Statistical & Mathematical Formulas
Richards Generalized Logistic Curve: Asymmetry Parameter (ν) & Inflection
Model asymmetric epidemic, crop, and market S-curves via F. J. Richards’ 4-parameter generalized logistic equation Y(t) = K * (1 + ν * EXP(-r * (t - t_m)))^(-1 / ν) and inflection height Y_inf = K * (1 + ν)^(-1 / ν).
One shape parameter ν continuously bridges Gompertz (36.8%), Logistic (50%), and late-peaking S-curves!
Fixed S-curve models lock their peak-velocity inflection point at a rigid percentage of carrying capacity K: Gompertz always inflects at K / e = 36.79%, while Verhulst logistic always inflects at K / 2 = 50.00%! Botanist F. J. Richards (Journal of Experimental Botany, 1959) introduced a single dimensionless asymmetry parameter ν > 0 inside the centered formulation Y(t) = K * (1 + ν * EXP(-r * (t - t_m)))^(-1 / ν), where t_m is the exact peak-growth inflection time!
At t = t_m, EXP(0) = 1 so inflection height equals Y_inf = K * (1 + ν)^(-1 / ν): as ν -> 0+, Y_inf -> K / e (36.8% Gompertz limit); at ν = 1, Y_inf = K / 2 (50.0% classic Verhulst logistic); at ν = 2, Y_inf = K / SQRT(3) = 57.74% (late-peaking epidemic wave or technology rush)! For K = 1,000, ν = 2.0, r = 0.25/day, and t_m = 12 days: Y(12) = 577.35 and Y(16) = 759.02!
| Richards Shape Parameter ν (K = 1,000, tm = 12 d) | Inflection Level Y_inf at t = tm | Trajectory Value at t = 16 d | Special-Case S-Curve Family |
|---|---|---|---|
| ν -> 0+ Limit (e.g., ν = 0.001) | 368.06 (~K / e = 36.8%) | 796.16 | Gompertz Early-Peak Limit |
| ν = 1.00 (Symmetric S-Curve) | 500.00 (= K / 2 = 50.0%) | 731.06 | Exact Verhulst Logistic |
| ν = 2.00 (Right-Skewed Surge) | 577.35 (= K / √3 = 57.7%) | 759.02 | Late-Inflecting Surge Wave |
Richards Generalized Logistic Trajectory Y(t)
Evaluates K * (1 + ν * EXP(-r * (t - t_m))) ^ (-1 / ν) for K > 0, ν > 0, and r > 0.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Richards Parameters", ROUND(B2 * ((1 + B3 * EXP(-B4 * (B6 - B5))) ^ (-1 / B3)), 2))Inflection-Point Level Y_inf = K * (1 + ν)^(-1 / ν) at t = t_m
Computes the exact level where growth rate dY/dt reaches its maximum.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Richards Parameters", ROUND(B2 * ((1 + B3) ^ (-1 / B3)), 2))Try the example
Use the original values or change them to test your understanding.