Modeling Population
The population of Tahoe was 15,000 in 1984 and now is 20,000. If we
use the model that the growth is proportional to the population, then
dP/dt = kp
or
dp/p = kdt,
dp/p =
kdt
lnP = kt + c, P = Ce^{kt}
since
P(0) = 15000, C = 15,000
so
P = 15000e^{kt}
since
P(15) = 20000,
20000 = 15000e^{k15}
or
k = .019
this model says that the population of Tahoe in 200 years will be
P(200) = 15000e^{.019(215)} = 891,629
or almost one million people.
We should hope that this is not a good model.
Alternatively, we can assume that the population will never become greater
than 50,000 people and that the rate of population growth is proportional
to the product of the population and 50,000 minus the population.
dP/dt = kP(50000 - P)
dP/[P(50000 - P)] = kdt
dP/P(50000 - P)
= kdt
Now use partial fractions:
A/P + B/(50000 - P) => A(50000 - P) + BP = 1
A = 1/50000, B = 1/50000
Hence
1/50000 [dP/P + dP/(50000 - P)] = kt + C
ln(P) - ln(50000 - P) = at + b
ln[P/(50000 - P) = at + b
P(0) = 15,000 gives
ln15/35 = b, b = ln3/7
P(15) = 20000 gives
ln2/3 = 15a + ln3/7 or
a = (ln(14/9)/15
Hence in 200 years we have
ln[P/(50000 - P] = at + b
or
P/(50000 - P) = e^{at+ b }= M
P = 50000M - MP, P = 50000M/(1 + M)
Now when
t = 215, M = 241.2
so that
P = 49794
Which is a more reasonable estimation.