Statistical & Econometric
Janoschek Growth Curve Formula: Bounded Sigmoidal Size & Inflection Age
Model nonzero-birth biological and forestry growth L(t) = L_∞ - (L_∞ - L_0) * EXP(-k * t^p) and compute exact analytical inflection age t_inf = ((p - 1) / (k * p))^(1 / p).
Why Janoschek’s Weibull-type growth law beats logistic curves when birth size L_0 is fixed!
Symmetric logistic and Gompertz curves cannot independently pin initial size L_0 at t = 0 while adjusting inflection skewness! Biometrician Adolf Janoschek (1957) introduced the Weibull-type bounded growth curve L(t) = L_∞ - (L_∞ - L_0) * EXP(-k * t^p), which satisfies L(0) = L_0 exactly, approaches asymptote L_∞ smoothly, reduces to monomolecular von Bertalanffy decay when p = 1, and develops a true S-shaped inflection point whenever p > 1 at exact analytical age t_inf = ((p - 1) / (k * p))^(1 / p)!
Track skeletal or timber growth from juvenile size L_0 = 8.0 cm toward mature asymptote L_∞ = 120.0 cm (rate k = 0.045, shape p = 2.2): peak growth velocity occurs at inflection age t_inf = 3.108 years; by t = 4.0 years the organism reaches L(4) = 76.69 cm, and by t = 8.0 years it approaches maturity at L(8) = 118.58 cm!
| Age t (L0 = 8, L∞ = 120, k = 0.045, p = 2.2) | Predicted Size L(t) | Inflection Point t_inf | Growth Phase |
|---|---|---|---|
| t = 3.108 yr (Peak Velocity Age) | 55.97 cm | 3.108 yr | Maximum dL/dt Acceleration Peak |
| t = 4.000 yr (Juvenile Surge) | 76.69 cm | 3.108 yr | Post-Inflection Deceleration |
| t = 8.000 yr (Near Asymptote) | 118.58 cm | 3.108 yr | 98.8% of Mature Ceiling L∞ |
Janoschek Bounded Sigmoidal Size L(t)
Evaluates L_∞ - (L_∞ - L_0) * EXP(-k * t^p) from initial size B2, asymptote B3, rate constant B4, shape exponent B5, and age B6.
=IF(OR(B3 <= B2, B4 <= 0, B5 <= 0, B6 < 0), "Invalid Janoschek Inputs", ROUND(B3 - (B3 - B2) * EXP(-B4 * (B6 ^ B5)), 2))Analytical Inflection Age t_inf = ((p - 1) / (k * p))^(1 / p)
Computes the exact inflection age when shape parameter p > 1, or flags monomolecular concave growth when p <= 1.
=IF(OR(B4 <= 0, B5 <= 0), "Invalid Janoschek Inputs", IF(B5 <= 1, "No Inflection (p <= 1)", ROUND(((B5 - 1) / (B4 * B5)) ^ (1 / B5), 3)))Try the example
Use the original values or change them to test your understanding.