Nutrigonometry III: curvature, area and differences between performance landscapes.

Nutrition is one of the underlying factors necessary for the expression of life-histories and fitness across the tree of life. In recent decades, the geometric framework (GF) has become a powerful framework to obtain biological insights through the construction of multidimensional performance landscapes. However, to date, many properties of these multidimensional landscapes have remained inaccessible due to our lack of mathematical and statistical frameworks for GF analysis. This has limited our ability to understand, describe and estimate parameters which may contain useful biological information from GF multidimensional performance landscapes. Here, we propose a new model to investigate the curvature of GF multidimensional landscapes by calculating the parameters from differential geometry known as Gaussian and mean curvatures. We also estimate the surface area of multidimensional performance landscapes as a way to measure landscape deviations from flat. We applied the models to a landmark dataset in the field, where we also validate the assumptions required for the calculations of curvature. In particular, we showed that linear models perform as well as other models used in GF data, enabling landscapes to be approximated by quadratic polynomials. We then introduced the Hausdorff distance as a metric to compare the similarity of multidimensional landscapes.