Demography and Survivorship

Of 100 adult birds banded at a colony, 60 are seen there the next year. Does that mean 40 died? Not necessarily. Some may be alive and unseen, and some may be breeding at another colony. Before you can say how fast a population grows, you need to know who survives and who reproduces at each age, and you need to know how good your counts are. This lesson builds that bookkeeping, the life table, and then asks where its numbers come from.

The idea: a life table is a cohort’s bookkeeping

A cohort is a group of individuals born together. Follow one from birth and record, at each exact age x, how many are still alive. That count is n_x. Everything else in a life table, a summary of how survival and reproduction change with age, is arithmetic on those counts.

Think of a ledger for a club whose members never join late: you start with a membership, you strike out names as they leave, and you note what each remaining member contributes each year. For a closed, completely observed cohort the columns are:

l_x = n_x / n_0 d_x = n_x - n_(x+1) q_x = d_x / n_x p_x = 1 - q_x

Survivorship, l_x, is the fraction of the original cohort alive at age x. d_x is the number dying between x and x + 1. q_x is the probability of dying in that interval given that you reached its start, and p_x is the probability of surviving it. Because surviving to age 3 means surviving year 0, then year 1, then year 2, l_x is a product: l_3 = p_0 x p_1 x p_2. That is the product rule for independent stages from probability and crosses, applied to survival.

Tables of this kind are usually built for females only, so the birth column, m_x, is the expected number of daughters born per female at age x. How births are divided among ages and offspring is the business of life histories; here you take m_x as measured.

From survival and births to R0 and generation time

Worked problem. In this made-up data set, a closed female cohort is followed with complete records: no migration, no censoring. Each surviving female gives birth in a single pulse at each exact age x, counted before the next interval’s deaths.

x (years)	n_x	l_x	d_x	q_x	p_x	m_x	l_x m_x	x l_x m_x
0	100	1.0	20	0.20	0.80	0	0	0
1	80	0.8	40	0.50	0.50	1	0.8	0.8
2	40	0.4	30	0.75	0.25	2	0.8	1.6
3	10	0.1	10	1.00	0	1	0.1	0.3
4	0	0	none	undefined	undefined	0	0	0

1. Survivorship. Divide each n_x by 100: l_x = 1.0, 0.8, 0.4, 0.1, 0.
2. Deaths. Subtract each count from the one before: d_x = 20, 40, 30, 10. They add to 100, so every death is accounted for.
3. Death and survival probabilities. Divide each d_x by the number alive at the start of its interval, not by the original 100. q_0 = 20/100 = 0.20, q_1 = 40/80 = 0.50, q_2 = 30/40 = 0.75, q_3 = 10/10 = 1.00. Then p_x = 0.80, 0.50, 0.25, 0. Check: l_2 = 0.8 x 0.5 = 0.4.
4. Net reproductive rate. Multiply each l_x by m_x: 0, 0.8, 0.8, 0.1. Add them:

R0 = sum of l_x m_x = 0.8 + 0.8 + 0.1 = 1.7 daughters per newborn female

Check it by counting: 80 females have 1 daughter each, 40 have 2, and 10 have 1, which is 80 + 80 + 10 = 170 daughters from 100 founders, or 1.7. R0 is per generation, not per year.

5. Generation time. Weight each age by its contribution to R0: x l_x m_x = 0.8, 1.6, 0.3, which sum to 2.7 daughter-years per newborn female.

T = sum of x l_x m_x / R0 = 2.7 / 1.7 = 1.59 years

T is the mean age of mothers when their daughters are born, in this cohort.

6. A rate per year. Growth per year needs the intrinsic rate of increase, r:

r is approximately ln(R0) / T = ln(1.7) / 1.59 = 0.530 / 1.59 = 0.334 per year

The yearly multiplier is lambda = e^r, about 1.40 per year. Both are approximations. The exact r in a model with fixed rates comes from a different calculation, and T is not in general exactly ln(R0)/r. Population growth takes r and lambda from here.

7. Shape. q_x rises with age (0.20, 0.50, 0.75, 1.00), so this cohort is not Type II. Mortality concentrated at older ages puts it nearer Type I.

The lesson’s table Nearly constant q to age 2 Low survival

Alive at age 1, n₁ 80 of 100 Alive at age 2, n₂ 40 (at most n₁) Alive at age 3, n₃ 10 (at most n₂)

your cohort the lesson’s table straight line (constant q)

x	nx	lx	dx	qx	mx	lxmx	x·lxmx

R₀ = Σ l_xm_x

T = Σ x·l_xm_x / R₀

r ≈ ln(R₀) / T

λ = e^r

In this sketch: closed cohort of 100 newborn females, complete records, births in a pulse at each exact age, no female alive past age 3 (so q₃ = 1.00), r approximate. Nearly equal here means q within 0.03. A sketch of the logic, not data.

