Scales of association: hierarchical linear models and the measurement of ecological systems

A fundamental challenge to understanding patterns in ecological systems lies in employing methods that can analyse, test and draw inference from measured associations between variables across scales. Hierarchical linear models (HLM) use advanced estimation algorithms to measure regression relationships and variance–covariance parameters in hierarchically structured data. Although hierarchical models have occasionally been used in the analysis of ecological data, their full potential to describe scales of association, diagnose variance explained, and to partition uncertainty has not been employed. In this paper we argue that the use of the HLM framework can enable significantly improved inference about ecological processes across levels of organization. After briefly describing the principals behind HLM, we give two examples that demonstrate a protocol for building hierarchical models and answering questions about the relationships between variables at multiple scales. The first example employs maximum likelihood methods to construct a two-level linear model predicting herbivore damage to a perennial plant at the individual- and patch-scale; the second example uses Bayesian estimation techniques to develop a three-level logistic model of plant flowering probability across individual plants, microsites and populations. HLM model development and diagnostics illustrate the importance of incorporating scale when modelling associations in ecological systems and offer a sophisticated yet accessible method for studies of populations, communities and ecosystems. We suggest that a greater coupling of hierarchical study designs and hierarchical analysis will yield significant insights on how ecological processes operate across scales.     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.     

Interspecific Competition and Speciation in Endoparasitoids

Ecological speciation occurs when inherent reproductive barriers to gene flow evolve between populations as a result of divergent natural selection. Frequency dependent effects associated with intraspecific resource competition are thought to be one important source of divergent selection facilitating ecological speciation. Interspecific competition may also play an important role in promoting population divergence. Although evidence for interspecific competition in nature is ubiquitous, there is currently little empirical data supporting its role in the speciation process. Here, we discuss two general models in which interspecific competition among species can promote ecological speciation among populations within a species. In both models, interspecific competition is the source of divergent selection driving adaption to different portions of the resource distribution, generating ecological reproductive isolation from other conspecific populations. We propose that the biology of endoparasitoids that attack phytophagous insects make model systems for studying the role of interspecific competition in ecological speciation. We describe details for one such system, the community of endoparasitic braconid wasps attacking Rhagoletis fruit flies, as a potential model for investigating competitive speciation. We conclude by hypothesizing that a model in which interspecific competition forces an inferior competitor to alternative fly hosts may be a common theme contributing to parasitoid diversification in the Rhagoletis-parasitoid system.     

Ecophysiological traits of terrestrial and aquatic carnivorous plants: are the costs and benefits the same?

Identification of tradeoffs among physiological and morphological traits and their use in cost–benefit models and ecological or evolutionary optimization arguments have been hallmarks of ecological analysis for at least 50 years. Carnivorous plants are model systems for studying a wide range of ecophysiological and ecological processes and the application of a cost–benefit model for the evolution of carnivory by plants has provided many novel insights into trait‐based cost–benefit models. Central to the cost–benefit model for the evolution of botanical carnivory is the relationship between nutrients and photosynthesis; of primary interest is how carnivorous plants efficiently obtain scarce nutrients that are supplied primarily in organic form as prey, digest and mineralize them so that they can be readily used, and allocate them to immediate versus future needs. Most carnivorous plants are terrestrial – they are rooted in sandy or peaty wetland soils – and most studies of cost–benefit tradeoffs in carnivorous plants are based on terrestrial carnivorous plants. However approximately 10% of carnivorous plants are unrooted aquatic plants. Here we ask whether the cost–benefit model applies equally well to aquatic carnivorous plants and what general insights into tradeoff models are gained by this comparison. Nutrient limitation is more pronounced in terrestrial carnivorous plants, which also have much lower growth rates and much higher ratios of dark respiration to photosynthetic rates than aquatic carnivorous plants. Phylogenetic constraints on ecophysiological tradeoffs among carnivorous plants remain unexplored. Despite differences in detail, the general cost–benefit framework continues to be of great utility in understanding the evolutionary ecology of carnivorous plants. We provide a research agenda that if implemented would further our understanding of ecophysiological tradeoffs in carnivorous plants and also would provide broader insights into similarities and differences between aquatic and terrestrial plants of all types.     

