A fish population in a closed pond satisfies the differential equation dtdy=0.04y(8000−y), where y(t) is the number of fish and t is measured in years. At time t=0, the population is y(0)=200.
a. Derive an explicit expression for y(t) that satisfies the given initial condition.
b. Using the expression from part (a), determine the time T (in years) at which the population growth rate is greatest.
c. Calculate the instantaneous growth rate dtdy at t=T. Give your answer in fish per year.
d. Explain, in the context of the pond ecosystem, why the time T you found in part (b) represents a moment of particularly rapid ecological change.