Question

The population P(t) of a species satisfies the logistic differential equation dP/dt=P(2-P/5000), where the initial population P(0)=3000 and t is the time in years. What is limit of P(t) as t approaches infinity?

Answer 1

@dumbcow @ganeshie8 @iambatman

Answer 2

separate variables and solve the DE first

Answer 3

\[\dfrac{dP}{dt} = P\left(2-\dfrac{P}{5000}\right) \]

\[ \int \dfrac{dP}{P\left(2-\dfrac{P}{5000}\right)} =\int dt \]

Answer 4

partial fractions on left side + solving explicitly gives u :

\[P = \dfrac{10000 e^{2t}}{C + e^{2t}}\]


take the limits as \(t \to \infty\)

Answer 5

separate variables
\[\frac{dP}{P(10,000-P)} = \frac{dt}{5000}\]
integrate
\[\frac{1}{2}(\ln P - \ln (10,000-P)) = \frac{t}{5000} + C\]
\[\frac{P}{10,000-P} = C e^{t/2500}\]
solve for C using initial value
C = 3/7
\[P = \frac{(30,000/7 e^{t/2500}}\]

Answer 6

Thanks.

Answer 7

for some reason there's this rule that it's always the denominator which here is 5000