Idealized, Inaccurate but Successful: A Pragmatic Approach to Evaluating Models in Theoretical Ecology

Ecologists attempt to understand the diversity of life with mathematical models. Often, mathematical models contain simplifying idealizations designed to cope with the blooming, buzzing confusion of the natural world. This strategy frequently issues in models whose predictions are inaccurate. Critics of theoretical ecology argue that only predictively accurate models are successful and contribute to the applied work of conservation biologists. Hence, they think that much of the mathematical work of ecologists is poor science. Against this view, I argue that model building is successful even when models are predictively inaccurate for at least three reasons: models allow scientists to explore the possible behaviors of ecological systems; models give scientists simplified means by which they can investigate more complex systems by determining how the more complex system deviates from the simpler model; and models give scientists conceptual frameworks through which they can conduct experiments and fieldwork. Critics often mistake the purposes of model building, and once we recognize this, we can see their complaints are unjustified. Even though models in ecology are not always accurate in their assumptions and predictions, they still contribute to successful science.     

Environmental Variables Explain Genetic Structure in a Beetle-Associated Nematode

The distribution of a species is a complex expression of its ecological and evolutionary history and integrating population genetic, environmental, and ecological data can provide new insights into the effects of the environment on the population structure of species. Previous work demonstrated strong patterns of genetic differentiation in natural populations of the hermaphroditic nematode Pristionchus pacificus in its La Réunion Island habitat, but gave no clear understanding of the role of the environment in structuring this variation. Here, we present what is to our knowledge the first study to statistically evaluate the role of the environment in shaping the structure and distribution of nematode populations. We test the hypothesis that genetic structure in P. pacificus is influenced by environmental variables, by combining population genetic analyses of microsatellite data from 18 populations and 370 strains, with multivariate statistics on environmental data, and species distribution modelling. We assess and quantify the relative importance of environmental factors (geographic distance, altitude, temperature, precipitation, and beetle host) on genetic variation among populations. Despite the fact that geographic populations of P. pacificus comprise vast genetic diversity sourced from multiple ancestral lineages, we find strong evidence for local associations between environment and genetic variation. Further, we show that significantly more genetic variation in P. pacificus populations is explained by environmental variation than by geographic distances. This supports a strong role for environmental heterogeneity vs. genetic drift in the divergence of populations, which we suggest may be influenced by adaptive forces.     

Resource allocation during ontogeny is influenced by genetic,developmental and ecological factors in the horned beetle,Onthophagus taurus

Resource allocation trade-offs arise when developing organs are in competition for a limited pool of resources to sustain growth and differentiation. Such competition may constrain the maximal size to which structures can grow and may force a situation in which the evolutionary elaboration of one structure may only be possible at the expense of another. However, recent studies have called into question both the consistency and evolutionary importance of resource allocation trade-offs. This study focuses on a well-described trade-off between the horns and eyes of Onthophagus beetles and assesses the degree to which it is influenced by genetic, developmental and ecological conditions. Contrary to expectations, we observed that trade-off signatures (i) were mostly absent within natural populations, (ii) mostly failed to match naturally evolved divergences in horn investment among populations, (iii) were subject to differential changes in F₁ populations derived from divergent field populations and (iv) remained largely unaffected by developmental genetic manipulations of horn investment. Collectively, our results demonstrate that populations subject to different ecological conditions exhibit different patterns of, and differential plasticity in, resource allocation. Further, variation in ecological conditions, rather than canalized developmental mechanisms, may determine whether and to what degree morphological structures engage in resource allocation trade-offs.     

Oscillations in continuous culture populations of Streptococcus pneumoniae: population dynamics and the evolution of clonal suicide

Agents that kill or induce suicide in the organisms that produce them or other individuals of the same genotype are intriguing puzzles for ecologists and evolutionary biologists. When those organisms are pathogenic bacteria, these suicidal toxins have the added appeal as candidates for the development of narrow spectrum antibiotics to kill the pathogens that produce them. We show that when clinical as well as laboratory strains of Streptococcus pneumoniae are maintained in continuous culture (chemostats), their densities oscillate by as much as five orders of magnitude with an apparently constant period. This dynamic, which is unanticipated for single clones of bacteria in chemostats, can be attributed to population-wide die-offs and recoveries. Using a combination of mathematical models and experiments with S. pneumoniae, we present evidence that these die-offs can be attributed to the autocatalytic production of a toxin that lyses or induces autolysis in members of the clone that produces it. This toxin, which our evidence indicates is a protein, appears to be novel; S. pneumoniae genetic constructs knocked out for lytA and other genes coding for known candidates for this agent oscillate in chemostat culture. Since this toxin lyses different strains of S. pneumoniae as well as other closely related species of Streptococcus, we propose that its ecological role is as an allelopathic agent. Using a mathematical model, we explore the conditions under which toxins that kill members of the same clone that produces them can prevent established populations from invasion by different strains of the same or other species. We postulate that the production of the toxin observed here as well as other bacteria-produced toxins that kill members of the same genotype, ‘clonal suicide’, evolved and are maintained to prevent colonization of established populations by different strains of the same and closely related species.     

How general is cognitive ability in non-human animals? A meta-analytical and multi-level reanalysis approach

General intelligence has been a topic of high interest for over a century. Traditionally, research on general intelligence was based on principal component analyses and other dimensionality reduction approaches. The advent of high-speed computing has provided alternative statistical tools that have been used to test predictions of human general intelligence. In comparison, research on general intelligence in non-human animals is in its infancy and still relies mostly on factor-analytical procedures. Here, we argue that dimensionality reduction, when incorrectly applied, can lead to spurious results and limit our understanding of ecological and evolutionary causes of variation in animal cognition. Using a meta-analytical approach, we show, based on 555 bivariate correlations, that the average correlation among cognitive abilities is low (r = 0.185; 95% CI: 0.087–0.287), suggesting relatively weak support for general intelligence in animals. We then use a case study with relatedness (genetic) data to demonstrate how analysing traits using mixed models, without dimensionality reduction, provides new insights into the structure of phenotypic variance among cognitive traits, and uncovers genetic associations that would be hidden otherwise. We hope this article will stimulate the use of alternative tools in the study of cognition and its evolution in animals.     

Graph models of habitat mosaics

Graph theory is a body of mathematics dealing with problems of connectivity, flow, and routing in networks ranging from social groups to computer networks. Recently, network applications have erupted in many fields, and graph models are now being applied in landscape ecology and conservation biology, particularly for applications couched in metapopulation theory. In these applications, graph nodes represent habitat patches or local populations and links indicate functional connections among populations (i.e. via dispersal). Graphs are models of more complicated real systems, and so it is appropriate to review these applications from the perspective of modelling in general. Here we review recent applications of network theory to habitat patches in landscape mosaics. We consider (1) the conceptual model underlying these applications; (2) formalization and implementation of the graph model; (3) model parameterization; (4) model testing, insights, and predictions available through graph analyses; and (5) potential implications for conservation biology and related applications. In general, and for a variety of ecological systems, we find the graph model a remarkably robust framework for applications concerned with habitat connectivity. We close with suggestions for further work on the parameterization and validation of graph models, and point to some promising analytic insights.     

Chaos in three species food chains

We study the dynamics of a three species food chain using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of the equilibrium where the top species in the food chain is absent. The goal of our study is to demonstrate the presence of chaos in a class of ecological models, rather than just in a specific model. This work extends earlier numerical studies of a particular system by Hastings and Powell (1991) by showing that chaos occurs in a class of ecological models. The mathematical techniques we use are based on work by Guckenheimer and Holmes (1983) on co-dimension two bifurcations. However, restrictions on the equations we study imposed by ecological assumptions require a new and somewhat different analysis.     

Cooperation in Microbial Populations: Theory and Experimental Model Systems

Cooperative behavior, the costly provision of benefits to others, is common across all domains of life. This review article discusses cooperative behavior in the microbial world, mediated by the exchange of extracellular products called public goods. We focus on model species for which the production of a public good and the related growth disadvantage for the producing cells are well described. To unveil the biological and ecological factors promoting the emergence and stability of cooperative traits we take an interdisciplinary perspective and review insights gained from both mathematical models and well-controlled experimental model systems. Ecologically, we include crucial aspects of the microbial life cycle into our analysis and particularly consider population structures where ensembles of local communities (subpopulations) continuously emerge, grow, and disappear again. Biologically, we explicitly consider the synthesis and regulation of public good production. The discussion of the theoretical approaches includes general evolutionary concepts, population dynamics, and evolutionary game theory. As a specific but generic biological example, we consider populations of Pseudomonas putida and its regulation and use of pyoverdines, iron scavenging molecules, as public goods. The review closes with an overview on cooperation in spatially extended systems and also provides a critical assessment of the insights gained from the experimental and theoretical studies discussed. Current challenges and important new research opportunities are discussed, including the biochemical regulation of public goods, more realistic ecological scenarios resembling native environments, cell-to-cell signaling, and multispecies communities.     

