The Euler Number as an Index of Spatial Integrity of Landscapes: Evaluation and Proposed Improvement

Abstract

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 (ε) is a numerical measure of spatial integrity, based upon the difference |n _f −n _p | between the number of fragments (n _f ) and the number of perforations (n _p ). In this contribution, ε is evaluated, and an improvement is presented as a new index ε*, which is a combination of two metrics (ε _i , ε _d ) based on n _f and n _p . The term ε _i quantifies the intensity of perforation and/or fragmentation. The term ε _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 ε*, calculated as ε*= (n _f +n _p )⁻¹√[1 +n _f ²]. Use, sensitivity, and application of ε*, ε _i , and ε _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 ε*, ε _i , and ε _d ; (2) the range of ε*, ε _i , and ε _d is fixed; (3) process dominance and intensity are both assessed; (4) ε*, ε _i , and ε _d are easy to calculate and to interpret; and (5) ε* is not only based upon |n _f −n _p |, as ε is. Guidelines for practical use by means of a biplot of ε _i and ε _d are given.