Carrying Capacity Calculator (Logistic Growth)
Use the logistic growth model to find how fast a population is growing at its current size, its per-capita growth rate and the maximum growth rate at half the carrying capacity.
How it works
dN/dt = r · N · (1 − N/K) · Per-capita rate = r · (1 − N/K) · Maximum growth rate = r·K / 4 at N = K/2
The logistic model adds a ceiling — the carrying capacity K — to exponential growth. As the population N approaches K, the term (1 − N/K) shrinks toward zero, so growth slows and eventually stops. Growth is not fastest when the population is small or when it is near K; it peaks at exactly half the carrying capacity, where dN/dt = rK/4. Above K the term goes negative and the population declines back toward the equilibrium.
Worked example
With r = 0.5, K = 1,000 and N = 300: per-capita rate = 0.5 × (1 − 300/1000) = 0.35 per time, so dN/dt = 0.35 × 300 = 105 per time. The fastest possible growth, at N = 500, would be 0.5 × 1000 / 4 = 125 per time.
Sources: Logistic function — Wikipedia
Frequently asked questions
At what population size does a species grow fastest?
Under logistic growth the absolute growth rate peaks at half the carrying capacity, N = K/2. That is why sustainable-harvest models often target keeping a population near K/2.
What happens if N is larger than K?
The factor (1 − N/K) becomes negative, so dN/dt is negative and the population declines back toward K. The calculator shows this and notes the population is above capacity.
What does the intrinsic growth rate r represent?
r is the maximum per-capita growth rate a population could achieve with unlimited resources — births minus deaths per individual per unit time. Real per-capita growth is always r scaled down by (1 − N/K).
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This calculator is for educational purposes. Double-check important results before acting on them.