Dynamics of Simple Structures: S-Shaped Growth Jayendran - - PowerPoint PPT Presentation
Dynamics of Simple Structures: S-Shaped Growth Jayendran Venkateswaran IEOR, IITBombay Growth limited by capacity p Consider scenario + + Net Birth Rate Population + + - + Effective Fractional Birth Rate - Population/ Carrying
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
p Consider scenario p What will be expected behavior of system? p Plot the expected rate-level graph p How can we expect Fractional birth rate to
Population Net Birth Rate Carrying Capacity Population/ Carrying Capacity Fractional Birth Rate Effective Fractional Birth Rate +
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IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
p Assume carrying capacity is fixed (C=1000) p P0=2; b = 0.2 p Net Rate = b.(1-P/C).P
Population Net Birth Rate Carrying Capacity Population/ Carrying Capacity Fractional Birth Rate Effective Fractional Birth Rate +
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Population (P) Net Birth Rate fractional birth rate (b) Carrying Capacity (C) Population Capacity Ratio Effective frac birth rate + + + +
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
pSimulate the SFD model
n What behavior pattern do you observe? n What is the stable population size? n When does population reach stability? n When is the ‘inflection point’?
p Tips: Make Custom Graphs in Vensim p Click Control Panel à Graph
n Rate-Level graph (Double click to view) n Eff_Brate graph (Double click to view)
p What happens when P0 = 2000?
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
p Let’s include Death Rate p What will be expected behavior of system? p Plot the expected Rate-Level graph p How can we expect Fractional Rates to change with
Population Net Birth Rate Carrying Capacity Population/ Carrying Capacity Fractional Birth Rate Effective Fractional Birth Rate +
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Death Rate
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IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
p Incorporate Deaths in your SFD model
n Fractional death rate, d = 0.07 n Death Rate = d.P
p Simulate the SFD model
n What behavior pattern do you observe? n What is the stable population size? n When does population reach stability? n When is the ‘inflection point’?
p Model NetRate = Births - Deaths p In Vensim, use Control Panel à Graph to
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
p Suppose other limiting factors, such as
p What will be expected behavior of system? p Plot the expected Rate-Level graph p How can we expect Fractional Rates to
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
p Update your SFD model
n Create new variable effective-fractional-death-rate n Fractional death rate, d = 0.07 n Death Rate = d.(P/C).P
p Simulate the SFD model
n What behavior pattern do you observe? n What is the stable population size? n When does population reach stability? n When is the ‘inflection point’?
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
p The behavior we disucssed is known as S-
p S-shaped pattern exhibited by
n Population trends of many animals & plants n Learning curves n Diffusion of news, riots, epidemics, & rumors n Growth of new products, & other socio-
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
Source: Sterman, John D. Business Dynamics (Fig 4-9)
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
Source: http://www.micro.ute xas.edu/courses/levin /bio304/com&pop/co mmunities.html
Tasmanian Sheep
http://www.life.umd. edu/classroom/biol1 06h/L28/L28_popeco .html
IEOR, IIT Bombay IE 604: System Dynamics Modelling & Analysis Jayendran Venkateswaran
p S-shaped or logistic or sigmoid growth
n The time path includes two distinct regions n General behavior
p Two types of generic structures can result in
n First structure used to model growth with
n Second structure derived from systems involving