Table 3 Models fitted to the allometry between horn length and pronotum width of male Onthophagus taurus

Sigmoidal Richards’ growth function

- 983.889

0

$Y(x)=A{[1+v\times exp(1+v)\times exp(\mu {(1+v)}^{1+1/v}\times (\lambda -\times )/A)]}^{(-1/v)}$

A (asymptote) = 0.70836

μ (maximum slope) = 18.52390

λ (lag-phase) = 0.75852

v (shape parameter) 2.70687

Sigmoidal Weibull growth function

- 981.318

2.571

$Y(x)=A-Drop\times exp(-exp(lrc)\times x^{\mathit{pwr}})$

A (asymptote) = 0. 700975

Drop (asymptote minus y intercept) = 0. 684813

lrc (ln rate constant) = 13. 622203

pwr (power x is raised to) = 55. 655926

Four-parameter logistic

- 981.127

2.762

$Y(x)=lA+(\mu A-lA)/(1+exp((xmid-\times )/scal))$

lA (lower asymptote) = 0. 0447065

uA (upper asymptote) = 0. 7220289

xmid (x value for inflection point) = 0. 7787523

scal (scale parameter) = 0. 0091249

Three-parameter logistic

- 977.034

6.855

$Y(x)=A/1+exp((xmid-x)/scal)$

A (asymptote) = 0.7306256

xmid (x value for inflection point) = 0.7774247

scal (scale parameter) = 0.0102605

Linear

- 721.143

262.746

$Y\left(x\right)=a+bx$

a (intercept) = −6.3043

b (slope) = 8.6193