A fisheries officer stocks a lake with 400 fish…

A fisheries officer stocks a lake with $400$ fish. The lake can support at most $5000$ fish, and she models the number $P$ of fish at time $t$ years by the logistic equation $\dfrac{dP}{dt} = kP\left(1 - \dfrac{P}{5000}\right)$. After $3$ years the count is $800$ fish. She writes the solution as $P = \dfrac{5000}{1 + Ae^{-kt}}$, where $A$ comes from the starting number. From the reading at $3$ years she finds $k$. She then wants two more numbers for the management plan: the time at which the population reaches $2500$ fish, which is half the capacity, and the largest rate of growth of the population over its whole history, in fish per year. Find $k$ in per year, the time in years to reach $2500$ fish, and the largest growth rate in fish per year.