Logistic Growth

Core Sentence / 核心句

Logistic growth is growth proportional to both the current population and the fraction of capacity still available.

$$\frac{dP}{dt}=kP\left(1-\frac{P}{K}\right).$$

$K$ is the carrying capacity. For $0<P<K$, growth is fastest at $P=K/2$, where the solution changes from concave up to concave down.

Watch Out / 易错点

Carrying capacity comes from the equilibrium value, not from the coefficient in front. A model written $kP(K-P)$ has the same capacity $K$.

Connected Notes / 连接

• 微积分BC第10次正课-累积函数与微分方程
• 微积分BC第11次正课-积分应用与截面体积
• Differential Equations
• Initial Conditions