Population Growth Calculator

Calculate exponential and logistic population growth with carrying capacity.

Model

Initial population N0

Growth rate r (per year)

Carrying capacity K

Time t (years) (years)

Second count N2 (for r estimate)

Please check your inputs and try again.

Exponential Versus Logistic

Exponential growth assumes unlimited resources — per capita growth rate stays constant no matter how crowded the population becomes. Logistic growth adds a limiting term that slows growth as the population approaches carrying capacity.

Exponential: N = N₀e^rt Logistic: dN/dt = rN(1 − N/K)

The bracketed term is the whole difference. When N is small it is close to 1 and growth is nearly exponential; as N approaches K it approaches zero and growth halts. This produces the characteristic S-shaped curve.

N relative to K	(1 − N/K)	Growth rate
N = 0.1K	0.90	Nearly exponential
N = 0.5K	0.50	Maximum absolute growth
N = 0.9K	0.10	Slowing sharply
N = K	0	Zero — equilibrium

Note that maximum absolute growth occurs at half carrying capacity, not at low density. Below K/2 there are too few individuals reproducing; above it, crowding suppresses the rate. This is the theoretical basis of maximum sustainable yield in fisheries management — harvesting to hold a stock near K/2 maximises the surplus available.

Doubling Time and Life History

t_double = ln 2 / r = 0.693 / r

	r-selected	K-selected
Offspring	Many, small	Few, large
Parental care	Minimal	Extensive
Lifespan	Short	Long
Population	Near or below K rarely; boom and bust	Stable near K
Examples	Insects, weeds, bacteria	Elephants, whales, humans

The MSY model is theoretically clean and practically fragile. It assumes accurate knowledge of K and r, no environmental variability, and no age structure — assumptions that have contributed to several major fishery collapses when harvest was set at a theoretical maximum with no margin.

Worked Examples

Example 1: Exponential: N0=100, r=0.3, t=10

N=100xe^3=2009

Result: Doubling every 2.31 years

Unlimited resources — rarely sustainable

Example 2: Logistic: N0=100, r=0.5, K=1000, t=10

N=1000/(1+9xe^-5)=943

Result: Approaching K — S-shaped curve

Density-dependent regulation

Example 3: Where growth is fastest

K = 1,000, comparing N = 100, 500 and 900

Result: Fastest at N = 500

Growth rate is rN(1 − N/K). At N = 500 this is maximised — the product of population size and remaining capacity peaks at the midpoint.

Example 4: Doubling time

r = 0.3 per year

Result: 2.31 years

t = 0.693/0.3. A modest annual rate still doubles a population within about two and a half years while resources remain unlimited.

Common Mistakes

⚠️

Applying exponential growth to real populations

Unlimited growth cannot persist. Exponential models describe only the early phase before resource limitation takes effect.

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Assuming maximum growth occurs at low density

Absolute growth peaks at half carrying capacity. Below that there are too few reproducing individuals; above it crowding suppresses the rate.

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Treating maximum sustainable yield as safe

MSY assumes accurate K and r estimates and no environmental variability. Harvesting at the theoretical maximum leaves no margin for error and has contributed to fishery collapses.

⚠️

Ignoring age structure

Both models treat all individuals as equivalent. Real populations have age-dependent reproduction and mortality that these equations cannot capture.

Frequently Asked Questions

r-selected vs K-selected?

r-selected: high r, low K, fast reproduction, boom-bust (insects, rodents). K-selected: low r, high K, long-lived, stable near K (elephants, whales).

Maximum sustainable yield?

MSY = rK/4 at N=K/2 (inflection point of logistic curve). Fisheries target MSY but complex in practice.

What is the difference between exponential and logistic growth?

Exponential assumes unlimited resources and constant per capita growth. Logistic includes a term that slows growth as the population approaches carrying capacity.

Where is population growth fastest?

At half the carrying capacity. Below that, too few individuals reproduce; above it, crowding suppresses the rate.

What is maximum sustainable yield?

The largest harvest a population can sustain indefinitely, theoretically at K/2 where surplus production peaks. In practice it is fragile because it assumes accurate parameters and no variability.

How do I calculate doubling time?

Divide 0.693 by the intrinsic growth rate r. This applies during exponential growth, before resource limitation takes hold.