Logistic Growth Calculator
Model population growth with carrying capacity using the logistic equation.
Compute population at any time, growth rate, and inflection point.
Logistic Growth
The logistic model is the standard description of population growth that levels off as resources become limiting. It captures three regimes in a single equation: exponential rise, slowing growth, and a stable carrying capacity.
Formula
P(t) = K / (1 + ((K − P₀) / P₀) × e^(−r × t))
Where:
- P₀ = initial population
- K = carrying capacity (maximum sustainable population)
- r = intrinsic growth rate (per time unit)
- t = elapsed time
Three Phases
| Phase | Behavior |
|---|---|
| P ≪ K | Nearly exponential growth at rate r |
| P ≈ K/2 | Maximum absolute growth (inflection point) |
| P → K | Growth slows asymptotically to zero |
Inflection Point
The inflection — where growth speed peaks — always occurs at P = K/2. The time of inflection is:
t_inflection = ln((K − P₀) / P₀) / r
Past this point, growth decelerates even though the population continues to rise.
Worked Example — Bacterial Culture
Start with 100 cells in a flask with carrying capacity 10⁶ cells, growth rate r = 1.5 per hour.
- After 6 h: P ≈ 10⁶ / (1 + 9999 × e⁻⁹) ≈ 10⁶ / (1 + 1.23) ≈ 4.5 × 10⁵
- After 8 h: ≈ 9.4 × 10⁵
- After 12 h: essentially K = 10⁶
The classic S-curve — a “lag” of slow visible change, a steep middle, then a plateau.
Where Logistic Growth Appears
| Field | Use |
|---|---|
| Ecology | Population of fish, deer, bacteria |
| Epidemiology | Cumulative disease cases (early stages) |
| Marketing | Product adoption (Bass diffusion model relative) |
| Resource economics | Sustainable harvest yield |
| Machine learning | Sigmoid activation function |
Comparison to Exponential Growth
Exponential growth assumes unlimited resources: P(t) = P₀ × e^(r × t), unbounded. The logistic model adds a (1 − P/K) brake that becomes effective only as P approaches K. For the first part of any logistic curve, the two models are indistinguishable.
Caveats
The logistic equation is deterministic and continuous — real populations are stochastic and integer-valued. For small populations, demographic noise and Allee effects (low-density depression) can dominate. For epidemics, individual heterogeneity, mixing patterns, and interventions all push real curves away from a perfect S.
Connection to the Verhulst Equation
In differential form: dP/dt = r × P × (1 − P/K). This is the original 1838 Verhulst formulation, named after Belgian mathematician Pierre-François Verhulst, and is sometimes called the Verhulst-Pearl equation after Raymond Pearl re-derived it in the 1920s for human population modelling.