A neutral theory with environmental stochasticity explains static and dynamic properties of ecological communities

High response diversity and conspecific density-dependence, not species interactions, drive dynamics of coral reef fish communities

Species-to-species and species-to-environment interactions are key drivers of community dynamics. Disentangling these drivers in species-rich assemblages is challenging due to the high number of potentially interacting species (the 'curse of dimensionality'). We develop a process-based model that quantifies how intraspecific and interspecific interactions, and species' covarying responses to environmental fluctuations, jointly drive community dynamics. We fit the model to reef fish abundance time series from 41 reefs of Australia's Great Barrier Reef. We found that fluctuating relative abundances are driven by species' heterogenous responses to environmental fluctuations, whereas interspecific interactions are negligible. Species differences in long-term average abundances are driven by interspecific variation in the magnitudes of both conspecific density-dependence and density-independent growth rates. This study introduces a novel approach to overcoming the curse of dimensionality, which reveals highly individualistic dynamics in coral reef fish communities that imply a high level of niche structure.

Processes governing species richness in communities exposed to temporal environmental stochasticity: A review and synthesis of modelling approaches

Research into the processes governing species richness has often assumed that the environment is fixed, whereas realistic environments are often characterised by random fluctuations over time. This temporal environmental stochasticity (TES) changes the demographic rates of species populations, with cascading effects on community dynamics and species richness. Theoretical and applied studies have used process-based mathematical models to determine how TES affects species richness, but under a variety of frameworks. Here, we critically review such studies to synthesise their findings and draw general conclusions. We first provide a broad mathematical framework encompassing the different ways in which TES has been modelled. We then review studies that have analysed models with TES under the assumption of negligible interspecific interactions, such that a community is conceptualised as the sum of independent species populations. These analyses have highlighted how TES can reduce species richness by increasing the frequency at which a species becomes rare and therefore prone to extinction. Next, we review studies that have relaxed the assumption of negligible interspecific interactions. To simplify the corresponding models and make them analytically tractable, such studies have used mean-field theory to derive fixed parameters representing the typical strength of interspecific interactions under TES. The resulting analyses have highlighted community-level effects that determine how TES affects species richness, for species that compete for a common limiting resource. With short temporal correlations of environmental conditions, a non-linear averaging effect of interspecific competition strength over time gives an increase in species richness. In contrast, with long temporal correlations of environmental conditions, strong selection favouring the fittest species between changes in environmental conditions results in a decrease in species richness. We compare such results with those from invasion analysis, which examines invasion growth rates (IGRs) instead of species richness directly. Qualitative differences sometimes arise because the IGR is the expected growth rate of a species when it is rare, which does not capture the variation around this mean or the probability of the species becoming rare. Our review elucidates key processes that have been found to mediate the negative and positive effects of TES on species richness, and by doing so highlights key areas for future research.

A tiny fraction of all species forms most of nature: Rarity as a sticky state

Using data from a wide range of natural communities including the human microbiome, plants, fish, mushrooms, rodents, beetles, and trees, we show that universally just a few percent of the species account for most of the biomass. This is in line with the classical observation that the vast bulk of biodiversity is very rare. Attempts to find traits allowing the tiny fraction of abundant species to escape rarity have remained unsuccessful. Here, we argue that this might be explained by the fact that hyper-dominance can emerge through stochastic processes. We demonstrate that in neutrally competing groups of species, rarity tends to become a trap if environmental fluctuations result in gains and losses proportional to abundances. This counter-intuitive phenomenon arises because absolute change tends to zero for very small abundances, causing rarity to become a "sticky state", a pseudoattractor that can be revealed numerically in classical ball-in-cup landscapes. As a result, the vast majority of species spend most of their time in rarity leaving space for just a few others to dominate the neutral community. However, fates remain stochastic. Provided that there is some response diversity, roles occasionally shift as stochastic events or natural enemies bring an abundant species down allowing a rare species to rise to dominance. Microbial time series spanning thousands of generations support this prediction. Our results suggest that near-neutrality within niches may allow numerous rare species to persist in the wings of the dominant ones. Stand-ins may serve as insurance when former key species collapse.

Universal abundance fluctuations across microbial communities, tropical forests, and urban populations

The growth of complex populations, such as microbial communities, forests, and cities, occurs over vastly different spatial and temporal scales. Although research in different fields has developed detailed, system-specific models to understand each individual system, a unified analysis of different complex populations is lacking; such an analysis could deepen our understanding of each system and facilitate cross-pollination of tools and insights across fields. Here, we use a shared framework to analyze time-series data of the human gut microbiome, tropical forest, and urban employment. We demonstrate that a single, three-parameter model of stochastic population dynamics can reproduce the empirical distributions of population abundances and fluctuations in all three datasets. The three parameters characterizing a species measure its mean abundance, deterministic stability, and stochasticity. Our analysis reveals that, despite the vast differences in scale, all three systems occupy a similar region of parameter space when time is measured in generations. In other words, although the fluctuations observed in these systems may appear different, this difference is primarily due to the different physical timescales associated with each system. Further, we show that the distribution of temporal abundance fluctuations is described by just two parameters and derive a two-parameter functional form for abundance fluctuations to improve risk estimation and forecasting.

