Dynamics of neutral biodiversity

Hubbell's neutral model has become a major paradigm in ecology. Whereas the steady-state structure is well understood, results about the dynamical aspects of the model are scarce. Here we derive dynamical equations for the Simpson diversity index. Both mean and variance of the diversity are proven to satisfy stable linear system dynamics. We show that in the stationary limit we indeed recover previous results, and we supplement this with numerical simulations to validate the dynamical part of our analytical computations. These findings are especially relevant for experiments in microbial ecology, where the Simpson diversity index can be accurately measured as a function of time.     

The zero-sum assumption in neutral biodiversity theory

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.     

Sampling Hubbell's neutral theory of biodiversity

In the context of neutral theories of community ecology, a novel genealogy‐based framework has recently furnished an analytic extension of Ewens’ 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 (θ > 1) but clearly departs from it for species‐poor metacommunities (θ < 1). Our theoretical framework should be helpful in the better assessment of diversity and testing of the neutral theory by using abundance data.     

The influence of spatial inhomogeneities on neutral models of geographical variation : I. Formulation

The influence of spatial variation in the carrying capacity and migration rate of a geographical barrier on the one-dimensional stepping-stone model is studied. The monoecious, diploid population is subdivided into an infinite linear array of panmictic colonies that exchange gametes. In each deme, the rate of self-fertilization is equal to the reciprocal of the number of individuals in that deme. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. In the diffusion approximation, a partial differential equation that incorporates spatial (and temporal) variation in the carrying capacity and migration rate is derived for the probability of identity. Transition conditions that simultaneously take into account discontinuities in the carrying capacity and migration rate are established: the probability of identity is continuous, but its partial derivatives are not, their ratio being a simple function of the carrying capacities and migrational variance on the two sides of the inhomogeneity. The partial derivatives of the probability of identity are continuous across a geographical barrier, whereas the probability of identity itself has a discontinuity proportional to the partial derivative at the barrier, the constant of proportionality being a measure of the difficulty of crossing the barrier.     

Regression models for spatial prediction: their role for biodiversity and conservation

This paper is an introduction to a Special Issue on regression models for spatial predictions published in Biodiversity and Conservation following an international workshop held in Switzerland in 2001 (http://leba.unige.ch/workshop). This introduction describes how the exponential growth in computing power has improved our ability to reach spatially explicit assessment of biodiversity and to develop cost-effective conservation management. New questions arising from these modern approaches are listed, while papers presenting examples of applications are briefly introduced.     

Speciation and the neutral theory of biodiversity

The neutral theory of biodiversity purports that patterns in the distribution and abundance of species do not depend on adaptive differences between species (i.e. niche differentiation) but solely on random fluctuations in population size (“ecological drift”), along with dispersal and speciation. In this framework, the ultimate driver of biodiversity is speciation. However, the original neutral theory made strongly simplifying assumptions about the mechanisms of speciation, which has led to some clearly unrealistic predictions. In response, several recent studies have combined neutral community models with more elaborate speciation models. These efforts have alleviated some of the problems of the earlier approaches, while confirming the general ability of neutral theory to predict empirical patterns of biodiversity. However, the models also show that the mode of speciation can have a strong impact on relative species abundances. Future work should compare these results to diversity patterns arising from non‐neutral modes of speciation, such as adaptive radiations.     

The unified neutral theory of biodiversity and biogeography at age ten

A decade has now passed since Hubbell published The Unified Neutral Theory of Biodiversity and Biogeography. Neutral theory highlights the importance of dispersal limitation, speciation and ecological drift in the natural world and provides quantitative null models for assessing the role of adaptation and natural selection. Significant advances have been made in providing methods for understanding neutral predictions and comparing them with empirical data. In this review, we describe the current state-of-the-art techniques and ideas in neutral theory and how these are of relevance to ecology. The future of neutral theory is promising, but its concepts must be applied more broadly beyond the current focus on species-abundance distributions.     

The neutral theory is dead. The current significance and standing of neutral and nearly neutral theories

Comparative studies of DNA sequences provide opportunities for testing the neutral and the selection theories of molecular evolution. In particular, the separate estimation of the numbers of synonymous and nonsynonymous substitutions is a powerful tool for detecting selection of the latter. The difference in the patterns of these two types of substitutions of mammalian genes turned out to be in accord with the slightly deleterious or nearly neutral mutation theory for nonsynonymous changes. Interaction systems at the amino acid level were suggested to be responsible for such nearly neutral, or very weak, selection. Synonymous substitutions are not strictly neutral, but because of their minute effect, random drift predominates such that the rate of substitution is only slightly less than the completely neutral prediction. It was concluded that the strictly neutral theory has not held up as well as the nearly neutral theory, yet remains invaluable as a null hypothesis for detecting selection. On the other hand, the main difference between the nearly neutral and the traditional selection theories is that the former predicts rapid evolution in small populations, whereas the latter predicts rapid evolution in large populations.     

Random sampling does not exclude spatial dependence: The importance of neutral models for ecological hypothesis testing

Lájer (2007) notes that, to investigate phytosociological and ecological relationships, many authors apply traditional inferential tests to sets of relevés obtained by non-random methods. Unfortunately, this procedure does not provide reliable support for hypothesis testing because non-random sampling violates the assumptions of independence required by many parametric inferential tests. Instead, a random sampling scheme is recommended. Nonetheless, random sampling will not eliminate spatial autocorrelation. For instance, a classical law of geography holds that everything in a piece of (biotic) space is interrelated, but near objects are more related than distant ones. Because most ecological processes that shape community structure and species coexistence are spatially explicit, spatial autocorrelation is a vital part of almost all ecological data. This means that, independently from the underlying sampling design, ecological data are generally spatially autocorrelated, violating the assumption of independence that is generally required by traditional inferential tests. To overcome this drawback, randomization tests may be used. Such tests evaluate statistical significance based on empirical distributions generated from the sample and do not necessarily require data independence. However, as concerns hypothesis testing, randomization tests are not the universal remedy for ecologists, because the choice of inadequate null models can have significant effects on the ecological hypotheses tested. In this paper, I emphasize the need of developing null models for which the statistical assumptions match the underlying biological mechanisms.     

Assessing and improving the transferability of current global spatial prediction models

Aim

Global-scale maps of the environment are an important source of information for researchers and decision makers. Often, these maps are created by training machine learning algorithms on field-sampled reference data using remote sensing information as predictors. Since field samples are often sparse and clustered in geographic space, model prediction requires a transfer of the trained model to regions where no reference data are available. However, recent studies question the feasibility of predictions far beyond the location of training data.

Innovation

We propose a novel workflow for spatial predictive mapping that leverages recent developments in this field and combines them in innovative ways with the aim of improved model transferability and performance assessment. We demonstrate, evaluate and discuss the workflow with data from recently published global environmental maps.

Main conclusions

Reducing predictors to those relevant for spatial prediction leads to an increase of model transferability and map accuracy without a decrease of prediction quality in areas with high sampling density. Still, reliable gap-free global predictions were not possible, highlighting that global maps and their evaluation are hampered by limited availability of reference data.     

A new sampling formula for neutral biodiversity

The neutral model of biodiversity, proposed by Hubbell (The Unified Neutral Theory of Biodiversity and Biogeography, Princeton University Press, Princeton, NJ, 2001) to explain the diversity of functionally equivalent species, has been subject of hot debate in community ecology. Whereas Hubbell studied the model mostly by simulations, recently analytical treatments have yielded expressions of the expected number of species of a particular abundance in a local community with dispersal limitation. Moreover, a formula has been offered for the joint likelihood of observing a given species‐abundance dataset in a local community with dispersal limitation, but this formula is too complicated to allow practical applications. Here, I present a much simplified expression that can be regarded as an enhanced version of the famous Ewens sampling formula. It can be used in maximum likelihood methods for quick estimation of the model parameters, using all information in the data, and for model comparison. I also show how to rapidly generate examples of species‐abundance distributions for a given set of model parameters and how to calculate Simpson's diversity index.     

