Algebra
How To Calculate Logistic Growth
Logistic growth is exponential growth with a ceiling. Everything grows quickly at first, then the growth rate fades as the population presses against what the environment can support.
Quick Answer
P(t) = K / (1 + ((K - P0) / P0) x e^(-rt))
- P(t)
- The population at time t
- K
- The carrying capacity, the ceiling the growth approaches
- P0
- The starting population
- r
- The intrinsic growth rate per unit of time
Divide the gap ratio by the exponential decay term, add one, and divide the carrying capacity by the result. Starting at 100 with a ceiling of 1000 and a rate of 50 percent, the population at t = 5 is 575.1209, which is 57.51 percent of capacity.
What Is Logistic Growth?
Logistic growth describes a quantity that grows fast when it is small and slows down as it approaches a limit. Plot it and you get the famous S curve, flat at the start, steep in the middle, flat again at the top.
Exponential growth has no ceiling. A population growing at a fixed percentage doubles forever, which works for a while and then collides with a world that does not grow with it.
The logistic model fixes that by making the growth rate depend on how full the system already is. The rate is r times the current size times one minus the current size over the capacity.
That one minus P over K term is the whole idea. When P is tiny it is nearly one and growth looks exponential. As P approaches K it falls towards zero and growth stops.
The carrying capacity K is the ceiling. It is the maximum the environment, the market or the system can hold, and the curve approaches it without ever quite touching it.
The fastest growth happens at exactly half the capacity. Below K over 2 there is not enough population to multiply; above it there is not enough room left, so the peak sits dead in the middle.
The closed form solution divides K by one plus a decaying exponential. The exponential term starts large when the population is far below capacity and decays towards zero as time runs on.
The constant inside that exponential measures how much room there is left to grow. It is the gap between the capacity and the start, divided by the start, which is 9 for a start of 100 against a ceiling of 1000.
Populations, epidemics, product adoption and market saturation all follow the shape. A new product spreads slowly, then quickly, then stalls as the remaining untapped buyers run out.
The early phase is genuinely exponential. With a start far below capacity, the doubling time in the early phase is ln(2) divided by r, which is 1.3863 time units for a rate of 50 percent.
The time to reach half the capacity is a clean formula: the natural log of the gap ratio divided by the rate. For a gap ratio of 9 and a rate of 0.5 that is 4.3944 time units.
The curve is symmetric around its midpoint. Whatever the value is at half the capacity plus some time, the value at half the capacity minus that time adds up to K, which is why the S looks balanced.
Start above the capacity and the model runs backwards: the population declines towards K instead of growing towards it. The same formula handles both directions without any change.
The rate r controls how quickly the curve turns over, not how high it goes. Doubling r halves the time to every milestone but leaves the ceiling exactly where it was.
Real systems are messier than the model. Capacities move, there are time lags, and populations overshoot and crash, so treat the S curve as a first approximation that captures the saturation idea.
Formula
P(t) = K / (1 + ((K - P0) / P0) x e^(-rt))
Formula page →The closed form solution, giving the population directly at any time.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P(t) | Population at time t | units of population | The size after t units of time. |
| K | Carrying capacity | units of population | The ceiling the population approaches. |
| P0 | Starting population | units of population | The size at time zero. |
| r | Growth rate | per unit time | The intrinsic rate per unit of time. |
dP/dt = r x P x (1 - P/K)
Formula page →The differential equation behind the curve, and the fastest way to read the current momentum.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| dP/dt | Growth rate | units per unit time | How fast the population is rising right now. |
| r | Intrinsic rate | per unit time | The unconstrained rate when the population is tiny. |
| 1 - P/K | Room left | unitless | The fraction of capacity still unused. |
t = ln( ((K - P0) / P0) / (K/P - 1) ) / r
Formula page →Inverts the curve, so you can ask when a milestone will be reached.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| t | Time | unit of time | How long until the target is reached. |
| P | Target population | units of population | The level you want to know the date of. |
| K | Carrying capacity | units of population | The ceiling of the model. |
How To Calculate Logistic Growth
- 1
Set the capacity and the starting value
The ceiling is 1000 and the start is 100. Knowing both is what separates the logistic model from plain exponential growth, which needs only a rate.
- 2
Work out the gap ratio
The room left to grow is 1000 minus 100, which is 900, and 900 over the starting 100 is 9. This single number says the population can multiply nine times over before it is full.
