Verhulst's Logistic Growth Model , an equation to model population growth or decay:
$$\dfrac{dP}{dt}=KP(P_m-P)$$
One way to do it is by partial fractions,
I get:
$\int \frac{1}{P(P_m-P)} \frac{dP}{dt} dt = \int K dt$
By using partial fraction decomposition
$\int \frac{1/P_m}{P} + \frac{1/P_m}{(P_m-P)} dP = \int K dt$
This then gives :
$e^{(P_mKt)}e^{CP_m} = |P||P_m-P|$
But the solution (moment 33:11) is :$$\dfrac{AP_m}{A+e^{-kP_mt}}$$
What did I do wrong? Or what was a better way to do it?
Best Answer
The result of the integration is $$ \frac{1}{P_m} \left(\ln |P| - \ln|P_m-P|\right) = Kt + C$$ that is $$ e^{P_m(Kt+C)} = \left|\frac{P}{P_m-P}\right|$$ $$ \frac{P}{P_m-P} = A e^{P_mKt}$$ $$ \frac{P_m}{P_m-P} = 1+ \frac{P}{P_m-P} = 1 + A e^{P_mKt}$$ $$ \frac{P}{P_m} = \frac{A e^{P_mKt}}{1 + A e^{P_mKt}}$$ $$ P = \frac{A P_m}{A + e^{-P_mKt}}$$