system dynamics – 1zm65 lecture 4 september 23, 2014 dr. ir. n.p. dellaert
TRANSCRIPT
System Dynamics – 1ZM65
Lecture 4
September 23, 2014Dr. Ir. N.P. Dellaert
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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 3Stocks : integrating flows
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Stock (t ) = [Inflow (s ) - Outflow (s )]ds + Stock (t 0)t0
t
In mathematical terms, stocks are an integration of the flowsBecause of the step size, Vensim is in fact using a summation in stead of an integration:
0( ) /
0 0 00
( ) [inflow( ) outflow( )] ( )t t step
n
stock t t n step t n step step stock t
Integration is in fact equivalent to finding the area of a region
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5 10 15 20
Inflows and outflows for a hypothetical stock recap
Challenge p. 239© J.S. Sterman, MIT, Business Dynamics, 2000
•sketch
•analytically
•VENSIM
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Analytical Integration of flows recap Stock ( t ) = [ Inflow ( u ) - Outflow ( u )] du + Stock ( t 0 )
t 0
t
( )
( )
( ) sin( )
( ) a
flow u c
flow u c u
flow u c u
flow u c u
01
0
02
0
)1/()(
)cos()(
2/)(
)(
satctstock
stcctstock
stctstock
stctstock
a
Quadratic versus cosine function
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Quadratic versus cosine function
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© J.S. Sterman, MIT, Business Dynamics, 2000
Chapter 8: Growth and goal seeking: structure and behavior
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© J.S. Sterman, MIT, Business Dynamics, 2000
First order, linear positive feedback system: structure and examples
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positive feedback rabbits
growth=birthrate*population
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analytical expression positive feedback
( ) ( )flow t g stock t
gteSS
ctgS
dtgS
dS
gdtS
dS
0
)ln(
d(Stock)/dt = Net Change in Stock = Inflow(t) – Outflow(t)
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0
1000
0 200 400 600 800 1000
Time Horizon = 10td
0
1 10 30
0 2000 4000 6000 8000 10000
Time Horizon = 100td
0
2
0 2 4 6 8 10
Time Horizon = 0.1td
0
2
0 20 40 60 80 100
Time Horizon = 1td
© J.S. Sterman, MIT, Business Dynamics, 2000Exponential growth over different time horizons
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© J.S. Sterman, MIT, Business Dynamics, 2000
First-order linear negative feedback: structure and examples
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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© J.S. Sterman, MIT, Business Dynamics, 2000
Phase plot for exponential decay via linear negative feedback
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analytical expression negative feedback
( ) ( )flow t g stock t
gteSS
ctgS
dtgS
dS
gdtS
dS
0
)ln(
d(Stock)/dt = Net Change in Stock = Inflow(t) – Outflow(t)
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-5
0
0 20 40 60 80 100
Structure
State of System (units)
t = 0
t = 10
t = 20
t = 3 0
t = 40
0
50
100
0
5
10
0 20 40 60 80 100
Behavior
State of the System(left scale)
Net Inflow(right scale)
Time
© J.S. Sterman, MIT, Business Dynamics, 2000
Exponential decay: structure (phase plot) and behavior (time plot)
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© J.S. Sterman, MIT, Business Dynamics, 2000
First-order linear negative feedback system with explicit goals
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© J.S. Sterman, MIT, Business Dynamics, 2000
Phase plot for first-order linear negative feedback system with explicit goal
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analytical expression negative feedbackwith explicit goal
*( ) ( ( )) /flow t S stock t AT /
1 2
/2
*1
*2 0
* * /0
:
then the flow would be :
( 1 ) /
:
( )
( )
t AT
t AT
t AT
Suppose S c c e
cdSe c S AT
dt AT
so c S
and c S S
or S S S S e
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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.
0
100
200
0 20 40 60 80 100
© J.S. Sterman, MIT, Business Dynamics, 2000
Exponential approach towards a goal
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© J.S. Sterman, MIT, Business Dynamics, 2000
Relationship between time constant and the fraction of the gap remaining
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© J.S. Sterman, MIT, Business Dynamics, 2000
Sketch the trajectory for the workforce and net hiring rate
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© J.S. Sterman, MIT, Business Dynamics, 2000
A linear first-order system can generate only growth, equilibrium, or decay
Example First order differential equation
• Suppose the behaviour of a population is described as:
P’+3P=12
What can you say about P?
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• 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?
Example First Order DEHow to model this in Vensim?
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Populationinflow outflow
constant (12)
3*Population
P’+3P=12
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© J.S. Sterman, MIT, Business Dynamics, 2000
Diagram for population growth in a capacitated environment
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0
Population/Carrying Capacity(dimensionless)
Fractional Birth Rate Fractional
Death Rate
Fractional Net Birth Rate
0 1
© J.S. Sterman, MIT, Business Dynamics, 2000
Nonlinear relationship between population density and the fractional growth rate.
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0
Population/Carrying Capacity(dimensionless)
Birth Rate
Death Rate
Net Birth Rate
0 Stable EquilibriumUnstable
Equilibrium
Positive Feedback Dominant
•• (P/C)inf 1
Negative FeedbackDominant
© J.S. Sterman, MIT, Business Dynamics, 2000
Phase plot for nonlinear population system
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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
dt
dPPC
PgRateBirthNet )1( *
dtgPC
PdP
*
)1(
dtgdP
PCP*11
)exp()1(1)(
)exp(
)ln()ln()ln()ln(
*
0
0
*0
00*
tgPC
CtP
PC
tgP
PC
P
PCPtgPCP
First order non-linear model
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Making partial fractions
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Simulation of logistic model
PNet Birth Rate
C
g*
Net Birth Rate= g* (1-P/C) * P
Graph for P100
75
50
25
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
P : Current
Graph for Net Birth Rate4
3
2
1
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
Net Birth Rate : Current
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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 S-shaped 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.
0
Population/Carrying Capacity(dimensionless)
0 1
g*
0
Population/Carrying Capacity(dimensionless)
Stable EquilibriumUnstable
Equilibrium
Positive Feedback Dominant
•• (P/C)inf
= 0.5
NegativeFeedback Dominant
0 1
0.0
0.5
1.0
-4 -2 0 2 40
0.25
Population(Left Scale)
Net Growth Rate(Right Scale)
Time
PC
= 11 + exp[-g*(t - h)]
g* = 1, h = 0
Logistic growth in action
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Instruction
Week 4 26-Sep 15:45-17:30
PAV B2
Vensim Tutorial
Mohammadreza Zolfagharian MSc
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Questions?
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