Intrinsic Rate of Increase Calculators

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The intrinsic rate of increase (r, also called the per-capita rate of increase) is the maximum per-capita growth rate of a population when resources are unlimited — the inherent reproductive potential of a species. r = b − d, where b is the per-capita birth rate and d is the per-capita death rate under optimal conditions. In the exponential growth model: dN/dt = rN. In the logistic model: dN/dt = rN(1 − N/K). A positive r means the population grows; r = 0 means it is stable; r < 0 means it is declining. r_max is species-specific and reflects life history strategy: r-selected species have high r; K-selected species have low r.

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Intrinsic Rate of Increase Formula

r = b − d (per-capita birth rate minus per-capita death rate)

Exponential growth: N(t) = N₀ × e^(rt). Doubling time: t_d = ln(2)/r = 0.693/r.

Example: rabbit population with b = 2.5 offspring/individual/yr and d = 0.5 deaths/individual/yr: r = 2.5 − 0.5 = 2.0 yr⁻¹. Doubling time = 0.693/2.0 = 0.35 yr ≈ 4.2 months.

Calculating r from Age-Specific Data

Euler-Lotka equation: 1 = Σ e^(−rx) × lₓ × mₓ. lₓ = survivorship to age x; mₓ = fecundity at age x. Solve numerically for r (e.g., using Newton's method or iterative approach). r ≈ ln(R₀)/T_c, where R₀ = net reproductive rate (Σlₓmₓ) and T_c = generation time.

Typical r Values

  • Bacteria (E. coli): r ≈ 20–60 per day (2.08 hr⁻¹ in LB)
  • Insects (Drosophila): r ≈ 0.3–0.4 per day
  • Small rodents (mice): r ≈ 0.1 per day
  • Humans (current global): r ≈ 0.0095 per yr = 0.95%
  • Elephants: r ≈ 0.05–0.07 per yr

r vs. K selection

High r (r-selected): small body size; short generation time; many small offspring; colonizers of disturbed habitats. Low r (K-selected): large body size; long generation time; few large offspring; stable, resource-limited environments.

Glossary

Intrinsic Rate of Increase (r)
r = b − d; the per-capita population growth rate under optimal conditions; r > 0 = growing; r = 0 = stable; r < 0 = declining; determines doubling time (t_d = 0.693/r).
Net Reproductive Rate (R₀)
R₀ = Σ(lₓ mₓ); the mean number of female offspring per female over a lifetime; R₀ > 1 = population growing; r ≈ ln(R₀)/T_c; distinct from the epidemiological R₀.
r-Selection
Selection for high intrinsic rate of increase; typical of small, short-lived organisms with many small offspring; adapted to variable, unpredictable environments; contrasts with K-selection.

Frequently Asked Questions

The intrinsic rate of increase (r) is the per-capita rate of population growth under optimal conditions (unlimited resources, no density dependence). r = b − d (birth rate minus death rate per individual per time unit). Positive r: population grows. Zero r: stable (births equal deaths). Negative r: declining population. r represents the maximum inherent growth potential of a species — it reflects life history (generation time, litter size, survival). Exponential growth model: dN/dt = rN → N(t) = N₀e^(rt). Logistic model includes density dependence: dN/dt = rN(1−N/K). The doubling time = 0.693/r.

From age-specific birth and survival data: Calculate R₀ = Σ(lₓ × mₓ) = net reproductive rate (average number of daughters per female over lifetime). Calculate mean generation time: T_c = Σ(x × lₓ × mₓ) / R₀. Approximate r ≈ ln(R₀) / T_c. Example: Σlₓmₓ = 3.5; T_c = 2.5 years: r ≈ ln(3.5)/2.5 = 1.253/2.5 = 0.50 yr⁻¹. For more precise r: use the Euler-Lotka equation 1 = Σ e^(−rx) lₓ mₓ solved numerically. R package 'popbio' and similar tools calculate r from life tables.

R₀ (basic reproductive rate in ecology, not to be confused with epidemiological R₀): the net reproductive rate = mean number of female offspring produced per female over her lifetime = Σ(lₓ × mₓ). R₀ > 1: population growing. R₀ = 1: stable. R₀ < 1: declining. r (intrinsic rate of increase): the per-capita instantaneous growth rate = ln(R₀)/T_c. R₀ answers: how many times does the population multiply per generation? r answers: how fast does the population grow per unit time? A population might have R₀ = 2 (doubles each generation) but small r if generations are very long (T_c is large). Both are needed to fully characterize population growth.

r-selected species are characterized by high r due to life history features that maximize per-capita birth rate and minimize generation time: Early reproduction (short T_c): smaller organisms typically mature earlier → more generations per unit time → higher r. Large litter sizes: more offspring per reproductive event → higher b. Shorter lifespan: resources allocated to current reproduction rather than longevity. Small body size: lower maintenance costs → more energy available for reproduction. Examples: bacteria (r ≈ 20/day), Drosophila (r ≈ 0.3/day), mice (r ≈ 0.1/day). K-selected species (elephants, humans, whales) have low r due to long generation times, small litter sizes, and high parental investment — they thrive in stable environments near carrying capacity, not by maximizing population growth rate.