exponentialgrowthcalculator.com

Logistic Growth Calculator

Enter a starting value, a growth rate, and a carrying capacity to see the S-curve's population, percent of capacity, and time to 90% of capacity update as you type.

Logistic growth calculator

The quantity at time zero
%
Intrinsic rate per period
The ceiling the curve approaches
Any unit, days, years, cycles
Population at Time t 1,583.81 after 20 periods at 25% growth, K = 2,000
% of Carrying Capacity 79.2% of K reached
Time to 90% of K 23.44 periods to reach 90% of capacity

Period-by-Period Values

Period tPopulation P(t)% of K
0 50.00 2.50%
1 63.75 3.19%
2 81.12 4.06%
3 102.97 5.15%
4 130.32 6.52%
5 164.29 8.21%
6 206.14 10.31%
7 257.16 12.86%
8 318.57 15.93%
9 391.35 19.57%
10 476.04 23.80%

What the table shows: at the default settings, 50 units growing at 25% per period toward a carrying capacity of 2,000, the curve still looks exponential through period 10: the gain in period 1 is 13.75 units, and by period 10 each period is still adding more than before, 84.70 units, because the population has only reached 23.80% of capacity. The visible bend toward the ceiling doesn't begin until closer to the inflection point at t ≈ 14.65 (where P = K/2 = 1,000); by the centerpiece's t = 20, the population reaches 1,583.81, 79.2% of K, and the per-period gains have already started shrinking as growth throttles down toward the ceiling.

Definition

What Is Logistic Growth?

Logistic growth is a growth model in which a quantity increases according to a fixed rate, but that rate is progressively throttled by a carrying capacity K that caps the maximum size the quantity can reach.

Where plain exponential growth multiplies by the same factor forever and climbs without any limit, logistic growth starts out behaving almost identically, then bends. As the quantity climbs toward K, the room left to grow shrinks, and so does the size of each period's increase, until the curve flattens out just short of the ceiling.

Plotted against time, this produces the characteristic S-shape (also called a sigmoid curve), unlike the ever-steepening J-shape of unconstrained exponential growth. The S-curve has three visible phases: a slow start near P₀, a fast middle stretch where growth looks almost exponential, and a flattening top where the curve settles in near K.

The hinge between the fast middle and the flattening top is the inflection point, the single moment where the curve is rising fastest. It always occurs at exactly half the carrying capacity, P = K/2, never earlier, never later, regardless of the growth rate or the starting value. Before that point the population is still accelerating; after it, growth keeps happening but each period adds less than the one before.

This single throttling idea, growth that slows as it nears a limit, is what separates logistic growth from the unconstrained model used elsewhere on this site, and it is why the two curves need two different formulas even though they start from the same place.

Equation

The Logistic Growth Formula

The logistic growth formula is P(t) = K / (1 + ((K − P₀)/P₀) × e^(−rt)), the closed-form solution to a population that grows at rate r but is damped by the term (1 − P/K). One equation describes the rate of change; the other solves it for any point in time.

Rate of change: the defining equation
dP/dt = rP(1 − P/K)

Growth at any instant equals the intrinsic rate r times the current population P, scaled down by the damping factor (1 − P/K), which shrinks toward zero as P approaches K.

Solved for time t: used by the calculator above
P(t) = K / (1 + ((K − P0)/P0) × e−rt)

This closed form gives the population directly at any t, without having to step through the rate equation period by period.

Variables in the logistic growth equation
SymbolNameWhat it representsExample
P(t)Population at time tThe quantity's value after t periods have passed.1,583.81
P0Initial valueThe quantity at t = 0, before any growth.50
rIntrinsic growth rateThe per-period growth rate as a decimal, before throttling is applied.0.25 (that is 25%)
KCarrying capacityThe ceiling the quantity approaches and never crosses.2,000
tElapsed timeHow many periods have passed.20 periods
1 − P/KDamping factorStarts near 1 when P is small and falls toward 0 as P nears K, which is what bends the curve away from pure exponential growth.0.2081

What the table shows: six symbols cover the whole model. P₀, r, K, and t are entered by hand; P(t) is the computed result. The damping factor is never typed in. It falls out of P and K automatically, reading 0.2081 at the centerpiece's t = 20, meaning growth there is running at only 20.8% of its unthrottled speed.

Method

How to Calculate Logistic Growth

To calculate logistic growth, find the ratio (K − P₀)/P₀, multiply it by e raised to the power of −rt, add 1, then divide the carrying capacity K by that total.

  1. Write down P₀, r, K, and t

    Identify the starting value, the intrinsic growth rate as a decimal, the carrying capacity, and how much time has elapsed.

  2. Compute the ratio (K − P₀)/P₀

    Subtract P₀ from K, then divide by P₀. This captures how much room is left to grow relative to the starting size.

  3. Compute e^(−rt)

    Multiply r by t, negate it, and raise e to that power.

  4. Multiply the ratio by e^(−rt) and add 1

    This total is the denominator of the logistic formula.

  5. Divide K by the denominator

    The result is P(t), the population at time t.

Worked example: wolves reintroduced to a protected habitat

A conservation program reintroduces 40 wolves into a habitat with an estimated carrying capacity of 600 wolves and an intrinsic growth rate of 35% per year. The logistic formula gives the expected population after 8 years.

