> [!Note] Theorem > Let $p'(t)=kp(t)\left( 1-\frac{p(t)}{p_{m}} \right)\tag{1}$be the [[Logistic Population Growth Model|logistic population growth model]]. > > Then $p(t) = \frac{p_{0}p_{m}}{(p_{m}-p_{0})e^{-kt} + p_{0}}$ **Proof**: By [[Nondimensionalising Population Growth Model]], $\tilde{p}'(\tilde{t})=\tilde{p}(\tilde{t})(1-\tilde{p}(\tilde{t})) \tag{1}$where $\tilde{p}(\tilde{t})=p(t)/p_{m}$ and $\tilde{t}= kt.$ By [[Implicit Solution to Initial Value Problem for Separable Equation]] with $p(0)=p_{0},$ $\int_{\tilde{p}_{0}}^{\tilde{p}(\tilde{t})} \frac{1}{\bar{p}(1-\bar{p})} \, d\bar{p} = \int_{0}^{\bar{t}} \, \tilde{d} $Note that $\frac{1}{\bar{p}(1-\bar{p})}= \frac{1}{\bar{p}}+ \frac{1}{1-\bar{p}}$Thus $\log\left( \frac{\tilde{p}(\tilde{t})}{1-\tilde{p}(\tilde{t})} \right) - \log\left( \frac{\tilde{p}_{0}}{1-\tilde{p}_{0}} \right)= \tilde{t}$Which gives $\tilde{p}(\tilde{t})= \frac{\tilde{p}_{0}}{(1-\tilde{p}_{0})e^{-t} +\tilde{p}_{0}}.$