Population Growth: The Two Models and the Rule of 70

You’ll be able to

Distinguish exponential from logistic growth and identify each on a graph
Compute population growth rate and doubling time
Explain overshoot, dieback, and the conditions that produce them

Two curves, two conditions

Exponential (J-shaped) growth occurs when resources are effectively unlimited: growth is proportional to current size, so the curve steepens indefinitely. Logistic (S-shaped) growth occurs when a carrying capacity K limits the population: growth is fastest at K/2, then slows as the population approaches K and levels off. The distinction is not merely descriptive, because the two models predict opposite things about the future — and real populations frequently show exponential growth first, when a species enters new habitat or a limiting factor is removed, and logistic behavior later once a limit binds.

Growth rate and doubling time

Population growth rate combines four flows: r = (births + immigration) − (deaths + emigration), expressed per capita and often as a percentage. To convert a percentage growth rate into a doubling time, use the rule of 70: doubling time in years ≈ 70 ÷ (percent growth rate per year). A country growing at 2% per year doubles in about 35 years; one growing at 0.5% doubles in about 140. The rule is an approximation to 70/(100·ln 2 ≈ 69.3), and it is on the AP formula sheet — but knowing why it works is what lets you use it in reverse to find a growth rate from a doubling time.

Overshoot and dieback

A population does not always level off politely at K. If it grows fast enough that the limiting resource is consumed before the feedback takes effect, it overshoots — exceeds carrying capacity — and then dies back, sometimes catastrophically, because the overshoot has damaged the resource base and lowered K itself. Reindeer introduced to islands with abundant lichen and no predators are the standard case: rapid growth, overshoot, crash to below the original carrying capacity because the lichen took decades to recover. Two exam-relevant features: overshoot requires a lag between resource depletion and its effect on births and deaths, and the post-crash carrying capacity can be permanently lower than the original, which is the mechanism behind desertification and fishery collapse.

Growth rate and the rule of 70

growth rate = [(births + immigration) − (deaths + emigration)] / population doubling time ≈ 70 / (% growth per year)

Immigration and emigration are frequently omitted by students and are frequently in the data.

Worked example

A population of 50,000 has 1,200 births, 600 deaths, 300 immigrants and 400 emigrants in one year. Find the growth rate as a percentage and the doubling time.

1.Additions: births + immigration = 1,200 + 300 = 1,500.
2.Subtractions: deaths + emigration = 600 + 400 = 1,000.
3.Net change = 1,500 − 1,000 = 500 individuals.
4.Growth rate = 500 / 50,000 = 0.01 = 1.0% per year.
5.Doubling time ≈ 70 / 1.0 = 70 years.

Answer: The growth rate is 1.0% per year and the doubling time is about 70 years. Note that omitting migration would have given (1,200 − 600)/50,000 = 1.2% and a doubling time of about 58 years — a substantial error from dropping two of the four flows.

Watch out

Include immigration and emigration whenever the data supplies them. Students routinely compute births minus deaths and stop, which is the natural increase rate rather than the growth rate. The exam supplies migration figures precisely to see whether they get used.

Checkpoint

A country grows at 3.5% per year. Its approximate doubling time is

Checkpoint

A population overshoots its carrying capacity and then crashes. The most environmentally significant consequence is often that

Answer the 2 checkpoints as you read.