# Limiting factors and models of population growth (2 of 2)

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## Objective

Students will be able to 1) distinguish between an exponential and logistic growth model; 2) draw exponential and logistic growth curves; and 3) identify exponential and logistic growth in populations of American Bison and African Lions.

#### Big Idea

Growth models help ecologists understand population changes. How might we examine the growth of actual and theoretical populations to define exponential and logistic growth models?

## FRAME: How do we model population growth?

How do we use mathematical models to describe population growth? Students have just developed an essential vocabulary of population ecology. In "Limiting factors and models of population growth" students will explore the cause and effect relationships among these terms. Students will also learn about how these populations grow over time and how environmental scientists model this growth. Why don't population sizes become infinitely large? How can population density work against population growth?

In the first lesson, students explore factors the limit population growth. They will look at the West Nile Virus, dense cities, and natural disasters. From these examples, students will develop a definition of carrying capacity. Finally, students will use online modeling software to identify cause and effect relationships among limiting factors and population growth.

In the second, students expand their understanding of population growth to include mathematical representations of exponential and logistic growth. Students will graph real historical data of American bison and work through a case study of the population dynamics of African lions. Students will also respond to a series of assessment questions to check for understanding.

Over the course of two lesson, successful students will have met the following objectives:

1. categorize factors that limit population growth as density-dependent or density-independent;
2. define carrying capacity
3. accurately predict how density-dependent factors will limit the growth of populations
4.  distinguish between an exponential and logistic growth model
5. draw exponential and logistic growth curves
6. identify exponential and logistic growth in populations of American Bison and African Lions.

## ENGAGE: Graphical expressions

5 minutes

What is the purpose of this section?

Students learn about the difference between logistic growth and exponential growth. By the end of this section students should be able to identify the shape of a logistic growth curve and the shape of an exponential growth curve.

What will students do?

Students go on a video scavenger hunt.

First students find graphs with two different shapes in clip. What is the name of each graph? Why does its shape mean?

Next, students draw each graph in their shared document and label each as "density dependent" or "density independent." Explain your labeling with evidence.

What will the teacher do?

The teacher facilitate a brief discussion of each activity and explicitly explains that the "J shape" is exponential growth and happens when populations are not subject to density dependent growth factors. The "S shape" is logistic growth; this shows carrying capacity and occurs when populations face density dependent limiting factors.

## EXPLORE: American Bison

15 minutes

What is the purpose of this section?

Students graph real-world data that exemplifies a population growth model using an online graphing platform. By the end of this section students should have drawn a logistic growth curve and explained the meaning of this curve.

What will students do?

Students will complete a number of tasks in this section.

TASK 1:  Here is footage of bison in Yellowstone National Park.  Do these Bison appear to have grown without limiting factors?  Explain?

TASK 2: Plot historical population data of American Bison in Yellowstone over time using plot.ly.

TASK 4: Describe your graph. What is its shape?  Why does it have this shape? How would you describe this growth in words using vocabulary that we have developed during this unit?

What will teachers do?

Teachers will need to support students' use of plot.ly and students explanations of the shape of graph. The plot.ly interface will be intuitive to many students; I like to pair tech savvy student with students that need support. As for the graph, many students will be able to identify it as an example of logistic growth based on the ENGAGE section. Teachers will want to push students to explain what logistic growth means. Why is the shape an S? What does this mean about the size of the population over time? Why does this particular shape occur with populations?

## EXPLAIN: Exponential versus logistic

10 minutes

What is the purpose of this section?

Students formally define exponential growth and logistic growth from the EXPLORE sections and then compare definitions to a textbook definition. By the end of this section students should be able to clearly articulate the differences between the exponential model of growth and the logistic model of growth.

What will students do?

TASK 1: Describe exponential and logistic growth using the conceptual graph below. What are the key differences between the two models of growth?  Develop a response using the vocabulary that we have developed in this unit so far.

TASK 2:  Compare your ideas with the following descriptions from an Environmental Science AP TExtbook (Friedland and Relyea, 2012).  To do this, highlight the space after “exponential growth” and change the text color so that you are able to read the definition. Currently all definitions are written in white text so they are invisible.

What will teachers do?

Teachers will primarily assist students with articulating how the definitions that they developed differ from the provided textbook definitions. Students should be able to reconcile these differences; teachers should push students to modify their original definitions using the textbook definition in language the student understands and uses.

## ELABORATE: African Lions

20 minutes

What is the purpose of this activity?

Students synthesize ideas from this lesson sequence through an interactive, self-paced "population lab."

What will students do?

OBJECTIVE: Students will examine population growth through an interactive activity.

TASK 1: Complete this virtual lab that examines growth in an African lion population and record all activities in a graphic organizer. This virtual lab walks students through the population ecology narrative developed over this lesson sequence and asks a formative assessment question at each step. Students have multiple chances to self-assess understanding and practice graphing and interpretation skills.

What will the teacher do?

Teachers will want to spend most of this time working with small groups and individual students. This is an opportunity to provide effective correct feedback to students that are struggling with a concept. One strategy I used for managing this work was to define themed help stations at the beginning of this task. If you need help with logistic growth, go to this table; if you need help with density-dependent limiting factors go to that table.

ITERATION NOTE:

RESOURCE NOTE: A copy of student work for this assignment is attached. One modification I would make to this graphic organizer is to change the "what I find confusing prompt..." to "one connection to my life or this class that I can make" and/or "something I am still curious about is..." As written, the proficient students does not have to do any reflective work if there is no confuses. This prompt was designed as something that would encourage students to reflect on learning; it failed in this iteration.

## EVALUATE: Problems

5 minutes

What will students do to demonstrate proficiency?

As with the previous EXIT, students again have a short problem set to complete. (This is available at the end of the PROTOTYPE ACTIVITY GUIDE.) Also, the format remains the same. Students must use evidence to explain why answers are correct and incorrect. This is an effective assessment strategy because it allows educators to better understand students' misconceptions. A student might be able to identify a graph as exponential through force of repetition. However, if that students cannot explain the meaning of the shape, or if the student cannot explain why invalid interpretations of the graph are incorrect ("it gets bigger because people get bigger" is an example), then the student has not mastered that skill.