Statistical methods in spatial genetics

The joint analysis of spatial and genetic data is rapidly becoming the norm in population genetics. More and more studies explicitly describe and quantify the spatial organization of genetic variation and try to relate it to underlying ecological processes. As it has become increasingly difficult to keep abreast with the latest methodological developments, we review the statistical toolbox available to analyse population genetic data in a spatially explicit framework. We mostly focus on statistical concepts but also discuss practical aspects of the analytical methods, highlighting not only the potential of various approaches but also methodological pitfalls.     

Spatial predictions at the community level: from current approaches to future frameworks

A fundamental goal of ecological research is to understand and model how processes generate patterns so that if conditions change, changes in the patterns can be predicted. Different approaches have been proposed for modelling species assemblage, but their use to predict spatial patterns of species richness and other community attributes over a range of spatial and temporal scales remains challenging. Different methods emphasize different processes of structuring communities and different goals. In this review, we focus on models that were developed for generating spatially explicit predictions of communities, with a particular focus on species richness, composition, relative abundance and related attributes. We first briefly describe the concepts and theories that span the different drivers of species assembly. A combination of abiotic processes and biotic mechanisms are thought to influence the community assembly process. In this review, we describe four categories of drivers: (i) historical and evolutionary, (ii) environmental, (iii) biotic, and (iv) stochastic. We discuss the different modelling approaches proposed or applied at the community level and examine them from different standpoints, i.e. the theoretical bases, the drivers included, the source data, and the expected outputs, with special emphasis on conservation needs under climate change. We also highlight the most promising novelties, possible shortcomings, and potential extensions of existing methods. Finally, we present new approaches to model and predict species assemblages by reviewing promising ‘integrative frameworks’ and views that seek to incorporate all drivers of community assembly into a unique modelling workflow. We discuss the strengths and weaknesses of these new solutions and how they may hasten progress in community‐level modelling.     

Comparing approximations to spatio-temporal models for epidemics with local spread

Analytical methods for predicting and exploring the dynamics of stochastic, spatially interacting populations have proven to have useful application in epidemiology and ecology. An important development has been the increasing interest in spatially explicit models, which require more advanced analytical techniques than the usual mean-field or mass-action approaches. The general principle is the derivation of differential equations describing the evolution of the expected population size and other statistics. As a result of spatial interactions no closed set of equations is obtained. Nevertheless, approximate solutions are possible using closure relations for truncation. Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. This study is made in the context of an SIS model for plant-disease epidemics introduced in Filipe and Gibson (1998, Studying and approximating spatio-temporal models for epidemic spread and control, Phil. Trans. R. Soc. Lond. B 353, 2153–2162) of which the contact process [Harris, T. E. (1974), Contact interactions on a lattice, Ann. Prob. 2, 969] is a special case. The various methods of approximation are derived and explained and their predictions are compared and tested against simulation. The merits and limitations of the various approximations are discussed. A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.     

Local Frequency Dependence and Global Coexistence

In sessile organisms such as plants, interactions occur locally so that important ecological aspects like frequency dependence are manifest within local neighborhoods. Using probabilistic cellular automata models, we investigated how local frequency-dependent competition influenced whether two species could coexist. Individuals of the two species were randomly placed on a grid and allowed to interact according to local frequency-dependent rules. For four different frequency-dependent scenarios, the results indicated that over a broad parameter range the two species could coexist. Comparisons between explicit spatial simulations and the mean-field approximation indicate that coexistence occurs over a broader region in the explicit spatial simulation.     

空间直观景观模型的验证方法

空间直观景观模型已是当前景观生态学研究的一大热点。空间景观模型模拟空间格局变化。其模拟结果包含非空间数据和空间数据。空间直观景观模型的验证除进行非空间数据的验证外,还需要进行空间数据的验证。本文回顾了空间直观模型发展历程,总结现有的空间直观模型验证方法。包括主观评价、图形比较、偏差分析、回归分析、假设检验、多尺度拟合度分析和景观指数分析,同时提出今后空间直观景观模型验证方法研究的重点方向。     

Universal scaling of species‐abundance distributions across multiple scales

The scale‐dependent species abundance distribution (SAD) is fundamental in ecology, but few spatially explicit models of this pattern have thus far been studied. Here we show spatially explicit neutral model predictions for SADs over a wide range of spatial scales, which appear to match empirical patterns qualitatively. We find that the assumption of a log‐series SAD in the metacommunity made by spatially implicit neutral models can be justified with a spatially explicit model in the large area limit. Furthermore, our model predicts that SADs on multiple scales are characterized by a single, compound parameter that represents the ratio of the survey area to the species’ average biogeographic range (which is in turn set by the speciation rate and the dispersal distance). This intriguing prediction is in line with recent empirical evidence for a universal scaling of the species‐area curve. Hence we hypothesize that empirical SAD patterns will show a similar universal scaling for many different taxa and across multiple spatial scales.     

