On the Statistical Mechanics of Alien Species Distribution

Abstract

Many species of plants are found in regions to which they are alien. Their global distributions are characterised by a family of exponential functions of the kind that arise in elementary statistical mechanics (an example in ecology is MacArthur's broken stick). We show here that all these functions are quantitatively reproduced by a model containing a single parameter-some global resource partitioned at random on the two axes of species number and site number. A dynamical model generating this equilibrium is a two-fold stochastic process and suggests a curious and interesting biological interpretation in terms of niche structures fluctuating with time and productivity, with sites and species highly idiosyncratic. Idiosyncrasy implies that attempts to identify a priori those species likely to become naturalised are unlikely to be successful. Although this paper is primarily concerned with a particular problem in population biology, the two-fold stochastic process may be of more general interest.

Endemics and Cosmopolitans: Application of Statistical Mechanics to the Dry Forests of Mexico

“…These distributions are consistent with cosmopolitan species (those capable of existing in more than one site) being distributed according to a statistical mechanics of division of some resource at random, and the number of species at sites of any given rank being drawn from an underlying exponential reflecting division among sites of that same resource. This is a second example of the structures first revealed, for alien species, in [5] and treated most completely in [6]. (Note: The reader familiar with [5] and [6] may recall that the population distribution over the sites is consistent with being drawn from an underlying exponential distribution that is not truncated.…”

“…If the same resource is also divided between the different sites in the same sort of way, the distribution of the number of sites with the number of species each site contains is, for our particular application, drawn from an underlying exponential distribution. These exponential distributions are themselves sufficient to generate the exponential distribution of the number of pairs (and higher multiplets) of sites with the number of species shared, without any further assumptions [6]. It is worth remarking that the placing of the cuts could be accomplished statically (as in MacArthur's broken stick) or dynamically, with species accepted and rejected from sites, and sites growing and contracting in receptivity; described by the appropriate master equations.…”

“…However, the species richness of the individual sites, Figure 2 upper panel, is not consistent with having been drawn from an underlying exponential probability. The most probable receptivity (or species richness) of a site of rank R is proportional to lnR 0 − lnR, where the richest site is rank 1, and for a simple exponential, R 0 is the number of sites +1, see [6]. For 20 sites and a pure exponential, lnR 0 = 3.04.…”

“…Beyond that, the number of the pairs of sites sharing p species is exponentially distributed with p [5]. The relationship between these various exponential probability distributions was elucidated in [6]. In [5] it is speculated that similar principles may apply more generally to community assembly, for there are examples of the number of species with the number of sites occupied being distributed exponentially in populations of heteroflagellates and of tree species in the seasonally dry tropical forests of Mexico.…”

“…In [5] it is speculated that similar principles may apply more generally to community assembly, for there are examples of the number of species with the number of sites occupied being distributed exponentially in populations of heteroflagellates and of tree species in the seasonally dry tropical forests of Mexico. Having clarified in [6] the roles of the alien species' exponentials, we have returned to the dry forests of Mexico, where the data [7] contain not only the exponential distribution of the number of species with the number of sites (which first drew our attention), but also list the number of species at each site, and beyond that the number of species common to each pair of sites. These three aspects of the data set are explained by a category of cosmopolitan species, distributed in accord with the model for alien species, and in addition, a category of specialists, endemic species found preferentially at high rank sites, i.e., those that are relatively species poor.…”