CPT 01: Modeling

CPT 01: ModelingExpected Educational ResultsDifferential EquationsModeling Population DecaySimulationPre-Experiment Discussion QuestionsConduct the Experimental ProcedureRun Python Code for SimulationDifference EquationsModeling Population Decay with ImmigrationSimulationPre-Experiment Discussion QuestionsConduct the Experimental ProcedureRun Python Code for SimulationModels for Population DecayDefinition:Mathematical ModelsDefinition:CC BY-NC-SA 4.0

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Differential Equations

Modeling Population Decay

We will conduct a simulation to construct model of a population. This population will consist of six-sided dice. In the activity, you will either gently roll all dice at the same time or simulate rolling N six-sided dice with Python.

Here is the code to simulate the rolling of 100 six-sided die. The comments explain the code.

Simulation

The simulation of population decay will consist of the following steps:

  1. Start with 100 dice;

  2. Roll or simulate the roll of 100 dice;

    • Any die that shows an odd number, then this individual has passed from the population and is removed from the population.
    • Record the number of individuals remaining in the population.
  3. Repeat Step 2 with the remaining die.

Pre-Experiment Discussion Questions

Prior to performing the experiment:

  1. Make a prediction, i.e., describe what you think will happen.
  2. Provide assumptions (simple, non-compound assumptions) that will support your prediction.
  3. Share your predictions and assumptions to the class.
  4. Critique the predictions and assumptions.

Conduct the Experimental Procedure

Record data in a two-column table similar to the table below:

Number of Rolls (Iterations)Number of Individuals Remaining in Population
  
  
  
  1. When should you stop the Experimental Procedure? Explain.

  2. Make a conclusion about the experiment. Explain.

    • Explain how you know your conclusion is plausible.
    • Compare your prediction with your results.
    • Indicate which assumptions actually played a role in the experiment.
  3. Present your data and conclusions to the class.

  4. Critique each group's data and conclusions.

  5. Create a mathematical model:

    • Let be the number of iterations.
    • Let be the number of dice, [individuals], remaining after each iteration.
    • Create the mathematical model.
    • Describe the reasonableness of your mathematical model for . Explain.
    • How would you measure how successful your mathematical model describes . Explain.
  6. Critique each group's model.

Run Python Code for Simulation

Make sure that you change Language to Python below the bottom-right portion of the code window.

The code will count the number of each outcome. For example,

Counter({3: 22, 1: 22, 4: 18, 2: 14, 6: 13, 5: 11})

states that there where 22 threes, 22 ones, 18 fours, 14 twos, 13 sixes, and 11 fives.


Difference Equations

Modeling Population Decay with Immigration

We will conduct a simulation to construct model of a population. This population will consist of six-sided dice. In the activity, you will either gently roll all dice at the same time or simulate rolling N six-sided dice with Python.

You will use the Python code to simulate the rolling of 100 six-sided die.

Simulation

The simulation of population decay will consist of the following steps:

  1. Start with 100 dice;

  2. Roll or simulate the roll of 100 dice;

    • Any die that shows an odd number, then this individual has passed from the population and is removed from the population.
    • Add 10 additional dice individuals who immigrate into the population.
    • Record the number of individuals remaining in the population.
  3. Repeat Step 2 with the remaining die.

Pre-Experiment Discussion Questions

Prior to performing the experiment:

  1. Make a prediction, i.e., describe what you think will happen.
  2. Provide assumptions (simple, non-compound assumptions) that will support your prediction.
  3. Share your predictions and assumptions to the class.
  4. Critique the predictions and assumptions.

Conduct the Experimental Procedure

Record data in a two-column table similar to the table below:

Number of Rolls (Iterations)Number of Individuals in Population
  
  
  
  1. When should you stop the Experimental Procedure? Explain.

  2. Make a conclusion about the experiment. Explain.

    • Explain how you know your conclusion is plausible.
    • Compare your prediction with your results.
    • Indicate which assumptions actually played a role in the experiment.
  3. Present your data and conclusions to the class.

  4. Critique each group's data and conclusions.

  5. Create a mathematical model:

    • Let be the number of iterations.
    • Let be the number of dice, [individuals], remaining after each iteration.
    • Create the mathematical model, i.e., .
    • Describe the reasonableness of your mathematical model for . Explain.
    • How would you measure how successful your mathematical model describes . Explain.
  6. Critique each group's model.

Run Python Code for Simulation

Make sure that you change Language to Python below the bottom-right portion of the code window.

The code will count the number of each outcome. For example,

Counter({3: 22, 1: 22, 4: 18, 2: 14, 6: 13, 5: 11})

states that there where 22 threes, 22 ones, 18 fours, 14 twos, 13 sixes, and 11 fives.


Models for Population Decay

Definition:

Exponential Growth and Decay: An exponential model is used when the rate of change of the population is proportional to the population size.

Compartmental Model: A compartmental model is used when there is an inflow and outflow to a system.

Mathematical Models

Definition:

Difference Equation: A difference equation is in the form: and is useful for compartmental models.

Differential Equation (DE): A model (i.e., equation) that contains derivatives (e.g., rate of change, acceleration, etc.) of an unknown function.

CC BY-NC-SA 4.0

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Last Modified: Sunday, 23 August 2020 19:57 EDT