Carrying Capacity Calculator

Enter the population size, intrinsic growth rate, and the average population change over the same time period to estimate the carrying capacity using the logistic growth model.

Carrying Capacity Calculator

Basic Logistic Forecast Grazing Capacity

Enter any 3 values to calculate the missing variable. Use the same time period for r and population change (e.g., per year).

Intrinsic Growth Rate r (% per time period) Population Size N Individuals Thousands Millions Population Change Rate CP (individuals per time period) Individuals Thousands Millions Carrying Capacity K Individuals Thousands Millions

Logistic growth: N(t) = K / (1 + ((K – N₀)/N₀)·e^(−r·t))

Initial Population N₀ Individuals Thousands Millions Intrinsic Growth Rate r (% per selected time unit) per year per month per day Carrying Capacity K Individuals Thousands Millions Time t years months days Species preset (illustrative; fills r and units) None Rabbit (example r≈40% per year) Fast-growing microbes (example r≈200% per day) Humans (example r≈1% per year) Deer (example r≈20% per year) Target of K to reach (%) percent

Estimate pasture carrying capacity using AUs and AUEs

Pasture Area (acres) Forage Standing Crop (lbs/acre) Preset vegetation Upland loamy (~2400) Upland sandy (~2400) Upland clayey (~2200) Lowland overflow (~3500) Wet meadow (~4500) Utilization Factor (%) 25% 35% 50% Grazing Period (days) Daily Forage Demand per AU (lbs/day) Livestock Class (AUE) Beef cow (AU = 1.0) Cow-calf pair (AUE = 1.4) Yearling (AUE = 0.6) Ewe (AUE = 0.2) Goat (AUE = 0.17) Horse (AUE = 1.25) Calculate Reset

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Carrying Capacity Formula

The logistic population growth model is:

\frac{dN}{dt} = rN\left(1-\frac{N}{K}\right)

If you know the current population size N, the intrinsic growth rate r (per time), and the average change in population per time period (CP ≈ ΔN/Δt), you can rearrange to estimate carrying capacity K:

K = \frac{N}{1-\frac{CP}{rN}}

• Where K is the carrying capacity (individuals)
• r is the intrinsic growth rate (as a decimal per unit time; e.g., 50% per year = 0.50/year)
• N is the current population size (individuals)
• CP is the average change in population per unit time (CP ≈ ΔN/Δt, in individuals per unit time)

To estimate the carrying capacity, convert r from percent to a decimal, compute the fraction CP/(rN), then calculate K = N / (1 − CP/(rN)). The time unit used for r and CP must match (for example, both “per year”).

Carrying Capacity Definition

Carrying capacity (often written as K) is the maximum population size of a species that an environment can sustainably support given available resources and other constraints. In simple models it is treated as constant, but in real ecosystems it can change over time as conditions change.

Carrying Capacity Example

How to calculate a carrying capacity?

First, determine the intrinsic growth rate r over a specific time unit. In this example, r = 50% per year, so r = 0.50/year.

Next, determine the current population size. For this example, N = 5000 individuals.

Next, determine the average change in population over the same time unit. In this case, CP = 100 individuals per year (i.e., ΔN/Δt = 100/year).

Finally, calculate the carrying capacity using the formula above:

K = N / (1 − CP/(rN))

K = 5000 / (1 − 100/(0.50·5000))

K = 5000 / (1 − 0.04) = 5000 / 0.96 = 5208.33

Estimated carrying capacity: K ≈ 5208 individuals

FAQ

What is carrying capacity?

Carrying capacity (K) is the maximum population size an environment can sustainably support. It is a key parameter in the logistic population growth model.