Restricted and Logistic Growth: Overview of Growth Processes
Restricted growth approaches a limit, and logistic growth follows an S-shaped curve that starts almost exponentially and then levels off at a maximum called the carrying capacity. In brief: linear growth adds a constant amount, exponential growth multiplies by a constant factor, restricted growth approaches an upper bound S with f(t) = S − (S − f0)at for 0 < a < 1, and logistic growth is f(t) = K/(1 + Ae−rt).
Pure exponential growth cannot continue indefinitely because of limited food, space, money or customers. Real processes therefore slow down. This guide compares the main growth types, explains restricted (bounded) growth and the logistic model, gives a table with the numbers for each type and shows how to recognise each pattern in data.
Which growth processes are there?
| Type | Rule | Formula | Long-term behavior |
|---|---|---|---|
| Linear | adds a constant amount | f(t) = f0 + dt | grows without bound, steadily |
| Exponential | multiplies by a constant factor | f(t) = f0 · bt | grows without bound, ever faster |
| Restricted | change proportional to the gap to a limit | f(t) = S − (S − f0)at | approaches S |
| Logistic | growth rate proportional to size and to remaining room | f(t) = K/(1 + Ae−rt) | S-curve approaching K |
What is restricted growth?
In restricted (or bounded) growth, a quantity approaches a fixed limit S, and the remaining gap S − f(t) shrinks by a constant factor a (0 < a < 1) at each step. The formula is f(t) = S − (S − f0) · at. The increase per step is largest at the beginning and gets smaller. The graph is an exponential decay curve turned upside down, with a horizontal asymptote y = S.
Example 1: learning
A student can memorise at most 100 words in a list. Each day, 30% of the words not yet known are learned. Starting from 0: f(t) = 100 − 100 · 0.7t. After 1 day: 30 words. After 2 days: 51. After 5 days: 100 − 100 · 0.16807 ≈ 83.2. After 10 days: 100 − 2.82 ≈ 97.2. The limit 100 is never exceeded.
Example 2: cooling
A drink at 90 °C is placed in a room at 20 °C. The temperature difference falls by 15% per minute: T(t) = 20 + 70 · 0.85t. After 10 minutes: 20 + 70 · 0.1969 ≈ 33.8 °C. The temperature approaches 20 °C but does not go below it. This is Newton’s law of cooling, an example of exponential decay of a difference.
What is logistic growth?
The logistic model describes growth in which the rate is proportional both to the current size and to the unused capacity. At small sizes it looks exponential, in the middle it is fastest, and near the capacity K it slows down. The formula is f(t) = K/(1 + Ae−rt), with A = (K − f0)/f0 and a growth parameter r > 0.
Properties: f(0) = K/(1 + A) = f0; the function increases and approaches K; the inflection point, where the growth is fastest, is at f = K/2, and happens at t = ln A/r.
Example 3: a population
A fish population in a lake follows f(t) = 1000/(1 + 9e−0.5t), t in years. Then f(0) = 1000/10 = 100. After 2 years: e−1 = 0.3679, so f = 1000/(1 + 3.311) ≈ 232. After 4 years: e−2 = 0.1353, so f = 1000/(1 + 1.218) ≈ 451. The half-capacity 500 is reached at t = ln 9/0.5 ≈ 4.39 years. After 10 years: e−5 = 0.00674, so f ≈ 1000/1.0606 ≈ 943.
What does the S-curve look like?
The sketch shows the two phases: an initial phase with accelerating growth and a later phase with decelerating growth. The change occurs at the inflection point, at half of K.
Growth types compared in a table
| t | Linear 100 + 50t | Exponential 100 · 1.3t | Restricted 1000 − 900 · 0.8t | Logistic 1000/(1 + 9e−0.5t) |
|---|---|---|---|---|
| 0 | 100 | 100 | 100 | 100 |
| 2 | 200 | 169 | 424 | 232 |
| 4 | 300 | 286 | 631 | 451 |
| 6 | 400 | 483 | 764 | 691 |
| 8 | 500 | 816 | 849 | 858 |
| 10 | 600 | 1379 | 903 | 943 |
The exponential column exceeds the limit 1000 at around t = 9, which is impossible for a quantity with capacity 1000. The restricted and logistic columns stay below 1000 and flatten out.
Where do these models come from?
The logistic equation was introduced by the Belgian mathematician Pierre François Verhulst in 1838 as a correction to Malthus’s unlimited exponential growth. The name “logistic” was given by Verhulst, and the model was rediscovered in the twentieth century in studies of populations, by Raymond Pearl and Lowell Reed among others. Restricted growth follows the same law as cooling, studied by Isaac Newton.
Where are they used today?
Ecologists model animal populations with carrying capacity. Marketers describe the adoption of a new product, which starts slowly, speeds up and saturates when most potential customers have bought it. Epidemiologists use logistic-type curves for the cumulative number of cases in a wave. Learning curves, battery charging and the growth of a plant are often well described by restricted or logistic growth.
Which mistakes are common?
Extending an exponential trend too far
Early growth data can look exponential even when the process is logistic. Predictions beyond the middle of the S-curve are unreliable without information about the capacity.
Confusing the capacity with the initial value
In f(t) = K/(1 + Ae−rt), the starting value is K/(1 + A), not K.
Expecting the maximum growth at the end
The fastest increase in the logistic model occurs when f = K/2.
Mixing the bound and the base
In restricted growth, the factor a is below 1 and applies to the gap S − f, not to f.