Logistic population growth

Concept

Logistic population growth

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Statement

Growth slows as a population nears carrying capacity.

Why it matters

Unconstrained exponential growth predicts a population growing without bound, which no real population sustains indefinitely because resources — space, food, nutrients — are always finite. Logistic growth is the standard correction, introducing a resource ceiling, carrying capacity, that slows growth as a population approaches it, and it underlies lotka-volterra-predation's more elaborate two-species model as the single-species starting point.

competitive-exclusion's treatment of two species sharing a limiting resource is a direct two-species extension of the same resource-limitation logic developed here for a single species.

Hypotheses

The environment sustains, at most, a fixed carrying capacity \(K\), and per-capita growth rate declines linearly as population size \(N\) approaches \(K\) from below.This is a simplifying assumption about exactly how growth rate declines, not the only mathematically possible relationship, but the standard, tractable one. The intrinsic, resource-unconstrained per-capita growth rate \(r\) is constant, reflecting a fixed birth-rate-minus-death-rate difference under otherwise ideal, uncrowded conditions.If \(r\) itself varied with age structure or environmental fluctuation, the clean closed-form logistic curve would no longer follow exactly. The model treats the population as a single, unstructured number \(N\), with no age structure, spatial structure, or time lag between a change in density and the corresponding change in growth rate — a real population's response to crowding is often delayed, producing overshoot and oscillation around \(K\) rather than the smooth logistic model's direct approach.

Proof

\frac{dN}{dt}=rN \ \Rightarrow\ \frac{1}{N}\frac{dN}{dt}=r \text{ (constant, regardless of } N\text{)}

Exponential growth alone gives unbounded growth, since per-capita growth rate never declines as the population grows — unrealistic for any population with finite resources. A

\frac{1}{N}\frac{dN}{dt}=r\left(1-\frac{N}{K}\right)

A linear resource-limitation term (Hypothesis 1) makes per-capita growth rate decline toward zero as \(N\to K\), recovering the unconstrained rate \(r\) exactly when \(N\ll K\). A

\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right), \qquad N(t)=\frac{K}{1+\left(\frac{K-N_0}{N_0}\right)e^{-rt}}

Multiplying Step2 through by \(N\) gives the logistic differential equation, whose solution is the S-shaped, sigmoid logistic curve. B

\frac{dN}{dt} \text{ is maximal at } N=K/2

Absolute growth rate, as a function of \(N\), is a downward parabola reaching its peak exactly halfway to carrying capacity, even though per-capita growth rate is falling steadily and monotonically throughout. A

N=K \text{ is a stable equilibrium: } dN/dt>0 \text{ for } 0K

A population perturbed slightly above or below \(K\) is driven back toward \(K\) by the model's own dynamics. A

Result

\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)

Reading. Growth rate is exponential when the population is far below carrying capacity and slows smoothly to zero as it approaches carrying capacity, producing a characteristic S-shaped growth curve rather than unbounded exponential growth.

Scope. Describes a single, unstructured population's smooth approach to a fixed carrying capacity; real populations frequently overshoot and oscillate around \(K\) instead, particularly with a delay between density and its effect on growth rate (Hypotheses, t3).

Corollaries & converses

lotka-volterra-predation's predator-prey model reduces, in the absence of any predator, to exactly this single-species logistic equation for the prey population, making logistic growth the single-species building block the two-species model is constructed from.
competitive-exclusion's treatment of two species competing for a shared limiting resource is a natural two-species extension of this same resource-limitation logic.
Converse: observing a population's absolute growth rate peak partway through its growth trajectory and then decline back toward zero, rather than slowing monotonically from the start, is a signature consistent with logistic dynamics and specifically the \(N=K/2\) maximum-growth-rate point (Step4).

