The Euler number as an index of spatial integrity of landscapes: evaluation and proposed improvement.

The spatial integrity of a habitat or landscape is determined by the occurrence of habitat fragments and of perforations inside them. A landscape is said to have less spatial integrity with increasing numbers of fragments and perforations. The Euler number (epsilon) is a numerical measure of spatial integrity, based upon the difference (nf-np) between the number of fragments (nf) and the number of perforations (np). In this contribution, epsilon is evaluated, and an improvement is presented as a new index epsilon*, which is a combination of two metrics (epsilon j, epsilon d) based on nf and np. The term epsilon j quantifies the intensity of perforation and/or fragmentation. The term epsilon d measures the extent to which fragmentation predominates perforation, and vice versa. The intensity and dominance measures are combined into an Euclidean distance measure, generating the new ensemble value epsilon*, calculated as epsilon* = (nf + np)-1 square root of [1 + nf2]. Use, sensitivity, and application of epsilon*, epsilon j, and epsilon d are illustrated using percolation maps. Application of the new metrics by environmental scientists is encouraged because (1) no negative values can be generated with epsilon*, epsilon j, and epsilon d; (2) the range of epsilon*, epsilon j, and epsilon d is fixed; (3) process dominance and intensity are both assessed; (4) epsilon*, epsilon j, and epsilon d are easy to calculate and to interpret; and (5) epsilon* is not only based upon (nf - np), as epsilon is. Guidelines for practical use by means of a biplot of epsilon j and epsilon d are given.