Contents lists available at ScienceDirect Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel Computational geometry applied to develop new metrics of road and edge effects and their performance to understand the distribution of small mammals in an Atlantic forest landscape Simone R. Freitas a, ⁎ , Everton Constantino b , Marcos M. Alexandrino c a Universidade Federal do ABC, Santo André, SP, Brazil b Universidade Estadual de Campinas, Campinas, SP, Brazil c Department of Mathematics, Institute of Mathematics and Statistics, Universidade de São Paulo, São Paulo, SP, Brazil ARTICLE INFO Keywords: Differential calculus Edge effect New software Road ecology Tropical forest Wildlife ABSTRACT Roads negatively affect many vertebrate species, whereas edge effect may favor some generalist species. This study aims to: 1) present a new way to calculate "line integral effects", represented by LIE and AVLIE, through new computer software, making this concept accessible to a broad audience of researchers interested in the study of Road Ecology and Tropical Forest Ecology; and, 2) test the performance of LIE and AVLIE indices, applied to road effect (LIE_road and AVLIE_road) and to edge effect (LIE_edge and AVLIE_edge), other road effect indices and forest area, using a data set on small mammal abundance in a human modified landscape in the Brazilian Atlantic Forest. Road and edge effects were represented by new metrics: Line Integral Effect (LIE) and Average Integral Effect (AVLIE), calculated using Line Integral from Differential Calculus of Several Variables through new free software developed by the second author. LIE_road and LIE_edge measure the total sum of the effect of roads (represented by lines) and edges (polygons), respectively, in relation to the forest fragment (point). AVLIE_road and AVLIE_edge measure the average of road and edge effect, respectively, in relation to the same sampling point. We used generalized linear regression models to explore the relationships between the abun- dance of the two groups of small mammals (forest specialists and habitat generalists) and the independent variables representing road, edge and forest effects. For forest specialists, the best model included AVLIE_road (negatively associated with abundance) and AVLIE_edge (negatively associated), while for habitat generalists, the best model included AVLIE_road (negatively associated) and LIE_edge (positively associated). Thus, there are more small mammals where road effect is lower. Forest fragments with higher edge effect showed more habitat generalists and less forest specialists. LIE and AVLIE could be useful metrics to explore edge effect separately to road effect on wildlife in forest fragments. 1. Introduction Line integral effect has been used in Freitas et al. (2012) and Malcolm (1994) to study road effect (Laurance et al., 2009) and edge effect (Murcia, 1995) respectively. It is defined as g (s,p)dl c where C is a curve in 2 that can model a road or edge of region and g : 4 is a function so that g (s,p) measures the effect of a point s C over a fixed point p in the studied region. Roughly speaking p LIE( ) is approximated by a sum of these effects multiplied by the length of parts of C . The challenge in using a line integral effects index in a systematic way is the issue of how to best integrate along so many curves that appear to naturally model roads and edges. In Freitas et al. (2012), these issues were approached through a simple numerical process that would allow us to approximate the desired integral by a finite sum. This numerical process (that has a controllable numerical error) requires some work from the user. More precisely users must chose which curves they want to integrate and which would make less sense from a biol- ogist point of view. They also have to collect length of roads in different discs (buffers) using GIS, making some calculations. All these steps in the numerical process require some time of the user, and some under- standing of the process itself and therefore the user could make mis- takes and introduce errors during the calculations. It was clearly de- sirable to have an automatic way to calculate this index with the least work as possible, and demanding less mathematical knowledge from the user than the previous procedure (Freitas et al., 2012). https://doi.org/10.1016/j.ecolmodel.2018.08.004 Received 20 April 2018; Received in revised form 2 August 2018; Accepted 3 August 2018 ⁎ Corresponding author at: Universidade Federal do ABC (UFABC), Avenida dos Estados, 5001, Bloco A, sala 631-3, 09210-580, Santo André, SP, Brazil. E-mail address: simonerfreitas.ufabc@gmail.com (S.R. Freitas). Ecological Modelling 388 (2018) 24–30 0304-3800/ © 2018 Elsevier B.V. All rights reserved. T