272. 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 .
272.1. Model
- : intrinsic growth rate (per unit time)
- : carrying capacity (max sustainable level)
- : current level
272.2. Closed-form solution
S-curve from asymptoting to .
272.3. Phases
| Phase | Behavior | Where you are |
|---|---|---|
| Exponential | , factor | |
| Inflection | peaks | , maximum growth rate |
| Saturation | decays | , growth tapers off |
272.4. Structurally
Reinforcing growth () plus balancing limit (). The R+B = S-curve archetype.
272.5. 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: (logistic + harvest)
- Tumor growth (Gompertz model is a variant)
272.6. Limitations
- Symmetric: growth and saturation phases are mirror images — real systems often asymmetric
- Static : real carrying capacities change (technology, environment)
- No overshoot: real systems often overshoot then collapse — see oscillation / collapse archetypes
272.7. Compared to Bass diffusion
| Logistic | Bass | |
|---|---|---|
| Initial growth | Exponential () | Innovation rate (, constant) |
| Driver | Word-of-mouth only ( 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.
272.8. Parameter estimation
Given time-series data :
- Plot — confirm S-shape
- Estimate from asymptote (or as a parameter to fit)
- Linearize: , fit a line; slope , intercept
Or just nonlinear-least-squares fit the closed form.
272.9. See also
- Bass Diffusion — refinement
- Feedback Loops — the R+B archetype
- Differential Equations — the parent topic