Logistic Growth
S-curve constrained growth.
What does Logistic Growth mean?
Logistic growth models how a population grows rapidly at first, then slows as it approaches a maximum sustainable size called the carrying capacity. Unlike exponential growth, which assumes unlimited resources, logistic growth produces an S-shaped curve that reflects real-world constraints such as food supply, habitat space, and competition. This model is widely used in biology, ecology, epidemiology, and business to predict adoption curves and market saturation.
How to calculate Logistic Growth
The logistic growth equation is P(t) = K / (1 + ((K - P0) / P0) * e^(-r*t)), where P0 is the initial population, K is the carrying capacity, r is the intrinsic growth rate, and t is time. For example, with P0 = 100, K = 10,000, r = 0.25 (25%), and t = 20, the population at time 20 is 10,000 / (1 + ((10,000 - 100) / 100) * e^(-0.25 * 20)) = approximately 8,808.
FAQ
Carrying capacity (K) is the maximum population size that an environment can sustain indefinitely given the available resources. In business contexts, it can represent total addressable market or maximum user base. As the population approaches K, growth rate slows to zero.
Exponential growth assumes unlimited resources and grows without bound. Logistic growth introduces a carrying capacity that limits growth — the population accelerates initially but decelerates as it nears the maximum. Exponential growth is a special case of logistic growth where the population is far below carrying capacity.
The growth rate r is the intrinsic rate of increase — how fast the population would grow if there were no resource constraints. A higher r means the population reaches carrying capacity faster. It is entered as a percentage and converted to a decimal internally (e.g., 25% becomes 0.25).
It is used in ecology to model animal and plant populations, in epidemiology to predict disease spread, in marketing to forecast product adoption (technology S-curves), and in business to estimate market saturation. Any scenario with initial rapid growth that plateaus fits this model.
If the initial population exceeds K, the model predicts the population will decrease toward K over time. In nature, exceeding carrying capacity leads to resource depletion, increased mortality, and population decline until a sustainable equilibrium is reached.
Related calculators
- Exponential Growth— Growth proportional to size.
- Doubling Time— Time required to double.
- Percentage Change— Relative increase or decrease.
- Rule of 72— Estimate doubling via interest rate.