Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(a)

MATH102 December 2019

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[hide] Question 08(a)

The Allee effect is a phenomenon in ecology by which the per capita reproduction rate of the population increases with increasing population density. In Canada the Allee effect can be observed in the Atlantic cod population in the Gulf of St.Lawrence, where cod is the preferred prey of grey seals. A simple model of the Atlantic cod population growth rate is a modified logistic equation: $\displaystyle f(N)=N({\frac {N}{2}}-1)(1-N)$ where N is measured in millions of kg of fish. We can assume $N\geq 0.$ (4 points) Sketch the function f(N) for $N\geq 0$. Include the N-values of any roots, extrema, inflection points, and horizontal asymptotes of the function f(N), should they exits. Make sure your graph is big enough to clearly show all intersecting points.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

[show] Hint

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[show] Solution

Whoops, no solution has been written yet. Be the first to submit a suggestion. The zeros of the functions are given by N = 0, 1, 2. The first derivative of the function is $f'(N)=-3{\frac {N^{2}}{2}}+3N-1$ thus the critical points are given by $N_{c}=1\pm {\frac {1}{\sqrt {3}}}$ Since the derivative is a downward quadratic it's easy to observe that $N_{1}=1-{\frac {1}{\sqrt {3}}}$ is minimum , while $N_{2}=1+{\frac {1}{\sqrt {3}}}$ is maximum. The second derivative is given by $f''(N)=3(1-N)$ and indeed since it's a linear function we must have a point of inflection at N = 1.