> [!Example] > 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]]. Determine the [[Stationary Points of Population Growth Model|stationary points]] and their [[Stable Stationary Point of First Order Autonomous Scalar Ordinary Differential Equation|stability]]. **Solution by phase line:** Let $f(p)=kp(1-p/p)_{m}.$ Then $(1)$ can be rewritten as $p'(t)=f(p(t)).$Now $f(p)=0$ gives $p\in\{ 0,p_{m} \}$ which are the stationary points. Note that for $p_{0}\in(0,p_{m}),$ $f(p_{0})>0$ so solutions will increase and move towards $p_{m}.$ Conclude that $p_{m}$ is stable will $0$ is unstable. **Solution by analytic condition**: Now $f'(p)=k(1-2p/p_{m}).$ Using [[Condition for Stability of Stationary Point of First Order Autonomous Scalar Ordinary Differential Equation]], $f'(0)=k>0\implies 0 \text{ is unstable}$and $f'(p_{m})=-k<0 \implies p_{m} \text{ is stable}.$ **Solution by solving IVP**: Again by [[Solution to Population Growth Model]], $p(t)=\frac{p_{0}p_{m}}{(p_{m}-p_{0})e^{-kt}+p_{0}}$which indeed converges to $p_{m}$ as $t\to \infty,$ so this is the stable point that attracts nearby solutions.