Logistic Model

$$\frac{dp}{dt}=kP\left(1-\frac{P}{L}\right)⟹P=\frac{L}{1+A{e}^{-kt}}$$

• Constants
  • L = limiting value (aka carrying capacity)
  • k = relative growth rate (when P is small relative to L)
  • $A=\frac{L-P_0}{P_0}$
• Graph
  • As $P\to 0,\frac{dp}{dt}\to kp$
  • As $P\to L,\frac{dp}{dt}\to 0$
• Important Facts
  • Slopes are…
    • Small near $P=0$ and $P=L$
    • Steepest near $P=\frac{L}{2}$ (the POI)
  • If $P>L$, the slope is negative