Two curves, and knowing which one you are looking at explains most of what happens in a population over time. Exponential has no ceiling and gives a J shape. Logistic has a ceiling and gives an S. The interesting part is that the S curve is slowest at both ends and fastest in the middle, which is the opposite of what people expect.
Exponential growth has no brakes, which is why it only holds while a population is small. Logistic growth is the same curve with a ceiling: the population grows fastest at half the carrying capacity and then flattens out as resources run short. Pick the model and this projects the population, the doubling time, and where the curve is bending hardest.
Exponential growth: the J curve
With unlimited resources, every individual adds at the same fractional rate, so the population multiplies:
N(t) = N₀ e^(rt)
Growth is proportional to how many are already there, which is why the curve is a J. Double the time and you roughly double the population. Bacteria in a flask do this for about a day before the flask runs out of something.
Because the rate is a fraction of the population, the time to double is constant regardless of size:
t(double) = ln 2 ÷ r
At r = 0.1 that is 6.93 time units. A population of 100 and a population of 10,000 double in exactly the same time. This is called the generation time and it is one of the more useful numbers in ecology.
Logistic growth: the S curve
Real populations run out of food, space and nesting sites. Carrying capacity, written K, is the population size the environment can sustain indefinitely, and once you hit it growth stops even though the birth rate has not changed.
N(t) = K ÷ (1 + ((K − N₀)/N₀) e^(−rt))
The population rises, bends over, and flattens out at K. It never crosses K — that is built into the equation, and this calculator will not report a number above K however far ahead you project.
The part that surprises people
Growth is fastest when the population is at half the carrying capacity, not at the start. At t = 0 the population is small and there are plenty of resources, so each individual reproduces easily — but there are few of them, so the absolute increase is small. Near K there are many individuals but each one is struggling, so the increase is small again. Halfway is where both are true at once.
That point is the inflection point of the S, and this tool reports the time at which it happens. A population on the way up should be managed hardest around here, which is exactly when a harvesting quota set for the start of the season goes wrong.
Linked and unlinked, in a different sense
If the population in the real world is not random — if individuals mate with nearby neighbours, or spread unevenly — it grows somewhere between the two curves. A model with a term for that sits between the J and the S.
The useful diagnostic is the doubling time. Compare the doubling time the model predicts with the one you actually observe. If the observed time is longer, some density-dependent factor is slowing the population down, and the J curve is over-predicting.
Two limits worth knowing
Carrying capacity is not a constant. It moves with weather, disease, and anything else that changes resource supply. A K of 1,000 last year says nothing about this year, and treating it as fixed is the most common way these models get misused.
The models describe, they do not predict. They assume a constant r and a constant K, and real populations have both varying. Use them to understand the shape of a curve and the meaning of a doubling time, not to forecast a number.
Frequently asked questions
What is the difference between exponential and logistic growth?
Exponential assumes unlimited resources and gives a J curve with no ceiling. Logistic adds carrying capacity and gives an S curve that levels off at K.
What is carrying capacity?
The population size the environment can sustain indefinitely. Growth stops there because resources, not birth rate, become the limit.
How do I calculate doubling time?
ln 2 divided by r, where r is the growth rate per unit time. It is independent of the starting population, because the rate is a fraction.
When is a population growing fastest?
At half the carrying capacity. The absolute increase is small at the start because there are few individuals, and small near K because each one is competing, so the peak sits in the middle.
What happens if I project a logistic population too far?
It approaches K and flattens. It will never exceed K, which is a property of the equation rather than something the tool enforces.
Why is the growth rate r usually so small?
Because it is a fraction per unit time, not a raw count. A population doubling in 70 days has an r of roughly 0.01 per day, which sounds tiny until you compound it.
What if the population is already at carrying capacity?
Then it stays there. Growth is zero, and the curve has no room left to move in either direction.
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