Environmental ScienceEnvironmental ScienceHigh school

Human Population

Growth rates and doubling time.

Population ecology studies how the number of individuals of a species changes over time. With unlimited resources a population grows exponentially, $\frac{dN}{dt} = rN$ , doubling in a fixed time. Real environments run short of food, space and other resources, so growth slows as the population approaches the carrying capacity $K$ .

The logistic model $\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)$ captures this S-shaped growth: nearly exponential at first, fastest at $\frac{K}{2}$ , and levelling off near $K$ . Births, deaths, immigration and emigration are the four ways a population changes. Human population grew slowly for most of history, then surged after 1800 as death rates fell before birth rates did, the pattern demographers call the demographic transition.

Key terms

Rule of 70: Approximate doubling time from a percentage growth rate.
$t_{\text{double}} \approx \frac{70}{r}$

Key ideas

Exponential growth: $N = N_0 e^{rt}$ ; doubling time $\approx \frac{0.693}{r}$ .
Density-dependent factors (competition, disease) strengthen as the population grows.
Density-independent factors (storms, fires) act regardless of population size.
r-selected species have many offspring with little care; K-selected species have few offspring with much care.

Worked example

A bacterial population of 500 grows exponentially with $r = 0.2$ per hour. How large is it after 10 hours?

$N = 500e^{0.2 \times 10} = 500e^2$ .
$e^2 \approx 7.389$ .
$N \approx 3695$ .

Answer: about $3695$

Common mistakes

Assuming populations grow exponentially forever.
Confusing carrying capacity with the maximum growth rate point ( $\frac{K}{2}$ ).
Treating density-independent factors as density-dependent.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Doubling time when $r = 0.05$ per year, to one decimal.
Show answer
$13.9$ years
2.In the logistic model with $K = 1000$ , at what population is growth fastest?
Show answer
$500$
3.A population has 120 births, 80 deaths, 10 immigrants and 20 emigrants. Net change?
Show answer
$+30$

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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