < M;
and
o Remaining constant when N = M.
13
Simple Goal-Seeking in STELLA
level
5. Goal-Seeking Behavior
(1) Simple Goal-Seeking
• A simple goal-seeking model may consist of 1
state variable, 1 biflow, and 2 constants.
• The state variable and the flow form a negative
feedback.
• A simple goal-seeking model may take the form:
dL
= c(G − L)
dt
where L is the level, G is the goal, and c is a constant.
• Apparently, exponential decay is a special case of
goal-seeking behavior, in which the goal is zero!
Inflow
constant
goal
discrepancy
level(t) = level(t - dt) + (Inflow) * dt
INIT level = 2
INFLOWS:
Inflow = constant*discrepancy
constant = 0.1
discrepancy = goal-level
goal = 100
1: level
(2) S-Shaped Growth
1:
200.00
• Also known as logistic growth or
sigmoidal growth.
• It is a first-order nonlinear system (see
1:
100.00 1
the Rate-Level graphs).
• The state variable and the flow form a
negative feedback that ultimately gives
rise to the goal seeking behavior (see
1:
0.00
0.00
the Forrester diagram).
• It may take different forms (uniflow or
biflow versions). [For example, in population
regulation, crowding may affect only the per
capita birth rate, or only the per capita death
rate, or both.]
2: level
3: level
2
2
1
2
1
3
3
1
2
3
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25.00
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Graph 1: Page 1 (Simple goa… Time
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level
Inflow
~
Rate Value
level(t) = level(t - dt) + (Inflow) * dt
INIT level = 300
INFLOWS:
Inflow = Rate_Value
Rate_Value = GRAPH(level)...
Rate-Level Graph
The behavior of this simple structure has three patterns:
1) Increasing and approaching the goal if simulation starts with a value of the level
smaller than the goal;
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2) Decreasing and approaching the goal if simulation starts with a value of the level
larger than the goal;
3) Remaining unchanged if simulation starts with the value of the level equal to the goal.
1:
1: level
300.00
2: level
3: level
3
2
1:
2
3
2
150.00
3
1
2
3
1
1
0.00 1
0.00
1:
12.50
25.00
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A familiar example of sigmoidal growth is the logistic equation:
dN
= rN(1 − N / K)
dt
This is a first-order nonlinear differential equation. The following is an equivalent simple
systems model.
N(t) = N(t - dt) +
(dN_dt) * dt
INIT N = 2
N
1: N v. dN dt
5.00
INFLOWS:
dN_dt = r*N*(K-N)/K
K = 100
r = 0.2
dN dt
r
2.50
0.00
K
0.00
50.00
1: N
1: N
1:
100.00
Graph 1: Page 2 (Untitled Graph) N
2: N
1:
2:
3: N
10:46 PM
4/10/97
2: dN dt
100.00
5.00
1
200.00
2
1
2
1:
2
3
100.00
3
2
3
1
2
1:
50.00
2:
2.50
2
3
1
1
2
1
1:
0.00
1
0.00
12.50
25.00
Graph 1: Page 3 (Untitled Graph)
Time
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1:
0.00
2:
0.00
2
1
0.00
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Dynamics from the Logistic Equation-Based Module
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