Lambda (Finite Rate of Increase) Calculators
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Lambda Definition
λ = N(t+1) / N(t)
Interpretation: λ = 1.2 means 20% increase per time step; λ = 0.8 means 20% decline per time step. Relationship to r (intrinsic rate of natural increase): λ = e^r, or r = ln(λ). λ > 1 ↔ r > 0 (growing); λ = 1 ↔ r = 0 (stable); λ < 1 ↔ r < 0 (declining).
Leslie Matrix
For age-structured populations, the Leslie matrix L contains: fecundity values (first row): F₁, F₂, ..., Fₙ; survival probabilities (subdiagonal): P₁, P₂, ..., Pₙ₋₁. Population vector N(t+1) = L × N(t). The dominant eigenvalue of L = λ, the long-term population growth rate. The corresponding eigenvector = stable age distribution.
Sensitivity and Elasticity
Sensitivity = ∂λ/∂aᵢⱼ — how much λ changes per unit change in matrix element aᵢⱼ. Elasticity = (aᵢⱼ/λ) × (∂λ/∂aᵢⱼ) — the proportional change in λ per proportional change in aᵢⱼ. Elasticity analysis identifies which vital rates most influence λ — used in conservation to target management actions (e.g., adult survival vs. juvenile recruitment).
Applications
- Conservation: λ < 1 signals population decline; management targets raising λ above 1
- Invasive species: λ > 1 and large λ predicts rapid spread
- Fisheries: λ analysis guides harvest rate setting to maintain λ ≥ 1
Glossary
Frequently Asked Questions
Lambda (λ) is the finite rate of increase: λ = N(t+1)/N(t). It is the multiplicative factor by which population size changes each time step. λ > 1: population growing. λ = 1: stable population. λ < 1: declining population. Related to intrinsic growth rate r by λ = e^r. In Leslie matrix models for age-structured populations, λ is the dominant eigenvalue of the projection matrix and determines the long-term growth trajectory and stable age distribution.
λ = e^r (for continuous-time equivalence). r = ln(λ). λ > 1 corresponds to r > 0 (growing); λ = 1 corresponds to r = 0 (stable); λ < 1 corresponds to r < 0 (declining). λ is used in discrete-time population models (seasonal breeders, annual censuses); r is used in continuous-time models (bacteria, continuously reproducing populations). Converting between them: if λ = 1.15 per year, r = ln(1.15) = 0.140 per year.
A Leslie matrix is a square matrix projecting an age-structured population: first row = age-specific fecundities (F₁, F₂, ... Fₙ); subdiagonal = age-specific survival probabilities (P₁, P₂, ... Pₙ₋₁). N(t+1) = L × N(t). Multiply the matrix by the population vector at each time step. Long-term: the population grows at rate λ = dominant eigenvalue of L; the corresponding eigenvector gives the stable age distribution. Calculated in R: eigen(L)$values[1] gives λ.
Elasticity of λ to matrix element aᵢⱼ = (aᵢⱼ/λ) × (∂λ/∂aᵢⱼ). It measures the proportional change in λ for a proportional change in the matrix element — making elasticities comparable across parameters on a 0–1 scale. All elasticities sum to 1. High elasticity for adult survival means protecting adult survival is the most effective strategy for increasing λ. In conservation of long-lived species (sea turtles, elephants, whales), adult survival typically has the highest elasticity — protecting adults has the most leverage on population growth.