Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (b)

MATH102 December 2013

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Question C 02 (b)

The population of fish in a particular lake is given by the function F(t) where F is measured in number of fish and t is measured in days. A company that manages fish stocks is hired to restock the lake, adding fish at a constant rate. Only N fishers are allowed to fish in the lake at a time. A simple model for this scenario is given by the equation: ${\frac {dF}{dt}}=I-\alpha NF$ Where I and $\alpha$ are constant and two cases for N are considered. Explain in words what the constant $\alpha$ represents.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Note the negative sign preceding the term containing $\alpha$; ${\tfrac {dF}{dt}}$ decreases as $\alpha$ increases. Note also that the term containing $\alpha$ is proportional to the product of fishermen, N, and fish, F. What do fishermen do to fish? ;)

Hint 2
The units for $\alpha$ are ${\frac {1}{{\text{fishermen}}\cdot {\text{days}}}}$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The term containing $\alpha$ tends to make the value of F decrease since it has a negative sign preceding it. Because of this, and the fact that this term depends on both the population of fish and the number of fishermen, we conclude that this term models the interaction between fishermen and the fish. The interaction of fishermen and fish results in fewer fish because they get caught by the fishermen. Thus the constant $\alpha$ represents the rate at which a single fisherman catches fish. The larger the value of $\alpha$ (i.e., the more effective a fisherman is at catching fish), the faster F decreases.