Improvements in anisotropic models of single tree effects in Cartesian coordinates

Abstract

For anisotropic density functions of e.g. fruit or leaf dispersal, most mathematical research is only done in polar coordinates. However, in software solutions aiming to derive inverse models for real world dispersal data, Cartesian coordinates may be preferred for several reasons. Thus, we introduce an anisotropic model in Cartesian coordinates following the approach in Wälder et al. (2009) with the von Mises approach. By introducing a correction factor, we thereby consider the fundamental attribute, that the integral over a density function with respect to the Cartesian coordinates has to be equal 1. It may have been overlooked so far that guaranteeing for this attribute needs different approaches whether working in polar or Cartesian coordinates. One result is that our approach can be used also for other anisotropic models rather than models from the von Mises approach.