Extinction time distributions of populations and genotypes

Ultimately, the eventual extinction of any biological population is an inevitable outcome. While extensive research has focused on the average time it takes for a population to go extinct under various circumstances, there has been limited exploration of the distributions of extinction times and the likelihood of significant fluctuations. Recently, Hathcock and Strogatz [D. Hathcock and S. H. Strogatz, Phys. Rev. Lett. 128, 218301 (2022)0031-900710.1103/PhysRevLett.128.218301] identified Gumbel statistics as a universal asymptotic distribution for extinction-prone dynamics in a stable environment. In this study we aim to provide a comprehensive survey of this problem by examining a range of plausible scenarios, including extinction-prone, marginal (neutral), and stable dynamics. We consider the influence of demographic stochasticity, which arises from the inherent randomness of the birth-death process, as well as cases where stochasticity originates from the more pronounced effect of random environmental variations. Our work proposes several generic criteria that can be used for the classification of experimental and empirical systems, thereby enhancing our ability to discern the mechanisms governing extinction dynamics. Employing these criteria can help clarify the underlying mechanisms driving extinction processes.

Pervasive within-species spatial repulsion among adult tropical trees

For species to coexist, performance must decline as the density of conspecific individuals increases. Although evidence for such conspecific negative density dependence (CNDD) exists in forests, the within-species spatial repulsion it should produce has rarely been demonstrated in adults. In this study, we show that in comparison to a null model of stochastic birth, death, and limited dispersal, the adults of dozens of tropical forest tree species show strong spatial repulsion, some to surprising distances of approximately 100 meters. We used simulations to show that such strong repulsion can only occur if CNDD considerably exceeds heterospecific negative density dependence-an even stronger condition required for coexistence-and that large-scale repulsion can indeed result from small-scale CNDD. These results demonstrate substantial niche differences between species that may stabilize species diversity.

Improving the realism of neutral ecological models by incorporating transient dynamics with temporal changes in community size

Neutral models in ecology assume that all species are demographically equivalent, such that their abundances differ ultimately because of demographic stochasticity rather than selection. In spite of their simplicity, neutral models have been found to accurately reproduce static patterns of biodiversity for diverse communities. However, the same neutral models have been found to exhibit species abundance dynamics that are far too slow compared to reality, resulting in poor fits to temporally dynamic patterns of biodiversity. Here, we show that one of the root causes of these slow dynamics is the additional assumption that a community has reached an equilibrium with a fixed community size, with species that have a net growth rate close to zero. We removed this additional assumption by constructing and analyzing a neutral model with an expected community size that can change over time and is not necessarily at equilibrium, which thus allows the historical formation of a community to be represented explicitly. Our analysis demonstrated that for the general scenario where a small community rapidly grows in size to a carrying capacity, representing recovery from ecological disturbance or assembly of a new community, the model produced much larger changes in species abundances and much shorter species ages than a neutral model at an equilibrium with fixed community size. In addition, the species abundance distribution was biphasic with a subset of abundant species arising from a founder effect. We confirmed these new results in applications of the new model to the specific scenario of recovery of the Amazon tree community after the end-Cretaceous bolide impact, which involved periods of increasing and decreasing community size. We conclude that incorporating transient dynamics in neutral models improves realism by allowing explicit consideration of how a community is formed over realistic time-scales.

Effect of compositional fluctuation on the survival of bet-hedging species

Understanding the coexistence of diverse species in a changing environment is an important problem in community ecology. Bet-hedging is a strategy that helps species survive in such changing environments. However, studies of bet-hedging have often focused on the expected long-term growth rate of the species by itself, neglecting competition with other coexisting species. Here we study the extinction risk of a bet-hedging species in competition with others. We show that there are three contributions to the extinction risk. The first is the usual demographic fluctuation due to stochastic reproduction and selection processes in finite populations. The second, due to the fluctuation of population growth rate caused by environmental changes, may actually reduce the extinction risk for small populations. Besides those two, we reveal a third contribution, which is unique to bet-hedging species that diversify into multiple phenotypes: The phenotype composition of the population will fluctuate over time, resulting in increased extinction risk. We compare such compositional fluctuation to the demographic and environmental contributions, showing how they have different effects on the extinction risk depending on the population size, generation overlap, and environmental correlation.

Spatial Distribution and Species Association of Dominant Tree Species in Huangguan Plot of Qinling Mountains, China

The spatial distribution pattern and population structure of trees are shaped by multiple processes, such as species characteristics, environmental factors, and intraspecific and interspecific interactions. Studying the spatial distribution patterns of species, species associations, and their relationships with environmental factors is conducive to uncovering the mechanisms of biodiversity maintenance and exploring the underlying ecological processes of community stability and succession. This study was conducted in a 25-ha Qinling Huangguan forest (warm-temperate, deciduous, broad-leaved) dynamic monitoring plot. We used univariate and bivariate g(r) functions of the point pattern analysis method to evaluate the spatial distribution patterns of dominant tree species within the community, and the intra- and interspecific associations among different life-history stages. Complete spatial randomness and heterogeneous Poisson were used to reveal the potential process of community construction. We also used Berman’s test to determine the effect of three topographic variables on the distribution of dominant species. The results indicated that all dominant species in this community showed small-scale aggregation distribution. When we excluded the influence of environmental heterogeneity, the degree of aggregation distribution of each dominant species tended to decrease, and the trees mainly showed random or uniform distribution. This showed that environmental heterogeneity significantly affects the spatial distribution of tree species. Dominant species mainly showed positive associations with one another among different life-history stages, while negative associations prevailed among different tree species. Furthermore, we found that the associations between species were characterized by interspecific competition. Berman’s test results under the assumption of complete spatial randomness showed that the distribution of each dominant species was mainly affected by slope and convexity.

