Fig. 4: A visual summary illustrating the relationship between the location of a species range relative to the focal point and its contribution to a phase of the SAR.

The top row comprises three plots exemplifying bivariate normal distributions at varying distances from the focal point, represented by the red ‘x’. a The distribution is centred on the focal point; b The focal point falls within a distance of the centre of distribution where \(2 < {\sigma }_{p} < 4\), where \({\sigma }_{p}^{2}\) is the variance; c The focal point is situated at a distance larger than \({\sigma }_{p} > 4\) (refer also to Fig. 1). d The black curves illustrate the corresponding distributions of distances: a Rayleigh distribution for the distribution of plot (a), and Rice distributions for the distributions of plots (b, c). The normal distribution serves as a good approximation for the Rice distribution when the location parameter is significantly larger than its scale parameter (distribution on (c)). The red curves represent the respective limiting distributions of the minima: Weibull distributions for the Rayleigh and Rice distributions, and a Gumbel-like distribution when the Rice becomes normal-like. Note that the distributions are not drawn to scale.