F. Population Dynamics · Population Dynamics

Fecundity, Mortality, Life Tables and Survivorship Curves

F2.1 F3.2 A1.6 A1.12About 70 minutes

Learning goals

I can define and correctly use fecundity, fertility, natality, mortality and life expectancy.
I can complete a life table (nₓ, lₓ, dₓ, qₓ) from survival data.
I can calculate the net reproductive rate R₀ from survivorship and fecundity, and interpret it.
I can sketch and interpret Type I, II and III survivorship curves and give examples of each.

Words to know

fecundity
The potential (maximum) reproductive capacity of an individual: the number of offspring it could produce in a lifetime or a breeding season.

fertility
The actual number of offspring produced. Usually lower than fecundity.

natality
The birth rate of a population: the number of births per unit time (often per 1000 individuals).

mortality
The death rate of a population: the number of deaths per unit time (often per 1000 individuals).

cohort
A group of individuals born during the same time period, followed through their lives.

life table
A table that follows a cohort and records how many survive and die at each age.

survivorship (lₓ)
The proportion of the original cohort still alive at the start of age x: lₓ = nₓ ÷ n₀.

survivorship curve
A graph of the number (or proportion) of a cohort still alive versus age, usually on a logarithmic vertical axis.

net reproductive rate (R₀)
The average number of female offspring produced by each female in her lifetime: R₀ = Σ lₓmₓ. R₀ > 1 means the population grows.

A thousand eggs, a handful of adults

A female Atlantic cod can release millions of eggs in a season, yet cod populations do not explode. A female black bear may have only a few cubs every second year, yet bear populations can be stable. The difference is who survives. In this lesson you will learn the vocabulary of births and deaths and two tools ecologists use to see survival patterns: the life table and the survivorship curve.

The words of births and deaths

Fecundity is the potential reproductive capacity: how many offspring an individual could produce under ideal conditions. A dandelion with 200 seeds per head has high fecundity.
Fertility is the actual number produced. Poor food, disease or a short season keep fertility below fecundity.
Natality is the birth rate of the whole population, for example 35 births per 1000 individuals per year.
Mortality is the death rate, for example 20 deaths per 1000 individuals per year.
Life expectancy is the average number of years an individual of a given age is expected to live.

The curriculum asks you to use these terms correctly. A common mistake is to use fertility when you mean fecundity: fecundity is what is possible; fertility is what happens.

Building a life table

To make a life table, ecologists follow a cohort (for example all the voles born one spring) and record how many are alive at the start of each age interval.

x = age interval (often years).
nₓ = number alive at the start of age x.
lₓ = nₓ ÷ n₀ = survivorship, the proportion of the cohort still alive.
dₓ = nₓ − nₓ₊₁ = number dying during age x.
qₓ = dₓ ÷ nₓ = age-specific mortality rate, the fraction of those alive at age x that die before age x + 1.

If you also know mₓ, the average number of female offspring born to each female of age x (a fecundity schedule), you can calculate the net reproductive rate:

R₀ = Σ lₓmₓ

R₀ is the average number of daughters a newborn female will produce over her whole life. If R₀ > 1 the population grows, if R₀ = 1 it is stable, and if R₀ < 1 it shrinks.

Worked example

Let's complete a life table for a cohort of meadow voles

A cohort of 1000 voles is followed. Numbers alive at the start of each age: age 0: 1000; age 1: 400; age 2: 300; age 3: 240; age 4: 120; age 5: 24; age 6: 0. The female offspring per female (mₓ) are: age 0: 0; age 1: 1.0; age 2: 2.0; age 3: 2.0; age 4: 1.0; age 5: 0.5. Complete lₓ, dₓ and qₓ, then find R₀.

lₓ = nₓ ÷ 1000: 1.000, 0.400, 0.300, 0.240, 0.120, 0.024.
dₓ = nₓ − nₓ₊₁: 1000 − 400 = 600; 400 − 300 = 100; 300 − 240 = 60; 240 − 120 = 120; 120 − 24 = 96; 24 − 0 = 24.
qₓ = dₓ ÷ nₓ: 600 ÷ 1000 = 0.60; 100 ÷ 400 = 0.25; 60 ÷ 300 = 0.20; 120 ÷ 240 = 0.50; 96 ÷ 120 = 0.80; 24 ÷ 24 = 1.00.
lₓmₓ: 1.000 × 0 = 0; 0.400 × 1.0 = 0.400; 0.300 × 2.0 = 0.600; 0.240 × 2.0 = 0.480; 0.120 × 1.0 = 0.120; 0.024 × 0.5 = 0.012.
R₀ = Σ lₓmₓ = 0 + 0.400 + 0.600 + 0.480 + 0.120 + 0.012 = 1.612.
Interpret: each newborn female will, on average, leave about 1.6 daughters, so R₀ > 1 and the population is growing (by about 61% per generation).