- 3
Evaluate the decay term
The exponential is e to the minus r times t, which is e to the minus 2.5 for a rate of 50 percent over 5 units of time, giving 0.082085.
- 4
Divide the capacity by one plus the product
Nine times 0.082085 is 0.738765, and one plus that is 1.738765. A thousand divided by 1.738765 is 575.1209, which is 57.51 percent of the ceiling.
Examples
Example 1: A culture starting at 100 with a ceiling of 1000
- Starting population
- 100
- Carrying capacity
- 1000
- Growth rate
- 50
- Time
- 5
| Step | Calculation | Result |
|---|---|---|
| Gap ratio | (1000 - 100) / 100 | 9 |
| Decay term | e^(-0.5 x 5) | 0.082085 |
| Denominator | 1 + 9 x 0.082085 | 1.738765 |
| Population at time 5 | 1000 / 1.738765 | 575.1209 |
Result: The gap ratio is 9 and the decay term is 0.082085, so the population at time 5 is 575.1209, which is 57.51 percent of the ceiling. Growth is still running at 122.1784 per unit of time.
Example 2: An app growing from 2000 users towards a 50000 market
- Starting population
- 2000
- Carrying capacity
- 50000
- Growth rate
- 80
- Time
- 3
| Step | Calculation | Result |
|---|---|---|
| Gap ratio | (50000 - 2000) / 2000 | 24 |
| Decay term | e^(-0.8 x 3) | 0.090718 |
| Denominator | 1 + 24 x 0.090718 | 3.177231 |
| Users after three months | 50000 / 3.177231 | 15736.9741 |
Result: With a gap ratio of 24 and a decay term of 0.090718, the app reaches 15736.9741 users after three months. That is 31.47 percent of the 50000 addressable market, so growth is still accelerating.
Calculator
Population at time t
575.1209
- Share of the capacity
- 57.51%
- Growth rate at that point
- 122.1784
- Time to reach half capacity
- 4.3944
- Population at twice the time
- 942.8256
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Logistic Growth calculator page.
Common Mistakes
Using exponential growth and sailing past the ceiling
Exponential growth has no limit, so it will happily report more than the market or the habitat can hold. The logistic model exists precisely to stop that.
Entering the rate as a percentage instead of a decimal
Fifty percent is 0.5 in the formula. Using 50 makes the decay term vanish and the answer jump straight to the capacity of 1000.
Assuming a constant doubling time
Doubling slows as the population fills up. The early doubling time of 1.3863 applies only while the population is far below capacity.
Starting above the capacity and expecting growth
The curve declines towards K when the start exceeds it. Reading that as a bug rather than the model working correctly is a common misread.
Inverting the gap ratio
The ratio is the gap over the start, not the start over the gap. Flipping it gives 0.1111 instead of 9 and pushes the answer above the true value.
Expecting the fastest growth at the end
Peak growth is at half the capacity, where there is both population and room. Approaching the ceiling, the growth rate falls towards zero.
FAQ
What is the logistic growth formula?
P(t) equals K divided by one plus the gap ratio times e to the minus r t, where the gap ratio is the capacity minus the start, divided by the start.
What is carrying capacity?
The maximum the system can sustain, written K. The curve approaches it asymptotically, so in theory the population never quite reaches it in finite time.
How is logistic growth different from exponential growth?
Exponential growth has a constant rate and no ceiling. Logistic growth multiplies that rate by the fraction of room left, so it slows down and settles at K.
Where is the growth fastest?
At exactly half the carrying capacity. The growth rate formula peaks there, because both the population and the remaining room are as large as they can be together.
Can the population exceed the carrying capacity?
Not in this model from a start below K. If the starting value is above K the curve declines towards it instead, which is the same formula running in reverse.
How long does it take to reach half the capacity?
The natural log of the gap ratio divided by the growth rate. For a gap ratio of 9 and a rate of 0.5 that is 4.3944 time units.
References
- [1]Wikipedia, Logistic function — https://en.wikipedia.org/wiki/Logistic_function
- [2]Wikipedia, Carrying capacity — https://en.wikipedia.org/wiki/Carrying_capacity
- [3]Wikipedia, Malthusian growth model — https://en.wikipedia.org/wiki/Malthusian_growth_model