A fish population in a closed pond satisfies the differential equation

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

\dfrac{dy}{dt}=0.04\,y\,(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

. (1) Derive an explicit expression for

y(t)

that satisfies the given initial condition. (2) Using the expression from part (a), determine the time

T

(in years) at which the population growth rate is greatest. (3) Calculate the instantaneous growth rate

\dfrac{dy}{dt}

t=T

. Give your answer in fish per year. (4) 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.