Biology · Population Ecology & Life-History Patterns
Births, deaths, age structure and life tables
A census begins with eighty living members and ends with ninety. Ten more are present, but that net change does not tell us how many were born or arrived. Demography opens the account: it follows the events and composition that can change a population.
Organisms, Environments & Ecological Niches introduces ecological scales, habitat and the distinction between observation and inference. Begin here with the counts and ordinary arithmetic. The mathematics explanations of functions and exponentials are optional preparation for the growth lesson; its multiplier-versus-slope explanation can be read before derivatives. Return to Animal Reproduction, Development & Life Histories and Evolutionary Processes & Adaptation for reproductive outcome stages and the evidence needed to connect demographic differences to selection.
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Balance a defined population interval and calculate age-specific contributions and cohort fractions with their correct conditioning sets.
Balance all changes across the same boundary
Over a fixed census interval, births and immigration add living members; deaths and emigration remove them. Immigration means entry across the chosen population boundary, and emigration means departure across it. Death is a change in living status rather than a geographic crossing. Count every component over the same interval and for the same species and membership definition.
The balance is ending N = starting N + births − deaths + immigration − emigration. These terms are event counts over an interval. A rate divides a count or change by a stated time or other denominator. Calling a count a rate without giving its interval can make comparisons misleading.
Equal births and deaths therefore do not guarantee unchanged local abundance when net migration is nonzero. Conversely, a population could have many births, deaths and movements while its ending total equals its starting total. A stable total is a net result, not evidence that nothing happened.
Four changes across a defined census interval
Read the diagram in words
The ending count is 80 + 18 − 10 + 7 − 5 = 90. Sixty-five initial individuals carry over, joined by eighteen births and seven arrivals. Net change ten divided by initial eighty is 0.125.
| Inside the fixed population boundary | Initial living population; Births inside the boundary; Deaths of initial members; Ending living population |
|---|---|
| Outside the population boundary | Arrivals from outside; Departures to outside |
| Initial living population | 80 |
|---|---|
| Births inside the boundary | 18 |
| Arrivals from outside | 7 |
| Ending living population | 90 |
| Deaths of initial members | 10 |
| Departures to outside | 5 |
| One defined census interval | Fixed geographic population boundary | Individuals |
|---|---|---|
| Initial members alive inside at the ending census | Initial living population → Ending living population | 65 |
| New living members born inside | Births inside the boundary → Ending living population | 18 |
| Members entering from outside | Arrivals from outside → Ending living population | 7 |
| Initial members that died | Initial living population → Deaths of initial members | 10 |
| Members leaving to outside | Initial living population → Departures to outside | 5 |
N(end) = N(start) + B − D + I − E
80 + 18 − 10 + 7 − 5 = 90
Change divided by initial abundance: 10/80 = 0.125
Death and boundary crossing are different events
Original complete ledger for one species inside a fixed boundary. Ten deaths and five departures remove distinct initial individuals; all eighteen new births and seven immigrants remain alive inside at the ending census. No additional events or detection errors.
A total hides the members able to contribute
Age structure describes how many members are in specified age groups. Stage structure can instead classify states such as seed, juvenile and reproductive adult. A group’s survival or reproductive contribution can differ from another’s, so populations with the same total can have different near-term changes.
In the supplied example, both populations start at 100. A has 50 younger,30 reproductive and 20 older members; B has 20, 60, 20. Applying the same assigned class-specific contributions gives different totals. The comparison does not say that age alone fixes births: the rates and outcome definitions are also supplied.
Survivors listed by starting age class are not being frozen at that age. They age during the interval. A forecast of the next age distribution would need transitions between classes as well as survival and recruits. Sex composition, reproductive state and environmental conditions can also matter; a simple total cannot substitute for those details.
The same total can have different demographic contributions
Read the diagram in words
Population A begins with 50, 30, 20 members by starting class and ends with 77 survivors plus 30 recruits, total 107. Population B begins with 20, 60, 20 and ends with 80 survivors plus 60 recruits, total 140. Both start at 100 and use the same assigned class-specific rates.
| Class at the starting census | Assigned fraction surviving the interval | Recruits per starting member of this class |
|---|---|---|
| Younger starting class | 0.8 | 0 |
| Reproductive starting class | 0.9 | 1 |
| Older starting class | 0.5 | 0 |
| Class at the starting census | Starting members | Survivors from this starting class | New recruits reaching the ending census |
|---|---|---|---|
| Younger starting class | 50 | 40 | 0 |
| Reproductive starting class | 30 | 27 | 30 |
| Older starting class | 20 | 10 | 0 |
| Starting members | 100 |
|---|---|
| Survivors from this starting class | 77 |
| New recruits reaching the ending census | 30 |
| All living members at the ending census | 77 + 30 = 107 |
| Class at the starting census | Starting members | Survivors from this starting class | New recruits reaching the ending census |
|---|---|---|---|
| Younger starting class | 20 | 16 | 0 |
| Reproductive starting class | 60 | 54 | 60 |
| Older starting class | 20 | 10 | 0 |
| Starting members | 100 |
|---|---|
| Survivors from this starting class | 80 |
| New recruits reaching the ending census | 60 |
| All living members at the ending census | 80 + 60 = 140 |
Assigned individuals
Age composition alone does not determine change without demographic rates
Original nonhuman teaching populations, complete starting classes and one interval. Class-average recruitment is counted per starting member; all recruits enter the ending census. No migration, inherited trait, sex-ratio inference, age-transition matrix or later trajectory. Surviving members are classified by their starting class, not assumed to remain in that age class.