Stochastic neutral drifts seem prevalent in driving human virome assembly: neutral, near-neutral and non-neutral theoretic analyses

It is estimated that human body is inhabited by approximately 380 trillions of viruses, which exist in the form of viral communities and are collectively termed as human virome. How virome is assembled and what kind of forces maintain the composition and diversity of viral communities is still an open question. The question is of obvious importance because of its implications to human health and diseases. Here we address the question by harnessing the power of Hubbell’s unified neutral theory of biodiversity (UNTB) in terms of three neutral models including standard Hubbell’s neutral model (HNM), Sloan’s near-neutral model (SNM) and Harris et al. (2017) multi-site neutral model (MSN), further augmented by Ning et al. (2019) normalized stochasticity ratio (NSR) and Hammal et al. (2015) power analysis for the neutral test (PNT). With the five models applied to 179 virome samples, we aim to obtain robust findings given both Type-I and Type-II errors are addressed and possible alternative, non-neutral processes are detected. It was found that stochastic neutral drifts seem prevalent: approximately 65–92% at metacommunity/landscape scales and 67–80% at virus species scale. The non-neutral selection is approximately 26–28% at community scale and 23% at species scale. The false negative rate is about 2–3%, which suggested rather limited confounding effects of non-neutral process on neutrality tests. We postulate that prevalence of neutrality in human virome is likely due to extremely simple structure of viruses (stands of DNA/RNA) and their inter-species homogeneities, forming the foundation of species equivalence—the hallmark of neutral theory.

Non-zero-sum neutrality test for the tropical rain forest community using long-term between-census data

For community ecologists, "neutral or not?" is a fundamental question, and thus, rejecting neutrality is an important first step before investigating the deterministic processes underlying community dynamics. Hubbell's neutral model is an important contribution to the exploration of community dynamics, both technically and philosophically. However, the neutrality tests for this model are limited by a lack of statistical power, partly because the zero-sum assumption of the model is unrealistic. In this study, we developed a neutrality test for local communities that implements non-zero-sum community dynamics and determines the number of new species (N sp) between observations. For the non-zero-sum neutrality test, the model distributed the expected N sp, as calculated by extensive simulations, which allowed us to investigate the neutrality of the observed community by comparing the observed N sp with distributions of the expected N sp derived from the simulations. For this comparison, we developed a new "non-zero-sum N sp test," which we validated by running multiple neutral simulations using different parameter settings. We found that the non-zero-sum N sp test rejected neutrality at a near-significance level, which justified the validity of our approach. For an empirical test, the non-zero-sum N sp test was applied to real tropical tree communities in Panama and Malaysia. The non-zero-sum N sp test rejected neutrality in both communities when the observation interval was long and N sp was large. Hence, the non-zero-sum N sp test is an effective way to examine neutrality and has reasonable statistical power to reject the neutral model, especially when the observed N sp is large. This unique and simple approach is statistically powerful, even though it only employs two temporal sequences of community data. Thus, this test can be easily applied to existing datasets. In addition, application of the test will provide significant benefits for detecting changing biodiversity under climate change and anthropogenic disturbance.

Competition with abundance-dependent fitness and the dynamics of heterogeneous populations in fluctuating environment

Species competition takes place in a fluctuating environment, so the selective forces on different populations vary through time. In many realistic situations the mean fitness and the amplitude of its temporal variations are abundance-dependent. Here we present a theory of two-species competition with abundance-dependent stochastic fitness variations and solve for the chance of ultimate fixation, the time to absorption and the time to fixation. We then examine the ability of this two-species system to serve as an effective model for high-diversity assemblages and to account for the presence of an intra-specific differential response to environmental variations. The effective model is shown to capture the main features of competition between composite populations.

The Dynamic Hypercube as a Niche Community Model

Different models of community dynamics, such as the MacArthur–Wilson theory of island biogeography and Hubbell’s neutral theory, have given us useful insights into the workings of ecological communities. Here, we develop the niche-hypervolume concept of the community into a powerful model of community dynamics. We describe the community’s size through the volume of the hypercube and the dynamics of the populations in it through the fluctuations of the axes of the niche hypercube on different timescales. While the community’s size remains constant, the relative volumes of the niches within it change continuously, thus allowing the populations of different species to rise and fall in a zero-sum fashion. This dynamic hypercube model reproduces several key patterns in communities: lognormal species abundance distributions, 1/f-noise population abundance, multiscale patterns of extinction debt and logarithmic species-time curves. It also provides a powerful framework to explore significant ideas in ecology, such as the drift of ecological communities into evolutionary time.

