Carrying Capacity Calculator — Logistic Population Growth
The logistic growth model predicts how a population grows when limited by resources. Enter the carrying capacity K, initial population size, intrinsic growth rate r and the time elapsed to calculate the current population and how close it is to the environmental limit.
years
Logistic growth: N(t) = K / (1 + ((K - N0) / N0) x e^(-rt))
- 1
Growth ratio A = (K − N₀) ÷ N₀
(1,000 − 50) ÷ 50 = 19 - 2
Decay factor e^(−r × t)
e^(−0.5 × 20) = 0.000045 - 3
Denominator 1 + A × e^(−rt)
1 + 19 × 0.000045 = 1.0009 - 4
Population N(t) = K ÷ denominator
1,000 ÷ 1.0009 = 999.1
How does this calculator work?
Population at time t = K / (1 + ((K - N0) / N0) x e^(-r*t)). Growth starts slow, accelerates to a maximum at population K/2, then decelerates as resources become limiting, asymptotically approaching the carrying capacity K.
Formula
How this is calculated
In nature, populations cannot grow exponentially forever — resources such as food, space and water impose an upper limit called the carrying capacity (K). The logistic model captures this by multiplying the exponential growth term by a braking factor (1 - N/K): growth is fastest when the population is small relative to K, slows as numbers increase, and approaches zero as the population nears K.
The formula N(t) = K / (1 + ((K - N0) / N0) x e^(-r*t)) gives the population at any time t, where N0 is the starting population, r is the intrinsic (maximum per-capita) growth rate, and e is approximately 2.718. The curve is S-shaped (sigmoidal): slow start, rapid middle growth, then levelling off. The inflection point — where growth rate is highest — occurs when the population reaches exactly K/2, at time t = ln((K - N0) / N0) / r.
The model assumes a closed population (no immigration or emigration), a constant K, and that all individuals are identical. Real ecosystems have stochastic variation, age structure, and non-constant K. The model is widely used in ecology, fisheries management, and epidemiology as a first approximation.
Frequently asked questions
Carrying capacity (K) is the maximum population size that an environment can sustain indefinitely, given available resources such as food, water, shelter and competition. It is not fixed — it changes if resource availability changes.
The intrinsic growth rate r is the theoretical maximum per-capita rate of increase when the population is far below K and resources are not limiting. For many mammals it is well below 1 per year; for bacteria or insects it can be very large.
In the logistic model, a population above K declines toward K, because the braking factor (1 - N/K) becomes negative. This calculator requires N0 < K — for overshoot scenarios, specialist ecological models are needed.
Also known as
TG we-Calculate Editorial Team. (2026). Carrying Capacity Calculator — Logistic Population Growth [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/carrying-capacity-calculator
TG we-Calculate Editorial Team. "Carrying Capacity Calculator — Logistic Population Growth." TG we-Calculate. 2026. https://we-calculate.com/calculator/carrying-capacity-calculator.
TG we-Calculate Editorial Team, "Carrying Capacity Calculator — Logistic Population Growth," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/carrying-capacity-calculator
@misc{wecalculate_carrying_capacity_calculator, title = {Carrying Capacity Calculator — Logistic Population Growth}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/carrying-capacity-calculator}}, year = {2026}, note = {TG we-Calculate} }
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