Keep conditional mortality separate from survivorship
A cohort is a group defined by a shared starting event, such as birth in a specified period. Following the same birth cohort records how many remain alive as it ages. A life table organises survival, death and sometimes reproductive information by age or interval. A table built from a followed cohort differs from an age snapshot containing members born in different years.
Conditional interval mortality divides deaths during an interval by the number alive at its start. Original-cohort survivorship divides the survivors at a census by the original starting number. The denominators answer different questions: among those who reached this interval, what fraction died; and of those present originally, what fraction remains?
For the original hundred-member cohort, surviving counts are 80, 40, 10, 0 after successive intervals. These are complete known fates. If an animal vanished from an observation record, disappearance could instead involve emigration or failure to detect it; the cohort arithmetic would then require additional information.
A conditional mortality fraction over one interval is not an instantaneous death hazard, and it is not a detection probability. Survivorship curves can describe early, approximately constant or later concentration of mortality under defined conditions. Such shapes summarise a record; they are not fixed laws assigned to every member of a broad taxonomic group.
Mortality and survivorship use different denominators
Read the diagram in words
Deaths are 20, 40, 30, 10 and sum 100. Conditional mortality is 20/100, 40/80, 30/40, 10/10. Survival from the original hundred at the four interval ends is 0.8, 0.4, 0.1, 0. Conditional survival products reproduce the original-cohort fractions.
| Assigned time intervals | Surviving members of the original cohort | Survivors divided by the original cohort |
|---|---|---|
| 0 | 100 | 100/100 = 1 |
| 1 | 80 | 80/100 = 0.8 |
| 2 | 40 | 40/100 = 0.4 |
| 3 | 10 | 10/100 = 0.1 |
| 4 | 0 | 0/100 = 0 |
| Assigned interval | Alive at the start of this interval | Deaths during this interval | Alive at the end of this interval | Deaths divided by interval entrants | Survivors divided by interval entrants | Survivors divided by the original cohort |
|---|---|---|---|---|---|---|
| 1 | 100 | 20 | 80 | 20/100 = 0.2 | 80/100 = 0.8 | 80/100 = 0.8 |
| 2 | 80 | 40 | 40 | 40/80 = 0.5 | 40/80 = 0.5 | 40/100 = 0.4 |
| 3 | 40 | 30 | 10 | 30/40 = 0.75 | 10/40 = 0.25 | 10/100 = 0.1 |
| 4 | 10 | 10 | 0 | 10/10 = 1 | 0/10 = 0 | 0/100 = 0 |
Fraction of the named conditioning set
Known fates are supplied; disappearance alone would not establish death
Original closed birth cohort with complete known fates, no new entrants, migration or undetected members. Four equal assigned time intervals; mortality fractions are interval-conditional, not instantaneous hazards or detection probabilities.
Worked example
Follow two different sets of denominators
The population ledger ends at 80 + 18 − 10 + 7 − 5 = 90. In this constructed account, ten deaths and five departures remove distinct initial members, leaving 65. The 18 births and 7 immigrants all remain, so 65 + 18 + 7 = 90.
In the separate cohort, the second interval starts with 80 and ends with 40. Its 40 deaths give conditional mortality 40/80 = 0.5. Forty survivors divided by the original 100 give survivorship 0.4. The first interval’s survival 0.8 multiplied by the second interval’s conditional survival 0.5 also gives 0.4.
Population A in the age example has 40 + 27 + 10 = 77 survivors and 30 recruits, total 107. B has 16 + 54 + 10 = 80 survivors and 60 recruits, total 140. Both began at 100; the larger reproductive starting class matters here because the demographic contributions were held fixed.
Check your understanding
A local population has equal birth and death counts but seven immigrants and five emigrants. Must its size stay unchanged?
Show reasoning
No. Births and deaths cancel, but net migration is+2. With no additional events, the ending population has two more members. The chosen boundary determines which movements count as migration.
Forty individuals die in an interval entered by eighty members of an original hundred-member cohort. What are interval mortality and ending original-cohort survivorship?
Show reasoning
Interval mortality is 40/80 = 0.5. The ending survivors are 40, so original-cohort survivorship is 40/100 = 0.4. The numerators happen to match here, but the conditioning sets differ.
Do the age-model counts show that the survivors remain in their starting age classes?
Show reasoning
No. Survivors are attributed to the class they occupied at the start. They can age or change stage. The model computes an aggregate ending total; the next age composition would require a transition account.
What to carry forward
A population total changes through a complete boundary ledger, while age composition and cohort denominators reveal different parts of that change.
References
Resources supporting this lesson’s explanations and examples.