Demographic analysis of Hubbell's neutral theory of biodiversity

Hubbell's neutral model is increasingly applied in both theoretical and empirical studies but so far little attention has been paid to the ecological mechanisms that determine species diversity in neutral communities. In this contribution we use a stochastic individual-based Markovian model to provide an explicit derivation of Hubbell's local community model from the fundamental processes of reproduction, mortality, and immigration, and show that such derivation provides important insights on the mechanisms regulating species diversity that cannot be obtained from the original model and its previous extensions. One important insight is that the basic parameters of Hubbell's model, community size (J) and the probability that a dying individual will be replaced by an immigrant (m), cannot be considered independent and that their interdependency leads to a counterintuitive trade-off between community size and species diversity. We further demonstrate that Hubbell's treatment of community size as a free parameter hides fundamental mechanisms that influence species diversity through their effect on the size of the community. For example, while in Hubbell's model immigration can only increase species diversity by promoting colonization rates, the demographic derivation shows that immigration can also promote species diversity by reducing extinction rates. Our demographic derivation also unifies previous contrasting predictions about the effect of reproduction on species diversity by showing that both positive and negative effects are possible, and that the balance between the two effects depends on the size of the community. The demographic derivation also reconciles an apparent contradiction between Hubbell's theory and patch occupancy theory, and integrates three previously proposed mechanisms of species diversity, the More Individuals Hypothesis, the rescue effect, and the dilution effect, within a single, unified framework.     

The robustness of neutral models of geographical variation

The approximation of diploid migration by gametic dispersion is studied. The monoecious, diploid population is subdivided into panmictic colonies that exchange migrants. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to a new allele at the same rate u. Diploid-migration models without self-fertilization and with selfing at the “random” rate (equal to the reciprocal of the deme size in each deme) are investigated; in the gametic-dispersion models, selfing occurs at the random rate. It is shown for the unbounded stepping-stone model in one and two dimensions, the circular stepping-stone model, and the island model that the probabilitities of identity in state at equilibrium for diploid migration are close to those for gametic dispersion if the mutation rate is small or the deme size is large. Explicit error bounds are presented in all the above cases. It is also proved that if the number of demes is finite and the migration matrix is arbitrary but time independent and ergodic, then in the strong-migration approximation the equilibrium and the ultimate rate and pattern of convergence of both diploid-dispersion models are close to the corresponding gametic-dispersion formulae. For the strong-migration approximation at equilibrium, migration must dominate both mutation and random drift; for the convergence results, it suffices that migration dominate random drift. All the results apply to a dioecious population if the migration pattern and mutation rate are sex independent.     

Relaxing the zero-sum assumption in neutral biodiversity theory

The zero-sum assumption is one of the ingredients of the standard neutral model of biodiversity by Hubbell. It states that the community is saturated all the time, which in this model means that the total number of individuals in the community is constant over time, and therefore introduces a coupling between species abundances. It was shown recently that a neutral model with independent species, and thus without any coupling between species abundances, has the same sampling formula (given a fixed number of individuals in the sample) as the standard model [Etienne, R.S., Alonso, D., McKane, A.J., 2007. The zero-sum assumption in neutral biodiversity theory. J. Theor. Biol. 248, 522-536]. The equilibria of both models are therefore equivalent from a practical point of view. Here we show that this equivalence can be extended to a class of neutral models with density-dependence on the community-level. This result can be interpreted as robustness of the model, i.e. insensitivity of the model to the precise interaction of the species in a neutral community. It can also be interpreted as a lack of resolution, as different mechanisms of interactions between neutral species cannot be distinguished using only a single snapshot of species abundance data.