Trout Pond

This investigation illustrates the use of iteration, recursion and algebra to model and analyze a changing fish population. Graphs, equations, tables, and technological tools are used to investigate the effect of varying parameters on the long-term population.

In the previous lessons of this unit, students found the population sizes, A(n), that correspond with different years, n. To explore this relationship further, it may be helpful for students to construct a graph that shows the population change over time.

The graph below shows values of n along the horizontal axis and A(n) along the vertical axis.

Year versus Population, n versus A(n) Click anywhere on the graph to redraw the curve.

You may wish to project the image above for students to consider, or you might want to have students generate the graph on their own. Either way, require students to analyze the graph and answer the following questions:

1. Describe the characteristics of the graph, and explain what information the characteristics give you about the trout population.
   
   
2. How is the long-term population shown in this graph?

As in the Numerical Analysis lesson, students should investigate changes that occur to the population if the Initial Population, the Growth Factor, or the Restock Amount should change. However, they should now investigate these changes graphically. Students can use the Affine Recurrence Plotter to investigate the same three questions from the previous lesson:

1. If the initial population doubles, what will happen to the long-term population?
   
   
2. If the annual restocking amount doubles, what will happen to the long-term population?
   
   
3. If the annual population decrease rate doubles, what will happen to the long-term population?

Allow students to create graphs for each of the three situations. Have them compare the graphs to the numerical results that they obtained previously, and allow them to experiment with the Affine Recurrence Plotter to see the graphical effect of changing the three parameters in this problem.

In addition to the standard graph obtained above, another type of graph that can provide a useful visual representation is a "cobweb graph" (also known as a "stairstep graph"). Students may benefit from seeing this type of graph.

Population at Year n versus Population at Year n + 1 Click anywhere on the graph to redraw the cobweb. Click on the "Cobweb Graph Hint" to redraw the graph with an explanation of each step in the lower right corner of the graph.

As with the standard graph, allow students an opportunity to think about and explain what is happening with the cobweb graph. Ask, "How is the long-term population shown in this graph?"

[After many years, the population from one year to the next changes very little. Therefore, the cobweb graph reaches a point at which there is no change. In the graph above, this occurs when the population in consecutive years approaches 5000.]

To conclude this lesson, conduct a whole-class discussion in which students elaborate on what they discovered. Make sure that students compare each graphical representation with the numerical results from the previous lesson.