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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:
where N is measured in millions of kg of fish. We can assume
The growth rate of the population is given by the differential equation Use the sketch from part (a) to draw the phase-line diagram(also called state-space) and classify them as stable or unstable. For this part, you can consider negative values of N in addition to the model domain.
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!
The steady state is stable if it is attractive, otherwise unstable.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
The steady state N = 0 is stable; N = 1 is unstable, and N = 2 is stable.