Report

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 4/8/2015 PAGE 1 Recap 3: Examples of stocks and flows with their units of measure ‘’the snapshot test’’ 4/8/2015 4/8/2015 PAGE 2 Recap 3 Stocks : integrating flows t Stock (t ) = [Inflow (s ) - Outflow (s )]ds + Stock (t 0) t0 In mathematical terms, stocks are an integration of the flows Because of the step size, Vensim is in fact using a summation in stead(t of an integration: t ) / step stock (t ) 0 n 0 [inflow(t0 n step) outflow(t0 n step)] step stock (t0 ) Integration is in fact equivalent to finding the area of a region 4/8/2015 4/8/2015 PAGE 3 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 4/8/2015 PAGE 4 Analytical Integration of flows recap t Stock (t ) = [ Inflow (u) - Outflow (u)] du + Stock (t 0) t0 flow(u ) c flow(u ) c u flow(u ) c sin(u ) stock(t ) c t s0 flow(u ) c u stock(t ) c t a 1 /( a 1) s0 a stock(t ) c t 2 / 2 s0 stock(t ) c c cos(t ) s0 4/8/2015 PAGE 5 Quadratic versus cosine function 1.2 1 0.8 0.6 quadratic Cosine 0.4 0.2 0 -1.5 -1 -0.5 0 0.5 1 1.5 -0.2 4/8/2015 PAGE 6 Quadratic versus cosine function 4/8/2015 PAGE 7 Chapter 8: Growth and goal seeking: structure and behavior © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 8 First order, linear positive feedback system: structure and examples © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 9 positive feedback rabbits growth=birthrate*population 4/8/2015 PAGE 10 analytical expression positive feedback d(Stock)/dt = Net Change in Stock = Inflow(t) – Outflow(t) flow(t ) g stock (t ) dS gdt S dS S gdt ln(S ) g t c S S 0 e gt 4/8/2015 PAGE 11 © J.S. Sterman, MIT, Business Dynamics, 2000 Exponential growth over different time horizons 4/8/2015 PAGE 12 First-order linear negative feedback: © J.S. Sterman, MIT, Business Dynamics, 2000 structure and examples 4/8/2015 PAGE 13 Phase plots Phase plots show relation between the state of a system and the rate of change Can be used to find equilibria 4/8/2015 PAGE 14 Phase plot for exponential decay via linear negative feedback © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 15 analytical expression negative feedback d(Stock)/dt = Net Change in Stock = Inflow(t) – Outflow(t) flow(t ) g stock (t ) dS gdt S dS S gdt ln(S ) g t c S S 0 e gt 4/8/2015 PAGE 16 Exponential decay: structure (phase plot) and behavior (time plot) © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 17 First-order linear negative feedback system with explicit goals © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 18 Phase plot for first-order linear negative feedback system with explicit goal © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 19 analytical expression negative feedback with explicit goal flow(t ) (S * stock (t )) / AT Suppose : S c1 c2 e t / AT then the flow would be : c2 t / AT dS e (c1 S ) / AT dt AT so : c1 S * and c2 ( S0 S * ) or S S * ( S0 S * )e t / AT 4/8/2015 PAGE 20 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. 4/8/2015 PAGE 21 Relationship between time constant and the fraction of the gap remaining © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 22 Sketch the trajectory for the workforce and net hiring rate © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 23 © J.S. Sterman, MIT, Business Dynamics, 2000 A linear firstorder system can generate only growth, equilibrium, or decay 4/8/2015 PAGE 24 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? 4/8/2015 PAGE 25 Example First Order DE How to model this in Vensim? P’+3P=12 Population inflow outflow constant (12) 3*Population 4/8/2015 PAGE 26 Diagram for population growth in a capacitated environment © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 27 Nonlinear relationship between population density and the fractional growth rate. © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 28 Phase plot for nonlinear population system © J.S. Sterman, MIT, Business Dynamics, 2000 4/8/2015 PAGE 29 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) PAGE 30 4/8/2015 Analysis logistic model dP Net BirthRate g * (1 P ) P C dt dP g *dt First order (1 P ) P C non-linear model 1 1 * P C P dP g dt Making partial fractions ln(P ) ln(C P ) g *t ln(P0 ) ln(C P0 ) P0 exp( g *t ) P CP C P0 P (t ) C 1 (C P0 1) exp( g *t ) PAGE 31 4/8/2015 Simulation of logistic model Net Birth Rate= g* (1-P/C) * P P Net Bir th Rat e C g* Graph for P Graph for Net Birth Rate 4 100 3 75 2 50 1 25 0 0 0 10 20 30 Net Bir th Rate : Curr ent 40 50 60 T im e (M on t h ) 70 80 90 100 0 10 20 30 40 50 60 T im e (M on t h ) 70 80 90 100 P : Curr en t PAGE 32 4/8/2015 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 Sshaped 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. PAGE 33 4/8/2015 Instruction Week 4 26-Sep 15:4517:30 PAV B2 Vensim Tutorial Mohammadreza Zolfagharian MSc 4/8/2015 PAGE 34 Questions? 4/8/2015 PAGE 35