Math Is Fun Forum

I missed the second half of my lecture due to an interview so it isn't that I'm not trying but that I don't have the notes and this math textbook only had 2 examples

First Problem:

Suppose a population is growing according to the logistic equation,

dP/dt = rP(1-P/K)

Prove that the rate at which the population is increasing is at its greatest when the population is at one-half of its carrying capacity. Hint:  Consider the second derivative of P.

Second Problem:

Consider a lake that is stocked with walleye pike and that the population of pike is governed by the logistic equation

P' = 0.1P(1-P/10)

where time is measured in days and P in thousands of fish.  Suppose that fishing is started in this lake and that 100 fish are removed each day.

(a) Modify the logistic model to account for the fishing

(b) Find and classify the equilibrium points for you model.

(c) Use quantitative analysis to completely discuss the fate of the fish population with this model.  In particular, if the initial fish population is 1000, what happens to the fish as time passes?  What will happen to an initial population having 2000 fish?

***Any Help Is Greatly Appreciated
***I'll do my part and help others as well