Read survivorship on a log axis: a straight line means the same fraction dies each interval, not the same number; R₀ is per generation, and r and λ are per year.

The model’s verdict, if these rates stayed fixed, is that R0 above 1 means the population grows in the long run, R0 below 1 means it declines, and R0 = 1 means it just replaces itself. That is a prediction from assumptions, not an observation: real vital rates change with density, weather and year.

Survivorship curves and age structure

Plot l_x against age and you have a survivorship curve. Always use a logarithmic y-axis. On a log scale, equal vertical drops mean equal proportional losses, which is what q_x measures.

• Type I: high survival through early and middle life, then heavy losses at old ages. The curve stays flat, then plunges.
• Type II: a roughly constant probability of death per interval. Because l_x = (1 - q)^x, log l_x falls in a straight line.
• Type III: very high mortality early, then better survival for the few that get through. The curve drops steeply, then flattens.

The commonest error is to read Type II as “the same number die each year.” It means the same fraction. Suppose 1,000 individuals face q = 0.5 every year: survivors go 1,000, 500, 250, 125, and deaths go 500, 250, 125. The number dying halves each year, while the log curve is a straight line. The three types are idealizations: real species mix them, often with Type III juvenile losses followed by Type I adult survival.

Low average life expectancy can mislead in the same way. Life expectancy at birth is a mean over everyone born, so heavy infant mortality drags it far down. In an invented population where 30% of newborns die within days of birth and the rest all live to exactly 50, life expectancy at birth is about 35 years, yet no adult dies at 35.

An age-structure diagram (an age pyramid) is a different object. It shows the proportion of a population in each age class now. Its shape depends on survival, but also on how many were born in each past year. A pyramid with a broad base is not a Type III curve; it may be a growing population with good juvenile survival. Even a population with fixed rates that grows exponentially can keep an unchanging age distribution, and the share in each class then depends on both l_x and the growth rate.

Age structure also carries demographic momentum. If fertility falls to replacement, a level at which each generation exactly replaces itself in the long run, a human population with many young people can keep growing for decades, because the number of women of reproductive age is still rising. The often quoted replacement figure of about 2.1 children per woman applies only where few children die before adulthood.

Where the numbers come from, and what they miss

Cohort follow-up gives the cleanest table but is slow: for a long-lived species, nobody lives long enough to watch one cohort die out. Static tables take a snapshot instead, ageing every individual alive, or every carcass, at one time and treating the age counts as one cohort. That works only if the population is stationary, with similar births and death rates for years. Mark-recapture follows marked animals that cannot be counted directly.

Each method has known weak points:

• Censoring. An animal still alive when the study ends, or lost to an accident, is right-censored: it counts as at risk for the time it was seen, never as a death. The 2016 primate tables, for example, removed half the censored animals from the number at risk in their interval.
• Few old individuals. Small samples at old ages make gradual changes in mortality hard to detect, so the tail of any table is its least certain part.
• Pooled years. Tables pooled across years, as the primate tables were, give average rates and hide good and bad years.
• Missing is not dead. An animal not seen may be alive and missed, or alive and gone. Mark-recapture models estimate detection and apparent survival separately, but apparent survival is true survival multiplied by site fidelity. Permanent emigration pulls it below true survival, a problem that spatial populations takes further.

The Hydra study shows how much conventions matter: its annual death probability near 0.006 holds for laboratory cohorts with catastrophic deaths censored. It is evidence against a universal rule of rising adult mortality, not evidence that Hydra are immortal.

Design problem

Return to the colony from the start of the lesson. Of 100 adult birds banded there, 60 are seen at the colony the next year. A student concludes that adult mortality is 40% a year. Evaluate the conclusion and design a study that would estimate adult survival better.

One way to do it

The 0.60 is an apparent return rate, not survival. It is lowered by three things besides death: birds alive at the colony but not seen, birds alive that moved elsewhere for good, and birds that lost their bands.

1. Detection. Search the colony on several occasions each season and for several years, and fit a Cormack-Jolly-Seber model to each bird’s encounter history. A bird missed one year and seen the next was alive and undetected, and these gaps let the model estimate detection probability and apparent survival separately.
2. Emigration. Apparent survival is still true survival times fidelity to the colony. Search neighbouring colonies, or fit a subset of birds with tracking devices, to estimate permanent emigration. In a study of Bonelli’s eagles in Catalonia, survival estimates for non-breeding females differed more than those for males between models that estimated survival jointly with dispersal and models that did not (Badia-Boher and colleagues, 2023).
3. Band loss. Double-mark some birds with two bands, or a band and a second mark, to estimate how often bands are lost.
4. Unknown fates. Record birds never seen again as unknown, censored at their last sighting, not as dead.

If detection is 0.9 and fidelity 0.8, an apparent return of 0.60 would point to true survival near 0.6/(0.9 x 0.8), about 0.83, and annual mortality nearer 17% than 40%. Those two values are invented to show the logic; only the study can supply the real ones.