Mapping the complexity of ecological models

We propose to define the complexity of an ecological model as the statistical complexity of the output it produces. This allows for a direct comparison between data and model complexity. Working with univariate time series, we show that this measure ‘blindly’ discriminates among the different dynamical behaviours a model can exhibit. We then search a model parameter space in order to segment it into areas of different dynamical behaviour and calculate the maximum complexity a model can generate. Given a time series, and the problem of choosing among a number of ecological models to study it, we suggest that models whose maximum complexity is lower than the time series complexity should be disregarded because they are unable to reconstruct some of the structures contained in the data. Similar reasoning could be used to disregard models’ subdomains as well as areas of unnecessary high complexity. We suggest that model complexity so defined better captures the difficulty faced by a user in managing and understanding the behaviour of an ecological model than measures based on a model ‘size’.     

Unravelling the dynamics of organisms in a changing world using ecological modelling

Understanding and predicting the dynamics of organisms is a central objective in ecology and conservation biology, and modelling provides a solution to tackling this problem. However, the complex nature of ecological systems means that for a thorough understanding of ecological dynamics at hierarchical scales, a set of modeling approaches need to be adopted. This review illustrates how modelling approaches can be used to understand the dynamics of organisms in applied ecological problems, focussing on mechanistic models at a local scale and statistical models at a broad scale. Mechanistic models incorporate ecological processes explicitly and thus are likely to be robust under novel conditions. Models based on behavioural decisions by individuals represent a typical example of the successful application of mechanistic models to applied problems. Considering the data-hungry nature of such mechanistic models, model complexity and parameterisation need to be explored further for a quick and widespread implementation of this model type. For broad-scale phenomena, statistical models play an important role in dealing with problems that are often inherent in data. Examples include models for quantifying population trends from long-term, large-scale data and those for comparative methods of extinction risk. Novel statistical approaches also allow mechanistic models to be parameterised using readily obtained data at a macro scale. In conclusion, the complementary use and improvement of multiple model types, the increased use of novel model parameterisation, the examination of model transferability and the achievement of wider biodiversity information availability are key challenges for the effective use of modelling in applied ecological problems.     

On the impossibility of coexistence of infinitely many strategies

We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.     

基于生态信息诊断框架的杉木人工林生物量生态预测研究

森林生物量会直接影响森林生态系统服务的评估。如何运用景感生态学,准确预测区域尺度下森林生物量的时空演变趋势,是关乎国家重大方针政策制定和生态产业体系建设的关键性战略课题。本研究目的是构建一套生态信息诊断框架,优化趋善化模型（3PG2模型）结构,解决由于模型结构设计所导致在森林景感营造过程中生态预测的不确定性。以杉木林分布广泛的福建南靖县为研究区域,选择合适的阈值范围和空间统计分析识别出模拟生物量的不确定性区域,构建包含Geogdetector软件、遗传技术和计算机程序3个部分组成的生态信息诊断框架,使用Geogdetector软件阐明多重因素交互作用对模型模拟的影响及机理,采用遗传技术优化模型结构以提升模拟精度,运用计算机程序和3PG2模型准确预测区域尺度杉木林生物量的时空演变趋势。结果表明：林龄是导致3PG2模型生物量模拟结果不确定性的主导因素。通过景感生态学（谜码数据和趋善化模型）构建的生态信息诊断框架可以准确预测森林生物量,实现区域尺度上的可持续森林管理。     

Modelling habitat associations with fingernail clam (Family: Sphaeriidae) counts at multiple spatial scales using hierarchical count models

