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**System Dynamics – 1ZM65 Lecture 4 September 23, 2014**

Dr. 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 4/15/2017 PAGE 1

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**Recap 3: Examples of stocks and flows with their units of measure**

‘’the snapshot test’’ 4/15/2017 4/15/2017 PAGE 2

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**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 4/15/2017 4/15/2017 PAGE 3

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**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 4/15/2017

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**Analytical Integration of flows recap**

4/15/2017

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**Quadratic versus cosine function**

4/15/2017

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**Quadratic versus cosine function**

4/15/2017

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**Chapter 8: Growth and goal seeking: structure and behavior**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**First order, linear positive feedback system: structure and examples**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**positive feedback rabbits**

growth=birthrate*population 4/15/2017

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**analytical expression positive feedback**

4/15/2017

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**Exponential growth over different time horizons**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**First-order linear negative feedback: structure and examples**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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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 4/15/2017

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**Phase plot for exponential decay via linear negative feedback**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**analytical expression negative feedback**

4/15/2017

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**Exponential decay: structure (phase plot) and behavior (time plot)**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**First-order linear negative feedback system with explicit goals**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**Phase plot for first-order linear negative feedback system with explicit goal**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**analytical expression negative feedback with explicit goal**

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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. 4/15/2017

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**Relationship between time constant and the fraction of the gap remaining**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**Sketch the trajectory for the workforce and net hiring rate**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**© J.S. Sterman, MIT, Business Dynamics, 2000**

A linear first-order system can generate only growth, equilibrium, or decay 4/15/2017

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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? 4/15/2017

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**Example First Order DE How to model this in Vensim?**

P’+3P=12 4/15/2017

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**Diagram for population growth in a capacitated environment**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**Nonlinear relationship between population density and the fractional growth rate.**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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**Phase plot for nonlinear population system**

© J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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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) 4/15/2017

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**Analysis logistic model**

First order non-linear model Making partial fractions 4/15/2017

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**Simulation of logistic model**

Net Birth Rate= g* (1-P/C) * P 4/15/2017

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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 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. 4/15/2017

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**Instruction Week 4 26-Sep 15:45-17:30 PAV B2 Vensim Tutorial**

Mohammadreza Zolfagharian MSc 4/15/2017

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Questions? 4/15/2017 PAGE 35

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