Fails without

Drop the resource ceiling / linear decline in per-capita growth rate (Hypothesis 1): the model reduces to \(dN/dt=rN\), plain exponential growth, in which the population grows without bound regardless of any real, finite limit on resources — a good short-term approximation only while \(N\) remains small relative to any true carrying capacity.
Drop the instantaneous, lag-free response of growth rate to density (Hypotheses, t3): if there is a meaningful delay between a population reaching a given density and that density's effect on birth or death rates being felt, the population characteristically overshoots \(K\) before growth rate falls, and can oscillate around \(K\) for an extended period rather than approaching it smoothly.

Common errors

Assuming carrying capacity \(K\) is a fixed, permanent property of an environment; it depends on resource availability, which can itself change with season, climate, or resource depletion by the population itself.
Confusing the point of maximum per-capita growth rate (which the model places at \(N\to0\)) with the point of maximum absolute growth rate (\(N=K/2\), Step4) — different quantities, occurring at different population sizes.
Assuming every real population follows a smooth logistic curve; many show overshoot and sustained or damped oscillation around \(K\) instead, particularly with any lag between density and its demographic effect.
Treating logistic growth as applicable to a population with no upper resource limit at all; in that regime a population is better approximated by simple exponential growth (Step1) until it approaches a genuine limit.

Discussion

The logistic growth equation was first proposed by Pierre-François Verhulst in 1838, explicitly as a correction to the unconstrained exponential growth model earlier described by Thomas Malthus in 1798, whose argument that population growth would inevitably outpace food production had already become hugely influential, including as a direct influence on both Darwin's and Wallace's independent formulation of natural selection by competition for limited resources.

In discrete-time versions of the logistic model, appropriate for organisms with non-overlapping generations, sufficiently high values of \(r\) can produce not smooth approach to \(K\) but sustained oscillation or even mathematically chaotic dynamics, a qualitatively richer behaviour than the smooth, continuous-time logistic curve derived here ever shows.

Common misconception: that a population sitting exactly at carrying capacity \(K\) has stopped changing altogether, with no births or deaths occurring. In reality \(K\) is a dynamic equilibrium at which births and deaths are occurring but happen to be equal in rate, exactly analogous to the dynamic equilibrium already described in island-biogeography for species richness.

Worked examples

\text{A population with } N_0 \ll K \text{ initially grows at a rate close to the unconstrained exponential rate } r.

While \(N\ll K\), the resource-limitation term \(\left(1-\frac{N}{K}\right)\approx1\), so growth is nearly exponential (Step2). A

\text{As } N \text{ climbs past roughly } K/2, \text{ the resource-limitation term becomes an increasingly important brake, and growth declines smoothly toward zero as } N\to K.

Absolute growth rate peaks near \(N=K/2\) (Step4) and then declines, tracing out the characteristic S-shaped curve. A

\text{near-exponential phase} \to \text{inflection at } N=K/2 \to \text{plateau as } N\to K

Reading. The full logistic trajectory passes through three qualitatively distinct phases governed by a single equation.

Scope. Applies wherever the underlying resource-limitation assumption of Hypothesis 1 holds reasonably well.

Problems

A population has \(r=0.4\) per year and \(K=2000\). At what population size is its absolute growth rate maximal, and what is that maximal growth rate?
Solution
Maximal at \(N=K/2=1000\) (Step4); maximal rate \(=rK/4=0.4\times2000/4=200\) individuals per year.
Explain why a population held for a long period well below its true carrying capacity, by a temporary resource shortage later resolved, is expected to resume growth at close to its full intrinsic rate \(r\) once the shortage ends.
Solution
With \(N\ll K\) restored (the shortage having kept \(N\) low relative to the now-recovered \(K\)), the resource-limitation term \(\left(1-\frac{N}{K}\right)\) is again close to 1, so growth proceeds at close to the unconstrained rate \(r\) (Step1–2), just as at the start of any logistic trajectory.
A population substantially overshoots its estimated carrying capacity before declining and oscillating around that value, rather than approaching it smoothly. What does this suggest about one of the model's Hypotheses?
Solution
It suggests the instantaneous, lag-free response assumed in the basic model (Hypotheses, t3) does not hold for this population; a delay between density and its effect on growth rate is a standard explanation for overshoot-and-oscillate dynamics not captured by the smooth logistic curve.