Exclusion of the fittest predicts microbial community diversity in fluctuating environments

Microorganisms live in environments that inevitably fluctuate between mild and harsh conditions. As harsh conditions may cause extinctions, the rate at which fluctuations occur can shape microbial communities and their diversity, but we still lack an intuition on how. Here, we build a mathematical model describing two microbial species living in an environment where substrate supplies randomly switch between abundant and scarce. We then vary the rate of switching as well as different properties of the interacting species, and measure the probability of the weaker species driving the stronger one extinct. We find that this probability increases with the strength of demographic noise under harsh conditions and peaks at either low, high, or intermediate switching rates depending on both species' ability to withstand the harsh environment. This complex relationship shows why finding patterns between environmental fluctuations and diversity has historically been difficult. In parameter ranges where the fittest species was most likely to be excluded, however, the beta diversity in larger communities also peaked. In sum, how environmental fluctuations affect interactions between a few species pairs predicts their effect on the beta diversity of the whole community.

Evaluating the Assembly Dynamics in the Human Vaginal Microbiomes With Niche-Neutral Hybrid Modeling

Using 2,733 longitudinal vaginal microbiome samples (representing local microbial communities) from 79 individuals (representing meta-communities) in the states of healthy, BV (bacterial vaginosis) and pregnancy, we assess and interpret the relative importance of stochastic forces (e.g., stochastic drifts in bacteria demography, and stochastic dispersal) vs. deterministic selection (e.g., host genome, and host physiology) in shaping the dynamics of human vaginal microbiome (HVM) diversity by an integrated analysis with multi-site neutral (MSN) and niche-neutral hybrid (NNH) modeling. It was found that, when the traditional “default” P-value = 0.05 was specified, the neutral drifts were predominant (≥50% metacommunities indistinguishable from the MSN prediction), while the niche differentiations were moderate (<20% from the NNH prediction). The study also analyzed two challenging uncertainties in testing the neutral and/or niche-neutral hybrid models, i.e., lack of full model specificity – non-unique fittings of same datasets to multiple models with potentially different mechanistic assumptions – and lack of definite rules for setting the P-value thresholds (also noted as P_t-value when referring to the threshold of P-value in this article) in testing null hypothesis (model). Indeed, the two uncertainties can be interdependent, which further complicates the statistical inferences. To deal with the uncertainties, the MSN/NNH test results under a series of P-values ranged from 0.05 to 0.95 were presented. Furthermore, the influence of P-value threshold-setting on the model specificity, and the effects of woman’s health status on the neutrality level of HVM were examined. It was found that with the increase of P-value threshold from 0.05 to 0.95, the overlap (non-unique) fitting of MSN and NNH decreased from 29.1 to 1.3%, whereas the specificity (uniquely fitted to data) of MSN model was kept between 55.7 and 82.3%. Also with the rising P-value threshold, the difference between healthy and BV groups become significant. These findings suggested that traditional single P-value threshold (such as the de facto standard P-value = 0.05) might be insufficient for testing the neutral and/or niche neutral hybrid models.

The characteristic time of ecological communities

A simple description of temporal dynamics of ecological communities may help us understand how community assembly proceeds, predict ecological responses to environmental disturbances, and improve the performance of biological conservation actions. Although community changes take place at multiple temporal scales, the variation of species composition and richness over time across communities and habitats shows general patterns that may potentially reveal the main drivers of community dynamics. We used the simplest stochastic model of island biogeography to propose two quantities to characterize community dynamics: the community characteristic time, as a measure of the typical time scale of species‐richness change, and the characteristic Jaccard index, as a measure of temporal β diversity, that is, the variation of community composition over time. In addition, the community characteristic time, which sets the temporal scale at which null, noninteracting species assemblages operate, allowed us to define a relative sampling frequency (to the characteristic time). Here we estimate these quantities across microbial and macroscopic species assemblages to highlight two related results. First, we illustrated both characteristic time and Jaccard index and their relation with classic time‐series in ecology, and found that the most thoroughly sampled communities, relative to their characteristic time, presented the largest similarity between consecutive samples. Second, our analysis across a variety of habitats and taxa show that communities span a large range of species turnover, from potentially very fast (short characteristic times) to rather slow (long characteristic times) communities. This was in agreement with previous knowledge, but indicated that some habitats may have been sampled less frequently than required. Our work provides new perspectives to explore the temporal component in ecological studies and highlights the usefulness of simple approximations to the complex dynamics of ecological communities.

Taming the diffusion approximation through a controlling-factor WKB method

The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. The DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (Wentzel-Kramers-Brillouin) large-deviations method, which requires only the logarithm of a given quantity to be smooth over its state space. Combining the WKB scheme with asymptotic matching techniques, we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations, applicable for much wider regimes of parameters. We also introduce a scalable (independent of population size) WKB-based numerical technique. The method is applied to a central problem in population genetics and evolution, finding the chance of ultimate fixation in a zero-sum, two-types competition.

Intraspecific variability in fluctuating environments: mechanisms of impact on species diversity

Recent studies have found considerable trait variations within species. The effect of such intraspecific trait variability (ITV) on the stability, coexistence, and diversity of ecological communities received considerable attention and in many models it was shown to impede coexistence and decrease species diversity. Here we present a numerical study of the effect of genetically inherited ITV on species persistence and diversity in a temporally fluctuating environment. Two mechanisms are identified. First, ITV buffers populations against varying environmental conditions (portfolio effect) and reduces variation in abundances. Second, the interplay between ITV and environmental variations tends to increase the mean fitness of diverse populations. The first mechanism promotes persistence and tends to increase species richness, while the second reduces the chance of a rare species population (which is usually homogeneous) to invade, thus decreasing species richness. We show that for large communities the portfolio effect is dominant, leading to ITV promoting species persistence and richness.

Macroecological laws describe variation and diversity in microbial communities

How the coexistence of many species is maintained is a fundamental and unresolved question in ecology. Coexistence is a puzzle because we lack a mechanistic understanding of the variation in species presence and abundance. Whether variation in ecological communities is driven by deterministic or random processes is one of the most controversial issues in ecology. Here, I study the variation of species presence and abundance in microbial communities from a macroecological standpoint. I identify three macroecological laws that quantitatively characterize the fluctuation of species abundance across communities and over time. Using these three laws, one can predict species' presence and absence, diversity, and commonly studied macroecological patterns. I show that a mathematical model based on environmental stochasticity, the stochastic logistic model, quantitatively predicts the three macroecological laws, as well as non-stationary properties of community dynamics.

Janzen-Connell Effects Are a Weak Impediment to Competitive Exclusion

AbstractA goal of ecology is to identify the stabilizing mechanisms that maintain species diversity in the face of competitive exclusion and drift. For tropical forest tree communities, it has been hypothesized that high diversity is maintained via Janzen-Connell effects, whereby host-specific natural enemies prevent any one species from becoming too abundant. Here we explore the plausibility of this hypothesis with theoretical models. We confirm a previous result that when added to a model with drift but no competitive exclusion-that is, a neutral model where intrinsic fitnesses are perfectly equalized across species-Janzen-Connell effects maintain very high species richness that scales strongly with community size. However, when competitive exclusion is introduced-that is, when intrinsic fitnesses vary across species-the number of species maintained by Janzen-Connell effects is substantially reduced and scales much less strongly with community size. Because fitness variation is pervasive in nature, we conclude that the potential of Janzen-Connell effects to maintain diversity is probably weak and that the mechanism does not yet provide a sufficient explanation for the observed high diversity of tropical forest tree communities. We also show that, surprisingly, dispersal limitation can further reduce the ability of Janzen-Connell effects to maintain diversity.

Coexistence of plant species under harsh environmental conditions: an evaluation of niche differentiation and stochasticity along salt marsh creeks

Background

Ecologists have achieved much progress in the study of mechanisms that maintain species coexistence and diversity. In this paper, we reviewed a wide range of past research related to these topics, focusing on five theoretical bodies: (1) coexistence by niche differentiation, (2) coexistence without niche differentiation, (3) coexistence along environmental stress gradients, (4) coexistence under non-equilibrium versus equilibrium conditions, and (5) modern perspectives.

Results

From the review, we identified that there are few models that can be generally and confidently applicable to different ecological systems. This problem arises mainly because most theories have not been substantiated by enough empirical research based on field data to test various coexistence hypotheses at different spatial scales. We also found that little is still known about the mechanisms of species coexistence under harsh environmental conditions. This is because most previous models treat disturbance as a key factor shaping community structure, but they do not explicitly deal with stressful systems with non-lethal conditions. We evaluated the mainstream ideas of niche differentiation and stochasticity for the coexistence of plant species across salt marsh creeks in southwestern Denmark. The results showed that diversity indices, such as Shannon–Wiener diversity, richness, and evenness, decreased with increasing surface elevation and increased with increasing niche overlap and niche breadth. The two niche parameters linearly decreased with increasing elevation. These findings imply a substantial influence of an equalizing mechanism that reduces differences in relative fitness among species in the highly stressful environments of the marsh. We propose that species evenness increases under very harsh conditions if the associated stress is not lethal. Finally, we present a conceptual model of patterns related to the level of environmental stress and niche characteristics along a microhabitat gradient (i.e., surface elevation).

Conclusions

The ecology of stressful systems with non-lethal conditions will be increasingly important as ongoing global-scale climate change extends the period of chronic stresses that are not necessarily fatal to inhabiting plants. We recommend that more ecologists continue this line of research.

Quantitative approximation of the discrete Moran process by a Wright-Fisher diffusion

The Moran discrete process and the Wright-Fisher model are the most popular models in population genetics. The Wright-Fisher diffusion is commonly used as an approximation in order to understand the dynamics of population genetics models. Here, we give a quantitative large-population limit of the error occurring by using the approximating diffusion in the presence of weak selection and weak immigration in one dimension. The approach is robust enough to consider the case where selection and immigration are Markovian processes, whose large-population limit is either a finite state jump process, or a diffusion process.

Population Dynamics in a Changing Environment: Random versus Periodic Switching

Environmental changes greatly influence the evolution of populations. Here, we study the dynamics of a population of two strains, one growing slightly faster than the other, competing for resources in a time-varying binary environment modeled by a carrying capacity switching either randomly or periodically between states of abundance and scarcity. The population dynamics is characterized by demographic noise (birth and death events) coupled to a varying environment. We elucidate the similarities and differences of the evolution subject to a stochastically and periodically varying environment. Importantly, the population size distribution is generally found to be broader under intermediate and fast random switching than under periodic variations, which results in markedly different asymptotic behaviors between the fixation probability of random and periodic switching. We also determine the detailed conditions under which the fixation probability of the slow strain is maximal.

