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