The Distribution of Abundance
in Neutral Communities
Graham Bell
Redpath Museum and Biology Department, McGill University, Montreal, Quebec H3A 2K6, Canada
THE AMERICAN NATURALIST, VOL. 155, NO. 5, MAY 2000, pp. 606-617.
abstract:
The patterns of abundance generated by a simple stochastic birth-death-immigration model are described in order to characterize the diversity of neutral communities of ecologically equivalent species. Diversity is described by species number S and the variance of frequency or log abundance q∼. The frequency distribution of abundance is very generally lognormal, skewed to the left by immigration and resembling descriptions of natural communities. Increased immigration and community size always cause S to increase. Their effect on q∼ is more complicated, but given biologically reasonable assumptions, S and q∼ will be positively correlated in most circumstances. Larger samples contain more species; the graph of log S on log individuals, equivalent to a species-area curve, is generally convex upward but becomes linear with a slope of about 10.25 when immigration is low and births exceed deaths. When individuals invade a new, vacant environment, both S and q∼ increase through time. Thus, a positive correlation between S and q∼ will usually be generated when sites of differing size or age are surveyed. At equilibrium, communities maintain roughly constant levels of S and q∼ but change in composition through time; composition may remain similar, however, for many generations. Many prominent patterns observed in natural communities can therefore be generated by a strictly neutral model. This does not show that community structure is determined exclusively by demographic stochasticity, but rather demonstrates the necessity for an appropriate null model when functional hypotheses are being tested.
Keywords: abundance, species diversity, biodiversity, species-area relationship, neutral model.
Regulation of Diversity
The variation of species number S with sample size n and community size K constitute two versions of the speciesarea relationship. Increasing sample size is akin to sampling successively larger areas of a mainland region, whereas varying community size is more nearly comparable with sampling islands of different size. In most cases, log S increases linearly with log area, so that the speciesarea relationship can be described by a power law; in other cases, however, S increases directly with log area (see Connor and McCoy 1979).
The log-series distributions generated by the models studied by Caswell (1976) give rise to semilog relationships of the form S = alog n + c, where a is a migration rate. Lognormal distributions yield a sigmoid curve, which is roughly linear in the central part of the plot (Preston 1948). In either case, the graph of log S on log n is convex upward. The NCM can give rise to linear log-log graphs, however, when birth rate is high and immigration rate low. In these circumstances a canonical lognormal is generated, and a power law with an exponent of about 0.27 follows, as expected (Preston 1962). Leitner and Rosenzweig (1997) have shown that this exponent should be about 0.77 for nested samples of a region within which species distributions have explicit spatial locations; this spatial structure is not present in the NCM.
Caswell's models predict a similar semilog relationship between species number S and community size K, so that the log-log graph will be convex upward. He suggests that a linear log-log plot might be the consequence of an increase of immigration rate with species number. Lognormal distributions of abundance generally give rise to linear log-log graphs. Under the NCM, the graph of log S on log K is convex upward for m > 1/s, and concave upward for m < i/s; for m = 1/s the graph is linear with a slope of about 0.3 (see fig. 3).
Neutral models are thus capable of generating speciesarea relationships that resemble those found in nature. In particular, the characteristic linearity of log-log plots emerges from the NCM with plausible combinations of parameter values. Moreover, the NCM can generate power laws with exponents close to those characteristic of natural systems. Indeed, the dependence of the shape of the relationship on the magnitude of immigration rate relative to birth rate can be given a simple and natural interpretation. In small areas the perimeter is large relative to the interior, and communities will contain a large proportion of recent immigrants; in these circumstances the log-log graph will be convex upward. In large areas the perimeter is much smaller relative to the interior, and most individuals will be natives, born within the area; the log-log graph will then tend to be linear. The neutral model thus gives rise to a biphasic species-area relationship that resembles that of natural regions within a single biogeographic province (Rosenzweig 1995).
More generally, an important lesson of the neutralistselectionist controversy in population genetics was that prolonged contemplation of frequency distributions is unlikely to provide decisive evidence about ecological or evolutionary mechanisms. It will be rare indeed that a particular pattern of abundance or diversity cannot be explained both by a neutral model, given the appropriate combination of parameters, and equally by some functional hypothesis. Sugihara's sequential-breakage model, for example, generates canonical lognormal abundance distributions through a procedure that is taken to represent an ecological mechanism of habitat partitioning (Sugihara 1980), and the self-similarity of species distributions may by itself lead to skewed lognormal distributions (Harte et al. 1999). One of the most important roles of neutral models in community ecology is merely cautionary. It might be that lognormal distributions of abundance indicate competition, that species-area relationships reveal the degree of community saturation, or that the spatial or temporal structure of diversity reflects patterns of heterogeneity and disturbance. These possibilities should be evaluated, however, in relation to the simpler explanations offered by chance and history.
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