Net Reproductive Rate Calculators
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Net Reproductive Rate Formula
R₀ = Σ(lx × mx)
Where:
- lx = survivorship at age x — probability of surviving from birth to age x
- mx = fecundity at age x — average number of female offspring produced per female of age x
- Sum is taken across all age classes x from birth to last reproductive age
Interpreting R₀
- R₀ = 1: Replacement-level reproduction. Each individual replaces itself exactly. Population is stable in the long run if maintained.
- R₀ > 1: Each individual produces more than one replacement offspring on average. Population will grow (given sufficient resources).
- R₀ < 1: Each individual fails to replace itself. Population will decline toward extinction.
Relationship to r and Generation Time (T)
r ≈ ln(R₀) / T
Where T (mean generation time) = Σ(x × lx × mx) / R₀
This approximation is exact at stable age structure and shows that a population's intrinsic growth rate depends on both how many offspring are produced (R₀) and how quickly generations turn over (T). A species with R₀ = 4 but T = 20 years grows more slowly than one with R₀ = 2 but T = 1 year.
Life Table Example
| Age (x) | lx | mx | lx × mx |
| 0 | 1.00 | 0 | 0 |
| 1 | 0.60 | 1 | 0.60 |
| 2 | 0.40 | 2 | 0.80 |
| 3 | 0.20 | 1 | 0.20 |
R₀ = 0 + 0.60 + 0.80 + 0.20 = 1.60 → population growing
R₀ in Epidemiology
In infectious disease epidemiology, R₀ is the basic reproduction number — the average number of secondary infections produced by one infected individual in a fully susceptible population. R₀ > 1 = epidemic spreads; R₀ < 1 = outbreak dies out. Herd immunity threshold = 1 − 1/R₀.
Glossary
Frequently Asked Questions
R₀ = Σ(lx × mx) — the sum of age-specific survival (lx) multiplied by age-specific fecundity (mx) across all ages. It represents the average number of offspring per individual over a lifetime, adjusted for survivorship. R₀ > 1 = population grows; R₀ = 1 = replacement; R₀ < 1 = population declines. It is calculated from life table data and is used alongside mean generation time to estimate the intrinsic growth rate r.
The crude birth rate counts births per individual per time period — it does not account for survival to reproductive age. R₀ integrates both survival (lx) and fertility (mx) across the entire lifespan: it asks 'of all the offspring a female produces, weighted by the probability of surviving to each reproductive age, how many are expected per lifetime?' R₀ is therefore a more complete measure of reproductive success in age-structured populations.
r ≈ ln(R₀) / T, where T = mean generation time = Σ(x × lx × mx) / R₀. If R₀ = 1, then r = 0 (stable). If R₀ > 1, then r > 0 (growing). The approximation works well when age structure is stable. Actual r requires solving the Euler-Lotka equation exactly, but the ln(R₀)/T approximation is useful for field estimates.
In epidemiology, R₀ (basic reproduction number) is the average number of secondary infections from one infectious individual in a fully susceptible population. R₀ > 1: each case produces more than one new case — epidemic grows. R₀ < 1: epidemic fades. R₀ = 1: endemic steady state. The herd immunity threshold (fraction that must be immune to stop spread) = 1 − 1/R₀. For measles (R₀ ≈ 15), ~93% population immunity is needed for herd immunity.