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

MATH102 December 2013

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Question C 02 (d)

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. Case 2: The agency that administers fishing permits decides that the number of fishers should be proportional to the number of fish currently in the lake, $N=F/3$. What is the population size $F_{*}$ to which F(t) approaches as $t\to \infty$ in this scenario?

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
For $t\rightarrow \infty$, the solution $F$ reaches its steady state.

Hint 2
By plugging in ${\frac {F}{3}}$ for $N$, the equation becomes ${\frac {dF}{dt}}=I-{\frac {\alpha }{3}}F^{2}$

Hint 3
Because we again search for the steady state of the equation, we set the derivative equal to zero an solve the resulting equation $0=I-{\frac {\alpha }{3}}F^{2}$ for $F$.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. Applying $N={\frac {F}{3}}$, the equation becomes ${\frac {dF}{dt}}=I-{\frac {\alpha }{3}}F^{2}$ For $t\rightarrow \infty$, the solution $F$ reaches its steady state $F_{*}$. Hence, we again set the left-hand side equal zero and solve for $F_{*}$. ${\begin{aligned}0=I-{\frac {\alpha }{3}}F_{*}^{2}\\\Rightarrow \quad F_{*}^{2}={\frac {3I}{\alpha }}\\\Rightarrow \quad F_{*}={\sqrt {\frac {3I}{\alpha }}}\end{aligned}}$ We only consider the positive root since the amount of fish can not be negative.