On the Simpson index for the Wright–Fisher process with random selection and immigration

Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the key parameters for ecologists who study for example, forest dynamics. Following ecological motivations, we will consider, here, the case, where there are various species with fitness and immigration parameters being random processes (and thus time evolving). The Simpson index is difficult to evaluate when the population is large, except in the neutral (no selection) case, because it has no closed formula. Our approach relies on the large population limit in the “weak” selection case, and thus to give a procedure which enables us to approximate, with controlled rate, the expectation of the Simpson index at fixed time. We will also study the long time behavior (invariant measure and convergence speed towards equilibrium) of the Wright–Fisher process in a simplified setting, allowing us to get a full picture for the approximation of the expectation of the Simpson index.

Stabilization of extensive fine-scale diversity by ecologically driven spatiotemporal chaos

It has recently become apparent that the diversity of microbial life extends far below the species level to the finest scales of genetic differences. Remarkably, extensive fine-scale diversity can coexist spatially. How is this diversity stable on long timescales, despite selective or ecological differences and other evolutionary processes? Most work has focused on stable coexistence or assumed ecological neutrality. We present an alternative: extensive diversity maintained by ecologically driven spatiotemporal chaos, with no assumptions about niches or other specialist differences between strains. We study generalized Lotka-Volterra models with antisymmetric correlations in the interactions inspired by multiple pathogen strains infecting multiple host strains. Generally, these exhibit chaos with increasingly wild population fluctuations driving extinctions. But the simplest spatial structure, many identical islands with migration between them, stabilizes a diverse chaotic state. Some strains (subspecies) go globally extinct, but many persist for times exponentially long in the number of islands. All persistent strains have episodic local blooms to high abundance, crucial for their persistence as, for many, their average population growth rate is negative. Snapshots of the abundance distribution show a power law at intermediate abundances that is essentially indistinguishable from the neutral theory of ecology. But the dynamics of the large populations are much faster than birth-death fluctuations. We argue that this spatiotemporally chaotic "phase" should exist in a wide range of models, and that even in rapidly mixed systems, longer-lived spores could similarly stabilize a diverse chaotic phase.

Neutral competition in a deterministically changing environment: Revisiting continuum approaches

Environmental variation can play an important role in ecological competition by influencing the relative advantage between competing species. Here, we consider such effects by extending a classical, competitive Moran model to incorporate an environment that fluctuates periodically in time. We adapt methods from work on these classical models to investigate the effects of the magnitude and frequency of environmental fluctuations on two important population statistics: the probability of fixation and the mean time to fixation. In particular, we find that for small frequencies, the system behaves similar to a system with a constant fitness difference between the two species, and for large frequencies, the system behaves similar to a neutrally competitive model. Most interestingly, the system exhibits nontrivial behavior for intermediate frequencies. We conclude by showing that our results agree quite well with recent theoretical work on competitive models with a stochastically changing environment, and discuss how the methods we develop ease the mathematical analysis required to study such models.

Temporal population variability in local forest communities has mixed effects on tree species richness across a latitudinal gradient

Among the local processes that determine species diversity in ecological communities, fluctuation-dependent mechanisms that are mediated by temporal variability in the abundances of species populations have received significant attention. Higher temporal variability in the abundances of species populations can increase the strength of temporal niche partitioning but can also increase the risk of species extinctions, such that the net effect on species coexistence is not clear. We quantified this temporal population variability for tree species in 21 large forest plots and found much greater variability for higher latitude plots with fewer tree species. A fitted mechanistic model showed that among the forest plots, the net effect of temporal population variability on tree species coexistence was usually negative, but sometimes positive or negligible. Therefore, our results suggest that temporal variability in the abundances of species populations has no clear negative or positive contribution to the latitudinal gradient in tree species richness.

Characterising extinction debt following habitat fragmentation using neutral theory

Habitat loss leads to species extinctions, both immediately and over the long term as 'extinction debt' is repaid. The same quantity of habitat can be lost in different spatial patterns with varying habitat fragmentation. How this translates to species loss remains an open problem requiring an understanding of the interplay between community dynamics and habitat structure across temporal and spatial scales. Here we develop formulas that characterise extinction debt in a spatial neutral model after habitat loss and fragmentation. Central to our formulas are two new metrics, which depend on properties of the taxa and landscape: 'effective area', measuring the remaining number of individuals and 'effective connectivity', measuring individuals' ability to disperse through fragmented habitat. This formalises the conventional wisdom that habitat area and habitat connectivity are the two critical requirements for long-term preservation of biodiversity. Our approach suggests that mechanistic fragmentation metrics help resolve debates about fragmentation and species loss.

