A sampling theory for asymmetric communities

We introduce the first analytical model of asymmetric community dynamics to yield Hubbell's neutral theory in the limit of functional equivalence among all species. Our focus centers on an asymmetric extension of Hubbell's local community dynamics, while an analogous extension of Hubbell's metacommunity dynamics is deferred to an appendix. We find that mass-effects may facilitate coexistence in asymmetric local communities and generate unimodal species abundance distributions indistinguishable from those of symmetric communities. Multiple modes, however, only arise from asymmetric processes and provide a strong indication of non-neutral dynamics. Although the exact stationary distributions of fully asymmetric communities must be calculated numerically, we derive approximate sampling distributions for the general case and for nearly neutral communities where symmetry is broken by a single species distinct from all others in ecological fitness and dispersal ability. In the latter case, our approximate distributions are fully normalized, and novel asymptotic expansions of the required hypergeometric functions are provided to make evaluations tractable for large communities. Employing these results in a Bayesian analysis may provide a novel statistical test to assess the consistency of species abundance data with the neutral hypothesis.     

Basing population genetic inferences and models of molecular evolution upon desired stationary distributions of DNA or protein sequences

Models of molecular evolution tend to be overly simplistic caricatures of biology that are prone to assigning high probabilities to biologically implausible DNA or protein sequences. Here, we explore how to construct time-reversible evolutionary models that yield stationary distributions of sequences that match given target distributions. By adopting comparatively realistic target distributions,evolutionary models can be improved. Instead of focusing on estimating parameters, we concentrate on the population genetic implications of these models. Specifically, we obtain estimates of the product of effective population size and relative fitness difference of alleles. The approach is illustrated with two applications to protein-coding DNA. In the first, a codon-based evolutionary model yields a stationary distribution of sequences, which, when the sequences are translated,matches a variable-length Markov model trained on human proteins. In the second, we introduce an insertion-deletion model that describes selectively neutral evolutionary changes to DNA. We then show how to modify the neutral model so that its stationary distribution at the amino acid level can match a profile hidden Markov model, such as the one associated with the Pfam database.     

Firing rate of a retinal neuron are not predictable from interspike interval statistics.

The intervals between successive action potentials (impulses, or "spikes") produced the maintained firing of a neuron (ISIs) are often treated as if they were independent on each other; that is, an impulse train is considered as a stationary renewal process. If this is so, the variability of the mean rate of firing impulses in a sequence of temporal windows should be predictable from the distribution of ISIs. This was found not to be the case for the maintained firing of retinal ganglion cells in goldfish. Although some evident nonstationarity sometimes resulted in greater variability of the observed rate distributions than those predicted (for relatively long temporal windows), as a general rule the observed rate distributions were considerable less dispersed than would be predicted by sampling of the ISI distributions. This was taken as evidence of long-term serial dependency between successive ISIs; however, two standard test for dependency (autocorrelations and serial correlograms failed to to reveal structure of sufficiently long duration to account for the effect noted.     

Neutral Models with Generalised Speciation

Hubbell’s neutral theory claims that ecological patterns such as species abundance distributions can be explained by a stochastic model based on simple assumptions. One of these assumptions, the point mutation assumption, states that every individual has the same probability to speciate. Etienne et al. have argued that other assumptions on the speciation process could be more realistic, for example, that every species has the same probability to speciate (Etienne, et al. in Oikos 116:241–258, 2007). They introduced a number of neutral community models with a different speciation process, and conjectured formulas for their stationary species abundance distribution. Here we study a generalised neutral community model, encompassing these modified models, and derive its stationary distribution, thus proving the conjectured formulas.     