A new view of radiation-induced cancer: integrating short- and long-term processes. Part I: Approach

Mathematical models of radiation carcinogenesis are important for understanding mechanisms and for interpreting or extrapolating risk. There are two classes of such models: (1) long-term formalisms that track pre-malignant cell numbers throughout an entire lifetime but treat initial radiation dose–response simplistically and (2) short-term formalisms that provide a detailed initial dose–response even for complicated radiation protocols, but address its modulation during the subsequent cancer latency period only indirectly. We argue that integrating short- and long-term models is needed. As an example of this novel approach, we integrate a stochastic short-term initiation/inactivation/repopulation model with a deterministic two-stage long-term model. Within this new formalism, the following assumptions are implemented: radiation initiates, promotes, or kills pre-malignant cells; a pre-malignant cell generates a clone, which, if it survives, quickly reaches a size limitation; the clone subsequently grows more slowly and can eventually generate a malignant cell; the carcinogenic potential of pre-malignant cells decreases with age.     

Estimates of Genetic Differentiation Measured by F(ST) Do Not Necessarily Require Large Sample Sizes When Using Many SNP Markers

Population genetic studies provide insights into the evolutionary processes that influence the distribution of sequence variants within and among wild populations. F_ST is among the most widely used measures for genetic differentiation and plays a central role in ecological and evolutionary genetic studies. It is commonly thought that large sample sizes are required in order to precisely infer F_ST and that small sample sizes lead to overestimation of genetic differentiation. Until recently, studies in ecological model organisms incorporated a limited number of genetic markers, but since the emergence of next generation sequencing, the panel size of genetic markers available even in non-reference organisms has rapidly increased. In this study we examine whether a large number of genetic markers can substitute for small sample sizes when estimating F_ST. We tested the behavior of three different estimators that infer F_ST and that are commonly used in population genetic studies. By simulating populations, we assessed the effects of sample size and the number of markers on the various estimates of genetic differentiation. Furthermore, we tested the effect of ascertainment bias on these estimates. We show that the population sample size can be significantly reduced (as small as n = 4–6) when using an appropriate estimator and a large number of bi-allelic genetic markers (k>1,000). Therefore, conservation genetic studies can now obtain almost the same statistical power as studies performed on model organisms using markers developed with next-generation sequencing.     

What do we gain from simplicity versus complexity in species distribution models?

Species distribution models (SDMs) are widely used to explain and predict species ranges and environmental niches. They are most commonly constructed by inferring species' occurrence–environment relationships using statistical and machine‐learning methods. The variety of methods that can be used to construct SDMs (e.g. generalized linear/additive models, tree‐based models, maximum entropy, etc.), and the variety of ways that such models can be implemented, permits substantial flexibility in SDM complexity. Building models with an appropriate amount of complexity for the study objectives is critical for robust inference. We characterize complexity as the shape of the inferred occurrence–environment relationships and the number of parameters used to describe them, and search for insights into whether additional complexity is informative or superfluous. By building ‘under fit’ models, having insufficient flexibility to describe observed occurrence–environment relationships, we risk misunderstanding the factors shaping species distributions. By building ‘over fit’ models, with excessive flexibility, we risk inadvertently ascribing pattern to noise or building opaque models. However, model selection can be challenging, especially when comparing models constructed under different modeling approaches. Here we argue for a more pragmatic approach: researchers should constrain the complexity of their models based on study objective, attributes of the data, and an understanding of how these interact with the underlying biological processes. We discuss guidelines for balancing under fitting with over fitting and consequently how complexity affects decisions made during model building. Although some generalities are possible, our discussion reflects differences in opinions that favor simpler versus more complex models. We conclude that combining insights from both simple and complex SDM building approaches best advances our knowledge of current and future species ranges.     

Estimating FST and kinship for arbitrary population structures

F_ST and kinship are key parameters often estimated in modern population genetics studies in order to quantitatively characterize structure and relatedness. Kinship matrices have also become a fundamental quantity used in genome-wide association studies and heritability estimation. The most frequently-used estimators of F_ST and kinship are method-of-moments estimators whose accuracies depend strongly on the existence of simple underlying forms of structure, such as the independent subpopulations model of non-overlapping, independently evolving subpopulations. However, modern data sets have revealed that these simple models of structure likely do not hold in many populations, including humans. In this work, we analyze the behavior of these estimators in the presence of arbitrarily-complex population structures, which results in an improved estimation framework specifically designed for arbitrary population structures. After generalizing the definition of F_ST to arbitrary population structures and establishing a framework for assessing bias and consistency of genome-wide estimators, we calculate the accuracy of existing F_ST and kinship estimators under arbitrary population structures, characterizing biases and estimation challenges unobserved under their originally-assumed models of structure. We then present our new approach, which consistently estimates kinship and F_ST when the minimum kinship value in the dataset is estimated consistently. We illustrate our results using simulated genotypes from an admixture model, constructing a one-dimensional geographic scenario that departs nontrivially from the independent subpopulations model. Our simulations reveal the potential for severe biases in estimates of existing approaches that are overcome by our new framework. This work may significantly improve future analyses that rely on accurate kinship and F_ST estimates.     

Fungal functional ecology: bringing a trait‐based approach to plant‐associated fungi

Fungi play many essential roles in ecosystems. They facilitate plant access to nutrients and water, serve as decay agents that cycle carbon and nutrients through the soil, water and atmosphere, and are major regulators of macro‐organismal populations. Although technological advances are improving the detection and identification of fungi, there still exist key gaps in our ecological knowledge of this kingdom, especially related to function . Trait‐based approaches have been instrumental in strengthening our understanding of plant functional ecology and, as such, provide excellent models for deepening our understanding of fungal functional ecology in ways that complement insights gained from traditional and ‐omics‐based techniques. In this review, we synthesize current knowledge of fungal functional ecology, taxonomy and systematics and introduce a novel database of fungal functional traits (Fun^Fun). Fun^Fun is built to interface with other databases to explore and predict how fungal functional diversity varies by taxonomy, guild, and other evolutionary or ecological grouping variables. To highlight how a quantitative trait‐based approach can provide new insights, we describe multiple targeted examples and end by suggesting next steps in the rapidly growing field of fungal functional ecology.     

The more the merrier: Multi-species experiments in ecology

A major objective in ecology is to find general patterns, and to establish the rules and underlying mechanisms that generate those patterns. Nevertheless, most of our current insights in ecology are based on case studies of a single or few species, whereas multi-species experimental studies remain rare. We underline the power of the multi-species experimental approach for addressing general ecological questions, e.g. on species environmental responses or on patterns of among- and within-species variation. We present simulations that show that the accuracy of estimates of between-group differences is increased by maximizing the number of species rather than the number of populations or individuals per species. Thus, the more species a multi-species experiment includes, the more powerful it is. In addition, we discuss some inevitable methodological challenges of multi-species experiments. While we acknowledge the value of single- or few-species experiments, we strongly advocate the use of multi-species experiments for addressing ecological questions at a more general level.     

Modeling gynodioecy: novel scenarios for maintaining polymorphism

Nuclear-cytoplasmic gynodioecy is a breeding system of plants in which females and hermaphrodites co-occur in populations, and gender is jointly determined by cytoplasmic male sterility (CMS) genes and nuclear restorers of male fertility. Persistent polymorphism at both CMS and nuclear-restorer loci is necessary to maintain this breeding system. Theoretical models have explained how nuclear-cytoplasmic gynodioecy can be stable for certain assumptions. However, recent advances in our understanding of the genetics, population biology, and molecular mechanisms of sex determination in nuclear-cytoplasmic gynodioecious species suggest the utility of new models with different underlying assumptions. In this article, we examine different negative pleiotropic fitness effects of nuclear restorers (costs of restoration) using genetic and population assumptions based on recent literature. Specifically, we model populations with two CMS types and separate nuclear restorer loci for each CMS type. Under these assumptions, both overdominance for fitness and frequency-dependent selection at nuclear-restorer loci can support nuclear-cytoplasmic gynodioecy. Costs of restoration can be either dependent or independent of the cytoplasmic background. Seed fitness costs are more vulnerable to fixation of CMS types than pollen costs. Survivorship costs are effective at maintaining polymorphism even when total reproductive effects are low. Overall, our models display differences in the stability of nuclear-cytoplasmic gynodioecy and predicted population sex ratios that should be informative to researchers studying gynodioecy in the wild.     

Evolution of quantitative traits in the wild: mind the ecology

Recent advances in the quantitative genetics of traits in wild animal populations have created new interest in whether natural selection, and genetic response to it, can be detected within long-term ecological studies. However, such studies have re-emphasized the fact that ecological heterogeneity can confound our ability to infer selection on genetic variation and detect a population''s response to selection by conventional quantitative genetics approaches. Here, I highlight three manifestations of this issue: counter gradient variation, environmentally induced covariance between traits and the correlated effects of a fluctuating environment. These effects are symptomatic of the oversimplifications and strong assumptions of the breeder''s equation when it is applied to natural populations. In addition, methods to assay genetic change in quantitative traits have overestimated the precision with which change can be measured. In the future, a more conservative approach to inferring quantitative genetic response to selection, or genomic approaches allowing the estimation of selection intensity and responses to selection at known quantitative trait loci, will provide a more precise view of evolution in ecological time.