Probability distributions of extinction times, species richness, and immigration and extinction rates in neutral ecological models

In community ecology, neutral models make the assumption that species are equivalent, such that species abundances differ only because of demographic stochasticity. Despite their ecological simplicity, neutral models have been found to give reasonable descriptions of expected patterns of biodiversity in communities with many species. Such patterns include the expected total number of species and species-abundance distributions describing the expected number of species in different abundance classes. However, the expected patterns represent only the central tendencies of the full distributions of possible outcomes. Thus, ecological inferences and conclusions based only on expected patterns are incomplete, and may be misleading. Here, we address this issue for the spatially implicit neutral model, by using classic results from birth-death processes to derive (1) the probability distribution of extinction time of a species with given abundance for the metacommunity; (2) the probability distributions of total species richness and number of species with given abundance for both the metacommunity and local community; and (3) the probability distributions of the average immigration and extinction rates in the local community, across different values of total species richness. We illustrate the utility of these probability distributions in providing greater ecological insight via statistical inference. Firstly, we show that under the neutral metacommunity model, there is only 2.65×10^-9 probability that the age of a common tree species in the Amazon is  ≤ 3  × 10⁸ yr, which is approximately the oldest estimated age of the first angiosperm. Thus, species ages from the model are unrealistically high. Secondly, for a tree community in a 50 ha plot at Barro Colorado Island in Panama, we show that the spatially implicit model can be fitted to observed species richness and an independent estimate of the immigration parameter, with the fitted model predicting a species-abundance distribution close to the observed distribution. Our results complement those using sampling formulae that specify the multivariate probability distribution of species abundances from neutral models.

Environmental spatial and temporal variability and its role in non-favoured mutant dynamics

Understanding how environmental variability (or randomness) affects evolution is of fundamental importance for biology. The presence of temporal or spatial variability significantly affects the competition dynamics in populations, and gives rise to some counterintuitive observations. In this paper, we consider both birth-death (BD) or death-birth (DB) Moran processes, which are set up on a circular or a complete graph. We investigate spatial and temporal variability affecting division and/or death parameters. Assuming that mutant and wild-type fitness parameters are drawn from an identical distribution, we study mutant fixation probability and timing. We demonstrate that temporal and spatial types of variability possess fundamentally different properties. Under temporal randomness, in a completely mixed system, minority mutants experience (i) higher than neutral fixation probability and a higher mean conditional fixation time, if the division rates are affected by randomness and (ii) lower fixation probability and lower mean conditional fixation time if the death rates are affected. Once spatial restrictions are imposed, however, these effects completely disappear, and mutants in a circular graph experience neutral dynamics, but only for the DB update rule in case (i) and for the BD rule in case (ii) above. In contrast to this, in the case of spatially variable environment, both for BD/DB processes, both for complete/circular graph and both for division/death rates affected, minority mutants experience a higher than neutral probability of fixation. Fixation time, however, is increased by randomness on a circle, while it decreases for complete graphs under random division rates. A basic difference between temporal and spatial kinds of variability is the types of correlations that occur in the system. Under temporal randomness, mutants are spatially correlated with each other (they simply have equal fitness values at a given moment of time; the same holds for wild-types). Under spatial randomness, there are subtler, temporal correlations among mutant and wild-type cells, which manifest themselves by cells of each type 'claiming' better spots for themselves. Applications of this theory include cancer generation and biofilm dynamics.

Comprehensive phase diagram for logistic populations in fluctuating environment

Population dynamics reflects an underlying birth-death process, where the rates associated with different events may depend on external environmental conditions and on the population density. A whole family of simple and popular deterministic models (such as logistic growth) supports a transcritical bifurcation point between an extinction phase and an active phase. Here we provide a comprehensive analysis of the phases of that system, taking into account both the endogenous demographic noise (random birth and death events) and the effect of environmental stochasticity that causes variations in birth and death rates. Three phases are identified: in the inactive phase the mean time to extinction T is independent of the carrying capacity N and scales logarithmically with the initial population size. In the power-law phase T∼N^{q}, and in the exponential phase T∼exp(αN). All three phases and the transitions between them are studied in detail. The breakdown of the continuum approximation is identified inside the power-law phase, and the accompanying changes in decline modes are analyzed. The applicability of the emerging picture to the analysis of ecological time series and to the management of conservation efforts is briefly discussed.

Dispersal increases ecological selection by increasing effective community size

Selection and drift are universally accepted as the cornerstones of evolutionary changes. Recent theories extend this view to ecological changes, arguing that any change in species composition is driven by deterministic fitness differences among species (enhancing selection) and/or stochasticity in birth and death rates of individuals within species (enhancing drift). These forces have contrasting effects on the predictability of ecological dynamics, and thus understanding the factors affecting their relative importance is crucial for understanding ecological dynamics. Here we test the hypothesis that dispersal increases the relative importance of ecological selection by increasing the effective size of the community (i.e., the size relevant for competitive interactions and drift). According to our hypothesis, dispersal increases the effective size of the community by mixing individuals from different localities. This effect diminishes the relative importance of demographic stochasticity, thereby reducing drift and increasing the relative importance of selective forces as drivers of species composition. We tested our hypothesis, which we term the "effective community size" hypothesis, using two independent experiments focusing on annual plants: a field experiment in which we manipulated the magnitude of dispersal and a mesocosm experiment in which we directly manipulated the effective size of the communities. Both experiments, as well as related model simulations, were consistent with the hypothesis that increasing dispersal increases the role of selective forces as drivers of species composition. This finding has important implications for our understanding of the fundamental forces affecting community dynamics, as well as the management of species diversity, particularly in patchy and fragmented environments.

