Double Richards selectivity as a function of length
sel_double_richards.RdA smooth, six-parameter curve formed by multiplying an ascending generalized-logistic (Richards) limb by one minus a descending Richards limb: $$(1 + \exp(-z_1))^{-1/\gamma_1} [1 - (1 + \exp(-z_2))^{-1/\gamma_2}].$$ Parameters are on the real line and transformed as follows:
par[1]Ascending 50% point:
mean(len) + par[1] * sd(len).par[2]Ascending width:
exp(par[2]) * sd(len).par[3]Ascending shape:
gamma1 = exp(par[3]).par[4]Descending 50% point is ascending 50% point plus
exp(par[4]) * sd(len).par[5]Descending width:
exp(par[5]) * sd(len).par[6]Descending shape:
gamma2 = exp(par[6]).
The Maunder parameterization is recovered with
alpha_k = L50_k + log(2^gamma_k - 1) / beta_k and
beta_k = log(19) / width_k. The peak can be below one when the limbs
overlap; it is intentionally not rescaled. As either shape parameter tends
to negative infinity, its limb tends to a Gompertz curve with the same 50%
point. Large shape values flatten the limb toward 0.5 (at par[3] = par[6] = 7, the curve is approximately 0.25). Mapping par[3] and
par[6] to zero gives a four-parameter double-logistic curve. Suggested
bounds for columns 3 and 6 are c(-5, 5); use the existing par_sel bounds
for the remaining columns.