1. Macroinvertebrate count data often exhibit nested or hierarchical structure. Examples include multiple measurements along each of a set of streams, and multiple synoptic measurements from each of a set of ponds. With data exhibiting hierarchical structure, outcomes at both sampling (e.g. within stream) and aggregated (e.g. stream) scales are often of interest. Unfortunately, methods for modelling hierarchical count data have received little attention in the ecological literature. 2. We demonstrate the use of hierarchical count models using fingernail clam (Family: Sphaeriidae) count data and habitat predictors derived from sampling and aggregated spatial scales. The sampling scale corresponded to that of a standard Ponar grab (0.052 m²) and the aggregated scale to impounded and backwater regions within 38–197 km reaches of the Upper Mississippi River. Impounded and backwater regions were resampled annually for 10 years. Consequently, measurements on clams were nested within years. Counts were treated as negative binomial random variates, and means from each resampling event as random departures from the impounded and backwater region grand means. 3. Clam models were improved by the addition of covariates that varied at both the sampling and regional scales. Substrate composition varied at the sampling scale and was associated with model improvements, and reductions (for a given mean) in variance at the sampling scale. Inorganic suspended solids (ISS) levels, measured in the summer preceding sampling, also yielded model improvements and were associated with reductions in variances at the regional rather than sampling scales. ISS levels were negatively associated with mean clam counts. 4. Hierarchical models allow hierarchically structured data to be modelled without ignoring information specific to levels of the hierarchy. In addition, information at each hierarchical level may be modelled as functions of covariates that themselves vary by and within levels. As a result, hierarchical models provide researchers and resource managers with a method for modelling hierarchical data that explicitly recognises both the sampling design and the information contained in the corresponding data.     

IS YOUR PHYLOGENY INFORMATIVE? MEASURING THE POWER OF COMPARATIVE METHODS

Phylogenetic comparative methods may fail to produce meaningful results when either the underlying model is inappropriate or the data contain insufficient information to inform the inference. The ability to measure the statistical power of these methods has become crucial to ensure that data quantity keeps pace with growing model complexity. Through simulations, we show that commonly applied model choice methods based on information criteria can have remarkably high error rates; this can be a problem because methods to estimate the uncertainty or power are not widely known or applied. Furthermore, the power of comparative methods can depend significantly on the structure of the data. We describe a Monte Carlo-based method which addresses both of these challenges, and show how this approach both quantifies and substantially reduces errors relative to information criteria. The method also produces meaningful confidence intervals for model parameters. We illustrate how the power to distinguish different models, such as varying levels of selection, varies both with number of taxa and structure of the phylogeny. We provide an open-source implementation in the pmc ("Phylogenetic Monte Carlo") package for the R programming language. We hope such power analysis becomes a routine part of model comparison in comparative methods.     

Study design and parameter estimability for spatial and temporal ecological models

The statistical tools available to ecologists are becoming increasingly sophisticated, allowing more complex, mechanistic models to be fit to ecological data. Such models have the potential to provide new insights into the processes underlying ecological patterns, but the inferences made are limited by the information in the data. Statistical nonestimability of model parameters due to insufficient information in the data is a problem too‐often ignored by ecologists employing complex models. Here, we show how a new statistical computing method called data cloning can be used to inform study design by assessing the estimability of parameters under different spatial and temporal scales of sampling. A case study of parasite transmission from farmed to wild salmon highlights that assessing the estimability of ecologically relevant parameters should be a key step when designing studies in which fitting complex mechanistic models is the end goal.     

A theory for species co-occurrence in interaction networks

The study of species co-occurrences has been central in community ecology since the foundation of the discipline. Co-occurrence data are, nevertheless, a neglected source of information to model species distributions and biogeographers are still debating about the impact of biotic interactions on species distributions across geographical scales. We argue that a theory of species co-occurrence in ecological networks is needed to better inform interpretation of co-occurrence data, to formulate hypotheses for different community assembly mechanisms, and to extend the analysis of species distributions currently focused on the relationship between occurrences and abiotic factors. The main objective of this paper is to provide the first building blocks of a general theory for species co-occurrences. We formalize the problem with definitions of the different probabilities that are studied in the context of co-occurrence analyses. We analyze three species interactions modules and conduct multi-species simulations in order to document five principles influencing the associations between species within an ecological network: (i) direct interactions impact pairwise co-occurrence, (ii) indirect interactions impact pairwise co-occurrence, (iii) pairwise co-occurrence rarely are symmetric, (iv) the strength of an association decreases with the length of the shortest path between two species, and (v) the strength of an association decreases with the number of interactions a species is experiencing. Our analyses reveal the difficulty of the interpretation of species interactions from co-occurrence data. We discuss whether the inference of the structure of interaction networks is feasible from co-occurrence data. We also argue that species distributions models could benefit from incorporating conditional probabilities of interactions within the models as an attempt to take into account the contribution of biotic interactions to shaping individual distributions of species.     

Absolute quality evaluation of protein model structures using statistical potentials with respect to the native and reference states

In protein structure prediction, it is crucial to evaluate the degree of native-likeness of given model structures. Statistical potentials extracted from protein structure data sets are widely used for such quality assessment problems, but they are only applicable for comparing different models of the same protein. Although various other methods, such as machine learning approaches, were developed to predict the absolute similarity of model structures to the native ones, they required a set of decoy structures in addition to the model structures. In this paper, we tried to reformulate the statistical potentials as absolute quality scores, without using the information from decoy structures. For this purpose, we regarded the native state and the reference state, which are necessary components of statistical potentials, as the good and bad standard states, respectively, and first showed that the statistical potentials can be regarded as the state functions, which relate a model structure to the native and reference states. Then, we proposed a standardized measure of protein structure, called native-likeness, by interpolating the score of a model structure between the native and reference state scores defined for each protein. The native-likeness correlated with the similarity to the native structures and discriminated the native structures from the models, with better accuracy than the raw score. Our results show that statistical potentials can quantify the native-like properties of protein structures, if they fully utilize the statistical information obtained from the data set.     

Using pattern-oriented modeling for revealing hidden information: a key for reconciling ecological theory and application

We suggest that the conscious use of information that is "hidden" in distinct structures in nature itself and in data extracted from nature (=pattern) during the process of modeling (=pattern-oriented modeling) can substantially improve models in ecological application and conservation. Observed patterns, such as time-series patterns and spatial patterns of presence/absence in habitat patches, contain a great deal of data on scales, site-history, parameters and processes. Use of these data provides criteria for aggregating the biological information in the model, relates the model explicitly to the relevant scales of the system, facilitates the use of helpful techniques of indirect parameter estimation with independent data, and helps detect underlying ecological processes. Additionally, pattern-oriented models produce comparative predictions that can be tested in the field. We developed a step-by-step protocol for pattern-oriented modeling and illustrate the potential of this protocol by discussing three pattern-oriented population models: (1) a population viability analysis for brown bears ( Ursus arctos ) in northern Spain using time-series data on females with cubs of the year to adjust unknown model parameters; (2) a savanna model for detecting underlying ecological processes from spatial patterns of tree distribution; and (3) the incidence function model of metapopulation dynamics as an example of process integration and model generalization. We conclude that using the pattern-oriented approach to its full potential will require a major paradigm shift in the strategies of modeling and data collection, and we argue that more emphasis must be placed on observing and documenting relevant patterns in addition to attempts to obtain direct estimates of model parameters.     

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.     

A predictive model of avian natal dispersal distance provides prior information for investigating response to landscape change

1. Informative Bayesian priors can improve the precision of estimates in ecological studies or estimate parameters for which little or no information is available. While Bayesian analyses are becoming more popular in ecology, the use of strongly informative priors remains rare, perhaps because examples of informative priors are not readily available in the published literature. 2. Dispersal distance is an important ecological parameter, but is difficult to measure and estimates are scarce. General models that provide informative prior estimates of dispersal distances will therefore be valuable. 3. Using a world-wide data set on birds, we develop a predictive model of median natal dispersal distance that includes body mass, wingspan, sex and feeding guild. This model predicts median dispersal distance well when using the fitted data and an independent test data set, explaining up to 53% of the variation. 4. Using this model, we predict a priori estimates of median dispersal distance for 57 woodland-dependent bird species in northern Victoria, Australia. These estimates are then used to investigate the relationship between dispersal ability and vulnerability to landscape-scale changes in habitat cover and fragmentation. 5. We find evidence that woodland bird species with poor predicted dispersal ability are more vulnerable to habitat fragmentation than those species with longer predicted dispersal distances, thus improving the understanding of this important phenomenon. 6. The value of constructing informative priors from existing information is also demonstrated. When used as informative priors for four example species, predicted dispersal distances reduced the 95% credible intervals of posterior estimates of dispersal distance by 8-19%. Further, should we have wished to collect information on avian dispersal distances and relate it to species' responses to habitat loss and fragmentation, data from 221 individuals across 57 species would have been required to obtain estimates with the same precision as those provided by the general model.     

