A population of fish in a closed pond evolves according to the differential equation…

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

Problem

A population

P(t)

of fish in a closed pond evolves according to the differential equation

\frac{dP}{dt}=\kappa P\Bigl(1-\frac{P}{500}\Bigr), \quad \kappa>0,

where

t

is measured in months and

P

in fish. At time

t=0

the population is

350

fish. After

6

months the population has risen to

400

fish. (a) Determine the value of the constant

\kappa

. Express

\kappa

exactly. (b) Find a formula for

P(t)

that satisfies the given conditions. (c) According to your model, how long in months will it take the population to reach

450

fish? Express your answer to the nearest tenth of a month. (d) Describe the limiting behavior of

P(t)

t\to\infty

and provide a brief justification based on your solution.