Modeling genome evolution with a diffusion approximation of a birth-and-death process

MOTIVATION: In our previous studies, we developed discrete-space birth, death and innovation models (BDIMs) of genome evolution. These models explain the origin of the characteristic Pareto distribution of paralogous gene family sizes in genomes, and model parameters that provide for the evolution of these distributions within a realistic time frame have been identified. However, extracting the temporal dynamics of genome evolution from discrete-space BDIM was not technically feasible. We were interested in obtaining dynamic portraits of the genome evolution process by developing a diffusion approximation of BDIM. RESULTS: The diffusion version of BDIM belongs to a class of continuous-state models whose dynamics is described by the Fokker-Plank equation and the stationary solution could be any specified Pareto function. The diffusion models have time-dependent solutions of a special kind, namely, generalized self-similar solutions, which describe the transition from one stationary distribution of the system to another; this provides for the possibility of examining the temporal dynamics of genome evolution. Analysis of the generalized self-similar solutions of the diffusion BDIM reveals a biphasic curve of genome growth in which the initial, relatively short, self-accelerating phase is followed by a prolonged phase of slow deceleration. This evolutionary dynamics was observed both when genome growth started from zero and proceeded via innovation (a potential model of primordial evolution), and when evolution proceeded from one stationary state to another. In biological terms, this regime of evolution can be tentatively interpreted as a punctuated-equilibrium-like phenomenon whereby evolutionary transitions are accompanied by rapid gene amplification and innovation, followed by slow relaxation to a new stationary state.     

On an approximate formula for the distribution of 2-locus 2-allele model with mutual mutations

An asymptotic approximation of the density function of 2-locus 2-allele model with mutual neutral mutations was obtained invoking the small disturbance asymptotic theory. It was shown by comparing the approximate formula with simulations that this asymptotic method gives a good approximation over the whole time evolution when the mutation rates are high, though it does not give good approximations near the stationary state when the mutation rates are low. On the stationary state, the squared standard linkage deviation made up by using the approximate formula was compared with the exact one obtained by Ohta and Kimura (1969b). It gave a good approximation when the recombination rate is high, even under low mutation rates. Furthermore, as an application of the asymptotic method, The Ancestral Recombination Graph (ARG) was considered.     

Prediction of site-specific amino acid distributions and limits of divergent evolutionary changes in protein sequences

We derive an analytic expression for site-specific stationary distributions of amino acids from the structurally constrained neutral (SCN) model of protein evolution with conservation of folding stability. The stationary distributions that we obtain have a Boltzmann-like shape, and their effective temperature parameter, measuring the limit of divergent evolutionary changes at a given site, can be predicted from a site-specific topological property, the principal eigenvector of the contact matrix of the native conformation of the protein. These analytic results, obtained without free parameters, are compared with simulations of the SCN model and with the site-specific amino acid distributions obtained from the Protein Data Bank. These results also provide new insights into how the topology of a protein fold influences its designability, i.e., the number of sequences compatible with that fold. The dependence of the effective temperature on the principal eigenvector decreases for longer proteins, as a possible consequence of the fact that selection for thermodynamic stability becomes weaker in this case.     

Metacommunity‐scale biodiversity regulation and the self‐organised emergence of macroecological patterns

There exist a number of key macroecological patterns whose ubiquity suggests that the spatio‐temporal structure of ecological communities is governed by some universal mechanisms. The nature of these mechanisms, however, remains poorly understood. Here, we probe spatio‐temporal patterns in species richness and community composition using a simple metacommunity assembly model. Despite making no a priori assumptions regarding biotic spatial structure or the distribution of biomass across species, model metacommunities self‐organise to reproduce well‐documented patterns including characteristic species abundance distributions, range size distributions and species area relations. Also in agreement with observations, species richness in our model attains an equilibrium despite continuous species turnover. Crucially, it is in the neighbourhood of the equilibrium that we observe the emergence of these key macroecological patterns. Biodiversity equilibria in models occur due to the onset of ecological structural instability, a population‐dynamical mechanism. This strongly suggests a causal link between local community processes and macroecological phenomena.     

Contribution of neutral processes to the assembly of gut microbial communities in the zebrafish over host development

Despite their importance to host health and development, the communities of microorganisms associated with humans and other animals are characterized by a large degree of unexplained variation across individual hosts. The processes that drive such inter-individual variation are not well understood. To address this, we surveyed the microbial communities associated with the intestine of the zebrafish, Danio rerio, over developmental time. We compared our observations of community composition and distribution across hosts with that predicted by a neutral assembly model, which assumes that community assembly is driven solely by chance and dispersal. We found that as hosts develop from larvae to adults, the fit of the model to observed microbial distributions decreases, suggesting that the relative importance of non-neutral processes, such as microbe-microbe interactions, active dispersal, or selection by the host, increases as hosts mature. We also observed that taxa which depart in their distributions from the neutral prediction form ecologically distinct sub-groups, which are phylogenetically clustered with respect to the full metacommunity. These results demonstrate that neutral processes are sufficient to generate substantial variation in microbiota composition across individual hosts, and suggest that potentially unique or important taxa may be identified by their divergence from neutral distributions.     

