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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