P0
40 wolves
r
35% → 0.35
K
600 wolves
t
8 years
P(t) = K / (1 + ((K − P0)/P0) × e−rt) P(8) = 600 / (1 + (600 − 40)/40 × e−0.35×8) P(8) = 600 / (1 + 14 × e−2.8) P(8) = 600 / (1 + 14 × 0.0608) P(8) = 600 / 1.8513 P(8) = 324.09 wolves

After 8 years the population reaches roughly 324 wolves, or 54.0% of the habitat's carrying capacity. That is just past the inflection point at t ≈ 7.54 years (P = K/2 = 300), so growth has already begun to slow even though the population is still rising quickly.

Real-World Limits

Why Unconstrained Exponential Growth Is Unrealistic

Every real population eventually runs into some resource limit, food, space, capital, market size, or physical material, and plain exponential growth has no way to represent that limit at all. Left alone, the exponential model just keeps multiplying by the same factor forever.

Run the unconstrained formula out far enough and it diverges to infinity: there is no K term to cap it, so a population, a balance, or a count of anything modeled this way grows past any resource that could realistically sustain it. A colony of bacteria doubling every hour would, left unconstrained, outweigh the planet within a few days. Obviously nothing like that happens, because the nutrients run out long before then.

The logistic curve fixes exactly this gap. Instead of diverging, it bends toward K and settles there, because the damping factor (1 − P/K) drives the growth rate toward zero as the ceiling is approached. The two models agree closely while the quantity is still small, that's the "still exponential-looking" phase covered above, and diverge sharply only once the quantity gets large enough for the ceiling to matter.

Anyone who has already used the exponential growth calculator on the homepage has effectively modeled the K → ∞ edge case of this same page's formula: no ceiling, no bend, just continued multiplication. For a version of this comparison built around unconstrained population counts specifically, with no capacity term at all, see the population growth calculator, which is the direct counterpart to the model described here.

Questions

Frequently Asked Questions

What is logistic growth?

Logistic growth is a growth model in which a quantity increases at a rate proportional to its own size, but that rate is throttled by a carrying capacity K that slows growth as the quantity approaches K. Instead of climbing forever like plain exponential growth, a logistic quantity levels off and stabilizes near its ceiling.

Plotted against time, logistic growth traces an S-shaped curve: slow at the start, fastest through the middle, then flattening near the top as it approaches K.

What is the logistic growth formula, and what does K mean?

The logistic growth formula is P(t) = K / (1 + ((K − P₀)/P₀) × e^(−rt)), where K is the carrying capacity, the maximum value the quantity can sustain and the ceiling the S-curve approaches but never crosses. P₀ is the starting value, r is the intrinsic growth rate, and t is elapsed time.

Removing K from consideration by letting it grow arbitrarily large turns the formula back into plain exponential growth, since the throttling term then has almost no effect.

What does the inflection point represent?

The inflection point is where the logistic curve switches from curving upward to curving toward the ceiling, and it always occurs at exactly half the carrying capacity, P = K/2. Before that point the population is still accelerating; after it, each period's increase gets smaller even though the total keeps rising.

For the calculator's default numbers above, K/2 = 1,000 and the inflection falls at t ≈ 14.65 periods, noticeably before the t = 20 shown in the centerpiece result.

How does logistic growth differ from plain exponential growth?

Logistic growth differs from plain exponential growth by including a carrying capacity K that bends the curve toward a ceiling, while plain exponential growth has no ceiling and increases without bound. Exponential growth is really logistic growth with K set to infinity. The throttling term (1 − P/K) is never small enough to matter.

The exponential growth calculator on the homepage models the unconstrained version directly, with no ceiling term at all.

Why does early logistic growth look exponential?

Early logistic growth looks exponential because when P is small relative to K, the damping factor (1 − P/K) is close to 1 and barely reduces the growth rate. At the calculator's defaults, a population of 50 against a carrying capacity of 2,000 is only 2.5% of the way to the ceiling, so the throttling is almost undetectable.

The gap only becomes visible once the quantity climbs past roughly a third of K, and it becomes obvious past the inflection point at K/2.

Can the carrying capacity K change over time?

In the standard logistic model K is held constant, but real systems sometimes have a K that shifts, a habitat losing food supply, or a market whose total size expands. Modeling a moving K requires re-running the formula with the new K from whatever time it changes, since the closed-form equation assumes a fixed ceiling throughout.

The calculator above always treats K as fixed for the full span from t = 0 to whatever t is entered.

Where is the logistic curve used in practice?

The logistic curve is used to model any quantity that grows toward a resource limit: wildlife populations bounded by habitat capacity, the cumulative spread of a technology or product bounded by the size of its addressable market, and, purely as a curve-fitting exercise, the cumulative case count in a compartmental epidemiology model bounded by the size of the affected group. In every case the same S-shape appears because the same throttling mechanism is at work.

The population growth calculator on this site handles the unconstrained side of that same modeling question.

Why does the calculator show a dash for some inputs?

The calculator shows a dash whenever the starting value P₀ is zero, negative, or greater than or equal to the carrying capacity K, because the formula has no meaningful result in those cases. A population cannot start at or above its own ceiling and still have room to grow toward it.

Enter any P₀ strictly between 0 and K to see the results return.