htl.html`The ModSkurt mean function, ${tex`\small f`}, that predicts average species abundance, ${tex`\small \mu`}, given environmental variable, ${tex`\small x`}, is defined as:

${tex`\displaystyle \mu_i = f(x_i|H,m,s,r,d,p) = H \left[r \exp\left(\frac{1}{p} \left[\frac{x_i-m}{sr} - d\right]\right) + (1 - r) \exp\left(\frac{1}{p}\left[d - \frac{x_i-m}{s\left(1 - r\right)} \right]\right) - \exp\left(-\frac{d}{p}\right) + 1\right] ^ {-p}, `}

with parameters

${tex`\small H \in \mathbb{R}_{>0}`}: the strictly positive maximum height of the curve; the peak in mean abundance along the environmental gradient,
${tex`\small m \in \mathbb{R}`}: the environmental location of peak mean abundance,
${tex`\small s \in \mathbb{R}_{>0}`}: that scales the width of the curve, describing how abundance decreases away from ${tex`\small m`},
${tex`\small r \in (0, 1)`}: that accounts for asymmetry, with ${tex`\small r=0.5`} indicating equal abundance in environmental conditions either side of ${tex`\small m`}, ${tex`\small r`} below ${tex`\small 0.5`} indicating more abundance below ${tex`\small m`}, and ${tex`\small r`} above ${tex`\small 0.5`} indicating more abundance above ${tex`\small m`}.,
${tex`\small d \in \mathbb{R}_{>0}`}: that controls the peakedness (${tex`\small d < 0`}) or broadness (${tex`\small d > 0`}, e.g. plateau-like shape) of the peak in mean abundance around ${tex`\small m`},
${tex`\small p > 0`}: persists higher-levels of abundance in relatively extreme conditions; this is a similar effect to increasing the tail weight of a distribution function.

Refer to {{link to thesis?}} for the ecological motivation and mathematical details behind each parameter.`