0. System Dynamics – 1ZM65 Lecture 4 September 23, 2014
Dr. Ir. N.P. Dellaert

1. 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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2. Recap 3: Examples of stocks and flows with their units of measure
‘’the snapshot
test’’
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3. Recap 3 Stocks : integrating flows
In mathematical terms, stocks are an integration of the flows
Because of the step size, Vensim is in fact using a summation
in stead of an integration:
Integration is in fact equivalent to finding the area of a region
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4. Inflows and outflows for a hypothetical stock recap Challenge p. 239
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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5. Analytical Integration of flows recap
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6. Quadratic versus cosine function
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7. Quadratic versus cosine function
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8. Chapter 8: Growth and goal seeking: structure and behavior
© J.S. Sterman, MIT, Business Dynamics, 2000
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9. First order, linear positive feedback system: structure and examples
© J.S. Sterman, MIT, Business Dynamics, 2000
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10. positive feedback rabbits
growth=birthrate*population
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11. analytical expression positive feedback
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12. Exponential growth over different time horizons
© J.S. Sterman, MIT, Business Dynamics, 2000
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13. First-order linear negative feedback: structure and examples
© J.S. Sterman, MIT, Business Dynamics, 2000
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14. 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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15. Phase plot for exponential decay via linear negative feedback
© J.S. Sterman, MIT, Business Dynamics, 2000
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16. analytical expression negative feedback
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17. Exponential decay: structure (phase plot) and behavior (time plot)
© J.S. Sterman, MIT, Business Dynamics, 2000
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18. First-order linear negative feedback system with explicit goals
© J.S. Sterman, MIT, Business Dynamics, 2000
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19. Phase plot for first-order linear negative feedback system with explicit goal
© J.S. Sterman, MIT, Business Dynamics, 2000
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20. analytical expression negative feedback with explicit goal
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21. 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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22. Relationship between time constant and the fraction of the gap remaining
© J.S. Sterman, MIT, Business Dynamics, 2000
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23. Sketch the trajectory for the workforce and net hiring rate
© J.S. Sterman, MIT, Business Dynamics, 2000
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24. © J.S. Sterman, MIT, Business Dynamics, 2000
A linear first-order system can generate only growth, equilibrium, or decay
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25. 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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26. Example First Order DE How to model this in Vensim?
P’+3P=12
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27. Diagram for population growth in a capacitated environment
© J.S. Sterman, MIT, Business Dynamics, 2000
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28. Nonlinear relationship between population density and the fractional growth rate.
© J.S. Sterman, MIT, Business Dynamics, 2000
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29. Phase plot for nonlinear population system
© J.S. Sterman, MIT, Business Dynamics, 2000
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30. 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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31. Analysis logistic model
First order
non-linear model
Making partial fractions
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32. Simulation of logistic model
Net Birth Rate= g* (1-P/C) * P
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33. 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 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.
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34. Instruction Week 4 26-Sep 15:45-17:30 PAV B2 Vensim Tutorial
Mohammadreza Zolfagharian MSc
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35. Questions?
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