Logistic Growth Calculator

How to Use the Logistic Growth Calculator

1. 1. Enter the model. N₀ is the population at time zero, K the carrying capacity (the size the environment can sustain) and r the intrinsic growth rate per unit of time. All three must be greater than zero.
2. 2. Set the time. Type the time t you want the population for, and the unit you measure it in. The unit only labels the answers; r must be in the same unit.
3. 3. Add a target (optional). Enter a population size to find the time at which the model reaches it.
4. 4. Read the results. You get N(t), the share of K, dN/dt, the inflection point, the working, the curve and a table of values you can copy.

The Logistic Growth Equation

Exponential growth assumes every individual keeps reproducing at the same rate. The logistic model, first proposed by Pierre-François Verhulst in 1838, makes the per-capita rate fall as the population fills its environment. It is written as a differential equation and has a closed-form solution:

The factor (1 − N ÷ K) is the share of the carrying capacity that is still unused. While N is small it is close to 1 and growth is nearly exponential, with a doubling time of ln 2 ÷ r. As N approaches K the factor goes to zero and growth stops. The growth rate dN/dt peaks halfway, at N = K ÷ 2, which is the curve's inflection point: before it the curve bends upward, after it the curve bends toward the ceiling.

Worked Example

A population of 50 animals is released into a reserve that can support 1,000, growing at r = 0.5 per year. These are the calculator's default inputs.

1. The offset is (1,000 − 50) ÷ 50 = 19.
2. After 10 years: N(10) = 1,000 ÷ (1 + 19 × e^(−5)) = 886.51 animals, which is 88.7% of K.
3. Growth then is dN/dt = 0.5 × 886.51 × (1 − 0.88651) = 50.306 animals per year, down from 23.75 per year at the start and 125 per year at the peak.
4. The inflection comes when N = 500, at t = ln 19 ÷ 0.5 = 5.8889 years. The maximum growth rate is rK ÷ 4 = 125 per year.
5. The time to reach 750 is ln(750 × 950 ÷ (50 × 250)) ÷ 0.5 = ln 57 ÷ 0.5 = 8.0861 years.

What the Numbers Mean and Where They Mislead

• r is the intrinsic rate. It is the per-capita growth rate at very low density (births minus deaths per individual), not the rate you observe once the population is crowded. The observed per-capita rate is r(1 − N ÷ K), shown in the results.
• K is not fixed in nature. It shifts with food, habitat, disease and season. The model treats it as a constant, so use it for a stable environment or for a classroom exercise.
• It assumes no time lag. Real populations respond to crowding late and can overshoot K and oscillate; this calculator does not model that. If you start above K, it shows the population falling back toward K from above.
• It describes one closed population. Immigration, emigration, harvesting and predation are not included.
• Targets can be unreachable. The curve approaches K but never reaches it, and it cannot cross K, so a target at or above K (when you start below it) has no solution.

Frequently Asked Questions

What is the formula for logistic growth?

The rate form is dN/dt = rN(1 − N ÷ K), and the population at time t is N(t) = K ÷ (1 + ((K − N₀) ÷ N₀) × e^(−rt)). Here N₀ is the starting size, K the carrying capacity and r the intrinsic growth rate.

What is carrying capacity?

Carrying capacity, K, is the largest population an environment can support indefinitely with the food, water, space and other resources available. In the logistic model, the population levels off at K and growth stops there.

Where is the inflection point of a logistic curve?

At N = K ÷ 2. The time it happens is t = ln((K − N₀) ÷ N₀) ÷ r, and the growth rate there is the maximum, rK ÷ 4. If the population starts at or above K ÷ 2, the curve has no inflection after time zero.

How is logistic growth different from exponential growth?

Exponential growth has a constant per-capita rate, so the population grows without limit along a J-shaped curve. Logistic growth lets the per-capita rate fall as N approaches K, producing an S-shaped curve that levels off. For small N the two are almost identical.

How do I find the growth rate r from data?

Early in growth, when N is far below K, ln N is close to a straight line in time with slope r. With two counts at times t₁ and t₂ in that phase, r ≈ (ln N₂ − ln N₁) ÷ (t₂ − t₁). For a whole S-curve, fit the logistic equation with a curve-fitting tool.

Can the population exceed the carrying capacity?

In the continuous logistic model, a population that starts below K never exceeds it. A population that starts above K declines toward it. In real populations, time lags and fluctuating conditions can push the numbers above K temporarily.