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Mathematics Workshop

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Mathematics Workshop

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    1. Angela Shiflet, Ph. D. Wofford College Robert Panoff, Ph. D. The Shodor Education Foundation Supercomputing 2007 November 12, 2007 Mathematics Workshop

    2. Modeling To analyze complex, real-world problems in order to predict what might happen Examples Behavior of disease in patient HIV Spread of disease Flu Ecological studies

    3. Mathematica in Calculus Notebooks Interactive lectures Sec. 2.1 - Instantaneous velocity Sec. 5.1 - How to measure distance traveled www.wofford.edu/ecs/ Link to text Link to Files to Download http://www.shodor.org/talks/ncsi/vensim/ http://www.shodor.org/talks/ncsi/excel/

    4. Systems dynamics modeling Change with time Rate of change Rate proportional to amount Unconstrained growth and decay Population growth with no constraints

    5. Example: Squirrels assuming No predators Unlimited resources

    6. Example: Population of squirrels if no predators & unlimited resources s - number of squirrels in the area t - time Rate of change of number of squirrels (ds/dt) proportional to number of squirrels (s)

    7. Vensim® To model dynamic systems

    8. Vensim model of unconstrained population growth with births and deaths

    9. Graph for unconstrained growth

    10. Table

    11. Modeling interactions Competition Predator-Prey Disease

    12. Predator-Prey Example

    13. Lotka-Volterra Model Alfred Lotka and Vito Volterra Mathematicians Proposed independently 1920s Simplifying assumptions Predator hunts only 1 prey No other animal eats prey

    14. If squirrels and hawks together Number of contacts between predators and prey proportional to s?h

    15. Model of squirrel population Squirrel population helped or hurt by contact? Deaths affected

    16. Model of hawk population Hawk population helped or hurt by contact? Births affected

    17. Parameters Initial squirrel population = 100 Initial hawk population = 15 squirrel birth fraction = 2 squirrel death fraction = 0.02 hawk birth fraction = 0.01 hawk death fraction = 1.06

    18. Vensim model

    19. Typical graphs of numbers of prey (red) and predators (blue)

    20. Interaction for spread of disease Spread of flu in boarding school How model? http://www.shodor.org/talks/ncsi/vensim/ http://www.shodor.org/talks/ncsi/excel/ www.wofford.edu/ecs/ Link to text Link to Files

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