Education Technology

# Activities

• ##### Subject Area

• Math: Calculus: Other Functions

9-12

60 Minutes

• TI-86
• ##### Other Materials
This is Activity 6 from the EXPLORATIONS Book:
Differential Equations With The TI-86

## 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.

• 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