Logistic
Growth
A model for a quantity that increases
quickly at first and then more slowly as the quantity approaches
an upper
limit. This model is used for such phenomena
as the increasing use of a new
technology, spread
of
a disease,
or saturation
of
a
market
(sales).
The equation for the logistic model is 
.
Here, t is time, N stands for the amount
at time t, N0 is
the initial amount (at time 0), K is the maximum amount
that can be sustained, and r is the rate of growth
when N is very small compared to K.
Note: The logistic growth model can be obtained by solving the
differential equation 
See
also
Exponential
growth, exponential decay
Worked Example
Problem: A biologist introduces 100 fish into a lake that can support at most 10,000 fish. The intrinsic growth rate is r = 0.5 per month. Find the fish population after 12 months.
Step 1: Identify the given values: initial population N₀ = 100, carrying capacity K = 10,000, growth rate r = 0.5 per month, and time t = 12 months.
N0=100,K=10,000,r=0.5,t=12 Step 2: Compute the constant (K − N₀)/N₀ that appears in the denominator.
N0K−N0=10010,000−100=1009,900=99 Step 3: Evaluate the exponential term e^(−rt).
e−rt=e−0.5×12=e−6≈0.002479 Step 4: Substitute into the logistic formula and compute the denominator.
1+99×0.002479=1+0.24539=1.24539 Step 5: Divide K by the denominator to find N(12).
N(12)=1.2453910,000≈8,030 Answer: After 12 months, the fish population is approximately 8,030.
Another Example
This example differs from the first by requiring you to find the unknown growth rate r from a data point before making a prediction — a common task in applied problems.
Problem: A rumor spreads through a school of 500 students. Initially, 5 students know the rumor. After 2 days, 50 students have heard it. Find the intrinsic growth rate r, then predict how many students will have heard the rumor after 5 days.
Step 1: Identify the known quantities: K = 500, N₀ = 5, N(2) = 50, t = 2 days.
N0K−N0=5500−5=99 Step 2: Substitute N(2) = 50 into the logistic equation and solve for e^(−2r).
50=1+99e−2r500⟹1+99e−2r=50500=10 Step 3: Isolate the exponential and solve for r.
99e−2r=9⟹e−2r=999=111⟹r=2ln11≈1.199 Step 4: Use the value of r to find N(5). First compute the exponential term.
e−1.199×5=e−5.995≈0.002497 Step 5: Substitute into the formula.
N(5)=1+99×0.002497500=1.2472500≈401 Answer: The intrinsic growth rate is approximately r ≈ 1.199 per day, and after 5 days about 401 students have heard the rumor.
Frequently Asked Questions
What is the difference between logistic growth and exponential growth?
Exponential growth assumes unlimited resources, so the quantity grows faster and faster without bound. Logistic growth includes a carrying capacity K that acts as a ceiling. Early on, logistic growth looks nearly exponential, but as the population nears K, growth slows and eventually levels off, producing the characteristic S-shaped (sigmoid) curve.
What is the carrying capacity in logistic growth?
The carrying capacity K is the maximum value the quantity can reach given the constraints of the environment or system. In population biology, it represents the largest population an ecosystem can sustain indefinitely. In the logistic formula, as t → ∞, N(t) approaches K.
What does the logistic growth curve look like?
The logistic growth curve is S-shaped (sigmoid). It starts with slow growth when the population is small, accelerates through a phase of rapid increase near the middle, then decelerates as it approaches the carrying capacity K. The inflection point — where growth is fastest — occurs at N = K/2.
Logistic Growth vs. Exponential Growth
| Logistic Growth | Exponential Growth |
|---|
| Definition | Growth that starts fast then levels off at a carrying capacity K | Growth where the quantity multiplies by a constant factor per unit time, without bound |
| Formula | N(t) = K / [1 + ((K − N₀)/N₀) e^(−rt)] | N(t) = N₀ · e^(rt) |
| Long-term behavior | Approaches a finite limit K | Grows without bound (→ ∞) |
| Curve shape | S-shaped (sigmoid) | J-shaped |
| When to use | When resources or capacity are limited (populations, market saturation, disease spread) | When growth is unconstrained (early-stage bacteria, compound interest) |
Why It Matters
Logistic growth appears in biology (population dynamics), epidemiology (modeling how diseases spread through a population), business (market saturation and technology adoption), and chemistry (autocatalytic reactions). In AP Calculus and college math courses, the logistic differential equation dN/dt = rN(1 − N/K) is a standard separable differential equation you are expected to solve. Understanding this model helps you recognize that real-world growth almost always faces limits, making logistic growth far more realistic than pure exponential growth.
Common Mistakes
Mistake: Confusing the intrinsic growth rate r with the actual growth rate at a given time.
Correction: The parameter r is the maximum per-capita growth rate when the population is very small. The actual growth rate decreases as N approaches K. The instantaneous growth rate is dN/dt = rN(1 − N/K), which equals zero when N = K.
Mistake: Forgetting to compute (K − N₀)/N₀ and instead using K/N₀ in the denominator.
Correction: The correct constant in the denominator is (K − N₀)/N₀, not K/N₀. Using K/N₀ will shift the entire curve and give the wrong initial value. You can verify by checking: at t = 0, the formula must return N(0) = N₀.
