F. Population Dynamics · Population Dynamics

Measuring Population Change: Births, Deaths, Immigration and Emigration

F2.1 F2.2 F3.2 A1.13About 65 minutes

Learning goals

I can explain how births, deaths, immigration and emigration change population size, and tell open from closed populations.
I can calculate the change in population size, the growth rate and the per capita growth rate.
I can convert between counts, rates per individual, rates per 1000 and percentages.
I can explain biotic potential and environmental resistance.

Words to know

natality (b)
The number of births in a population over a time period (or the birth rate).

mortality (d)
The number of deaths over a time period (or the death rate).

immigration (i)
Individuals moving into a population from elsewhere.

emigration (e)
Individuals leaving a population permanently.

open population
A population whose size is changed by births, deaths, immigration and emigration.

closed population
A population changed only by births and deaths (no movement in or out), for example on a remote island or in a sealed culture.

growth rate
The change in population size per unit time: gr = ΔN ÷ Δt.

per capita growth rate (cgr)
The change in population size relative to the starting size: cgr = ΔN ÷ N. Often given as a percentage.

biotic potential
The maximum rate at which a population could increase under ideal conditions (unlimited resources, no predators or disease).

environmental resistance
All the factors (limited food and space, predators, disease, weather) that keep a population from reaching its biotic potential.

Four ways to change a population

Only four things can change the size of a population:

Births (natality) add individuals.
Immigration adds individuals that arrive from elsewhere.
Deaths (mortality) remove individuals.
Emigration removes individuals that leave.

So the change in population size over a time period is

ΔN = (b + i) − (d + e)

If births plus immigration are greater than deaths plus emigration, the population grows. If they are equal, the population is in equilibrium (zero population growth). If they are less, it declines.

A population is like a bank account: births and immigration are deposits, deaths and emigration are withdrawals.

Open and closed populations

Most wild populations are open: animals and seeds move in and out. A population is closed when movement is impossible or negligible, so only b and d matter. Moose and wolves on Isle Royale, an island in Lake Superior, are a famous nearly closed system. A bacteria culture in a sealed flask is fully closed.

Human populations are open too. For Canada, immigration is now the main source of growth. In some recent quarters, Statistics Canada has reported that deaths almost equalled or slightly exceeded births, so natural increase (births minus deaths) was close to zero.

Rates: how fast and how fast per individual

The growth rate is the change per unit of time:

gr = ΔN ÷ Δt (units: individuals per year, per day, etc.)

But 200 new deer means something very different in a herd of 400 than in a herd of 40 000. So ecologists also use the per capita growth rate, which compares the change with the starting size:

cgr = ΔN ÷ N (often × 100% to give a percentage)

Similarly, birth and death rates are often given per capita (per individual) or per 1000: a birth rate of 0.032 per individual per year is the same as 32 per 1000 per year.

Worked example

Let's calculate the growth of a deer herd

A white-tailed deer herd has 2400 animals on January 1. During the year there are 760 births, 520 deaths, 40 deer move in and 80 move out. Calculate ΔN, the new population, the growth rate and the per capita growth rate.

ΔN = (b + i) − (d + e) = (760 + 40) − (520 + 80) = 800 − 600 = +200 deer.
New population: 2400 + 200 = 2600 deer.
Growth rate: gr = ΔN ÷ Δt = 200 deer ÷ 1 year = 200 deer/year.
Per capita growth rate: cgr = ΔN ÷ N = 200 ÷ 2400 = 0.083, or 8.3% per year.
Per capita birth and death rates: b = 760 ÷ 2400 = 0.317 per deer per year (317 per 1000); d = 520 ÷ 2400 = 0.217 (217 per 1000).

ΔN = +200, new N = 2600, gr = 200 deer/year, cgr ≈ 8.3% per year.

The deer herd's inputs (births and immigration) total 800 and its outputs (deaths and emigration) total 600, so the herd grows by 200.

Worked example

Let's compare two populations by per capita rate

Pond A's leopard frogs grow from 150 to 180 in one year. Pond B's grow from 1200 to 1290. Which population is growing faster?

Pond A: ΔN = 180 − 150 = 30. cgr = 30 ÷ 150 = 0.20 (20% per year).
Pond B: ΔN = 1290 − 1200 = 90. cgr = 90 ÷ 1200 = 0.075 (7.5% per year).
Pond B added more frogs (90 versus 30), but Pond A grew faster relative to its size.
Per capita rates let you compare populations of different sizes fairly.

Pond B has the larger growth rate (90 frogs/year), but Pond A has the larger per capita growth rate (20% versus 7.5%).

Biotic potential and environmental resistance

Biotic potential is the maximum rate a population could grow under ideal conditions: unlimited food and space, no predators, no disease. It depends on traits such as:

the number of offspring per reproduction (fecundity),
how often the organism reproduces,
the age at first reproduction,
how many offspring survive to reproduce.

A bacterium dividing every 20 minutes has an enormous biotic potential; a moose cow with one or two calves a year has a much smaller one.

