Education Technology

Biological Models with Differential Equations

Activity Overview

Students explore several models representing the growth (or decline) of a biological population. They study models of a single population that have a closed-form solution, and models of several interacting populations that have an open-form, numerical solution.

Before the Activity

  • See the attached PDF file for detailed instructions for this activity
  • Print pages 57 - 68 from the attached PDF file for your class
  • During the Activity

  • Distribute the pages to the class.
  • Follow the activity procedures:

  • Exponential and Logistic Regression:
  • Fit exponential and logistic growth models to the U.S. Census population data for years 1790 to 1990
  • Enter the data as a stat list
  • Compute the regression coefficients
  • View the resulting exponential function and the original data on the same graph
  • Calculate a logistic regression and graph it
  • Note the exponential model is not close to the actual data, while the logistic model fits the data better
  • Find the expected population for the year 2020 using the logistic model

  • Logistic Growth Models and Critical Depensation:
  • Graph and compare the Logistic Growth and Critical Depensation differential equation models
  • Change the initial conditions to see how the critical depensation differential equation model differs

  • Predator-Prey Model:
  • Enter the system of differential equations for a predator-prey model
  • Plot the slope field function and trace several possible graphs for the solution to the differential equation

  • Competitive Species Model:
  • Enter the system differential equations that models two similar species living in the same habitat and competing for the same resources
  • Study the direction field
  • Note that there are several solutions to the equations that move towards a limit where one population wins

  • SIR Model:
  • Enter 3 differential equations
  • Graph the equations together
  • Study the direction field and explore various solutions
  • After the Activity

    Students complete the problems on the exercise page.

  • Review student results:
  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary