Species abundance distributions (SADs) follow one of ecologyÕs oldest and most universal laws -every community shows a hollow curve or hyperbolic shape on a histogram with many rare species and just a few common species. Here, we review theoretical, empirical and statistical developments in the study of SADs. Several key points emerge. (i) Literally dozens of models have been proposed to explain the hollow curve. Unfortunately, very few models are ever rejected, primarily because few theories make any predictions beyond the hollow-curve SAD itself. (ii) Interesting work has been performed both empirically and theoretically, which goes beyond the hollow-curve prediction to provide a rich variety of information about how SADs behave. These include the study of SADs along environmental gradients and theories that integrate SADs with other biodiversity patterns. Central to this body of work is an effort to move beyond treating the SAD in isolation and to integrate the SAD into its ecological context to enable making many predictions. (iii) Moving forward will entail understanding how sampling and scale affect SADs and developing statistical tools for describing and comparing SADs. We are optimistic that SADs can provide significant insights into basic and applied ecological science.

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The role of stochasticity and its interplay with nonlinearity are central current issues in studies of the complex population patterns observed in nature, including the pronounced oscillations of wildlife and infectious diseases. The dynamics of childhood diseases have provided influential case studies to develop and test mathematical models with practical application to epidemiology, but are also of general relevance to the central question of whether simple nonlinear systems can explain and predict the complex temporal and spatial patterns observed in nature outside laboratory conditions. Here, we present a stochastic theory for the major dynamical transitions in epidemics from regular to irregular cycles, which relies on the discrete nature of disease transmission and low spatial coupling. The full spectrum of stochastic fluctuations is derived analytically to show how the amplification of noise varies across these transitions. The changes in noise amplification and coherence appear robust to seasonal forcing, questioning the role of seasonality and its interplay with deterministic components of epidemiological models. Childhood diseases are shown to fall into regions of parameter space of high noise amplification. This type of "endogenous" stochastic resonance may be relevant to population oscillations in nonlinear ecological systems in general.

A central problem in ecology is determining the processes that shape the complex networks known as food webs formed by species and their feeding relationships. The topology of these networks is a major determinant of ecosystems' dynamics and is ultimately responsible for their responses to human impacts. Several simple models have been proposed for the intricate food webs observed in nature. We show that the three main models proposed so far fail to fully replicate the empirical data, and we develop a likelihood-based approach for the direct comparison of alternative models based on the full structure of the network. Results drive a new model that is able to generate all the empirical data sets and to do so with the highest likelihood.

The importance of dispersal for biodiversity has long been recognized. However, it was never advertised as vigorously as Stephen Hubbell did in the context of his neutral community theory. After his book appeared in 2001, several scientists have sought and found analytical expressions for the effect of dispersal limitation on community composition, still in the neutral context. This has been done along two relatively independent lines of research that have a different mathematical approach and focus on different, yet related, types of results. Here, we study both types in a new framework that makes use of the sampling nature of the theory. We present sampling distributions that contain binomial or hypergeometric sampling on the one hand, and dispersal limitation on the other, and thus views dispersal limitation as ubiquitous as sampling effects. Further, we express the results of one line of research in terms of the other and vice versa, using the concept of subsamples. A consequence of our findings is that metacommunity size does not independently affect the outcome of neutral models in contrast to a previous assertion (Ecol. Lett., 7, 2004, p. 904) based on an incorrect formula (Phys. Rev. E, 68, 2003, p. 061902, eqns 11-14). Our framework provides the basis for development of a dispersal-limited non-neutral community theory and applies in population genetics as well, where alleles and mutation play the roles of species and speciation respectively.

Climate change impacts on malaria are typically assessed with scenarios for the long-term future. Here we focus instead on the recent past (1970–2003) to address whether warmer temperatures have already increased the incidence of malaria in a highland region of East Africa. Our analyses rely on a new coupled mosquito–human model of malaria, which we use to compare projected disease levels with and without the observed temperature trend. Predicted malaria cases exhibit a highly nonlinear response to warming, with a significant increase from the 1970s to the 1990s, although typical epidemic sizes are below those observed. These findings suggest that climate change has already played an important role in the exacerbation of malaria in this region. As the observed changes in malaria are even larger than those predicted by our model, other factors previously suggested to explain all of the increase in malaria may be enhancing the impact of climate change.

In the context of neutral theories of community ecology, a novel genealogy-based framework has recently furnished an analytic extension of EwensÕ sampling multivariate abundance distribution, which also applies to a random sample from a local community. Here, instead of taking a multivariate approach, we further develop the sampling theory of Hubbell's neutral spatially implicit theory and derive simple abundance distributions for a random sample both from a local community and a metacommunity. Our result is given in terms of the average number of species with a given abundance in any randomly extracted sample. Contrary to what has been widely assumed, a random sample from a metacommunity is not fully described by the Fisher log-series, but by a new distribution. This new sample distribution matches the log-series expectation at high biodiversity values (h > 1) but clearly departs from it for species-poor metacommunities (h < 1). Our theoretical framework should be helpful in the better assessment of diversity and testing of the neutral theory by using abundance data.

The neutral theory of biodiversity as put forward by Hubbell in his 2001 monograph has received much criticism for its unrealistic simplifying assumptions. These are the assumptions of functional equivalence among different species (neutrality), the assumption of point mutation speciation, and the assumption that resources are continuously saturated, such that constant resource availability implies constant community size (zero-sum assumption). Here we focus on the zero-sum assumption. We present a general theory for calculating the probability of observing a particular species-abundance distribution (sampling formula) and show that zero-sum and non-zero-sum formulations of neutral theory have exactly the same sampling formula when the community is in equilibrium. Moreover, for the non-zero-sum community the sampling formula has this same form, even out of equilibrium. Therefore, the term "zero-sum multinomial (ZSM)" to describe species abundance patterns, as coined by Hubbell [2001. The Unified Neutral Theory of Biodiversity and Biogeography, Princeton University Press, Princeton, NJ], is not really appropriate, as it also applies to non-zero-sum communities. Instead we propose the term "dispersal-limited multinomial (DLM)", thus making explicit one of the most important contributions of neutral community theory, the emphasis on dispersal limitation as a dominant factor in determining species abundances.