The Department of Resources has decided to reintroduce mountain sheep to a particular area of Arizona by placing 100 sheep at the start of the project. The population, P, of the sheep...

Question

The Department of Resources has decided to reintroduce mountain sheep to a particular area of Arizona by placing 100 sheep at the start of the project. The population, P, of the sheep is expected to grow according to the differential equation  , where P t is measured in months. When does the population of the sheep reach 900? Give your answer to the nearest month.

Answer 1

From description, P(0) = 100

P' = 0.08P(1-P/1000)

We can solve the differential equation by separating the variables

1000∫dP/(P(1000-P) = 0.08/1000 ∫dt = 0.08t +C

Concentrating on the left hand side of the equation

1000/(P(1000-P) = A/P + B/(1000-P) --> A = B = 1

1000∫1/P + 1/(1000-P) dP = 1000(ln(P) - ln(P-1000)) = 1000ln(P/(P-1000))

Putting left and right sides together

ln(P/(P-1000)) = 0.08t + C

When t = 0

C = ln(-100/(900)) = ln(-1/9)

Taking both sides to power of e

P/(P-1000) = e^{0.08t + ln(-1/9)} = -e^0.08t/9

e^0.08t=9P/(1000-P)

When P = 900

e^0.08t= 9(900/900)

0.08t = ln(9)

t = ln(9)/0.08 ≈ 28 months ≈ 2.3 years