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

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

Recap 3: Examples of stocks and flows with their units of measure ‘’the snapshot test’’ 4/15/2017 4/15/2017 PAGE 2

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

Inflows and outflows for a hypothetical stock recap Challenge p. 239 5 10 15 0 Inflows and outflows for a hypothetical stock recap Challenge p. 239 © J.S. Sterman, MIT, Business Dynamics, 2000 sketch analytically VENSIM 5 10 15 20 4/15/2017

Analytical Integration of flows recap 4/15/2017

Quadratic versus cosine function 4/15/2017

Quadratic versus cosine function 4/15/2017

Chapter 8: Growth and goal seeking: structure and behavior © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

First order, linear positive feedback system: structure and examples © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

positive feedback rabbits growth=birthrate*population 4/15/2017

analytical expression positive feedback 4/15/2017

Exponential growth over different time horizons © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

First-order linear negative feedback: structure and examples © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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

Phase plot for exponential decay via linear negative feedback © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

analytical expression negative feedback 4/15/2017

Exponential decay: structure (phase plot) and behavior (time plot) © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

First-order linear negative feedback system with explicit goals © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

Phase plot for first-order linear negative feedback system with explicit goal © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

analytical expression negative feedback with explicit goal 4/15/2017

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

Relationship between time constant and the fraction of the gap remaining © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

Sketch the trajectory for the workforce and net hiring rate © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

© J.S. Sterman, MIT, Business Dynamics, 2000 A linear first-order system can generate only growth, equilibrium, or decay 4/15/2017

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

Example First Order DE How to model this in Vensim? P’+3P=12 4/15/2017

Diagram for population growth in a capacitated environment © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

Nonlinear relationship between population density and the fractional growth rate. © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

Phase plot for nonlinear population system © J.S. Sterman, MIT, Business Dynamics, 2000 4/15/2017

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

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

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

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

Instruction Week 4 26-Sep 15:45-17:30 PAV B2 Vensim Tutorial Mohammadreza Zolfagharian MSc 4/15/2017

Questions? 4/15/2017 PAGE 35