Statistical & Mathematical Formulas
Von Bertalanffy Growth Function (VBGF): Asymptotic Length & Age
Predict organism length-at-age L(t) = L_∞ * (1 - EXP(-K * (t - t_0))) and invert catch length into biological age t(L) = t_0 - LN(1 - L / L_∞) / K for fisheries stock assessment and wildlife ecology.
How metabolic anabolism vs catabolism shapes universal vertebrate growth!
Austrian theoretical biologist Ludwig von Bertalanffy (1938) derived his iconic growth law by balancing surface-area-limited nutrient anabolism (~L^2) against volume-proportional tissue catabolism (~L^3), yielding an exponential approach toward asymptotic body length L_∞: L(t) = L_∞ * (1 - EXP(-K * (t - t_0))), where Brody growth coefficient K (yr^-1) sets curvature speed and theoretical zero-length age t_0 (< 0) accounts for nonzero larval hatching size!
Fisheries biologists routinely invert VBGF to convert trawler length-frequency samples into cohort ages: t(L) = t_0 - LN(1 - L / L_∞) / K! For North Atlantic cod (L_∞ = 120.0 cm, K = 0.18 yr^-1, t_0 = -0.35 yr): an age-5 cod reaches L(5) = 74.19 cm (climbing to 107.01 cm at age 12), and inverting a 74.19 cm catch specimen recovers age t = 5.00 years!
| Atlantic Cod Cohort Age (L∞ = 120 cm, K = 0.18/yr, t0 = -0.35) | Predicted Length L(t) | % of Asymptotic L∞ | Fisheries Maturity Stage |
|---|---|---|---|
| Juvenile Recruit (t = 2.0 yr) | 41.42 cm | 34.5% | Entering Trawl Selectivity |
| Spawning Adult (t = 5.0 yr) | 74.19 cm | 61.8% | Prime Spawning Stock Biomass |
| Trophy / BOFFFF Spawner (t = 12.0 yr) | 107.01 cm | 89.2% | Near Asymptotic Ceiling |
Von Bertalanffy Predicted Length-at-Age L(t) (cm)
Evaluates L_∞ * (1 - EXP(-K * (t - t_0))) for L_∞ > 0, K > 0, and t > t_0.
=IF(OR(B2 <= 0, B3 <= 0, B5 <= B4), "Invalid VBGF Parameters", ROUND(B2 * (1 - EXP(-B3 * (B5 - B4))), 2))Inverse Age-from-Length Cohort Estimator t(L) (years)
Inverts VBGF via t_0 - LN(1 - L / L_∞) / K while guarding specimens exceeding L_∞.
=IF(OR(B2 <= 0, B3 <= 0, B6 <= 0, B6 >= B2), "Length >= Linf or Invalid", ROUND(B4 - LN(1 - B6 / B2) / B3, 2))Try the example
Use the original values or change them to test your understanding.