# Differential equation of population growth or decay $\frac{dP}{dt}=KP(P_m-P)$

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?

## 2 Answers

The result of the integration is $$\frac{1}{P_m} \left(\ln |P| - \ln|P_m-P|\right) = Kt + C$$ that is $$e^{P_m(Kt+C)} = \left|\frac{P}{P_m-P}\right|$$ $$\frac{P}{P_m-P} = A e^{P_mKt}$$ $$\frac{P_m}{P_m-P} = 1+ \frac{P}{P_m-P} = 1 + A e^{P_mKt}$$ $$\frac{P}{P_m} = \frac{A e^{P_mKt}}{1 + A e^{P_mKt}}$$ $$P = \frac{A P_m}{A + e^{-P_mKt}}$$

When you integrated, you had a sign error:$$\int\frac{dP}{P_m-P}=\color{blue}{-}\ln|P_m-P|+C.$$