Analyzing tropical forest tree species abundance distributions using a nonneutral model and through approximate Bayesian inference

The neutral theory of biodiversity challenges the classical niche-based view of ecological communities, where species attributes and environmental conditions jointly determine community composition. Functional equivalence among species, as assumed by neutral ecological theory, has been recurrently falsified, yet many patterns of tropical tree communities appear consistent with neutral predictions. This may mean that neutral theory is a good first-approximation theory or that species abundance data sets contain too little information to reject neutrality. Here we present a simple test of neutrality based on species abundance distributions in ecological communities. Based on this test, we show that deviations from neutrality are more frequent than previously thought in tropical forest trees, especially at small spatial scales. We then develop a nonneutral model that generalizes Hubbell's dispersal-limited neutral model in a simple way by including one additional parameter of frequency dependence. We also develop a statistical method to infer the parameters of this model from empirical data by approximate Bayesian computation. In more than half of the permanent tree plots, we show that our new model fits the data better than does the neutral model. Finally, we discuss whether observed deviations from neutrality may be interpreted as the signature of environmental filtering on tropical tree species abundance distributions.     

Dispersal, edaphic fidelity and speciation in species-rich Western Australian shrublands: evaluating a neutral model of biodiversity

Over evolutionary time, the number of species in a community reflects the balance between the rate of speciation and the rate of extinction. Over shorter time‐scales local species richness is also affected by how often species move into and out of the local community. These processes are at the heart of Hubbell's ‘unified neutral theory of biodiversity’ ( Hubbell 2001 ). Hubbell's spatially implicit, dispersal‐limited neutral model is the most widely used of the many implementations of neutral theory and it provides an estimate of the rate of speciation in a metacommunity (if metacommunity size is known) and the rate at which species migrate into the local community from the wider metacommunity. Recently, this neutral model has been used to compare rates of speciation and migration in the species‐rich fynbos of South Africa and in neotropical forests. Here we use new analytical methods for estimating the neutral model's parameters to infer speciation and dispersal rates for three sites in species‐rich sclerophyll shrublands (equivalent to fynbos) in Western Australia (WA). Our estimates suggest that WA shrublands are intermediate between fynbos and tropical rainforest in terms of speciation and dispersal. Although a weak test, the model predicts species abundance distributions and species accumulation curves similar to those observed at the three sites. The neutral model's predictions also remain plausible when confronted with independent data describing: (1) known edaphic relationships between sites, (2) estimates of metacommunity species richness and (3) rates of speciation among resprouters and nonsprouters. Two of the site pairs, however, show species turnovers significantly different from those predicted by the spatially implicit form of the neutral model that we use. This suggests that non‐neutral processes, in this case probably edaphic specialisation, are important in the WA shrubland metacommunity. The neutral model predicts similar rates of speciation in resprouter and sprouter taxa, a finding supported by recent molecular phylogenies. Finally, when converted into temporally scaled speciation rates and species longevities, the estimates produced by the neutral model seem implausible. The apparent departure from neutrality in the turnover of species between some sites and the implausible temporal dynamics may be due to the particular model chosen and does not reduce the significance of our other results, which confirm that local dispersal limitation, coupled with broader scale edaphic fidelity, combine to structure this biodiverse metacommunity.     

Distance decay of similarity,effects of environmental noise and ecological heterogeneity among species in the spatio‐temporal dynamics of a dispersal‐limited community

The joint spatial and temporal fluctuations in community structure may be due to dispersal, variation in environmental conditions, ecological heterogeneity among species and demographic stochasticity. These factors are not mutually exclusive, and their relative contribution towards shaping species abundance distributions and in causing species fluctuations have been hard to disentangle. To better understand community dynamics when the exchange of individuals between localities is very low, we studied the dynamics of the freshwater zooplankton communities in 17 lakes located in independent catchment areas, sampled at end of summer from 2002 to 2008 in Norway. We analysed the joint spatial and temporal fluctuations in the community structure by fitting the two‐dimensional Poisson lognormal model under a two‐stage sampling scheme. We partitioned the variance of the distribution of log abundance for a random species at a random time and location into components of demographic stochasticity, ecological heterogeneity among species, and independent environmental noise components for the different species. Non‐neutral mechanisms such as ecological heterogeneity among species (20%) and spatiotemporal variation in the environment (75%) explained the majority of the variance in log abundances. Overdispersion relative to Poisson sampling and demographic stochasticity had a small contribution to the variance (5%). Among a set of environmental variables, lake acidity was the environmental variable that was most strongly related to decay of community similarity in space and time.     

The value of space‐for‐time substitution for studying fine‐scale microevolutionary processes

When the drivers of biological turnover in space are the same as those that drive turnover through time, space can be substituted for time to model how patterns of variation are predicted to change into the future. These space‐for‐time substitutions are widely used in ecological modeling but have only recently been applied to the study of microevolutionary processes, particularly over relatively fine spatial and temporal scales. Here, we review recent examples that have employed space‐for‐time substitution to study genetic patterns on stationary and non‐stationary landscapes and examine whether space can reliably substitute for time in studies of population divergence, genetic structure, and adaptive evolution. Although there are only a relatively few examples, several recent studies were excellently crafted to provide valuable insights into the conditions governing the validity of space‐for‐time substitutions applied to population genetic data. We found that, although caution should be taken, space‐for‐time substitutions appear valid for studying microevolutionary processes on both stationary and non‐stationary landscapes. Further studies can help to evaluate the conditions under which space‐for‐time substitutions are reliable. When these methods are reliable, they will play an important role in modeling genetic responses to environmental change, population viability on non‐stationary landscapes, and patterns of divergence and adaptation.     

Predictions of Taylor's power law, density dependence and pink noise from a neutrally modeled time series

There has recently been increasing interest in neutral models of biodiversity and their ability to reproduce the patterns observed in nature, such as species abundance distributions. Here we investigate the ability of a neutral model to predict phenomena observed in single-population time series, a study complementary to most existing work that concentrates on snapshots in time of the whole community. We consider tests for density dependence, the dominant frequencies of population fluctuation (spectral density) and a relationship between the mean and variance of a fluctuating population (Taylor's power law). We simulated an archipelago model of a set of interconnected local communities with variable mortality rate, migration rate, speciation rate, size of local community and number of local communities. Our spectral analysis showed ‘pink noise’: a departure from a standard random walk dynamics in favor of the higher frequency fluctuations which is partly consistent with empirical data. We detected density dependence in local community time series but not in metacommunity time series. The slope of the Taylor's power law in the model was similar to the slopes observed in natural populations, but the fit to the power law was worse. Our observations of pink noise and density dependence can be attributed to the presence of an upper limit to community sizes and to the effect of migration which distorts temporal autocorrelation in local time series. We conclude that some of the phenomena observed in natural time series can emerge from neutral processes, as a result of random zero-sum birth, death and migration. This suggests the neutral model would be a parsimonious null model for future studies of time series data.     

Asymmetry in species regional dispersal ability and the neutral theory

The neutral assumption that individuals of either the same or different species share exactly the same birth, death, migration, and speciation probabilities is fundamental yet controversial to the neutral theory. Several theoretical studies have demonstrated that a slight difference in species per capita birth or death rates can have a profound consequence on species coexistence and community structure. Whether asymmetry in migration, a vital demographic parameter in the neutral model, plays an important role in community assembly still remains unknown. In this paper, we relaxed the ecological equivalence assumption of the neutral model by introducing differences into species regional dispersal ability. We investigated the effect of asymmetric dispersal on the neutral local community structure. We found that per capita asymmetric dispersal among species could reduce species richness of the local community and result in deviations of species abundance distributions from those predicted by the neutral model. But the effect was moderate compared with that of asymmetries in birth or death rates, unless very large asymmetries in dispersal were assumed. A large difference in species dispersal ability, if there is, can overwhelm the role of random drift and make local community dynamics deterministic. In this case, species with higher regional dispersal abilities tended to dominate in the local community. However, the species abundance distribution of the local community under asymmetric dispersal could be well fitted by the neutral model, but the neutral model generally underestimated the fundamental biodiversity number but overestimated the migration rate in such communities.     

