Downscaling abundance from the distribution of species: Occupancy theory and applications

Abstract

One of the most important contributions to our understanding of how and why species distribute in landscapes is to document the significant correlation between abundance and distribution of species across a broad range of scales (Brown 1984, 1995, Gaston and Blackburn 2000). The correlation suggests that there is a general tendency that locally abundant species are more widely distributed in space than rare species, which forms a positive distribution-abundance (or occupancy-abundance) relationship. While the observed relationship of this macroecological pattern begs for ultimate biological accounts (Brown 1984, Hanski et al. 1993, Gaston 1994, Kolasa and Drake 1998, Gaston and Blackburn 2000), the mathematical forms of the relationship derived from physical, statistical and geometrical considerations have greatly advanced the study on the topics and have indeed provided a solid ground for fermenting biological explanation further (Maurer 1990, Wright 1991, Hanski et al. 1993, Leitner and Rosenzweig 1997, Hartley 1998, Kunin 1998, He and Gaston 2000, Kunin et al. 2000, Harte et al. 2001, He et al. 2002; see Holt et al. 2002 for a review). An important implication of the distribution-abundance correlation is to allow for the derivation of species abundance from information on species distribution, a downscaling process (Wu and Li, Chapters 1 and 2). Here we will follow this premise to derive abundance by examining the spatial distribution of species in landscapes based on the combinatorial theory of occupancy. The combinatorial theory of occupancy can date back as far as Pierre Laplace (Barton and David 1962) and has a long application in physics (Feller 1967). Laplace?s classical example of occupancy considers the following birth game. Assume that there are N births taking place within a year and that each birth has the same chance to occur in any of the 365 days. What Laplace wanted to know was how many days out of the 365 would have no births, i.e., the number of empty days. Similarly, in statistical mechanics physicists are interested in knowing how N particles occupy a space composed of M small cells. The most well-known models that describe the number of empty cells (without particles) include Maxwell- Botltzmann and Bose-Einstein models. In this chapter, however, we wanted to know the reverse: not how many cells are empty, but how many particles are there provided that the number of empty cells is known. Specifically, let?s consider a real example illustrated in Figure 5.1a in which a 50 ha plot in a rain forest of Malaysia is evenly divided into 800 cells of 25 ? 25 m each (Figure 5.1b). The distribution (or occurrence map, binary map, or atlas) is so generated that a cell is grey if the species is present and white if it is absent. Thus, a grey cell has at least one tree, but can have many more. Given such a map, we want to find out how many trees there are; of course, for Figure 5.1 we already know the number of trees and their locations in the plot. Note that real distribution maps are usually not as regularly bordered as Figure 5.1, but, for simplicity, statistical derivations dealt with in this study will be based on a map with assumed regular borders. It will become clear later that the models so derived are equally applicable to irregular maps. © 2006 Springer.