Logistic Growth

274. Logistic Growth

The simplest limits-to-growth model. A population (or any quantity) grows exponentially when small, then saturates as it approaches a carrying capacity $𝐾$ .

274.1. Model

\cdot 𝑁 = 𝑟 𝑁 (1 - \frac{𝑁}{𝐾})

$𝑟$ : intrinsic growth rate (per unit time)
$𝐾$ : carrying capacity (max sustainable level)
$𝑁$ : current level

274.2. Closed-form solution

𝑁 (𝑡) = \frac{𝐾}{1 + 𝐴 𝑒^{- 𝑟 𝑡}}, 𝐴 = \frac{𝐾 - 𝑁_{0}}{𝑁_{0}}

S-curve from $𝑁_{0}$ asymptoting to $𝐾$ .

274.3. Phases

Phase	Behavior	Where you are
Exponential	$\cdot 𝑁 \approx 𝑟 𝑁$	$𝑁 ≪ 𝐾$ , factor $(1 - \frac{𝑁}{𝐾}) \approx 1$
Inflection	$\cdot 𝑁$ peaks	$𝑁 = \frac{𝐾}{2}$ , maximum growth rate
Saturation	$\cdot 𝑁$ decays	$𝑁 \to 𝐾$ , growth tapers off

274.4. Structurally

Reinforcing growth ( $\cdot 𝑁 = 𝑟 𝑁$ ) plus balancing limit ( $1 - \frac{𝑁}{𝐾}$ ). The R+B = S-curve archetype.

274.5. Where it shows up

Population biology: limited food supply, predation
Bacterial growth: nutrient depletion
Market penetration: saturating consumer base
Technology adoption: total addressable market
Renewable resources under harvest: $\cdot 𝑅 = 𝑟 𝑅 (1 - \frac{𝑅}{𝐾}) - ℎ 𝑅$ (logistic + harvest)
Tumor growth (Gompertz model is a variant)

274.6. Limitations

Symmetric: growth and saturation phases are mirror images — real systems often asymmetric
Static $𝐾$ : real carrying capacities change (technology, environment)
No overshoot: real systems often overshoot $𝐾$ then collapse — see oscillation / collapse archetypes

274.7. Compared to Bass diffusion

	Logistic	Bass
Initial growth	Exponential ( $𝑟 𝑁$ )	Innovation rate ( $𝑝 𝑁$ , constant)
Driver	Word-of-mouth only ( $𝑟$ effectively combines)	Innovation + imitation separately
Use case	Pure peer-driven growth	New product with external + internal drivers

Bass is logistic-like but separates the two drivers — better for marketing analysis.

274.8. Parameter estimation

Given time-series data $(𝑡_{𝑖}, 𝑁_{𝑖})$ :

Plot $𝑁 (𝑡)$ — confirm S-shape
Estimate $𝐾$ from asymptote (or as a parameter to fit)
Linearize: $ln (𝐾 - 𝑁) - ln 𝑁 = ln 𝐴 - 𝑟 𝑡$ , fit a line; slope $= - 𝑟$ , intercept $= ln 𝐴$

Or just nonlinear-least-squares fit the closed form.

274.9. See also

Bass Diffusion — refinement
Feedback Loops — the R+B archetype
Differential Equations — the parent topic