Carrying Capacity Calculator (logistic K from growth rate or three counts)

Understanding the carrying capacity calculator — concept and principle

Carrying capacity K is the largest population an environment can support in the long run. This calculator estimates K for logistic growth, either from the current population, its growth rate and the intrinsic rate r, or from three counts taken at equal intervals, and projects the population forward.

How it is calculated

• Logistic growth: dN/dt = rN(K − N) ÷ K.
• From N, dN/dt and r: K = rN² ÷ (rN − dN/dt).
• From three equally spaced counts: K = N₁(N₀N₁ + N₁N₂ − 2N₀N₂) ÷ (N₁² − N₀N₂); r = ln[(K/N₀ − 1) ÷ (K/N₁ − 1)] ÷ Δt.
• Projection: N(t) = K ÷ (1 + ((K − N)/N) e^(−rt)).
• Fastest growth rK/4 occurs at N = K/2.

The principle

A small population with plenty of resources grows almost exponentially at rate r. As it grows, competition for food, space or other resources slows it down. The logistic model expresses this by multiplying exponential growth by the fraction of capacity still unused, (K − N) ÷ K. Growth is slow when the population is tiny, fastest at half the capacity, and stops at K.

Worked example

Default inputs (method: N, dN/dt and r): N = 400, dN/dt = 30, r = 0.15, projection 5 time units.

1. K = 0.15 × 400² ÷ (0.15 × 400 − 30) = 24,000 ÷ 30 = 800.
2. The population is at 50% of K, which is exactly where growth is fastest: rK/4 = 30 per time unit at N = 400.
3. After 5 time units: N = 800 ÷ (1 + 1 × e^(−0.75)) ≈ 543.3.
4. Time to reach 90% of K = ln 9 ÷ 0.15 ≈ 14.65 time units.
5. Intrinsic growth rate used = 0.15.

How to read the result

Managers of fisheries and wildlife sometimes aim to keep populations near K/2, where growth (and the sustainable harvest in this simple model) is greatest. Real populations fluctuate with weather, predators and disease, and K itself can change, so treat the estimate as a guide.

Common mistakes

• Using growth rates and counts measured in different time units.
• Applying the logistic model to populations with strong time lags or seasonal breeding.
• Treating K as fixed when habitat changes.
• Using three counts that are not equally spaced in time.

Frequently asked questions

What is carrying capacity in ecology?

The maximum population size an environment can sustain over time with its available resources.

When does a logistic population grow fastest?

At half the carrying capacity, N = K/2.

Can a population exceed its carrying capacity?

Temporarily yes, for example with time lags, but then it usually declines.

Sources: Verhulst, P.-F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance mathématique et physique, 10. · Gotelli, N. J., A Primer of Ecology, 4th ed. Sinauer