Example 1 — Exponential Projection
\(P_0 = 50{,}000\), \(r = 0.03\,\text{yr}^{-1}\), \(t = 10\,\text{yr}\).
Population Growth Calculator
Population Growth Calculator projects future size using exponential or logistic models, growth rates, doubling time, carrying capacity, and step-by-step formulas.
Choose the pattern. Enter any three variables below; we compute the missing one.
Population at the start (≥ 0). Supports math like 1e4.
Exponential: percent per period (allow negative for decay). Linear: absolute increase k per period. Doubling: time D to double.
Time periods (≥ 0). Can be fractional (e.g., 2.5).
Population after t periods (≥ 0). Leave exactly one field empty to solve it.
Equation Preview
Helping Notes
- Exponential: \(X_t = X_0 (1+r)^t\), r in decimal (we take your %/100).
- Linear: \(X_t = X_0 + k\,t\), k is absolute change each period.
- Doubling time: \(X_t = X_0 \cdot 2^{t/D}\), D is periods to double.
- Leave exactly one variable empty to solve it, just like the example site. Values accept math expressions.
Results
Solved Value
All Variables
Error
Steps
What Is a Population Growth Calculator?
A Population Growth Calculator estimates changes in a population over time using standard mathematical models. Exponential growth assumes the growth rate is proportional to the current population, while discrete compounding updates population period-by-period. Logistic growth accounts for environmental limits by introducing a carrying capacity \(K\). With initial population \(P_0\), continuous growth rate \(r\), discrete percentage \(g\), and carrying capacity \(K\), core formulas include:
Continuous exponential growth:
Discrete compounding:
Doubling time and rate conversions:
Logistic growth with carrying capacity \(K\):
About the Population Growth Calculator
This calculator supports continuous, discrete, and logistic growth models. Enter \(P_0\), a time horizon, and either a continuous rate \(r\) or discrete percentage \(g\). If only two measurements are known, the calculator can infer \(r\) from \(P_0\) and \(P_t\) using
or infer \(g\) via . For logistic growth, specify \(K\); outputs include midpoint timing, long-run limit \(\lim_{t\to\infty} P(t) = K\), and comparison to exponential growth. Steps display model choice, parameter conversion, substitutions, and labeled results. Units are flexible (people, cells, devices), and formulas render responsively.How to Use the Population Growth Calculator
- Select a model: Exponential (continuous), Discrete, or Logistic.
- Enter \(P_0\) and known parameters (\(r\) or \(g\)); include \(K\) for logistic growth.
- Provide the time horizon (\(t\) or number of periods \(n\)).
- Click calculate to view parameter conversions, substituted formulas, and projected \(P(t)\) or \(P_n\).
- Compare scenarios by adjusting \(r\), \(g\), or \(K\); copy steps for reports or homework.
Examples
Example 2 — Rate from Two Censuses
\(P_0 = 20{,}000\), \(P_5 = 25{,}000\).
Example 3 — Doubling Time
\(r = 0.02\,\text{yr}^{-1}\).
Example 4 — Logistic Saturation
\(P_0 = 100{,}000\), \(r = 0.05\), \(K = 1{,}000{,}000\), \(t = 10\).
FAQs
When should I use exponential versus logistic growth?
Exponential growth applies to early, resource-abundant phases. Logistic growth is appropriate when carrying capacity \(K\) limits population.
What is the difference between \(r\) and \(g\)?
\(r\) is a continuous growth rate, \(g\) is a per-period percentage. They relate via \(r = \ln(1+g)\) and \(g = e^r - 1\).
How do I estimate \(r\) from two measurements?
Use \(r = (1/t)\ln(P_t/P_0)\). For discrete periods, \(g = (P_n/P_0)^{1/n} - 1\).
Can the calculator compute doubling or halving times?
Yes. Doubling time: \(T_d = \ln 2 / r\). Halving time (negative \(r\)): \(T_h = \ln 2 / |r|\).
What if the carrying capacity \(K\) is unknown?
Use exponential growth or test different \(K\) values; calibrate using field data.
Does immigration/emigration change results?
Constant net flow \(M\) modifies the model: \(\frac{dP}{dt} = rP + M \Rightarrow P(t) = P_0 e^{rt} + \frac{M}{r}(e^{rt}-1)\).
How accurate are long-term projections?
Accuracy decreases over time; treat results as scenarios rather than precise forecasts.
Can I use percentages instead of decimals?
Yes; the tool converts automatically. Formulas internally use decimals for \(r\) and \(g\).
Why do discrete and continuous models differ slightly?
Discrete compounding uses \((1+g)^n\), continuous uses \(e^{rt}\); they match only if \(g = e^r - 1\).
What units should I use for time?
Use consistent units for \(r\) and \(t\) (years, months, etc.).
Can I model multi-phase growth?
Yes. Divide the time horizon into segments with different \(r\) or \(K\), applying formulas piecewise.