Biology · Population Biology

Logistic Growth

Q: What does the logistic growth model describe?

The logistic model describes growth that is slowed by a limited resource. The growth rate is dN/dt = r·N·(1 − N/K), with the population size N, the growth rate r and the carrying capacity K. As long as N is small compared with K, the population grows almost exponentially at rate r. As N approaches the capacity K, the braking factor (1 − N/K) becomes ever smaller and the growth slows down, until it stops completely at N = K. Example bacteria with r = 0.5/h, K = 10⁶ and N₀ = 1000: after 10 hours N is about 860 000, and in the long run N tends towards K. The typical S-shaped growth curve results.

Q: What does the carrying capacity K mean?

The carrying capacity K is the maximum population size an environment can sustain permanently, because resources such as food, space or oxygen are limited. In the logistic model K is the value towards which the population tends in the long run. When N reaches the capacity, the braking factor (1 − N/K) becomes zero and growth stops; the population stays constant. If N is above K, for example after overpopulation, the factor becomes negative and the population shrinks back towards K. K is therefore a stable equilibrium. It depends on the specific environmental conditions and can change if resources or habitat change. K is one of the two central quantities of the model besides the growth rate r.

Q: What is the difference between exponential and logistic growth?

Exponential growth describes a population that grows unchecked at a constant rate; the growth speed keeps rising and the population would theoretically become infinitely large. Logistic growth, by contrast, accounts for limited resources through the braking factor (1 − N/K), which slows the growth as the capacity K is approached. For small populations, when N is much smaller than K, both models are nearly identical, because the braking factor is then close to one. Only at larger population do they differ clearly: logistic growth flattens and tends towards K, exponential growth keeps growing without bound. In nature the logistic model is more realistic, because resources are always limited.

Q: When is the absolute growth rate the largest?

The absolute growth rate dN/dt is largest at half the capacity, that is at N = K/2. There the product of the existing population N and the free room (1 − N/K) is maximal. At smaller N there is much room but few individuals to reproduce; at larger N there are many individuals but hardly any room left. Exactly in the middle the balance is optimal. At this point the S-shaped growth curve has its inflection point, where the slope is steepest. After that the growth decreases again until it stops completely at N = K. This inflection point at K/2 is a characteristic feature of the logistic model.

Q: What is the logistic growth model used for in practice?

The logistic model describes many saturation processes beyond biology. In population ecology it models animal and plant populations with limited habitat. In microbiology it describes the growth of bacterial cultures that reach their limits in a nutrient medium. In epidemiology the spread of a disease in a population resembles the logistic course, because the number of not yet infected people decreases. The spread of new technologies or products also often follows an S-curve, because the market becomes saturated. Wherever initially fast growth is slowed by a natural upper limit, the logistic model is a fitting and widely used descriptive approach.

The differential equation of logistic growth describes population growth with a capacity limit and is a realistic model for biological populations.

UniversityExam-relevant

Study as an exam set I already have an account

Free · no credit card · in your study plan in 2 minutes

Formula

\frac{d N}{d t} = r \cdot N \cdot (1 - \frac{N}{K})

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

N in individuals, r in 1/time, K in individuals, t in time

Variables & units – Logistic Growth

Symbol	Meaning	Unit
N	Population size (number of individuals)	individuals
r	Intrinsic growth rate (birth rate minus death rate)	1/t
K	Carrying capacity of the habitat	individuals
t	Time	s, d or year

Derivation & background – Logistic Growth

Pierre François Verhulst (1838) corrected Malthus' exponential approach: at N << K growth is nearly exponential; at N → K, dN/dt → 0. The inflection point (steepest rate) occurs at N = K/2. Analytical solution: N(t) = K/(1 + ((K−N₀)/N₀)·e^(−rt)).

Exam blueprint

Validity range

Applies to growth with limited resources and constant carrying capacity K over the model period.

Derivation steps

Exponential growth is reduced by a braking factor that tends to zero when N approaches K.

1Without a limit, dN/dt = rN.
2The factor (1 - N/K) slows growth as N approaches K.

Rearrangements

Per-capita growth rate

\frac{1}{N} \frac{d N}{d t} = r (1 - \frac{N}{K})

At N = K/2 the absolute growth rate is maximal.

Task variant

What happens at N = K?

The factor (1 - N/K) becomes 0; growth stops in the model.

Common mistakes

Treating logistic growth as permanently exponential.

Only for N << K is it approximately exponential.

Exam context

Typical in population ecology, bacterial cultures, epidemiology and saturation models.

These mistakes cost points in real exams. The set drills them until they stick.

Train these mistakes in the set I already have an account

Formula cluster

Saturation models

Connects differential equations, biology and model limits.

v = \frac{V _{m a x} \cdot [ S ]}{K _{m} + [ S ]}

Michaelis-Menten Kinetics (Enzyme Kinetics)

P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}

Bayes' Theorem

f (x) = \frac{1}{σ 2 π} e^{- \frac{( x - μ ) ^{2}}{2 σ ^{2}}}

Normal Distribution (Gaussian Bell Curve)

Worked example

Bacteria: r = 0.5/h, K = 10⁶, N₀ = 1000. After t = 10 h: N ≈ 10⁶/(1 + 999·e⁻⁵) ≈ 860 000. As t → ∞: N → K.