Ecohydrodynamics of Cold-Water Coral Reefs: A Case Study of the Mingulay Reef Complex (Western Scotland)

Ecohydrodynamics investigates the hydrodynamic constraints on ecosystems across different temporal and spatial scales. Ecohydrodynamics play a pivotal role in the structure and functioning of marine ecosystems, however the lack of integrated complex flow models for deep-water ecosystems beyond the coastal zone prevents further synthesis in these settings. We present a hydrodynamic model for one of Earth''s most biologically diverse deep-water ecosystems, cold-water coral reefs. The Mingulay Reef Complex (western Scotland) is an inshore seascape of cold-water coral reefs formed by the scleractinian coral Lophelia pertusa. We applied single-image edge detection and composite front maps using satellite remote sensing, to detect oceanographic fronts and peaks of chlorophyll a values that likely affect food supply to corals and other suspension-feeding fauna. We also present a high resolution 3D ocean model to incorporate salient aspects of the regional and local oceanography. Model validation using in situ current speed, direction and sea elevation data confirmed the model''s realistic representation of spatial and temporal aspects of circulation at the reef complex including a tidally driven current regime, eddies, and downwelling phenomena. This novel combination of 3D hydrodynamic modelling and remote sensing in deep-water ecosystems improves our understanding of the temporal and spatial scales of ecological processes occurring in marine systems. The modelled information has been integrated into a 3D GIS, providing a user interface for visualization and interrogation of results that allows wider ecological application of the model and that can provide valuable input for marine biodiversity and conservation applications.     

Binary analysis and optimization-based normalization of gene expression data

MOTIVATION: Most approaches to gene expression analysis use real-valued expression data, produced by high-throughput screening technologies, such as microarrays. Often, some measure of similarity must be computed in order to extract meaningful information from the observed data. The choice of this similarity measure frequently has a profound effect on the results of the analysis, yet no standards exist to guide the researcher. RESULTS: To address this issue, we propose to analyse gene expression data entirely in the binary domain. The natural measure of similarity becomes the Hamming distance and reflects the notion of similarity used by biologists. We also develop a novel data-dependent optimization-based method, based on Genetic Algorithms (GAs), for normalizing gene expression data. This is a necessary step before quantizing gene expression data into the binary domain and generally, for comparing data between different arrays. We then present an algorithm for binarizing gene expression data and illustrate the use of the above methods on two different sets of data. Using Multidimensional Scaling, we show that a reasonable degree of separation between different tumor types in each data set can be achieved by working solely in the binary domain. The binary approach offers several advantages, such as noise resilience and computational efficiency, making it a viable approach to extracting meaningful biological information from gene expression data.     

Gene selection for oligonucleotide array: an approach using PM probe level data

MOTIVATION: Analysis of oligonucleotide array data, especially to select genes of interest, is a highly challenging task because of the large volume of information and various experimental factors. Moreover, interaction effect (i.e. expression changes depend on probe effects) complicates the analysis because current methods often use an additive model to analyze data. We propose an approach to address these issues with the aim of producing a more reliable selection of differentially expressed genes. The approach uses the rank for normalization, employs the percentile-range to measure expression variation, and applies various filters to monitor expression changes. RESULTS: We compare our approach with MAS and Dchip models. A data set from an angiogenesis study is used for illustration. Results show that our approach performs better than other methods either in identification of the positive control gene or in PCR confirmatory tests. In addition, the invariant set of genes in our approach provides an efficient way for normalization.     

