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*North Carolina School of Science and Mathematics *

*Distance Education and Extended Programs *

The Fish and Wildlife Division monitors the trout population in a stream that is under its jurisdiction. Its research indicates that natural predators, together with pollution and fishing, are causing the trout population to decrease at a rate of 20% per month. The Division proposes to introduce additional trout each month to replenish the stream. Assume the current population is 300. Use tables and graphs to investigate the following questions.

1.

What will happen to the trout population over the next 10 months with no replenishment program?** **

𝑻

𝟎

= 𝟑𝟎𝟎

𝑻 𝒏

= 𝑻 𝒏−𝟏

− 𝟎. 𝟐𝑻 𝒏−𝟏

**Where **

𝒏

** represents number of months from now. Since **

𝑻

𝟏𝟎

= 𝟑𝟐. 𝟐

**, the trout population decreases to approximately 32 fish. **

2.

What is the long-term result of introducing 100 trout into the stream each month?

𝑻

𝟎

= 𝟑𝟎𝟎

𝑻 𝒏

= 𝟎. 𝟖𝑻 𝒏−𝟏

+ 𝟏𝟎𝟎

**We find that the trout population stabilizes at 500. **

3.

Investigate the result of changing the number of trout introduced each month. What is the impact of the number of trout added each month?

**The impact is that the equilibrium changes. For example, if we add fewer than 60 trout each month, the equilibrium level is less than 300. **

4.

Investigate the impact of changing the initial population on the long-term behavior of the population. What is the effect of changing the initial population?

**The initial population affects whether the population increases or decreases towards the equilibrium. It does not have an effect on the equilibrium level. **

5.

What is the impact of the rate of decrease in the population during the replenishment program?

**The larger the rate of decrease, the more you need to add to get the **

**population to stabilize at the same level.**

*Adapted from Contemporary Precalculus through Applications by the North Carolina School of Science and *

*Mathematics, Everyday Learning Corporation, 2000 *