Population Growth Calculator and Simulator: Exponential vs Logistic

Calculate a future population, compare exponential and logistic models, find a growth rate or target time, and graph the result. This carrying capacity calculator also shows exact steps, population dynamics, and an accessible data table—all privately in your browser.

Enter population data

Calculation

Enter r as a percentage per selected time unit; 50% means r = 0.50, not 0.50%.

The rate and time must use this same unit.

Inputs
100

50%

1,000

15 years

Advanced simulation and export options

Δt and noise are disabled because exact closed-form curves do not use them.

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Answer

N(15 years) = 995.05

Logistic model · continuous rate

N(15) = 1,000 ÷ [1 + 9 × e−0.50×15] = 995.05

After 15 years, the model estimates about 995 individuals, or 99.5% of carrying capacity.

Assumes a closed population, constant r and K, identical individuals, continuous change, and no time delay.

99.50%
of K reached
4.95
remaining capacity
2.46/year
instantaneous dN/dt
4.39 years
logistic inflection time
125/year
maximum rK/4
1.39 years
doubling time
50%: 4.39 · 90%: 8.79
time to K thresholds
Show step-by-step solution

1. Identify values: N₀=100, r=0.50/year, t=15 years, and K=1,000.

2. Find A: A=(K−N₀)/N₀=(1,000−100)/100=9.

3. Evaluate the exponent: −rt=−(0.50)(15)=−7.50, so e−rt=0.000553.

Final: 1,000/[1+9(0.000553)]=995.05.

Rearranged logistic formulas

Rate: r = −ln[(K/N(t) − 1)/A] / t

Time: t = −ln[(K/N(t) − 1)/A] / r

Capacity: K = N(t)(1 − e−rt) / [1 − N(t)e−rt/N₀]

Explained population scenarios

Select a scenario to load its assumptions and expected behavior.

Compare exponential and logistic curves

Exponential (dashed blue)Logistic (solid red)K, K/2, inflection, and target markers

Pointer or touch the chart for exact values. With the chart focused, use ← and → to inspect points; Home and End jump to the first and last point.

Textual chart summary will appear after calculation.

Population over time data

The table is the complete non-canvas alternative. Change the interval under Advanced options; CSV export uses these same rows.

TimeExponential populationLogistic populationLogistic % of KLogistic dN/dt

How to use this population growth calculator

  1. Choose exponential growth for an unlimited short-run model or logistic growth when resources impose a carrying capacity.
  2. Select the unknown, time unit, and rate convention. Enter percentages clearly: 50 means 50%, while 0.5 means 0.5%.
  3. Calculate for the answer and steps, then inspect the comparable chart and exact table. Use Euler only to study numerical simulation.

Model choice and formulas

Continuous exponential: N(t)=N₀ert
Discrete exponential: N(t)=N₀(1+g)t
Logistic: N(t)=K/[1+Ae−rt], where A=(K−N₀)/N₀.

In discrete mode, this tool uses g for exponential compounding and converts it to a continuous-equivalent logistic rate with r=ln(1+g).

Variables and curve interpretation

N₀ is the starting population, N(t) the population at time t, r the continuous per-capita rate, g the discrete change per period, and K the logistic carrying capacity. When 0<N₀<K and r>0, logistic growth is S-shaped: early growth resembles exponential growth, the full term N(1−N/K) peaks at K/2, and the curve then approaches K.

Assumptions, limits, and applications

Both models assume fixed parameters, homogeneous individuals, and no migration, age structure, delay, predation, or changing environment. Use them for teaching, rough short-run comparisons, microbial or habitat examples, and hypothesis building—not as forecasts without fitting and validation.

Worked examples using the defaults

Defaults: N₀=100, continuous r=50% per year (0.50/year), K=1,000.

1. Calculate logistic N(10)

A=(1,000−100)/100=9. N(10)=1,000/[1+9e−0.50×10]=942.83. After 10 years, the population is about 94.3% of capacity.

2. Calculate exponential N(10)

N(10)=100e0.50×10=14,841.32. Without a resource limit, the estimate greatly exceeds the logistic capacity.

3. Find time to 90% of K

Target N=900. t=−ln[(1,000/900−1)/9]/0.50=8.79 years. The logistic population reaches 900 near the end of year 9.

4. Why early logistic growth resembles exponential growth

At year 1, exponential N=164.87 and logistic N=154.83. Because N is still small compared with K, (1−N/K) is close to 1; the curves separate as crowding strengthens.

Population growth calculator FAQ

Which population growth model should I use?

Use exponential growth for short periods with no meaningful resource limit. Use logistic growth when a defensible carrying capacity slows growth.

What does carrying capacity mean?

Carrying capacity K is the population level the logistic model can sustain under fixed conditions.

What does r represent?

In continuous mode r is the instantaneous per-capita rate per selected time unit. In discrete mode the entered percentage is converted with r = ln(1 + g).

Can a population exceed K?

Real populations can exceed an estimated capacity. The closed-form logistic model approaches K monotonically when N₀ is below K; numerical Euler settings can create artificial overshoot.

What happens when N₀ is greater than K?

With positive r, the closed-form logistic curve declines toward K from above. The result panel reports that starting domain rather than treating it as an error.

How does a negative rate behave?

Exponential growth becomes decay with half-life ln(2)/|r|. In the standard logistic equation, negative r reverses stability around K and can produce behavior that is usually not an ecological decline model.

Why is logistic growth S-shaped?

Growth begins almost exponentially, reaches its steepest point at N=K/2, then slows as the remaining capacity shrinks.

When is logistic growth fastest?

For positive r, the full term rN(1 − N/K) is largest at N = K/2, with maximum growth rK/4.

How do I find time to a target population?

Select “Time t,” enter the target, and calculate. A target at or beyond the logistic asymptote is reported as unreachable rather than as a dash.

Why does simulated overshoot occur?

Large Euler steps can jump over K and oscillate. That is numerical instability; genuine ecological overshoot requires biological mechanisms beyond this one-equation model.

Why may real populations not follow either model?

Migration, age structure, delays, interactions, shocks, and changing resources violate the models' fixed-rate and fixed-capacity assumptions.

Methodology, verification, and sources

Method: closed-form results use N=N₀ert and N=K/[1+((K−N₀)/N₀)e−rt]. Discrete exponential results use N=N₀(1+g)t. Optional simulation uses explicit Euler updates Nnext=N+f(N)Δt, with an optional illustrative multiplicative shock.

Verification: page logic was checked against r=0 (constant exponential population), N₀=K (constant logistic population), the default values above, N₀>K, negative exponential rates, inverse rate/time solutions, and unstable rΔt warnings. Extremely large exponentials are reported as outside the numeric range.

Reviewed: 17 July 2026 by Starlight Tools. No professional credential is claimed. Educational use only; results are not forecasts.

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