生态学中的尺度及尺度转换方法

尺度作为生态学的重要范式,已经引起了广泛重视,但对尺度问题的研究还不够成熟.尺度具有多维性特点,即功能尺度、空间尺度、时间尺度等,但生态学研究的重点是空间和时间尺度.并且时空尺度还具有复杂性、变异性特征.尺度研究的根本目的在于通过适宜的空间和时间尺度来揭示和把握复杂的生态学规律.为此,科学有效的尺度选择和尺度转换方法不可或缺.常见的尺度转换方法有图示法、回归分析、变异函数、自相关分析、谱分析、分形和小波变换,同时遥感和地理信息系统技术在尺度研究中也发挥着重要作用.结合实例对上述方法进行了分析和论述,认为各种方法都有其内在的优势和不足,新方法的引入和应用对于尺度转换方法体系的充实和完善非常重要.有关尺度的研究将进一步加强,研究的重点是尺度变异性、不同尺度间的相互作用机制以及尺度转换方法等.     

Spatial modelling of succession-disturbance dynamics in forest ecosystems: Concepts and examples

Over the last few decades it has become increasingly obvious that disturbance, whether natural or anthropogenic in origin, is ubiquitous in ecosystems. Disturbance-related processes are now considered to be important determinants of the composition, structure and function of ecological systems. However, because disturbance and succession processes occur across a wide range of spatio-temporal scales their empirical investigation is difficult. To counter these difficulties much use has been made of spatial modelling to explore the response of ecological systems to disturbance(s) occurring at spatial scales from the individual to the landscape and above, and temporal scales from minutes to centuries. Here we consider such models by contrasting two alternative motivations for their development and use: prediction and exploration, with a focus on forested ecosystems. We consider the two approaches to be complementary rather than competing. Predictive modelling aims to combine knowledge (understanding and data) with the goal of predicting system dynamics; conversely, exploratory models focus on developing understanding in systems where uncertainty is high. Examples of exploratory modelling include model-based explorations of generic issues of criticality in ecological systems, whereas predictive models tend to be more heavily data-driven (e.g. species distribution models). By considering predictive and exploratory modelling alongside each other, we aim to illustrate the range of methods used to model succession and disturbance dynamics and the challenges involved in the model-building and evaluation processes in this arena.     

Doubly robust multiple imputation using kernel‐based techniques

We consider the problem of estimating the marginal mean of an incompletely observed variable and develop a multiple imputation approach. Using fully observed predictors, we first establish two working models: one predicts the missing outcome variable, and the other predicts the probability of missingness. The predictive scores from the two models are used to measure the similarity between the incomplete and observed cases. Based on the predictive scores, we construct a set of kernel weights for the observed cases, with higher weights indicating more similarity. Missing data are imputed by sampling from the observed cases with probability proportional to their kernel weights. The proposed approach can produce reasonable estimates for the marginal mean and has a double robustness property, provided that one of the two working models is correctly specified. It also shows some robustness against misspecification of both models. We demonstrate these patterns in a simulation study. In a real‐data example, we analyze the total helicopter response time from injury in the Arizona emergency medical service data.     

Relevance of evolutionary history for food web structure

Explaining the structure of ecosystems is one of the great challenges of ecology. Simple models for food web structure aim at disentangling the complexity of ecological interaction networks and detect the main forces that are responsible for their shape. Trophic interactions are influenced by species traits, which in turn are largely determined by evolutionary history. Closely related species are more likely to share similar traits, such as body size, feeding mode and habitat preference than distant ones. Here, we present a theoretical framework for analysing whether evolutionary history--represented by taxonomic classification--provides valuable information on food web structure. In doing so, we measure which taxonomic ranks better explain species interactions. Our analysis is based on partitioning of the species into taxonomic units. For each partition, we compute the likelihood that a probabilistic model for food web structure reproduces the data using this information. We find that taxonomic partitions produce significantly higher likelihoods than expected at random. Marginal likelihoods (Bayes factors) are used to perform model selection among taxonomic ranks. We show that food webs are best explained by the coarser taxonomic ranks (kingdom to class). Our methods provide a way to explicitly include evolutionary history in models for food web structure.