Mathematics

Problem

Consider the differential equation

\dfrac{dy}{dt}=k\,y\,(a-y)

, where

k>0

and

a>0

are constants. a. Treating

\dfrac{dy}{dt}

as a function of

y

, compute its derivative with respect to

y

. Clearly show your work. b. Use your result from part (a) to determine the value of

y

that maximizes

\dfrac{dy}{dt}

, and verify that this critical point indeed corresponds to a maximum. c. Find an expression for the maximum possible growth rate in terms of

k

and

a

. Simplify your answer. d. Discuss the concavity of

\dfrac{dy}{dt}

as a function of

y

at the point of maximum growth and explain the practical significance of this concavity when modeling real-world population dynamics.