MATH101 April 2007
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Question 06 (b)
Full-Solution Problems. Justify your answers and show all your work. Simplification of answers is not required.
The population of fish in a lake is m million, where varies with time t measured in years. The number of fish is currently 2 million.
Suppose instead that (because of fishing by humans) m satisfies
Will the fish population ever equal 3 million? You must give justification for your answer.
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!
Find the steady states of this differential equation. Are steady states actually reached in finite time?
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We want to find the steady states of this differential equations. To do so, set , i.e.
and once more
and so the steady states are given by and . It is quickly checked, e.g. by plugging in , , and , that
At time we have . Hence, despite the fishing, the population of fish will grow initially. But the growth will slow down as the population size approaches the steady state value . A steady state can never be reached in finite time, hence the answer is No, the fish population will never equal 3 million.
Caption: We plot the rate of change
-axis) as function of population size
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MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Population dynamics, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag