Reinterpreting maximum entropy in ecology: a null hypothesis constrained by ecological mechanism

Simplified mechanistic models in ecology have been criticised for the fact that a good fit to data does not imply the mechanism is true: pattern does not equal process. In parallel, the maximum entropy principle (MaxEnt) has been applied in ecology to make predictions constrained by just a handful of state variables, like total abundance or species richness. But an outstanding question remains: what principle tells us which state variables to constrain? Here we attempt to solve both problems simultaneously, by translating a given set of mechanisms into the state variables to be used in MaxEnt, and then using this MaxEnt theory as a null model against which to compare mechanistic predictions. In particular, we identify the sufficient statistics needed to parametrise a given mechanistic model from data and use them as MaxEnt constraints. Our approach isolates exactly what mechanism is telling us over and above the state variables alone.     

Do lichens show latitudinal patterns of diversity?

No previous study has directly investigated whether lichens show latitudinal patterns of diversity. We used vouchered data and MaxEnt models to compile richness estimates (species, genera, and families) across the western coastal region of the US. Nonparametric multiplicative regression then sought the geographic factors or interactions of factors that explained the most variability in lichen richness. Collection density was the strongest predictor of raw estimates of richness at all taxonomic ranks. Latitude was the overall single-best predictor of MaxEnt modeled species, generic, and familial richness in all models. MaxEnt modeling was necessary to minimize collection bias, which otherwise obscured any other patterns of diversity. While geography explained a sizable portion of variance in lichen richness, it does not trend linearly with latitude. Instead, lichen diversity may be influenced by a compilation of regional and local factors including climate, disturbance, and competition.     

The performance of range maps and species distribution models representing the geographic variation of species richness at different resolutions

Aim The method used to generate hypotheses about species distributions, in addition to spatial scale, may affect the biodiversity patterns that are then observed. We compared the performance of range maps and MaxEnt species distribution models at different spatial resolutions by examining the degree of similarity between predicted species richness and composition against observed values from well‐surveyed cells (WSCs). Location Mexico. Methods We estimated amphibian richness distributions at five spatial resolutions (from 0.083° to 2°) by overlaying 370 individual range maps or MaxEnt predictions, comparing the similarity of the spatial patterns and correlating predicted values with the observed values for WSCs. Additionally, we looked at species composition and assessed commission and omission errors associated with each method. Results MaxEnt predictions reveal greater geographic differences in richness between species rich and species poor regions than the range maps did at the five resolutions assessed. Correlations between species richness values estimated by either of the two procedures and the observed values from the WSCs increased with decreasing resolution. The slopes of the regressions between the predicted and observed values indicate that MaxEnt overpredicts observed species richness at all of the resolutions used, while range maps underpredict them, except at the finest resolution. Prediction errors did not vary significantly between methods at any resolution and tended to decrease with decreasing resolution. The accuracy of both procedures was clearly different when commission and omission errors were examined separately. Main conclusions Despite the congruent increase in the geographic richness patterns obtained from both procedures as resolution decreases, the maps created with these methods cannot be used interchangeably because of notable differences in the species compositions they report.     

Do fitness-equalizing tradeoffs lead to neutral communities?

Neutral theory in ecology is aimed at describing communities where species coexist due to similarities rather than the classically posited niche differences. It assumes that all individuals, regardless of species identity, are demographically equivalent. However, Hubbell suggested that neutral theory may describe even niche communities because tradeoffs equalize fitness across species which differ in their traits. In fact, tradeoffs can involve stabilization as well as fitness equalization, and stabilization involves different dynamics and can lead to different community patterns than neutral theory. Yet the important question remains if neutral theory provides a robust picture of all fitness-equalized communities, of which communities with demographic equivalence are one special case. Here, I examine Hubbell’s suggestion for a purely fitness-equalizing interspecific birth–death tradeoff, expanding neutral theory to a theory describing this broader class of fitness-equalized communities. In particular, I use a flexible framework allowing examination of the influence of speciation dynamics. I find that the scaling of speciation rates with birth and death rates, which is poorly known, has large impacts on community structure. In most cases, the departure from the predictions of current neutral models is substantial. This work suggests that demographic and speciation complexities present a challenge to the future development and use of neutral theory in ecology as null model. The framework presented here will provide a starting point for meeting that challenge, and may also be useful in the development of stochastic niche models with speciation dynamics.     

A stochastic biodiversity model with overlapping niche structure

The niche is a fundamental ecological concept that underpins many explanations of patterns of biodiversity. The complexity of niche processes in ecological systems, however, means that it is difficult to capture them accurately in theoretical models of community assembly. In this study, we build upon simple neutral biodiversity models by adding the important ingredient of overlapping niche structure. Our model is spatially implicit and contains a fixed number of equal-sized habitats. Each species in the metacommunity arises through a speciation event; at which time, it is randomly assigned a fundamental niche or set of environments/habitats in which it can persist. Within each habitat, species compete with other species that have different but overlapping fundamental niches. Species abundances then change through ecological drift; each, however, is constrained by its maximum niche breadth and by the presence of other species in its habitats. Using our model, we derive analytical expressions for steady-state species abundance distributions, steady-state distributions of niche breadth across individuals and across species, and dynamic distributions of niche breadth across species. With this framework, we identify the conditions that produce the log-series species abundance distribution familiar from neutral models. We then identify how overlapping niche structure can lead to other species abundance distributions and, in particular, ask whether these new distributions differ significantly from species abundance distributions predicted by non-overlapping niche models. Finally, we extend our analysis to consider additional distributions associated with realized niche breadths. Overall, our results show that models with overlapping niches can exhibit behavior similar to neutral models, with the caveat that species with narrow fundamental niche breadths will be very rare. If narrow-niche species are common, it must be because they are in a non-overlapping niche or have countervailing advantages over broad-niche species. This result highlights the role that niches can play in establishing demographic neutrality.     

Community structure of vascular epiphytes: a neutral perspective

Vascular epiphytes form a diverse group of almost 30 000 species, yet theory concerning their community structure is still largely lacking. We therefore employed the simplest models of biodiversity, (near-)neutral models, to generate hypotheses concerning their community structure. With recently developed tools for (near-)neutral models we analyzed species abundance data from many samples in Central and South America which we divided into four metacommunities (Mesoamerica, Central America, Amazonia and Paraná), where for each metacommunity we considered two subsets differing in dispersal syndrome: an animal-dispersed guild and a wind-dispersed guild. We considered three models differing in the underlying speciation mode. Across all metacommunities, we found observed patterns to be indistinguishable from patterns generated by neutral or near-neutral processes. Furthermore, we found that subdivision in different dispersal guilds was often supported, with recruitment limitation being stronger for animal-dispersed species than for wind-dispersed species. This is the first time that (near-)neutral theory has been applied to epiphyte communities. Future efforts with additional data sets and more refined models are expected to further improve our understanding of community structure in epiphytes and will have to test the generality of our findings.     

Adding energy gradients and long‐distance dispersal to a neutral model improves predictions of Madagascan bird diversity

Macroecological patterns are likely the result of both stochastically neutral mechanisms and deterministic differences between species. In Madagascar, the simplest stochastically neutral hypothesis – the mid‐domain effects (MDE) hypothesis – has already been rejected. However, rejecting the MDE hypothesis does not necessarily refute the existence of all other neutral mechanisms. Here, we test whether adding complexity to a basic neutral model improves predictions of biodiversity patterns. The simplest MDE model assumes that: (1) species' ranges are continuous and unfragmented, (2) are randomly located throughout the landscape, and (3) can be stacked independently and indefinitely. We designed a simulation based on neutral theory that allowed us to weaken each of these assumptions incrementally by adjusting the habitat capacity as well as the likelihood of short‐ and long‐distance dispersal. Simulated outputs were compared to four empirical patterns of bird diversity: the frequency distributions of species richness and range size, the within‐island latitudinal diversity gradient, and the distance‐decay of species compositional similarity. Neutral models emulated empirical diversity patterns for Madagascan birds accurately. The frequency distribution of range size, latitudinal diversity gradient, and the distance‐decay of species compositional similarity could be attributed to stochastic long‐distance migration events and zero‐sum population dynamics. However, heterogenous environmental gradients improved predictions of the frequency distribution of species richness. Patterns of bird diversity in Madagascar can broadly be attributed to stochastic long‐distance migration events and zero‐sum population dynamics. This implies that rejecting simple hypotheses, such as MDE, does not serve as evidence against stochastic processes in general. However, environmental gradients were necessary to explain patterns of species richness and deterministic differences between species are probably important for explaining the distributions of narrow‐range and endemic species.     

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.     

Combining mechanism and drift in community ecology: a novel statistical mechanics approach

A key challenge for models of community ecology is to combine deterministic mechanism and stochastic drift in a systematic, transparent and tractable manner. Another challenge is to explain and unify different ecological patterns, hitherto modelled in isolation, within a single modelling framework. Here, we show that statistical mechanics provides an effective way to meet both challenges. We apply the statistical principle of maximum entropy (MaxEnt) to a simple resource-based, non-neutral model of a plant community. In contrast to previous ecological applications of MaxEnt, our use of MaxEnt emphasises its theoretical basis in the combinatorics of sampling frequencies, an approach that clarifies its ecological interpretation. In this approach, mechanism and drift are identified, respectively, with ecological resource constraints and entropy maximization. We obtain realistic predictions for species abundance distributions as well as contrasting stability-diversity relationships at community and population levels. The model also predicts critical behaviour that may provide a basis for understanding desertification and other ecological tipping points. Our results complement and extend previous ecological applications of MaxEnt to new areas of community ecology, and further illustrate MaxEnt as a powerful yet simple modelling tool for combining mechanism and drift in a way that unifies disparate ecological patterns.     

Neutral community dynamics, the mid-domain effect and spatial patterns in species richness

The mid‐domain effect (MDE) aims to explain spatial patterns in species richness invoking only stochasticity and geometrical constraints. In this paper, we used simulations to show that its main qualitative prediction, a hump‐shaped pattern in species richness, converges to the expectation of a spatially bounded neutral model when communities are linked by short‐distance migration. As these two models can be linked under specific situations, neutral theory may provide a mechanistic population level basis for MDE. This link also allows establishing in which situations MDE patterns are more likely to be found. Also, in this situation, MDE models could be used as a first approximation to understand the role of both stochastic (ecological drift and migration) and deterministic (adaptation to environmental conditions) processes driving the spatial structure of species richness.     

Niches versus neutrality: uncovering the drivers of diversity in a species-rich community

Ecological models suggest that high diversity can be generated by purely niche-based, purely neutral or by a mixture of niche-based and neutral ecological processes. Here, we compare the degree to which four contrasting hypotheses for coexistence, ranging from niche-based to neutral, explain species richness along a body mass niche axis. We derive predictions from these hypotheses and confront them with species body-mass patterns in a highly sampled marine phytoplankton community. We find that these patterns are consistent only with a mechanism that combines niche and neutral processes, such as the emergent neutrality mechanism. In this work, we provide the first empirical evidence that a niche-neutral model can explain niche space occupancy pattern in a natural species-rich community. We suggest this class of model may be a useful hypothesis for the generation and maintenance of species diversity in other size-structured communities.     

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.     

中国湿地物种多样性与生境面积关系及其生态学机理的模拟研究

生物群落中的物种多样性是群落生态学研究的一个基本问题。大量的实验研究表明物种多样性和生境面积的常数次幂成比例,这说明它们的对数数量级呈线性正相关。我们研究了中国境内15块湿地的高等植物和32块湿地鸟类多样性,发现它们分别和湿地面积在对数尺度上呈线性正相关．进一步支持了种数与面积的幂指数关系。我们还借助计算机模拟系统地讨论了产生这种简单规律的生态学机理,包括中性理论、集合种群动态和物种分布的自相似性。中性理论假设了群落中物种的个体之间只有竞争关系,忽略了其它的种间关系。集合种群动态理论考虑的是由多个亚群落构成的集合群落,在研究种数和面积关系时也忽略了种间关系,所以也是中性的。尽管物种分布的自相似性可导致这种面积幂指数关系,但在自然界中自相似性也可能不成立。     

The impact of species-neutral stage structure on macroecological patterns

Despite its radical assumption of ecological equivalence between species, neutral biodiversity theory can often provide good fits to species abundance distributions observed in nature. Major criticisms of neutral theory have focused on interspecific differences, which are in conflict with ecological equivalence. However, neutrality in nature is also broken by differences between conspecific individuals at different life stages, which in many communities may vastly exceed interspecific differences between individuals at similar stages. These within-species asymmetries have not been fully explored in species-neutral models, and it is not known whether demographic stage structure affects macroecological patterns in neutral theory. Here, we present a two-stage neutral model where fecundity and mortality change as an individual transitions from one stage to the other. We explore several qualitatively different scenarios, and compare numerically obtained species abundance distributions to the predictions of unstructured neutral theory. We find that abundance distributions are generally robust to this kind of stage structure, but significant departures from unstructured predictions occur if adults have sufficiently low fecundity and mortality. In addition, we show that the cumulative number of births per species, which is distributed as a power law with a 3/2 exponent, is invariant even when the abundance distribution departs from unstructured model predictions. Our findings potentially explain power law-like abundance distributions in organisms with strong demographic structure, such as eusocial insects and humans, and partially rehabilitate species abundance distributions from past criticisms as to their inability to distinguish between biological mechanisms.     

Ecological Niche Models using MaxEnt in Google Earth Engine: Evaluation,guidelines and recommendations

Google Earth Engine (GEE) has revolutionized geospatial analyses by fast-processing formerly demanding analyses from multiple research areas. Recently, maximum entropy (MaxEnt), the most commonly used method in ecological niche models (ENMs), was integrated into GEE. This integration can significantly enhance modeling efficiency and encourage multidisciplinary approaches of ENMs, but an evaluation assessment of MaxEnt in GEE is lacking. Herein, we present the first MaxEnt models in GEE, as well as its first statistical and spatial evaluation. We also identify the limitations of the approach, providing guidelines and recommendations for its easier applicability in GEE.We tested MaxEnt in GEE using 11 case studies. For each case, we used species of different taxa (insects, amphibians, reptiles, birds and mammals) distributed across global and regional extents. Each species occupied habitats with distinct environmental characteristics (nine terrestrial and two marine species) and within divergent ecoregions across five continents. The models were performed in GEE and Maxent software, and both approaches were contrasted for their model discrimination performance (assessed by eight evaluation metrics) and spatial consistency (correlation analyses and two measures of niche overlap/equivalency).MaxEnt in GEE allows setting several parameters, but important analyses and outputs are unavailable, such as automatic selection of background data, model replicates, and analyses of variable importance (concretely, jackknife analyses and response curves). GEE provided MaxEnt models with high discrimination performance (area under the curve mean between all species models of 0.90) and with spatial equivalency in relation to Maxent software outputs (Hellinger's I mean between all species models >0.90).Our work demonstrates the first application and assessment of MaxEnt in GEE at global and regional scales. We conclude that the GEE modeling method provides ENMs with high performance and reliable spatial predictions, comparable to the widely used Maxent software. We also acknowledge important limitations that should be integrated into GEE in the future, particularly those related to the assessment of variable importance. We expect that our guidelines, recommendations and potential solutions to surpass the identified limitations could help researchers easily apply MaxEnt in GEE across different research fields.     

Biological invasions and the neutral theory

Aim  The invasion of natural communities by alien species represents a serious threat, but creates opportunities to learn about community functions. Neutral theory proposes that the niche concept may not be needed to explain the assemblage and diversity of natural communities, challenging the classical view of community ecology and generating a lasting debate. Biological invasions, when considered as natural experiments, can be used to contrast some of the predictions of neutral and classic niche theories.
Location  Global.
Methods  We use data from biological invasions as natural experiments to contrast some of the fundamental predictions of neutral theory.
Results  Some emerging patterns did not differ from neutral model expectations (e.g. the relationship between native and exotic species richness, invasibility of resource-rich habitats, and the relationship between propagule release and invasion success). Nevertheless, other patterns (e.g. experimental evidence of the relationship between diversity and susceptibility to invasion, the invasion of communities with a low resource availability, invasiveness related to species traits) contrasted with the predictions that can be inferred from neutral theory.
Main conclusions  Neutral theory correctly highlights the need to include randomness in models of community structure. Biological invasion patterns show that neutral forces are important in structuring natural communities, but the patterns differ from those inferred from a complete neutral model. For biodiversity-conservation purposes, the implications of accepting or not accepting neutral theory as a process-based theory are very important.     

Optimal transportation theory for species interaction networks

1. Observed biotic interactions between species, such as in pollination, predation, and competition, are determined by combinations of population densities, matching in functional traits and phenology among the organisms, and stochastic events (neutral effects).
2. We propose optimal transportation theory as a unified view for modeling species interaction networks with different intensities of interactions. We pose the coupling of two distributions as a constrained optimization problem, maximizing both the system''s average utility and its global entropy, that is, randomness. Our model follows naturally from applying the MaxEnt principle to this problem setting.
3. This approach allows for simulating changes in species relative densities as well as to disentangle the impact of trait matching and neutral forces.
4. We provide a framework for estimating the pairwise species utilities from data. Experimentally, we show how to use this framework to perform trait matching and predict the coupling in pollination and host–parasite networks.

