Journal of Theoretical Biology 249 (2007) 791–803 Heterogeneous communities with lognormal species abundance distribution: Species–area curves and sustainability Steinar Engen à Department of Mathematical Sciences, Norwegian University for Science and Technology, N-7491 Trondheim, Norway Received 18 April 2007; received in revised form 12 August 2007; accepted 1 September 2007 Available online 12 September 2007 Abstract Heterogeneous species abundance models are models in which the dynamics differ between species, described by variation among parameters defining the dynamics. Using a dynamic and heterogeneous species abundance model generating the lognormal species abundance distribution it is first shown that different degrees of heterogeneity may result in equivalent species abundance distributions. An alternative to Preston’s canonical lognormal model is defined by assuming that reduction in resources, for example reduction in available area, increases the density regulation of each species. This leads to species–individual curves and species–area curves that are approximately linear in a double logarithmic plot. Preston’s canonical parameter g varies little along these curves and takes values in the neighborhood of one. Quite remarkably, the curves, which define the sensitivity of the community to area reductions, are independent of the heterogeneity among species for this model. As a consequence, the curves can be estimated from a single sample from the community using the Poisson lognormal distribution. It is shown how to perform sensitivity analysis with respect to over-dispersion in sampling relative to the Poisson distribution as well as sampling intensity, that is, the fraction of the community sampled. The method is exemplified by analyzing three simulated data sets. r 2007 Elsevier Ltd. All rights reserved. Keywords: Community dynamics; Species abundance distribution; Lognormal distribution; Species–area curves; Preston’s canonical lognormal; Heterogeneity; Dynamic model; Neutral model; Extinction; Extinction process; Conservation biology; Colonization; Invasion; Speciation; Sustainability 1. Introduction The International Union for Conservation of Nature and Natural Resources (IUCN) has listed a number of criteria for assessing risk of extinction of animal and plant populations (IUCN, 2001), one important parameter being the risk of extinction within a given time (Shaffer, 1981; Mace and Lande, 1991). As many species are naturally rare, and especially rare in samples, it is often difficult to estimate these probabilities. Taking all uncertainly into account and constructing population prediction intervals then often lead to extremely wide intervals for future population size as well as for time to extinction (Lande et al., 2003). Another problem is that computation of prediction intervals requires long time series which are often not available, especially for rare species. A totally different approach is based on studying the abundance distribution of the species in a community, as first done by Fisher et al. (1943) and Preston (1948, 1962), and derive so-called species–area curves. Such curves can be defined in different ways: they can express how the number of species will change as the area available for the community varies; how many species one will find within a sub-area of given size; or the number of species expected to be found when sampling randomly a fraction of a total area or a random fraction of the individuals. Fisher et al. (1943), using the well known limiting form of the gamma distribution for species abundances, derived the curve based on random sampling for the log-series model and found that the number of species was approximately proportional to the log number of individuals in the sample. Engen (1974, 1978) showed, however, that the gamma model with a negative value of the shape parameter gave almost exactly a linear relation in a double logarithmic plot, while Preston (1962) and May (1975) ARTICLE IN PRESS www.elsevier.com/locate/yjtbi 0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2007.09.005 à Tel.: +47 73 591747; fax: +47 73 591038. E-mail address: steinaen@math.ntnu.no