Cohort Analysis Calculators
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Life Table Structure
Columns: x = age class; nₓ = number alive at start of age x; lₓ = nₓ/n₀ (survivorship, 0–1); dₓ = nₓ − nₓ₊₁ (deaths in age class x); qₓ = dₓ/nₓ (mortality rate); mₓ = mean female offspring per female at age x; lₓmₓ = age-specific contribution to R₀; xlₓmₓ = used for T calculation.
Key Demographic Parameters
R₀ = Σlₓmₓ = net reproductive rate (average lifetime offspring per female). R₀ > 1: population growing. R₀ = 1: stable. R₀ < 1: declining. T = Σxlₓmₓ / R₀ = mean generation time (years). r ≈ ln(R₀) / T = intrinsic rate of increase (rough estimate; Euler equation is exact).
Survivorship Curves
- Type I (e.g., humans, large mammals): low early mortality; high late-life mortality; curve convex
- Type II (e.g., birds, lizards): constant mortality at all ages; linear semi-log curve
- Type III (e.g., fish, oysters, most invertebrates): very high early mortality; low late-life mortality; curve concave
Glossary
Frequently Asked Questions
A cohort life table tracks all individuals born in the same time period (a cohort) from birth to death, recording key demographic information. Standard columns: x = age class (e.g., years). nₓ = number alive at the start of age x. lₓ = survivorship = nₓ/n₀; proportion surviving from birth to age x. dₓ = deaths during age class x = nₓ − nₓ₊₁. qₓ = age-specific mortality rate = dₓ/nₓ. mₓ = fecundity = mean number of female offspring per female in age class x. lₓmₓ = contribution to lifetime reproduction. From these, calculate: R₀ (net reproductive rate), T (generation time), and r (intrinsic rate of increase).
R₀ = Σ lₓ × mₓ (sum over all age classes x). R₀ = the average number of female offspring produced per female over her lifetime. Interpretation: R₀ > 1: each female replaces herself more than once → population growing (N will multiply by R₀ each generation). R₀ = 1: exactly replacement → stable population. R₀ < 1: population declining (failing to replace). Example: species with lₓmₓ = 0 (age 0) + 0 (age 1) + 0.3 (age 2) + 0.8 (age 3) + 0.6 (age 4) + 0.2 (age 5) + 0 (age 6): R₀ = 0 + 0 + 0.3 + 0.8 + 0.6 + 0.2 = 1.9. Each female produces 1.9 daughters over her lifetime → population growing.
Survivorship curves plot lₓ (proportion surviving) vs. age, typically on a semi-log scale. Type I (convex): low age-independent mortality throughout most of life → high survival to old age → sudden senescence-related death. Examples: humans, elephants, many large mammals. Characteristic of species with intensive parental care. Type II (straight line on semi-log): constant mortality rate at all ages → lₓ decreases exponentially. Examples: many birds, adult lizards, some rodents. qₓ is age-independent. Type III (concave): extremely high mortality in early life → few survivors → those who survive early stage live to old age. Examples: fish, oysters, trees, many invertebrates, most plants. Investment in many offspring; no parental care. Most species show intermediate patterns between these three types.
Cohort (dynamic) life table: follows a real cohort of individuals born at the same time through their entire lives. Advantages: direct observation of age-specific survival and reproduction; no assumptions about steady-state demographics. Disadvantages: takes many years to complete for long-lived organisms; conditions may change over the cohort's lifetime (environmental variation). Static (time-specific) life table: takes a snapshot of the population at one moment in time; estimates lₓ from the age structure at that time (assumes population is at stable age distribution). Advantages: quick to produce. Disadvantages: requires assumption of stable age distribution; if population is growing or declining, age structure biases the estimates. For short-lived organisms (insects, annual plants): cohort tables are feasible. For long-lived species (trees, whales): static tables or mark-recapture survival estimates are practical alternatives.