Which Equation Represents the Logistic Growth Rate of a Population?

The equation that represents the logistic growth rate of a population is dN/dt = rN(1 − N/K), where N is population size, t is time, r is the intrinsic rate of increase, and K is the carrying capacity. This differential equation gives the population’s growth rate at any moment: growth starts out nearly exponential when N is small and slows toward zero as N approaches K. So if you are asking which equation represents the logistic growth rate of a population, that expression — sometimes rearranged as dN/dt = rN(K − N)/K — is the standard answer used in ecology and population biology.

The Logistic Growth Equation

The standard form is:

dN/dt = rN(1 − N/K)

Each symbol has a specific meaning:

• dN/dt — the instantaneous rate of change in population size, or how many individuals are added (or lost) per unit of time.
• N — the current population size (number of individuals).
• r — the intrinsic rate of increase, the maximum per-capita growth rate under ideal conditions with unlimited resources.
• K — the carrying capacity, the maximum population size the environment can sustain given available food, space, and other resources.

The term (1 − N/K) is the density-dependent factor, or environmental resistance term. It ranges from nearly 1 when N is tiny to 0 when N equals K. The equation is commonly credited to the Belgian mathematician Pierre-François Verhulst, who published it in the 1830s; it was later popularized by Raymond Pearl and Lowell Reed, which is why it is sometimes called the Verhulst–Pearl equation.

A quick worked example: suppose r = 0.5 per year, K = 1,000 individuals, and N = 200. Then dN/dt = 0.5 × 200 × (1 − 200/1,000) = 0.5 × 200 × 0.8 = 80 individuals per year. At N = 900, dN/dt = 0.5 × 900 × 0.1 = 45 per year — growth has slowed sharply as the population crowds K.

Why This Equation Works

Logistic growth is built on a simple piece of ecological logic: populations have the potential to grow exponentially, but real environments impose limits. The equation captures both forces in one expression.

The first factor, rN, is exactly the exponential growth term. If resources were unlimited, a population would grow at a constant per-capita rate r, producing the familiar J-shaped curve. In exponential growth, the per-capita growth rate never changes no matter how large the population becomes.

The second factor, (1 − N/K), is what makes growth density-dependent. Density dependence means the per-capita growth rate itself changes as population size changes. Here is how it behaves:

• When N is much smaller than K, N/K is close to zero, so (1 − N/K) is close to 1. The population grows almost exponentially.
• When N is half of K, the factor equals 0.5, so the population grows at half its maximum potential rate.
• When N equals K, the factor becomes 0, so dN/dt = 0. The population stops growing.
• If N somehow exceeds K, the factor turns negative, and the population declines back toward K.

This produces the characteristic S-shaped (sigmoid) curve: slow initial growth, a rapid acceleration through the middle, then a gradual leveling off at K. Nature Education’s Scitable library describes this as growth that is “density-dependent,” because the birth and death rates respond to crowding (Nature Education).

An important subtlety: the maximum number of new individuals added per unit time does not occur at N = K. It occurs at N = K/2, where the product N(1 − N/K) is largest. This matters in conservation and harvesting management — for example, when setting sustainable fishing or hunting quotas.

Common Equation Variations

Because textbooks and instructors format the same relationship differently, it helps to recognize the equivalent forms. All of the following describe identical logistic growth:

Discrete-time versions also exist for populations with non-overlapping generations — for instance, the Ricker model, N(t+1) = N(t) · exp[r(1 − N(t)/K)], and simpler discrete analogues like N(t+1) = N(t) + rN(t)(1 − N(t)/K). These are used when you count a population in annual pulses rather than continuously. Khan Academy’s population growth material covers both differential and discrete treatments, noting that continuous logistic growth produces the smooth sigmoid curve (Khan Academy).

If an exam or quiz presents the formula with K − N in the numerator, subtracts an rN²/K term, or uses lambda notation for discrete steps, you are still looking at the logistic model — just a rearranged or discretized version.

Exponential vs. Logistic

The most common point of confusion is telling the two growth models apart:

• Exponential growth: dN/dt = rN. No K term. Growth is density-independent, the per-capita rate stays constant, and the curve is a J shape that rises without bound. It models idealized conditions — a new species in an empty habitat, or bacteria in fresh culture medium.
• Logistic growth: dN/dt = rN(1 − N/K). Includes the carrying-capacity term. Growth is density-dependent, the per-capita rate falls as N rises, and the curve is an S shape that levels off at K.

So which equation answers the query? If the formula includes r, N, and a term subtracting N/K (or any equivalent), it is the logistic equation. If it contains only r and N, with no ceiling, it is exponential. Encyclopaedia Britannica’s treatment of population growth notes that logistic models assume resources are finite, which is precisely why a limiting term appears (Encyclopaedia Britannica).

In real ecosystems, few populations follow the smooth logistic curve exactly — environmental stochasticity, time lags, and Allee effects all complicate the picture. But as a first approximation for resource-limited growth, the logistic equation remains the standard model in ecology courses and textbooks.

Key Takeaways

• The logistic growth rate equation is dN/dt = rN(1 − N/K), where r is the intrinsic rate of increase and K is the carrying capacity.
• The (1 − N/K) factor makes growth density-dependent: fast when N is small, zero when N reaches K, negative if N overshoots K.
• Equivalent forms include dN/dt = rN(K − N)/K and dN/dt = rN − (r/K)N²; discrete versions like the Ricker model are also used.
• Maximum absolute growth occurs at N = K/2, not at K.
• Exponential growth (dN/dt = rN) lacks the carrying-capacity term and produces an unbounded J curve; logistic growth produces the S-shaped curve that levels off.
• Logistic models are approximations of well-established theory, not exact predictions for every real population — for applied management decisions, consult a qualified ecologist or wildlife professional.