Life Tables

A life table is a tabulation of survival and reproduction for a cohort — a group of individuals of the same age or born at the same time — broken down age class by age class. The central columns are: x, the age class; nₓ, the number of individuals alive at the start of age x; lₓ, the proportion of the original cohort surviving to age x; dₓ, the number dying during age interval x to x+1; qₓ, the age-specific probability of dying during that interval; and mₓ, the age-specific fecundity — the average number of offspring produced per surviving individual of age x (in many treatments only female offspring per female are counted, which simplifies the population's growth arithmetic). There are two fundamentally different ways to build one. A cohort (dynamic) life table follows one real group of individuals — all born in the same season, in the same population — from birth until the last one dies, recording deaths and births as the cohort ages in real time. This is the gold standard because it directly observes each individual's fate, but it can take decades for long-lived species and is often logistically impossible. A static (time-specific, or vertical) life table instead takes a single census across an entire population at one moment, counts how many individuals are currently in each age class, and treats that age distribution as if it were the survival trajectory of one hypothetical cohort. This substitution is only valid if birth rates, death rates, and population size have been roughly constant for a long time (a stable age distribution) — an assumption that is frequently violated and is the single biggest source of error in static life tables. Once the lₓ and mₓ columns are in hand, a life table yields quantities far beyond simple survival counts. The net reproductive rate, R₀ = Σ lₓ*mₓ, gives the average number of offspring produced by an individual over its entire lifetime, accounting for the fact that many individuals die before reproducing. The <strong>generation time</strong>, T* ≈ (Σ x lₓ*mₓ) / (Σ l*ₓmₓ), gives the mean age of parents when their offspring are born. And R₀ combined with T gives an approximate intrinsic rate of increase, r ≈ ln(R₀)/T, connecting life tables directly to the exponential growth equations used to project population size — the bridge that lets demography feed straight into population dynamics.

Step 1 — Define the cohort. Choose N₀ individuals, all effectively the same age (e.g., all eggs laid in one week, or all seeds germinating in one spring). Set l₀ = 1.0 (or, equivalently, start with n₀ = 1000 and express later columns as proportions of it). This starting choice matters: life tables describe relative survival, not absolute population size, so the same table applies whether the cohort started at 50 individuals or 50,000.

Step 2 — Census at regular intervals. At each subsequent age class x = 1, 2, 3 … (days, months, years, or life stages, depending on the organism), count how many of the original cohort are still alive: nₓ. Age classes should be chosen to match the biology — instars for an insect, size classes for a tree, breeding seasons for a bird — not an arbitrary calendar unit.

Step 3 — Compute survivorship, lₓ. lₓ = nₓ / n₀ This is a monotonically non-increasing sequence: survivorship can never rise, because death is irreversible and no new individuals are added to a closed cohort. lₓ starts at 1.0 and falls toward zero as the last individuals die.

Step 4 — Compute the death schedule, dₓ and qₓ. dₓ = lₓ − lₓ₊₁ (proportion of the original cohort dying in that interval) qₓ = dₓ / lₓ (probability of dying in the interval, given you were alive at the start of it — the age-specific mortality rate) qₓ is the more biologically informative number because it is conditioned on survival to that age: a qₓ of 0.5 at age 10 means half of the individuals who made it to age 10 die before age 11, regardless of how many died earlier. This is exactly the logic insurance actuaries use to price a policy for a 70-year-old versus a 20-year-old.

Step 5 — Compute age-specific fecundity, mₓ. For each age class, record the average number of offspring produced per individual alive at that age (again, often counted as daughters per female, to make the arithmetic of population replacement cleaner). Pre-reproductive and post-reproductive age classes have mₓ = 0.

Step 6 — Combine survival and fecundity to get lifetime reproduction. The product lₓ*mₓ at each age gives the expected number of offspring produced by age class x*, per original cohort member — it discounts fecundity by the chance of even surviving to use it. Summing this product over all ages gives: R₀ = Σ lₓ*mₓ If R*₀ > 1, each individual more than replaces itself on average, and (all else equal) the population grows; if R₀ < 1, it shrinks; R₀ = 1 marks replacement-level, zero-growth demography. This is the same logic used later, with age structure explicitly included, in Age Structure and Population Pyramids.

Step 7 — Static life tables reconstruct these same columns from a single snapshot. Instead of watching a cohort age, a static table counts how many individuals currently occupy each age class in the whole population right now and treats the relative sizes of those age classes as a proxy for lₓ — assuming that the population's birth and death rates have not changed for as long as the oldest individual has been alive. Any recent change in mortality (a bad winter, a new predator, a hunting ban) distorts a static table, because it mixes cohorts born under different conditions rather than tracking one cohort's true fate.

Step 8 — Interpreting the shape. The pattern of qₓ across ages — whether mortality is concentrated early in life (many species with tiny, numerous offspring), spread flat across ages (species where risk is age-independent, like many small birds), or concentrated late in life (species that protect their young, like large mammals) — determines the shape of the survivorship curve, formally introduced in Survivorship Curves, and reflects the life-history strategy shaped by natural selection acting on the trade-off between current and future reproduction, a theme picked up in Reproductive Trade-offs.

John Graunt and the parish bills of mortality (1662). Long before ecology existed as a discipline, Graunt analyzed London's weekly "Bills of Mortality" — records that Anglican parishes had kept for over a century, listing christenings and burials, ostensibly to track plague outbreaks. Graunt reorganized these raw counts into what he called a "Table of Ages," estimating that of 100 children born, only 25 survived to age 26, and only 3 survived to age 66. This was the first documented life table for any species, and it established the core logic still used today: start with a cohort, track deaths, compute the proportion surviving at each age. Graunt's work directly inspired the actuarial tables that make modern life insurance mathematically possible, and ecologists borrowed the method wholesale two and a half centuries later.

Raymond Pearl's laboratory rat and Drosophila colonies (1920s). Pearl, working at Johns Hopkins, ran some of the first controlled cohort life tables for non-human animals, raising rats and fruit flies under constant laboratory conditions and recording survival and fecundity at each age with a precision impossible in field studies. Because he controlled temperature, food, and density, Pearl could show that the lₓ*mₓ schedule — not just the birth rate or death rate alone — determined how fast a population grew, directly validating the arithmetic behind R*₀ and providing the first experimental life tables used to test early population growth models.

Edward Deevey's landmark synthesis (1947). In a paper titled "Life tables for natural populations of animals," published in the Quarterly Review of Biology, Deevey compiled cohort life-table data across a taxonomically wide range of species — including Utida's flour beetles, the Yellowstone Dall mountain sheep data originally gathered by Olaus Murie, and hydra, rotifers, and songbirds — and plotted lₓ on a logarithmic scale against age for each. This is the paper that established the now-classic three-type classification of survivorship curves (convex Type I, diagonal Type II, concave Type III), a scheme built entirely out of life-table data and covered in detail in Survivorship Curves. Deevey's key methodological contribution was showing that wildly different organisms — an oyster and a sheep — could be compared on the same axes once their raw counts were converted into the standardized lₓ column, because lₓ strips away absolute population size and leaves only the shape of the mortality schedule.

Olaus Murie's Dall mountain sheep skulls (1930s–40s, analyzed by Deevey and later Caughley). Murie collected 608 skulls of Dall mountain sheep (Ovis dalli) that had died of natural causes in Mount McKinley (Denali) National Park. Because sheep horns grow in annual rings, Murie could determine the age at death of each skull found on the landscape — effectively reconstructing a static life table from a graveyard, since each skull represented one individual's age at death rather than a living census. This is a particularly elegant example of a life table built neither from following a live cohort nor from a standard age census, but from the age-at-death distribution of carcasses accumulated over many years — a method still used today for species (many ungulates and carnivores) whose skeletal remains or teeth carry legible age markers.

Joseph Connell's barnacle cohort study (1961, 1970). Connell followed marked cohorts of the barnacle Balanus balanoides (now Semibalanus balanoides) settling on rock surfaces in Scotland, individually tagging newly settled larvae and recording their survival and reproduction over subsequent years — a genuine, direct cohort life table for a wild marine invertebrate. Crucially, Connell also manipulated the environment (removing a competing barnacle species, Chthamalus, from some plots) and showed that the qₓ schedule changed depending on competitive and predatory pressure, demonstrating experimentally that life-table parameters are not fixed species constants but depend on ecological context — a result that anticipated much of modern life-history theory.

Example 1 — A cohort life table for an annual plant, computing R₀ and generation time. Suppose ecologists follow a cohort of 200 seeds of an annual wildflower from germination through death, censusing every month during the single growing season, and recording seed output per surviving plant at each age.

Walking through the arithmetic: lₓ is simply nₓ/200 at each age (e.g., l₂ = 60/200 = 0.30). dₓ is the drop in lₓ from one age to the next (l₀ − l₁ = 1.00 − 0.45 = 0.55). qₓ divides that drop by the lₓ it started from (d₀/l₀ = 0.55/1.00 = 0.55 at age 0, but d₃/l₃ = 0.10/0.25 = 0.40 at age 3 — mortality risk given survival to age 3 is lower than the raw drop suggests, because lₓ was already small). Notice q₄ = 1.00: by definition, all survivors die by the end of the season, since this is an annual plant. Net reproductive rate: R₀ = Σ lₓ*mₓ = 0 + 0 + 1.20 + 3.00 + 0.90 = <strong>5.10 seeds per original seed</strong>. Generation time: T* ≈ Σ(x·*lₓm*ₓ) / Σ(lₓ*mₓ) = [(2×1.20)+(3×3.00)+(4×0.90)] / 5.10 = (2.40+9.00+3.60)/5.10 = 15.00/5.10 ≈ <strong>2.94 months</strong>. Intrinsic rate of increase: r* ≈ ln(R₀)/T = ln(5.10)/2.94 ≈ 1.629/2.94 ≈ 0.554 per month. Interpretation: because R₀ = 5.10 is well above 1, each seed that germinates leaves, on average, over five seeds behind — despite the fact that fully 85% of the cohort dies before ever reproducing (l₂ = 0.30 means only 30% even reach the first reproductive age class). This is the essential lesson of a life table: enormous early mortality is entirely compatible with strong population growth, provided the survivors reproduce prolifically enough — exactly the Type III survivorship pattern that will be formalized in Survivorship Curves.

Example 2 — Comparing two populations with equal R₀ but different age schedules of mortality (a conservation scenario). A wildlife agency is deciding how to allocate limited conservation funding between two declining seabird populations, Population A and Population B, both currently estimated to have R₀ ≈ 1.0 (stable, replacement-level reproduction) based on static life tables built from banding-return data. Population A's qₓ schedule shows very high mortality in the first year (q₀ = 0.70) but low, roughly constant mortality among adults (qₓ ≈ 0.05 per year from age 3 onward), consistent with a species that produces many chicks but loses most to predation before fledging — a Type III-leaning pattern. Population B's qₓ schedule shows the opposite: low juvenile mortality (q₀ = 0.15) but a steep late-life mortality increase after age 15 (qₓ rising from 0.05 to 0.40), consistent with a long-lived species whose survival is high until senescence sets in — a Type I pattern. Reasoning it through: even though both populations currently show R₀ ≈ 1.0, a management intervention that reduces adult mortality (say, bycatch reduction in fishing nets) will have a much larger effect on Population B's R₀, because a large fraction of Population B's expected lifetime reproduction (lₓ*mₓ) is concentrated in older age classes that a fishing-net policy can protect. In Population A, most of the "action" in the life table is in the l*ₓ column between ages 0 and 1 — the lₓ*mₓ product for older ages is already small because so few birds reach maturity in the first place, so reducing bycatch of adults will help, but a nest-predator-control program aimed at the q*₀ term will produce a proportionally larger jump in R₀. This is exactly the kind of quantitative triage that life tables make possible: without decomposing mortality by age, both populations would look identical (same R₀, same overall decline), but the correct conservation intervention is entirely different for each.

John Graunt was not a scientist by training or profession — he ran a haberdashery shop in London and served as a city councilman — but he had a compulsive habit of collecting and cross-checking numbers. Beginning in the 1650s, he obtained access to the weekly "Bills of Mortality," parish-level tallies of christenings and burials that London churches had compiled since the 16th century, chiefly to give early warning of plague. Where earlier readers had used the bills only to track disease outbreaks week to week, Graunt asked a different, more ambitious question: if you followed an imaginary cohort of newborns forward through the burial records, what fraction would still be alive at each age? In 1662 he published Natural and Political Observations Made upon the Bills of Mortality, in which he presented his "Life Table" — a set of estimates suggesting that of 100 children born, roughly 36 died before age 6, and only about 1 in 100 survived past age 76. His numbers were rough by modern standards (he lacked precise ages at death for most burials and had to infer age structure from causes of death listed in the bills), but the method — track a hypothetical cohort's survival age class by age class — was exactly right, and it is the same method, refined with far better data, that underlies every life table used in ecology today. Graunt was elected a Fellow of the Royal Society in 1663 on the strength of this work, reportedly on the personal recommendation of King Charles II, who insisted that a tradesman with such original ideas deserved the honor regardless of his lack of formal scientific credentials.

Conservation biology and endangered species management rely on life tables to identify which age class's survival most limits population growth (a "sensitivity" or "elasticity" analysis built directly on the lₓ*m*ₓ schedule) — this is precisely how agencies decide whether to invest in protecting nesting sites (juvenile survival) or in reducing adult bycatch, as in the sea turtle and albatross conservation programs that use age-structured life tables to prioritize turtle-excluder devices on shrimp trawlers versus beach nest protection.

Fisheries management builds age-structured life tables from catch data (using scale rings, otoliths, or length-frequency distributions as age markers, much like Murie's horn rings) to set sustainable harvest quotas; a fishery that removes individuals before they reach lₓ*m*ₓ peak reproductive ages can collapse a stock even while total catch numbers look stable in the short term.

Public health and human demography still use Graunt's basic logic, refined into modern actuarial life tables that insurance companies and national statistics agencies use to calculate life expectancy, price life insurance and annuities, and project the age structure of national populations decades ahead — the same lₓ column, computed now from vital-registration systems processing millions of records per year rather than parish bills.

Agricultural pest management applies insect life tables (often built stage-by-stage rather than day-by-day, since insects pass through discrete instars) to identify which life stage's mortality is most sensitive to a proposed control measure — biological control programs are typically evaluated by their effect on the qₓ of the target pest's most vulnerable stage, often the egg or early larval instar.

Invasive species risk assessment and reintroduction biology use life tables built from the source population to predict R₀ and r for a species establishing in a new range or being reintroduced after local extinction, informing both how aggressively an invasive species should be controlled and how many founder individuals a reintroduction program needs to release to have a reasonable chance of establishing a self-sustaining population.

Ask the tutor: How do ecologists use elasticity or sensitivity analysis on a life table to decide which age class's survival matters most for conservation priorities?