A model for the population p (t) in a suburb of a large city is given by the initial-value problem dp dt p (101 2 107p), p (0) 5000, where t is measured in months. what is the limiting value of the population? at what time will the population be equal to one-half of this limiting value?

Given - dp/dt = P (10^-1 - 10^-7P), P (0) = 5000, Solution - a = 10^-1, b = 10^-7,

dp/dt = P (a-bP),

P (t) = aP/{bP + (a-bP) e^-at}

P (t) = 500/{0.0005+0.0995e^-0.1t}

on limiting t--infinite -

limiting value P = 1000000

P (t) = 500000 = 500/{0.0005+0.0995e^-0.1t}

on solving ... t = 52.9 months.