lx (Survivorship) Calculators

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In population ecology, lₓ (survivorship) is the proportion of individuals from a cohort that survive from birth to the beginning of age class x: lₓ = nₓ / n₀, where nₓ is the number alive at the start of age class x and n₀ is the initial cohort size. lₓ ranges from 1.0 at birth (x = 0) to 0 when all individuals have died. Combined with mₓ (age-specific fecundity = mean female offspring per female in age class x), lₓ forms the life table — the foundation of demographic analysis. Net reproductive rate R₀ = Σlₓmₓ; generation time T = Σxlₓmₓ / R₀.

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Life Table Variables

lₓ = nₓ/n₀ (survivorship, 0–1). dₓ = lₓ − lₓ₊₁ (proportion dying in age x). qₓ = dₓ/lₓ (age-specific mortality rate = probability of dying in age class x). mₓ = age-specific fecundity (female offspring per female). lₓmₓ = age-specific contribution to R₀. xlₓmₓ = for generation time calculation.

Key Demographic Parameters

R₀ = Σlₓmₓ (net reproductive rate). R₀ > 1: growing. R₀ = 1: stable. R₀ < 1: declining. T = Σxlₓmₓ / R₀ (mean generation time). r ≈ ln(R₀) / T (intrinsic rate of increase).

Survivorship Curve Types

  • Type I (convex): low early mortality, high late mortality; humans, large mammals
  • Type II (linear on semi-log): constant mortality at all ages; many birds, lizards
  • Type III (concave): very high early mortality, low late mortality; fish, oysters, trees

Plotting

Semi-log: plot ln(lₓ) vs. x. Type I → convex. Type II → straight line (constant slope = constant mortality rate). Type III → concave (steep early slope, then flat).

Glossary

lx (Survivorship)
lₓ = nₓ/n₀; proportion of cohort surviving to age x; l₀ = 1.0; decreases to 0; used with mₓ (fecundity) to calculate R₀ = Σlₓmₓ (net reproductive rate).
Net Reproductive Rate (R₀)
R₀ = Σlₓmₓ; average female offspring per female over lifetime accounting for survival; R₀ > 1 growing; = 1 stable; < 1 declining; generation time T = Σxlₓmₓ/R₀.
Survivorship Curve Types
Type I (convex, human): low early mortality; Type II (linear semi-log, birds): constant mortality; Type III (concave, fish): very high early mortality; plotted as ln(lₓ) vs. age.

Frequently Asked Questions

lₓ = survivorship = the proportion of the original cohort still alive at the beginning of age class x. lₓ = nₓ / n₀. nₓ = number alive at start of age class x. n₀ = initial cohort size at birth (x = 0). l₀ = 1.0 always (100% alive at birth by definition). lₓ decreases monotonically from 1.0 toward 0 as age increases. Example: cohort of 1,000 organisms. At age 0: n₀ = 1000 → l₀ = 1.00. At age 1: n₁ = 800 → l₁ = 0.80. At age 2: n₂ = 600 → l₂ = 0.60. At age 3: n₃ = 200 → l₃ = 0.20. qₓ (age-specific mortality): q₀ = (l₀−l₁)/l₀ = (1.0−0.8)/1.0 = 0.20; q₁ = (0.8−0.6)/0.8 = 0.25.

R₀ = Σ lₓ × mₓ (sum over all age classes). mₓ = mean number of female offspring produced per female in age class x. lₓmₓ = the contribution of age class x to R₀. R₀ = average number of female offspring produced by a female over her lifetime, accounting for survival probability. Interpretation: R₀ > 1: population growing (each female more than replaces herself). R₀ = 1: exactly stable. R₀ < 1: declining. Example: l₂×m₂ = 0.60 × 2 = 1.20 (contribution from age 2); l₃×m₃ = 0.20 × 3 = 0.60. If ΣlₓMₓ = 1.8 → R₀ = 1.8 → population growing (each female produces 1.8 daughters on average).

Survivorship curves (lₓ vs. age, with lₓ on log scale): Type I (convex): low constant mortality throughout most of life → most individuals survive to old age → then high mortality near maximum lifespan. Examples: humans (with modern medicine), elephants, large ungulates. Characteristic of species with intensive parental care and few offspring. Type II (straight line on semi-log scale): constant mortality rate at all ages → qₓ is constant → ln(lₓ) decreases linearly with age. Examples: many birds, adult lizards, some rodents. Type III (concave): extremely high early mortality → few survivors → but survivors live long. Examples: fish, oysters, trees, most marine invertebrates, most plants. Produces many offspring; provides no parental care; most die young. Most real populations show intermediate patterns or different types at different life stages.

Life tables provide the demographic foundation for population viability analysis (PVA): Identify critical life stages: which age class has the highest elasticity (proportional contribution to λ = finite rate of increase)? High elasticity at adult survival → management should focus on reducing adult mortality. High elasticity at juvenile survival → focus on improving juvenile habitat and reducing early mortality. Minimum viable population (MVP): using matrix population models built from life table data → project extinction probability. Harvest management: set safe harvest rates based on measured adult survival and fecundity. Example: leatherback sea turtle — high adult survival, low juvenile survival (Type III) → extremely high elasticity for adult survival → even small increases in adult bycatch mortality can drive the population to decline. Management: reduce adult bycatch (TEDs — turtle excluder devices) rather than focusing only on nest protection.