Population Growth & Carrying Capacity Calculator

A population with plenty of food, water and space grows faster the bigger it gets, because more individuals have more offspring. That cannot last: as the population fills its environment, each individual gets less, births fall and deaths rise. The carrying capacity K is the population the environment can support over the long run, where births and deaths balance.

K belongs to the environment, not to the population. It changes when the environment does: a dry year, a new predator, better farming or more nesting sites all move it.

• Exponential, N(t) = N₀e^(rt): growth in proportion to size, with no limit. It doubles every ln 2 / r. Real populations follow it only while resources are plentiful. The Exponential Growth & Decay Calculator works it out and explains the number e.
• Logistic, N(t) = K / (1 + ((K − N₀)/N₀)e^(−rt)): starts almost exponential, slows as N nears K, and levels off at K. Its graph is an S-shaped curve. This calculator uses it.
• Logarithmic, such as N(t) = a + b·ln t: keeps growing ever more slowly, with no ceiling. It is a different function, and it is not the logistic model, although the names sound alike.

Tick Also show unlimited (exponential) growth to see how far the two curves part.

The logistic model is the differential equation dN/dt = rN(1 − N/K): the growth rate is the exponential rate rN, scaled down by the share of the capacity still free, 1 − N/K.

• Equilibria. dN/dt = 0 when N = 0 (nothing to grow) or N = K (no room left). Below K the population rises; above K, 1 − N/K is negative, so it falls back toward K.
• Fastest growth. rN(1 − N/K) is a downward parabola in N with its peak at N = K/2, where the rate is rK/4.
• Inflection point. The curve bends upward (concave up) while N is below K/2 and bends over (concave down) above it. It has an inflection point only if it starts below K/2: at t = ln((K − N₀)/N₀) / r.
• Solving it. Separating variables and using partial fractions gives the closed form above.

The logistic model fits populations with steady resources and no large movements in or out: yeast or bacteria in a flask, a species in a closed pond, a herd on a fenced reserve. It explains why growth slows, and it gives a first answer to "how many can this place support?".

It assumes K and r stay fixed, every individual is alike, and the population responds instantly to crowding. Real populations overshoot and crash, follow seasons, migrate, are hunted or harvested, and live with K that shifts with weather and management. Real carrying capacities are uncertain estimates, not fixed numbers.

For decisions about land, water or people, a single K from a ratio of resources is a starting point for discussion, never a forecast on its own. This page draws no conclusions about human populations.