A population (in individuals) grows according to the logistic differential equation…

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

Problem

A population

P(t)

(in individuals) grows according to the logistic differential equation

\frac{dP}{dt}=0.2P\Bigl(1-\frac{P}{1000}\Bigr),

where

t

is measured in weeks. At time

t=0

the population is

200

individuals. (a) Solve the differential equation to obtain an explicit formula for the general solution

P(t)

. (b) Determine the particular solution satisfying the initial condition

P(0)=200

. (c) Compute

\displaystyle\lim_{t\to\infty}P(t)

and interpret the result in the context of the population model.