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

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PAGE 1 5/11/2015 PAGE 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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PAGE 2 5/11/2015 Recap 3: Examples of stocks and flows with their units of measure ‘’the snapshot test’’ 5/11/2015PAGE 2

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PAGE 3 5/11/2015 Recap 3 Stocks : integrating flows PAGE 35/11/2015 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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PAGE 4 5/11/ 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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PAGE 5 5/11/2015 Analytical Integration of flows recap

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Quadratic versus cosine function PAGE 6 5/11/2015

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Quadratic versus cosine function PAGE 7 5/11/2015

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PAGE 8 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Chapter 8: Growth and goal seeking: structure and behavior

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PAGE 9 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 First order, linear positive feedback system: structure and examples

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PAGE 10 5/11/2015 positive feedback rabbits growth=birthrate*population

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PAGE 11 5/11/2015 analytical expression positive feedback

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PAGE 12 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Exponential growth over different time horizons

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PAGE 13 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 First-order linear negative feedback: 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 PAGE 14 5/11/2015

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PAGE 15 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Phase plot for exponential decay via linear negative feedback

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PAGE 16 5/11/2015 analytical expression negative feedback

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PAGE 17 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Exponential decay: structure (phase plot) and behavior (time plot)

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PAGE 18 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 First-order linear negative feedback system with explicit goals

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PAGE 19 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Phase plot for first-order linear negative feedback system with explicit goal

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PAGE 20 5/11/2015 analytical expression negative feedback with explicit goal

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PAGE 21 5/11/2015 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. © J.S. Sterman, MIT, Business Dynamics, 2000 Exponential approach towards a goal

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PAGE 22 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Relationship between time constant and the fraction of the gap remaining

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PAGE 23 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Sketch the trajectory for the workforce and net hiring rate

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PAGE 24 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 A linear first- order 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? PAGE 25 5/11/2015 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? PAGE 26 5/11/2015 P’+3P=12

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PAGE 27 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Diagram for population growth in a capacitated environment

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PAGE 28 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Nonlinear relationship between population density and the fractional growth rate.

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PAGE 29 5/11/2015 © J.S. Sterman, MIT, Business Dynamics, 2000 Phase plot for nonlinear population system

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5/11/2015 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*P 2 /C Maximum growth P inf =C/2 ( differentiating over P) PAGE 30

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5/11/2015 Analysis logistic model First order non-linear model PAGE 31 Making partial fractions

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5/11/2015 Simulation of logistic model Net Birth Rate= g* (1-P/C) * P PAGE 32

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5/11/2015 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. Logistic growth in action PAGE 33

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PAGE 34 5/11/2015 Instruction Week 426-Sep15:45- 17:30 PAV B2Vensim Tutorial Mohammadreza Zolfagharian MSc

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PAGE 35 5/11/2015 Questions? PAGE 35

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