Logistic Growth Models

By the end of this lesson, you will be able to describe logistic growth behavior, identify the carrying capacity and inflection point, and use logistic models to solve population and spread problems.

The Big Idea

Exponential growth assumes unlimited resources — unrealistic in most real scenarios. Logistic growth adds a ceiling: the carrying capacity $L$. Population grows exponentially initially, then slows as it approaches $L$, creating the characteristic S-shaped curve.

Logistic models describe: biological populations, disease spread (S-curves of epidemics), market saturation, and technology adoption.

The Logistic Function

$$P(t) = \frac{L}{1 + Ce^{-kt}}$$

• $L$ = carrying capacity (maximum sustainable population)
• $k > 0$ = growth rate
• $C$ = parameter determined by initial conditions
• $C = \dfrac{L - P_0}{P_0}$ where $P_0 = P(0)$ is the initial value

Key Features of the Logistic Curve

Behavior:

• $P(0) = L/(1+C)$ — initial value
• As $t \to \infty$: $P(t) \to L$ (approaches carrying capacity)
• As $t \to -\infty$: $P(t) \to 0$
• Inflection point at $P = L/2$ (fastest growth rate) — occurs at $t = \frac{\ln C}{k}$

Growth rate is fastest at half the carrying capacity.

Example 1 — Population of Bacteria

Bacteria in a Petri dish grow logistically with $L = 5000$, $k = 0.3$, and initial population $P(0) = 500$.

$C = \dfrac{5000 - 500}{500} = 9$

$P(t) = \dfrac{5000}{1 + 9e^{-0.3t}}$

Find $P(10)$:

$P(10) = \dfrac{5000}{1 + 9e^{-3}} = \dfrac{5000}{1 + 9(0.0498)} = \dfrac{5000}{1.448} \approx 3453$

When does population reach 4000?

$4000 = \dfrac{5000}{1 + 9e^{-0.3t}} \implies 1 + 9e^{-0.3t} = 1.25 \implies e^{-0.3t} = 0.0278$

$-0.3t = \ln(0.0278) \approx -3.585 \implies t \approx 11.95$ time units.

Example 2 — Inflection Point

From Example 1, when does the population grow fastest?

Inflection at $P = L/2 = 2500$. Time: