Enter your initial population, growth rate (%), and time period to calculate projected population size using exponential or logistic growth models. For logistic growth, add a carrying capacity. You get back the final population, total growth, and a year-by-year growth table.
Projected Population
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Total Growth
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Total Growth (%)
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Doubling Time
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Exponential growth assumes unlimited resources, so the population grows at a constant percentage rate indefinitely. Logistic growth is more realistic — it accounts for a carrying capacity (K), the maximum population an environment can sustain. As the population approaches K, the growth rate slows, producing the characteristic S-shaped curve.
Use the formula x(t) = x₀ × (1 + r/100)ᵗ, where x₀ is the initial population, r is the annual growth rate as a percentage, and t is time in years. For example, a population of 1,000 growing at 2% per year for 10 years yields 1,000 × (1.02)¹⁰ ≈ 1,219.
Carrying capacity (K) is the maximum population size that an environment can support indefinitely given available resources like food, space, and water. In the logistic model, growth is fastest when the population is at K/2 and slows to near zero as the population approaches K.
Yes. A negative growth rate models population decline or exponential decay. For example, a growth rate of -1.5% represents a population shrinking by 1.5% per year. Note that rates below -100% are not meaningful, as a population cannot decline by more than 100%.
Doubling time is how long it takes a population to double in size at a given growth rate. It is estimated using the Rule of 70: doubling time ≈ 70 / r, where r is the annual growth rate in percent. For a 2% growth rate, the doubling time is approximately 35 years.
Population growth models are used in ecology to study wildlife populations, in epidemiology to track disease spread, in economics to project labor force size, in urban planning to forecast city growth, and in biology for bacterial culture studies. The exponential model is also used in radiocarbon dating and compound interest calculations.
If you know the initial population (P₀) and final population (Pₜ) over t years, the annual growth rate is: r = [(Pₜ / P₀)^(1/t) - 1] × 100. For example, if a population grew from 500 to 800 over 5 years, r = [(800/500)^(1/5) - 1] × 100 ≈ 9.86% per year.
When the population equals the carrying capacity (K), the growth term (1 - P/K) equals zero, so the net growth rate becomes zero and the population stabilizes. In reality, populations may oscillate around K or overshoot it temporarily depending on environmental conditions.