Biology,Methodology or Chance? The Degree Distributions of Bipartite Ecological Networks

The distribution of the number of links per species, or degree distribution, is widely used as a summary of the topology of complex networks. Degree distributions have been studied in a range of ecological networks, including both mutualistic bipartite networks of plants and pollinators or seed dispersers and antagonistic bipartite networks of plants and their consumers. The shape of a degree distribution, for example whether it follows an exponential or power-law form, is typically taken to be indicative of the processes structuring the network. The skewed degree distributions of bipartite mutualistic and antagonistic networks are usually assumed to show that ecological or co-evolutionary processes constrain the relative numbers of specialists and generalists in the network. I show that a simple null model based on the principle of maximum entropy cannot be rejected as a model for the degree distributions in most of the 115 bipartite ecological networks tested here. The model requires knowledge of the number of nodes and links in the network, but needs no other ecological information. The model cannot be rejected for 159 (69%) of the 230 degree distributions of the 115 networks tested. It performed equally well on the plant and animal degree distributions, and cannot be rejected for 81 (70%) of the 115 plant distributions and 78 (68%) of the animal distributions. There are consistent differences between the degree distributions of mutualistic and antagonistic networks, suggesting that different processes are constraining these two classes of networks. Fit to the MaxEnt null model is consistently poor among the largest mutualistic networks. Potential ecological and methodological explanations for deviations from the model suggest that spatial and temporal heterogeneity are important drivers of the structure of these large networks.     

Departures from neutrality induced by niche and relative fitness differences

Breaking the core assumption of ecological equivalence in Hubbell’s “neutral theory of biodiversity” requires a theory of species differences. In one framework for characterizing differences between competing species, non-neutral interactions are said to involve both niche differences, which promote stable coexistence, and relative fitness differences, which promote competitive exclusion. We include both in a stochastic community model in order to determine if relative fitness differences compensate for changes in community structure and dynamics induced by niche differences, possibly explaining neutral theory’s apparent success. We show that species abundance distributions are sensitive to both niche and relative fitness differences, but that certain combinations of differences result in abundance distributions that are indistinguishable from the neutral case. In contrast, the distribution of species’ lifetimes, or the time between speciation and extinction, differs under all combinations of niche and relative fitness differences. The results from our model experiment are inconsistent with the hypothesis of “emergent neutrality” and support instead a hypothesis that relative fitness differences counteract effects of niche differences on distributions of abundance. However, an even more developed theory of interspecific variation appears necessary to explain the diversity and structure of non-neutral communities.     

Mathematical modeling of evolution. Solved and open problems

Evolution is a highly complex multilevel process and mathematical modeling of evolutionary phenomenon requires proper abstraction and radical reduction to essential features. Examples are natural selection, Mendel’s laws of inheritance, optimization by mutation and selection, and neutral evolution. An attempt is made to describe the roots of evolutionary theory in mathematical terms. Evolution can be studied in vitro outside cells with polynucleotide molecules. Replication and mutation are visualized as chemical reactions that can be resolved, analyzed, and modeled at the molecular level, and straightforward extension eventually results in a theory of evolution based upon biochemical kinetics. Error propagation in replication commonly results in an error threshold that provides an upper bound for mutation rates. Appearance and sharpness of the error threshold depend on the fitness landscape, being the distribution of fitness values in genotype or sequence space. In molecular terms, fitness landscapes are the results of two consecutive mappings from sequences into structures and from structures into the (nonnegative) real numbers. Some properties of genotype–phenotype maps are illustrated well by means of sequence–structure relations of RNA molecules. Neutrality in the sense that many RNA sequences form the same (coarse grained) structure is one of these properties, and characteristic for such mappings. Evolution cannot be fully understood without considering fluctuations—each mutant originates form a single copy, after all. The existence of neutral sets of genotypes called neutral networks, in particular, necessitates stochastic modeling, which is introduced here by simulation of molecular evolution in a kind of flowreactor.