Population Growth Calculator
Calculate population growth with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
Exponential: N(t) = N0 x e^(rt) | Logistic: N(t) = K / (1 + ((K-N0)/N0) x e^(-rt))
N(t) is the population at time t, N0 is the initial population, r is the intrinsic growth rate, K is the carrying capacity, and e is Euler's number (~2.718). Exponential growth assumes unlimited resources. Logistic growth adds the factor (1 - N/K) which slows growth as the population approaches carrying capacity.
Worked Examples
Example 1: Bacterial Colony Growth
Problem:A bacterial colony starts with 100 cells and has an intrinsic growth rate of 0.30 per hour. Model exponential growth over 24 hours.
Solution:Using N(t) = N0 x e^(rt)\nN(24) = 100 x e^(0.30 x 24) = 100 x e^7.2\nN(24) = 100 x 1,339.43 = 133,943 cells\nDoubling time = ln(2) / 0.30 = 2.31 hours\nNumber of doublings in 24 hours = 24 / 2.31 = 10.4
Result:133,943 cells after 24 hours | Doubling time: 2.31 hours | Growth factor: 1,339x
Example 2: Deer Population with Carrying Capacity
Problem:A deer population of 50 in a forest with carrying capacity K=500 and growth rate r=0.20/year. Model logistic growth over 30 years.
Solution:N(t) = K / (1 + ((K-N0)/N0) x e^(-rt))\nN(30) = 500 / (1 + (450/50) x e^(-0.20 x 30))\nN(30) = 500 / (1 + 9 x e^(-6))\nN(30) = 500 / (1 + 9 x 0.00248) = 500 / 1.0223 = 489\nTime to reach K/2: ln(9) / 0.20 = 10.99 years\nMax growth rate at K/2: 0.20 x 500/4 = 25 deer/year
Result:489 deer after 30 years | Half-K reached at year 11 | Max growth: 25 deer/year
Frequently Asked Questions
What is the difference between exponential and logistic growth?
Exponential growth occurs when a population grows at a constant per-capita rate without any resource limitations, producing a J-shaped curve described by N(t) = N0 * e^(rt). This model assumes unlimited resources and space, which rarely occurs in nature for extended periods. Logistic growth incorporates a carrying capacity (K), producing an S-shaped (sigmoid) curve where growth slows as the population approaches K. The logistic model is more realistic because all environments have finite resources. In the logistic equation, the term (1 - N/K) acts as a brake on growth, reducing the growth rate to zero when N reaches K.
What is the intrinsic growth rate (r)?
The intrinsic rate of natural increase (r) represents the maximum per-capita growth rate of a population under ideal conditions with unlimited resources. It is calculated as the difference between birth rate and death rate (r = b - d). Species with high r values (r-selected species) like bacteria, insects, and rodents reproduce rapidly but have short lifespans. Species with low r values (K-selected species) like elephants and whales reproduce slowly but invest heavily in offspring survival. The value of r determines how quickly a population can grow; a population with r = 0.05 doubles approximately every 14 time periods, while r = 0.10 doubles every 7 periods.
How do population growth models work?
Exponential growth follows dN/dt = rN, producing a J-shaped curve with unlimited resources. Logistic growth follows dN/dt = rN(K-N)/K, producing an S-shaped curve that levels off at carrying capacity (K). Real populations typically follow logistic growth with fluctuations around K.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy