The logistic model
Exponential growth cannot last forever: food, space or other resources run out. The logistic model, introduced by Pierre-François Verhulst in 1838, adds a carrying capacity K, the maximum size the environment can support. Growth is fast at first, slows as the population approaches K, and levels off, tracing an S-shaped curve.
The formula
P(t) = K ÷ (1 + ((K − P0) ÷ P0) × e^(−r t))
- P0: initial population;
- K: carrying capacity;
- r: intrinsic growth rate per time unit;
- t: time.
The growth rate at any moment is dP/dt = r × P × (1 − P ÷ K).
Key properties
- Inflection point: growth is fastest when P = K ÷ 2, at time t* = ln((K − P0) ÷ P0) ÷ r.
- Maximum growth rate: r × K ÷ 4.
- Time to reach a target P: t = ln((K − P0) ÷ P0 × P ÷ (K − P)) ÷ r.
- When P is much smaller than K, the model behaves like exponential growth.
Example
P0 = 100, K = 1,000, r = 0.2 per year:
- at t = 10 years: P ≈ 451, 45% of capacity;
- growth at that moment: 0.2 × 451 × 0.549 ≈ 49.5 per year, close to the maximum of 50;
- inflection point at t ≈ 11 years, when P = 500;
- 900 is reached after about 22 years.
Where it is used
- Ecology: populations of animals, plants, bacteria in a culture.
- Epidemiology: the early cumulative curve of an outbreak in a closed population.
- Business: adoption of a new product or technology (the S-curve of diffusion), market saturation.
- Machine learning: the logistic (sigmoid) function underpins logistic regression.
Limits of the model
- Real populations overshoot, oscillate or crash, especially with time lags in reproduction.
- K can change with the environment, seasons or technology.
- Predators, competition and migration need richer models, such as Lotka–Volterra.
- Estimating r and K from early data is unreliable: many different S-curves fit the first part of the curve equally well.