Angela Shiflet, Ph. D.

1
Mathematics Workshop

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

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

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Example Squirrels assuming

• No predators
• Unlimited resources

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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)

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Vensim

• To model dynamic systems

• Dynamic systems are usually
• Very complex
• Many components
• Involved relationships

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Vensim model of unconstrained population growth
with births and deaths

• ModelSettings
• Time Step 0.25
• Final Time 12

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Graph for unconstrained growth
Control Panel - top far right Graph New or
Modify Name Title Sel Variables As Table
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Table
Control Panel - top far right Graph New or Modify
or Copy Name Title Sel Variables As Table
11
Modeling interactions

• Competition
• Predator-Prey
• Disease

12
Predator-Prey Example

• Hawks and squirrels

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

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If squirrels and hawks together

• Number of contacts between predators and prey
  proportional to s?h

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Model of squirrel population

• Squirrel population helped or hurt by contact?
• Deaths affected

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Model of hawk population

• Hawk population helped or hurt by contact?
• Births affected

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Parameters
ModelSettings Time Step 0.0078125 Final Time
12
Input Output Device Constant Set to left of DB
name

• 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

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Vensim model
hawks- squirrels final
19
Typical graphs of numbers of prey (red) and
predators (blue)
Control Panel on top far right
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