Analyze the logistic differential equation \(dP/dt = kP(1-P/L)\).
Identify the carrying capacity (L) as a horizontal asymptote.
Locate the inflection point where the growth rate is maximized (P = L/2).
Key Equations
\( \frac{dP}{dt} = k P \left(1 - \frac{P}{L} \right) \)
\( P(t) = \frac{L}{1 + C e^{-kt}} \)
\( C = \frac{L - P_0}{P_0} \)
Why It Matters
Unlike unbounded exponential growth, logistic models account for environmental limits—like resource scarcity. As the population P approaches carrying capacity L, the growth rate smoothly declines to zero.
Tags
logistic growthdifferential equationscarrying capacityAP Calculus BC