Population Dynamics: Example1

Example: Let P(t) be the population of a certain animal species. Assume that P(t) satisfies the logistic growth equation

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1.
Is the above differential equation separable?
2.
Is the differential equation autonomous?
3.
Is the differential equation linear?
4.
Without solving the differential equation, give a sketch of the graph of P(t).
5.
What is the long-term behavior of the population P(t)?
6.
Show that the solution is of the form

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Find A and B.
Hint: Use the initial condition and the result of 5.

7.
Where is the solution's inflection point?
Hint: This can be done without using the answer of 6.
8.
What is special about the growth rate of the population P(t) at the inflection point (found in 7)?

Solution:

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