System Dynamics – 1ZM65
Lecture 4
September 23, 2014
Dr. Ir. N.P. Dellaert
Agenda
• Recap of Lecture 3
• Dynamic behavior of basic systems
• exponential growth
• growth towards a limit
• S-shaped growth
• If time left, some Vensim
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Recap 3:
Examples of stocks and flows with their units of
measure
‘’the snapshot
test’’
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Recap 3
Stocks : integrating flows
t
Stock (t ) =
[Inflow (s ) - Outflow (s )]ds + Stock (t 0)
t0
In mathematical terms, stocks are an integration of the flows
Because of the step size, Vensim is in fact using a summation
in stead(t of
an integration:
t ) / step
stock (t ) 

0
n 0
[inflow(t0  n  step)  outflow(t0  n  step)]  step  stock (t0 )
Integration is in fact equivalent to finding the area of a region
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Inflows and outflows for a hypothetical stock recap
Challenge p. 239
© J.S. Sterman, MIT, Business Dynamics, 2000
•sketch
•analytically
•VENSIM
5
10
15
20
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Analytical Integration of flows recap
t
Stock (t ) =
[ Inflow (u) - Outflow (u)] du + Stock (t 0)
t0
flow(u )  c
flow(u )  c  u
flow(u )  c  sin(u )
stock(t )  c  t  s0
flow(u )  c  u
stock(t )  c  t a 1 /( a  1)  s0
a
stock(t )  c  t 2 / 2  s0
stock(t )  c  c  cos(t )  s0
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Quadratic versus cosine function
1.2
1
0.8
0.6
quadratic
Cosine
0.4
0.2
0
-1.5
-1
-0.5
0
0.5
1
1.5
-0.2
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Quadratic versus cosine function
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Chapter 8: Growth and goal seeking:
structure and behavior
© J.S. Sterman, MIT, Business Dynamics, 2000
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First order, linear positive feedback system:
structure and examples
© J.S. Sterman, MIT, Business Dynamics, 2000
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positive feedback rabbits
growth=birthrate*population
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analytical expression positive feedback
d(Stock)/dt = Net Change in Stock = Inflow(t) – Outflow(t)
flow(t )  g  stock (t )
dS
 gdt
S
dS
 S   gdt
ln(S )  g  t  c
S  S 0  e gt
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© J.S. Sterman, MIT, Business Dynamics, 2000
Exponential growth over different time horizons
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First-order linear negative feedback:
© J.S. Sterman, MIT, Business Dynamics, 2000
structure and examples
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Phase plots
Phase plots show relation between the state of a
system and the rate of change
Can be used to find equilibria
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Phase plot for exponential decay via linear
negative feedback
© J.S. Sterman, MIT, Business Dynamics, 2000
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analytical expression negative feedback
d(Stock)/dt = Net Change in Stock = Inflow(t) – Outflow(t)
flow(t )   g  stock (t )
dS
  gdt
S
dS
 S    gdt
ln(S )   g  t  c
S  S 0  e  gt
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Exponential decay: structure (phase plot) and
behavior (time plot)
© J.S. Sterman, MIT, Business Dynamics, 2000
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First-order linear negative feedback system
with explicit goals
© J.S. Sterman, MIT, Business Dynamics, 2000
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Phase plot for first-order linear negative
feedback system with explicit goal
© J.S. Sterman, MIT, Business Dynamics, 2000
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analytical expression negative feedback
with explicit goal
flow(t )  (S *  stock (t )) / AT
Suppose : S  c1  c2 e  t / AT
then the flow would be :
c2  t / AT
dS

e
 (c1  S ) / AT
dt
AT
so : c1  S *
and c2  ( S0  S * )
or
S  S *  ( S0  S * )e  t / AT
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Exponential approach towards a goal
© J.S. Sterman, MIT, Business Dynamics, 2000
The goal is 100 units. The upper curve begins with S(0) = 200; the lower curve
begins with s(0) = 0. The adjustment time in both cases is 20 time units.
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Relationship between time constant and the
fraction of the gap remaining
© J.S. Sterman, MIT, Business Dynamics, 2000
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Sketch the trajectory for the workforce and net
hiring rate
© J.S. Sterman, MIT, Business Dynamics, 2000
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© J.S. Sterman, MIT, Business Dynamics, 2000
A linear firstorder system
can generate
only growth,
equilibrium, or
decay
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Example First order differential equation
• Suppose the behaviour of a population is described
as:
P’+3P=12
What can you say about P?
• For solving mathematically you first solve homogeneous equation
P’+3P=0 and then adapt the constants
• For finding an explicit solution more information is needed:
P(0) !
• Without solving explicitly we can say something about the limiting behavior:
the population will (neg.) exponentially grow to 12/3=4!
How to model this in Vensim?
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Example First Order DE
How to model this in Vensim?
P’+3P=12
Population
inflow
outflow
constant (12)
3*Population
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Diagram for population growth in a
capacitated environment
© J.S. Sterman, MIT, Business Dynamics, 2000
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Nonlinear relationship between population
density and the fractional growth rate.
© J.S. Sterman, MIT, Business Dynamics, 2000
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Phase plot for nonlinear population system
© J.S. Sterman, MIT, Business Dynamics, 2000
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Logistic growth model (Ch 9)
• General case:
• Fractional birth and death rate are functions of ratio
population P and carrying capacity C
• Example b(t)=aP*(1-0.25P/C) en d(t)= b*P*(1+P/C).
• Logistic growth is special case with:
• Net growth rate=g*P-g*(P/C)*P  g*P-g*P2/C
• Maximum growth
Pinf=C/2
(differentiating over P)
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Analysis logistic model
dP
Net BirthRate  g * (1  P ) P 
C
dt
dP
 g *dt
First order
(1  P ) P
C
non-linear model
1 
1
*
 P  C  P  dP  g dt
Making partial fractions






ln(P )  ln(C  P )  g *t  ln(P0 )  ln(C  P0 )
P0 exp( g *t )
P

CP
C  P0
P (t ) 
C
1  (C
P0
 1) exp( g *t )
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Simulation of logistic model
Net Birth Rate= g* (1-P/C) * P
P
Net Bir th Rat e
C
g*
Graph for P
Graph for Net Birth Rate
4
100
3
75
2
50
1
25
0
0
0
10
20
30
Net Bir th Rate : Curr ent
40
50
60
T im e (M on t h )
70
80
90
100
0
10
20
30
40
50
60
T im e (M on t h )
70
80
90
100
P : Curr en t
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Logistic growth in action
Figure 9-1 Top: The fractional growth rate declines linearly as population grows. Middle: The
phase plot is an inverted parabola, symmetric about (P/C) = 0.5 Bottom: Population follows an Sshaped curve with inflection point at (P/C) =0.5; the net growth rate follows a bell-shaped curve
with a maximum value of 0.25C per time period.
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Instruction
Week 4
26-Sep
15:4517:30
PAV B2 Vensim
Tutorial
Mohammadreza
Zolfagharian MSc
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Questions?
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