Noise-induced stabilization and fixation in fluctuating environment

The dynamics of a two-species community of N competing individuals are considered, with an emphasis on the role of environmental variations that affect coherently the fitness of entire populations. The chance of fixation of a mutant (or invading) population is calculated as a function of its mean relative fitness, the amplitude of fitness variations and their typical duration. We emphasize the distinction between the case of pairwise competition and the case of global competition; in the latter a noise-induced stabilization mechanism yields a higher chance of fixation for a single mutant. This distinction becomes dramatic in the weak selection regime, where the chance of fixation for a single deleterious mutant is an N-independent constant for global competition and decays like (ln N)⁻¹ in the pairwise competition case. A Wentzel-Kramers-Brillouin (WKB) technique yields a general formula for the chance of fixation of a deleterious mutant in the strong selection regime. The possibility of long-term persistence of large [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr{O}}$$\end{document}O(N)] suboptimal (and extinction-prone) populations is discussed, as well as its relevance to stochastic tunneling between fitness peaks.

Theory of time-averaged neutral dynamics with environmental stochasticity

Competition is the main driver of population dynamics, which shapes the genetic composition of populations and the assembly of ecological communities. Neutral models assume that all the individuals are equivalent and that the dynamics is governed by demographic (shot) noise, with a steady state species abundance distribution (SAD) that reflects a mutation-extinction equilibrium. Recently, many empirical and theoretical studies emphasized the importance of environmental variations that affect coherently the relative fitness of entire populations. Here we consider two generic time-averaged neutral models; in both the relative fitness of each species fluctuates independently in time but its mean is zero. The first (model A) describes a system with local competition and linear fitness dependence of the birth-death rates, while in the second (model B) the competition is global and the fitness dependence is nonlinear. Due to this nonlinearity, model B admits a noise-induced stabilization mechanism that facilitates the invasion of new mutants. A self-consistent mean-field approach is used to reduce the multispecies problem to two-species dynamics, and the large-N asymptotics of the emerging set of Fokker-Planck equations is presented and solved. Our analytic expressions are shown to fit the SADs obtained from extensive Monte Carlo simulations and from numerical solutions of the corresponding master equations.

What drives the spatial distribution and dynamics of local species richness in tropical forest?

Understanding the structure and dynamics of highly diverse tropical forests is challenging. Here we investigate the factors that drive the spatio-temporal variation of local tree numbers and species richness in a tropical forest (including 1250 plots of 20 × 20 m²). To this end, we use a series of dynamic models that are built around the local spatial variation of mortality and recruitment rates, and ask which combination of processes can explain the observed spatial and temporal variation in tree and species numbers. We find that processes not included in classical neutral theory are needed to explain these fundamental patterns of the observed local forest dynamics. We identified a large spatio-temporal variability in the local number of recruits as the main missing mechanism, whereas variability of mortality rates contributed to a lesser extent. We also found that local tree numbers stabilize at typical values which can be explained by a simple analytical model. Our study emphasized the importance of spatio-temporal variability in recruitment beyond demographic stochasticity for explaining the local heterogeneity of tropical forests.

Biodiversity maintenance may be lower under partial niche differentiation than under neutrality

Niche differentiation is normally regarded as a key promoter of species coexistence in competitive systems. One might therefore expect that relative to neutral assemblages, niche-differentiated communities should support more species with longer persistence and lower probability of extinction. Here we compare stochastic niche and neutral dynamics in simulated assemblages, and find that when local dynamics combine with immigration from a regional pool, the effect of niches can be more complex. Trait variation that lessens competition between species will not necessarily give all immigrating species their own niche to occupy. Such partial niche differentiation protects certain species from local extinction, but precipitates exclusion of others. Differences in regional abundances and intrinsic growth rates have similar impacts on persistence times as niche differentiation, and therefore blur the distinction between niche and neutral dynamical patterns-although niche dynamics will influence which species persist longer. Ultimately, unless the number of niches available to species is sufficiently high, niches may actually heighten extinction rates and lower species richness and local persistence times. Our results help make sense of recent observations of community dynamics, and point to the dynamical observations needed to discern the influence of niche differentiation.

Impact of environmental colored noise in single-species population dynamics

Variability on external conditions has important consequences for the dynamics and the organization of biological systems. In many cases, the characteristic timescale of environmental changes as well as their correlations play a fundamental role in the way living systems adapt and respond to it. A proper mathematical approach to understand population dynamics, thus, requires approaches more refined than, e.g., simple white-noise approximations. To shed further light onto this problem, in this paper we propose a unifying framework based on different analytical and numerical tools available to deal with “colored” environmental noise. In particular, we employ a “unified colored noise approximation” to map the original problem into an effective one with white noise, and then we apply a standard path integral approach to gain analytical understanding. For the sake of specificity, we present our approach using as a guideline a variation of the contact process—which can also be seen as a birth-death process of the Malthus-Verhulst class—where the propagation or birth rate varies stochastically in time. Our approach allows us to tackle in a systematic manner some of the relevant questions concerning population dynamics under environmental variability, such as determining the stationary population density, establishing the conditions under which a population may become extinct, and estimating extinction times. We focus on the emerging phase diagram and its possible phase transitions, underlying how these are affected by the presence of environmental noise time-correlations.