How this instrument works
A population growing in an unlimited world would expand exponentially forever, but real environments cap how many individuals they can sustain — food, space, nesting sites, and other resources run out. The logistic growth model, first formalized by Pierre-François Verhulst in the 1830s, captures that ceiling with a single parameter, carrying capacity (K): the population size an environment can sustain indefinitely without further growth.
The model's equation, dN/dt = rN(1 − N/K), starts from simple exponential growth (rN) and multiplies it by a braking term, (1 − N/K), that shrinks toward zero as the population N approaches K. When N is small relative to K, that braking term is close to 1 and growth looks nearly exponential; as N climbs toward K, the term shrinks toward zero and growth rate falls with it, reaching exactly zero once the population equals its carrying capacity.
This produces the characteristic S-shaped logistic curve ecologists use to describe everything from bacterial colonies in a petri dish to deer populations in a bounded habitat: slow growth at low population, a period of fast growth through the middle, and growth tapering off as the population saturates its environment's capacity.
- Enter Intrinsic growth rate (r) — the population's maximum per-capita growth rate under ideal, unconstrained conditions.
- Enter Current population (N) — how many individuals are in the population right now.
- Enter Carrying capacity (K) — the maximum population size the environment can sustain long-term.
- Read Population growth rate (dN/dt) — how many individuals are being added to the population per unit time, right now, at this N.
Worked example — a population at half its carrying capacity
A population has an intrinsic growth rate r = 0.1, currently sits at N = 500 individuals, and its environment has a carrying capacity K = 1,000. The braking term is (1 − 500/1000) = 0.5, so dN/dt = 0.1 × 500 × 0.5 = 25.
A growth rate of 25 individuals per unit time is the maximum this population will ever reach under this model — N = 500 is exactly half of K, which is where the logistic curve's growth rate peaks. Below that halfway point growth is still accelerating; above it, growth rate starts falling even though the population itself keeps rising, all the way to zero once N reaches K.
Questions
Why does growth rate peak exactly at half of carrying capacity?
Because dN/dt = rN(1 − N/K) is the product of a rising term (N) and a falling term (1 − N/K), and that product is maximized at the midpoint, N = K/2 — the same shape as any quantity that's a product of one increasing and one decreasing linear factor. Below K/2, the rising N term dominates and growth accelerates; above K/2, the shrinking braking term dominates and growth decelerates, even while the population itself continues to increase.
What happens to dN/dt once the population reaches carrying capacity?
It reaches exactly zero — when N equals K, the braking term (1 − N/K) becomes (1 − 1) = 0, so growth rate is zero regardless of how large r is. The population has stopped growing and, in an idealized logistic model, stays stable at K indefinitely unless something in the environment changes.
What does a negative dN/dt mean?
It means the population is shrinking, which happens whenever N exceeds K — the braking term (1 − N/K) turns negative once population overshoots carrying capacity. This models a population that has temporarily exceeded what its environment can sustain (through a resource boom or migration, for example), and the model predicts it will decline back toward K rather than keep growing.
How is the intrinsic growth rate r determined for a real population?
Field ecologists estimate r from observed birth and death rates under conditions close to unconstrained growth — typically early in a population's establishment, when resources are abundant relative to population size and the (1 − N/K) braking term is close to 1. It varies enormously by species: bacteria and insects can have r values many times per day, while large mammals with slow reproduction often have r well under 1 per year.
Is the logistic model an exact description of real population growth?
No — it's a simplified, widely used starting model that assumes a fixed carrying capacity and instantaneous, smooth response to population density, while real populations respond to resource limits with time lags, environmental fluctuation, and factors the model doesn't capture, like predation or disease. It remains a standard first approximation in ecology precisely because it captures the core dynamic — growth that slows as a population approaches its environment's limit — with just two parameters.