lₓ: 1.000, 0.400, 0.300, 0.240, 0.120, 0.024; dₓ: 600, 100, 60, 120, 96, 24; qₓ: 0.60, 0.25, 0.20, 0.50, 0.80, 1.00; R₀ ≈ 1.61 (growing).

Age x	nₓ	lₓ	dₓ	qₓ	mₓ	lₓmₓ
0	1000	1.000	600	0.60	0	0
1	400	0.400	100	0.25	1.0	0.400
2	300	0.300	60	0.20	2.0	0.600
3	240	0.240	120	0.50	2.0	0.480
4	120	0.120	96	0.80	1.0	0.120
5	24	0.024	24	1.00	0.5	0.012

The completed vole life table. The last column adds up to R₀ = 1.612.

Survivorship (lₓ) for the vole cohort. The steepest drop is in the first year, when 60% of the young die.

Survivorship curves

Plot nₓ against age and you get a survivorship curve. Ecologists usually use a logarithmic vertical axis (1, 10, 100, 1000 equally spaced). On a log axis, a constant percentage of deaths each year gives a straight line, so the shape of the curve tells you when in life most deaths happen. To compare species with very different life spans, the horizontal axis is often percent of maximum life span.

Type I: low mortality through most of life, then many deaths in old age. Few offspring, lots of parental care. Examples: humans in countries with good health care, Dall sheep, elephants, many large mammals.
Type II: a roughly constant death rate at every age (a straight line on the log graph). Examples: many songbirds, some lizards, hydras, and squirrels in some studies.
Type III: very high mortality among the young, then low mortality for the few survivors. Huge numbers of offspring, little or no parental care. Examples: oysters, cod and most marine fish, most trees and many plants (millions of seeds, few seedlings survive).

Real species are often in between, and a species' curve can change; for example, human survivorship has shifted toward Type I as infant mortality fell.

The three idealized survivorship curves on a log scale. Type I loses few until old age, Type II loses a constant fraction at every age (straight line), and Type III loses most individuals very young.

Read a log graph carefully

On a log scale, the gap from 1000 to 100 is the same height as the gap from 100 to 10. Each step down is a 90% loss. So a line that drops one gridline per equal time step means the same fraction dies every interval, which is Type II.

Worked example

Let's identify a survivorship type from data

A cohort of 10 000 eastern white pine seeds is followed: 10 000 at year 0, 300 seedlings at year 1, 120 at year 5, 80 at year 20, 60 at year 50, 40 at year 100. Which survivorship type is this, and what does it suggest about reproduction?

Year 0 to 1: 10 000 → 300 means 9700 died, so the first-year mortality q = 9700 ÷ 10 000 = 0.97 (97%).
Later years: from year 20 to year 100 (80 years), only 80 − 40 = 40 die, so the average loss is small and slow.
Shape: very high early mortality, then low mortality for survivors, which is Type III.
Reproduction: a Type III species typically produces very many offspring with little parental investment in each, so a few survive by chance to adulthood.

Type III. The pine makes huge numbers of seeds, almost all die young, and the few established trees survive a long time.

Just for fun

A Type III survivorship curve in one sentence, from a cod's point of view: "I had two million brothers and sisters, and I'm the one who got invited to the family reunion."

Hands-on activity

Build a survivorship curve with dice

You need: 30 dice (or 30 coins, or a free online dice roller), paper, a ruler, a calculator.

Each die is one animal in a cohort.