The coupling between species in a species interaction network can be modeled using optimal transportation. As an application of the MaxEnt principle, it jointly maximizes interaction utility and entropy. This allows for anticipating how the interaction coupling can change when species abundances change.     

Opportunities for improved distribution modelling practice via a strict maximum likelihood interpretation of MaxEnt

Maximum entropy (MaxEnt) modelling, as implemented in the Maxent software, has rapidly become one of the most popular methods for distribution modelling. Originally, MaxEnt was described as a machine‐learning method. More recently, it has been explained from principles of Bayesian estimation. MaxEnt offers numerous options (variants of the method) and settings (tuning of parameters) to the users. A widespread practice of accepting the Maxent software's default options and settings has been established, most likely because of ecologists’ lack of familiarity with machine‐learning and Bayesian statistical concepts and the ease by which the default models are obtained in Maxent. However, these defaults have been shown, in many cases, to be suboptimal and exploration of alternatives has repeatedly been called for. In this paper, we derive MaxEnt from strict maximum likelihood principles, and point out parallels between MaxEnt and standard modelling tools like generalised linear models (GLM). Furthermore, we describe several new options opened by this new derivation of MaxEnt, which may improve MaxEnt practice. The most important of these is the option for selecting variables by subset selection methods instead of the ?₁‐regularisation method, which currently is the Maxent software default. Other new options include: incorporation of new transformations of explanatory variables and user control of the transformation process; improved variable contribution measures and options for variation partitioning; and improved output prediction formats. The new options are exemplified for a data set for the plant species Scorzonera humilis in SE Norway, which was analysed by the standard MaxEnt procedure in a previously published paper. We recommend that thorough comparisons between the proposed alternative options and default procedures and variants thereof be carried out.     

Spatial autocorrelation analysis allows disentangling the balance between neutral and niche processes in metacommunities

One of the most popular approaches for investigating the roles of niche and neutral processes driving metacommunity patterns consists of partitioning variation in species data into environmental and spatial components. The logic is that the distance decay of similarity in communities is expected under neutral models. However, because environmental variation is often spatially structured, the decay could also be attributed to environmental factors that are missing from the analysis. Here, we use a spatial autocorrelation analysis protocol, previously developed to detect isolation‐by‐distance in allele frequencies, to evaluate patterns of species abundances under neutral dynamics. We show that this protocol can be linked with variation partitioning analyses. Moreover, in an attempt to test the neutral model, we derive three predictions to be applied both to original species abundances and to abundances predicted by a pure spatial model species abundances will be uncorrelated; Moran's I correlograms will reveal similar short‐distance autocorrelation patterns; an increasing degree of non‐neutrality will tend to generate patterns of correlation among abundances within groups of species with similar correlograms (i.e. within species with neutral and non‐neutral dynamics). We illustrate our protocol by analyzing spatial patterns in abundance of 28 terrestrially breeding anuran species from Central Amazonia. We recommend that researchers should investigate spatial autocorrelation patterns of abundances predicted by pure spatial models to identify similar patterns of spatial autocorrelation at short distances and lack of correlation between species abundances. Therefore, the hypothesis that spatial patterns in abundances are primarily due to pure neutral dynamics (rather than to missing spatiallystructured environmental factors) can be confirmed after taking environmental variables into account.     

Emergent neutrality or hidden niches?

Emergent neutrality is the idea in community ecology that species interactions may drive a system in a direction where some species become so similar that this similarity will be the primary cause for their coexistence instead of niche differentiation. A recent, widely cited model of emergent neutrality is by Scheffer and van Nes, later applied to species abundance distribution patterns by Vergnon et al. We take issue with the ecological interpretation of this model, demonstrating that it in fact presupposes important differences between superficially similar‐looking species. We argue that the temptation to interpret the model as one of emergent neutrality stems from the fact that these differences are unmodeled and therefore hidden, obscuring the underlying coexistence mechanisms. We therefore claim that the model is actually one of hidden niches, and present several alternative ways to make its hidden portions more explicit. These alterations to the model also make its proper interpretation as one of hidden niches more transparent. We also polemize with the claim of Vergnon et al. that multimodality in species abundance distributions is support for their emergent neutrality model: we demonstrate that appropriate stochastic versions of classical resource partitioning or even neutral models can lead to such patterns in a robust way. Observation of these patterns is therefore inconclusive as to the underlying mechanisms that generate them.