In nature, a population almost never reaches its biotic potential. Environmental resistance (food shortage, predators, disease, competition, harsh weather) holds it back. The balance between biotic potential and environmental resistance is the subject of the next three lessons: Geometric and Exponential Growth Models, Logistic Growth and Carrying Capacity and Limiting Factors.

Watch the signs and the starting value

ΔN can be negative (a declining population). Per capita rates always divide by the starting N for the period, not the final N. And keep units: "200 deer/year" and "8.3% per year" are different kinds of numbers.

Just for fun

Why did the emigrating caribou refuse to fill out the census form? It said it was just passing through, and technically it was right: it counts in emigration, not in N.

Hands-on activity

Track a "population" at home with pennies

You need: about 100 pennies or small tokens, two cups (one labelled "population", one labelled "outside world"), a die, paper and a calculator.

Put 40 tokens in the population cup. This is N at year 0.
Each "year", roll the die and apply these rules:

Births: add tokens equal to N × 0.25, rounded to the nearest whole number.
Deaths: remove tokens equal to N × 0.15, rounded.
Immigration: add the number shown on the die.
Emigration: if the die shows an even number, remove 4 tokens.

Record b, d, i, e, ΔN, the new N and cgr = ΔN ÷ N each year in a table for 8 years.
Graph N against year.
Change one rule (for example, deaths become N × 0.30) and run it again.

What to look for: with b = 0.25 and d = 0.15, births exceed deaths by 10% of N each year, so the population tends to grow faster as it gets bigger, a first glimpse of exponential growth. Immigration and emigration add a little randomness. With deaths at 0.30, births are smaller than deaths, so the population shrinks unless immigration rescues it.

Your turn

Write the equation for the change in population size, and say what each letter means.

A beaver colony of 12 has 4 births, 2 deaths, 0 immigrants and 3 emigrants in a year. Find ΔN and the new N.

A population of 5000 has a per capita birth rate of 0.040 and a per capita death rate of 0.025 per year (no migration). Find ΔN for one year.

A population grows from 800 to 920 in a year. Find the per capita growth rate as a percentage.

What is the difference between an open and a closed population? Give an example of each.

Define biotic potential and give two traits that increase it.

Express a death rate of 0.012 per individual per year as a rate per 1000.

More practice: warm-up

Which two processes add individuals to a population?

What is the formula for growth rate?

If b + i = d + e, what is happening to the population?

What is environmental resistance?

A herd changes from 300 to 270. What is ΔN?

More practice: on your own

A lake trout population of 3600 has 540 births and 450 deaths over 3 years, with no migration. Find ΔN, the growth rate per year and the per capita growth rate over the whole period.

A town of 25 000 has a crude birth rate of 9 per 1000 and a crude death rate of 11 per 1000. Net migration is +300 people. Find ΔN for one year.

Two populations each grow by 500 in a year. One started at 2000, the other at 50 000. Compare their per capita growth rates and explain why the difference matters.

A population of 400 has a per capita growth rate of 10% per year. Assuming that rate stays the same, find N after 1 year and after 2 years. Why is the increase in year 2 larger?

Explain why an ecologist studying a moose population on an island in a large lake might treat it as closed, and when that assumption might fail.

Challenge

A population of 1500 snowshoe hares has per capita rates b = 1.20, d = 0.95, i = 0.05 and e = 0.10 per year. (a) Find ΔN. (b) Find cgr. (c) If all four rates stay constant, what will N be after 2 years?

A researcher counts a bird colony each spring: 2022: 620; 2023: 700; 2024: 760; 2025: 790; 2026: 800. Calculate cgr for each year, describe the trend and suggest a biological explanation.

A fisheries manager wants a population of 10 000 fish to stay constant. Births are 2200 per year and natural deaths are 1700 per year. There is no migration. (a) What maximum harvest per year keeps N constant? (b) What assumption is built into this calculation, and why might it be risky?

Check your understanding

Quiz

A population starts at 900. There are 120 births, 70 deaths, 10 immigrants and 20 emigrants. What is ΔN?

In the same population, what is the per capita growth rate?

Which population is closed?

A death rate of 0.008 per individual per year equals:

Biotic potential is:

Which factor is part of environmental resistance?

For parents and teachers

Big idea (F2.1, F2.2): only births, deaths, immigration and emigration change a population: ΔN = (b + i) − (d + e). Growth rate (ΔN ÷ Δt) and per capita growth rate (ΔN ÷ N) describe how fast it changes. Biotic potential is the ideal maximum; environmental resistance holds populations back.

How to help: use your own community's numbers if you like (Statistics Canada publishes births, deaths and migration for every province), or run the penny activity together. Ask your teen to explain why a big population can add more individuals yet grow more slowly per capita.

Common difficulties:

Sign errors when deaths or emigration are larger.
Dividing by the final N instead of the starting N.
Confusing 0.008 with 8% or 8 per 1000.

Sources: Statistics Canada quarterly demographic estimates (2026 releases) for the note on natural increase; the rest is standard population ecology. Example data are invented for teaching.