Applications

Ecology (population dynamics), epidemiology (SIR models), bacterial growth, tumor growth, market saturation

Quanta exam set

Curated exam set for "Logistic Growth":

Question (front)

Which formula describes Logistic Growth?

Answer in your set

Question (front)

How do you rearrange dN/dt = r·N·(1 − N/K) for Per-capita growth rate?

Answer in your set

Question (front)

Which common mistake happens with Logistic Growth?

Answer in your set

+ 7 more cards: units, variables, derivation, example, exam task

These 10 cards are ready. One click and they sit in your deck, FSRS schedules the reviews until exam day.

Take the set, free I already have an account

Scientific sources

[1]Verhulst, P.F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique, 10, 113–121.

Common notations & search queries

dN/dt = r*N*(1-N/K)N(t)=K/(1+e^(-r(t-t0)))logistisches Wachstum FormelPopulationswachstum Formellogistic growth formulaKapazitätsgrenze WachstumWachstumsgleichung Biologie

Related formulas

f (x) = \frac{1}{σ 2 π} e^{- \frac{( x - μ ) ^{2}}{2 σ ^{2}}}

Normal Distribution (Gaussian Bell Curve)

P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}

Bayes' Theorem

v = \frac{V _{m a x} \cdot [ S ]}{K _{m} + [ S ]}

Michaelis-Menten Kinetics (Enzyme Kinetics)

N = \frac{M \cdot C}{R}

Lincoln Index (Mark-Recapture Method)

More Biology formulas

v = \frac{V _{m a x} \cdot [ S ]}{K _{m} + [ S ]}

Michaelis-Menten Kinetics (Enzyme Kinetics)

p^{2} + 2 pq + q^{2} = 1

Hardy-Weinberg Equilibrium

N (t) = N_{0} \cdot e^{r \cdot t}

Exponential Growth

χ^{2} = \sum \frac{( B - E ) ^{2}}{E}

Chi-Square Test (Goodness of Fit)

Spaced-repetition apps

Compare scheduling workflows

Study evidence

Read methods, results and limits

Compare flashcard apps

FSRS support and study features

Create a formula exam set

Free in Quanta

Frequently asked questions about Logistic Growth

What does the logistic growth model describe?+

The logistic model describes growth that is slowed by a limited resource. The growth rate is dN/dt = r·N·(1 − N/K), with the population size N, the growth rate r and the carrying capacity K. As long as N is small compared with K, the population grows almost exponentially at rate r. As N approaches the capacity K, the braking factor (1 − N/K) becomes ever smaller and the growth slows down, until it stops completely at N = K. Example bacteria with r = 0.5/h, K = 10⁶ and N₀ = 1000: after 10 hours N is about 860 000, and in the long run N tends towards K. The typical S-shaped growth curve results.

What does the carrying capacity K mean?+

The carrying capacity K is the maximum population size an environment can sustain permanently, because resources such as food, space or oxygen are limited. In the logistic model K is the value towards which the population tends in the long run. When N reaches the capacity, the braking factor (1 − N/K) becomes zero and growth stops; the population stays constant. If N is above K, for example after overpopulation, the factor becomes negative and the population shrinks back towards K. K is therefore a stable equilibrium. It depends on the specific environmental conditions and can change if resources or habitat change. K is one of the two central quantities of the model besides the growth rate r.

What is the difference between exponential and logistic growth?+

Exponential growth describes a population that grows unchecked at a constant rate; the growth speed keeps rising and the population would theoretically become infinitely large. Logistic growth, by contrast, accounts for limited resources through the braking factor (1 − N/K), which slows the growth as the capacity K is approached. For small populations, when N is much smaller than K, both models are nearly identical, because the braking factor is then close to one. Only at larger population do they differ clearly: logistic growth flattens and tends towards K, exponential growth keeps growing without bound. In nature the logistic model is more realistic, because resources are always limited.

When is the absolute growth rate the largest?+

The absolute growth rate dN/dt is largest at half the capacity, that is at N = K/2. There the product of the existing population N and the free room (1 − N/K) is maximal. At smaller N there is much room but few individuals to reproduce; at larger N there are many individuals but hardly any room left. Exactly in the middle the balance is optimal. At this point the S-shaped growth curve has its inflection point, where the slope is steepest. After that the growth decreases again until it stops completely at N = K. This inflection point at K/2 is a characteristic feature of the logistic model.

What is the logistic growth model used for in practice?+

The logistic model describes many saturation processes beyond biology. In population ecology it models animal and plant populations with limited habitat. In microbiology it describes the growth of bacterial cultures that reach their limits in a nutrient medium. In epidemiology the spread of a disease in a population resembles the logistic course, because the number of not yet infected people decreases. The spread of new technologies or products also often follows an S-curve, because the market becomes saturated. Wherever initially fast growth is slowed by a natural upper limit, the logistic model is a fitting and widely used descriptive approach.

Retain Logistic Growth for exams

Create a curated FSRS exam set for dN/dt = r·N·(1 − N/K): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Study as an exam set Already registered? Continue directly

Free · curated formula set · LaTeX · FSRS spaced repetition

How do you calculate with Logistic Growth?

Here is how to work through a typical Logistic Growth (dN/dt = r·N·(1 − N/K)) task step by step:

1
Task
What happens at N = K?
Solution path
The factor (1 - N/K) becomes 0; growth stops in the model.