Type II model: roll all 30 dice. Every die showing a 6 "dies"; remove it. Record how many are left (n₁). Roll the survivors again and remove the 6s. Repeat until none are left or you reach 15 rounds.
Type III model: start with 30 dice. In round 1 only, remove every die showing 1, 2, 3, 4 or 5 (only 6s survive). After that, survivors die only on a 6.
Type I model: start with 30 dice. For rounds 1 to 5, nobody dies. From round 6 on, a die dies if it shows 1, 2, 3 or 4.
Make a table of nₓ for each model. Plot all three on one graph of nₓ (vertical) against round number (horizontal). If you can, also plot log₁₀(nₓ) for each model, skipping any round where nₓ = 0 (log 0 is undefined).

What to look for: Type II loses about one sixth each round, so it declines smoothly and, on the log graph, is close to a straight line. Type III drops to about 5 in round 1 (30 × 1/6), then declines slowly. Type I stays at 30 for five rounds and then crashes within two or three rounds. Your exact numbers will vary because dice are random, just as real cohorts are.

Your turn

What is the difference between fecundity and fertility?

A town of 12 000 has 156 births and 96 deaths in a year. Calculate natality and mortality per 1000.

A cohort of 500 starts at age 0; 350 are alive at age 1. Find l₁, d₀ and q₀.

Which survivorship type matches each: (a) maple keys and seedlings, (b) humans in Canada, (c) a bird that loses about 40% of adults every year regardless of age?

A species has lₓmₓ values of 0, 0.3, 0.4, 0.2. Find R₀ and say whether the population is growing.

Why do ecologists use a log scale for survivorship curves?

More practice: warm-up

What does nₓ mean in a life table?

Which survivorship type has the most parental care?

What value of R₀ means a stable population?

Define cohort.

Is the death rate of a population called natality or mortality?

More practice: on your own

Complete lₓ and qₓ for a cohort of 200 frogs: age 0: 200; age 1: 20; age 2: 12; age 3: 6; age 4: 0.

For the frogs above, mₓ (female offspring per female) is 0 at age 0, 30 at age 1, 40 at age 2 and 40 at age 3. Find R₀.

A Type II bird population loses 50% of its members each year. Starting with 800, how many are left after 4 years? Why does this graph as a straight line on a log scale?

Explain why the human survivorship curve in Canada has become more strongly Type I over the past 150 years.

A conservation biologist finds that a turtle species has Type III survivorship but adults normally live 60+ years. Why is killing even a few adult turtles (for example on roads) so harmful?

Challenge

A cohort of 1000 sockeye salmon eggs gives 1000 (age 0), 50 (age 1), 10 (age 2), 4 (age 3), 2 (age 4, when they spawn and die), 0 (age 5). Each spawning female produces about 3000 eggs, half of them female. (a) Find R₀ (consider only the age 4 term). (b) Interpret it. (c) What does this suggest about how sensitive salmon populations are to small changes in early survival?

Two populations of the same deer species have the same fecundity. Population A has qₓ = 0.10 for adults; population B, in an area with heavy predation on fawns, has qₓ = 0.10 for adults but q₀ = 0.70 for fawns compared with 0.30 in A. Sketch (describe) how their survivorship curves differ and predict which has the lower R₀.

Explain how the life table method could be misleading if (a) the cohort is very small, and (b) the population is studied using a "static" snapshot of ages at one time instead of following a real cohort through time.

Check your understanding

Quiz

Fecundity is best defined as:

A cohort of 800 has 200 survivors at age 1. What is q₀?

Which organism is the best example of a Type III survivorship curve?

On a survivorship graph with a log scale, a straight sloping line means:

A population's R₀ is 0.8. The population is:

lₓ in a life table stands for:

For parents and teachers

Big idea (F2.1, F3.2): fecundity, fertility, natality and mortality describe births and deaths. Life tables (nₓ, lₓ, dₓ, qₓ) and survivorship curves (Types I, II and III) show when in life most deaths happen, and R₀ = Σ lₓmₓ tells whether a population is replacing itself.

How to help: the dice activity takes about 30 minutes; let your teen do the arithmetic and you roll. Ask: "Which type of curve do humans have, and why has it changed?"

Common difficulties:

Mixing up dₓ (a number) and qₓ (a fraction).
Dividing by the wrong n: qₓ divides by nₓ (those alive at the start of that age), not by n₀.
Reading log graphs: each gridline is ten times the one below.
Fecundity versus fertility.

Sources: life table notation and survivorship types from standard university ecology texts (for example, Krebs, Ecology). Example data are invented for teaching.