Estimating and interpreting migration of Amazonian forests using spatially implicit and semi‐explicit neutral models

With many sophisticated methods available for estimating migration, ecologists face the difficult decision of choosing for their specific line of work. Here we test and compare several methods, performing sanity and robustness tests, applying to large‐scale data and discussing the results and interpretation. Five methods were selected to compare for their ability to estimate migration from spatially implicit and semi‐explicit simulations based on three large‐scale field datasets from South America (Guyana, Suriname, French Guiana and Ecuador). Space was incorporated semi‐explicitly by a discrete probability mass function for local recruitment, migration from adjacent plots or from a metacommunity. Most methods were able to accurately estimate migration from spatially implicit simulations. For spatially semi‐explicit simulations, estimation was shown to be the additive effect of migration from adjacent plots and the metacommunity. It was only accurate when migration from the metacommunity outweighed that of adjacent plots, discrimination, however, proved to be impossible. We show that migration should be considered more an approximation of the resemblance between communities and the summed regional species pool. Application of migration estimates to simulate field datasets did show reasonably good fits and indicated consistent differences between sets in comparison with earlier studies. We conclude that estimates of migration using these methods are more an approximation of the homogenization among local communities over time rather than a direct measurement of migration and hence have a direct relationship with beta diversity. As betadiversity is the result of many (non)‐neutral processes, we have to admit that migration as estimated in a spatial explicit world encompasses not only direct migration but is an ecological aggregate of these processes. The parameter m of neutral models then appears more as an emerging property revealed by neutral theory instead of being an effective mechanistic parameter and spatially implicit models should be rejected as an approximation of forest dynamics.     

Isolation by distance,resistance and/or clusters? Lessons learned from a forest‐dwelling carnivore inhabiting a heterogeneous landscape

Landscape genetics provides a valuable framework to understand how landscape features influence gene flow and to disentangle the factors that lead to discrete and/or clinal population structure. Here, we attempt to differentiate between these processes in a forest‐dwelling small carnivore [European pine marten (Martes martes)]. Specifically, we used complementary analytical approaches to quantify the spatially explicit genetic structure and diversity and analyse patterns of gene flow for 140 individuals genotyped at 15 microsatellite loci. We first used spatially explicit and nonspatial Bayesian clustering algorithms to partition the sample into discrete clusters and evaluate hypotheses of ‘isolation by barriers’ (IBB). We further characterized the relationships between genetic distance and geographical (‘isolation by distance’, IBD) and ecological distances (‘isolation by resistance’, IBR) obtained from optimized landscape models. Using a reciprocal causal modelling approach, we competed the IBD, IBR and IBB hypotheses with each other to unravel factors driving population genetic structure. Additionally, we further assessed spatially explicit indices of genetic diversity using sGD across potentially overlapping genetic neighbourhoods that matched the inferred population structure. Our results revealed a complex spatial genetic cline that appears to be driven jointly by IBD and partial barriers to gene flow (IBB) associated with poor habitat and interspecific competition. Habitat loss and fragmentation, in synergy with past overharvesting and possible interspecific competition with sympatric stone marten (Martes foina), are likely the main factors responsible for the spatial genetic structure we observed. These results emphasize the need for a more thorough evaluation of discrete and clinal hypotheses governing gene flow in landscape genetic studies, and the potential influence of different limiting factors affecting genetic structure at different spatial scales.     

Coupled map lattice approximations for spatially explicit individual-based models of ecology

Spatially explicit individual-based models are widely used in ecology but they are often difficult to treat analytically. Despite their intractability they often exhibit clear temporal and spatial patterning. We demonstrate how a spatially explicit individual-based model of scramble competition with local dispersal can be approximated by a stochastic coupled map lattice. The approximation disentangles the deterministic and stochastic element of local interaction and dispersal. We are thus able to understand the individual-based model through a simplified set of equations. In particular, we demonstrate that demographic noise leads to increased stability in the dynamics of locally dispersing single-species populations. The coupled map lattice approximation has general application to a range of spatially explicit individual-based models. It provides a new alternative to current approximation techniques, such as the method of moments and reaction-diffusion approximation, that captures both stochastic effects and large-scale patterning arising in individual-based models.     

The Urban Funnel Model and the Spatially Heterogeneous Ecological Footprint

Urban ecological systems are characterized by complex interactions between the natural environment and humans at multiple scales; for an individual urban ecosystem, the strongest interactions may occur at the local or regional spatial scale. At the regional scale, external ecosystems produce resources that are acquired and transported by humans to urban areas, where they are processed and consumed. The assimilation of diffuse human wastes and pollutants also occurs at the regional scale, with much of this process occurring external to the urban system. We developed the urban funnel model to conceptualize the integration of humans into their ecological context. The model captures this pattern and process of resource appropriation and waste generation by urban ecosystems at various spatial scales. This model is applied to individual cities using a modification of traditional ecological footprint (EF) analysis that is spatially explicit; the incorporation of spatial heterogeneity in calculating the EF greatly improves its accuracy. The method for EF analysis can be further modified to ensure that a certain proportion of potential ecosystem services are left for in situ processes. Combining EF models of human appropriation with ecosystem process models would help us to learn more about the effects of ecosystem service appropriation. By comparing the results for food and water, we were able to identify some of the potentially limiting ecological factors for cities. A comparison of the EFs for the 20 largest US cities showed the importance of urban location and interurban competition for ecosystem services. This study underscores the need to take multiple scales and spatial heterogeneity into consideration to expand our current understanding of human–ecosystem interactions. The urban funnel model and the spatially heterogeneous EF provide an effective means of achieving this goal. Received 17 October 2000; accepted 31 May 2001.     

Simple models for complex systems: exploiting the relationship between local and global densities

Simple temporal models that ignore the spatial nature of interactions and track only changes in mean quantities, such as global densities, are typically used under the unrealistic assumption that individuals are well mixed. These so-called mean-field models are often considered overly simplified, given the ample evidence for distributed interactions and spatial heterogeneity over broad ranges of scales. Here, we present one reason why such simple population models may work even when mass-action assumptions do not hold: spatial structure is present but it relates to global densities in a special way. With an individual-based predator–prey model that is spatial and stochastic, and whose mean-field counterpart is the classic Lotka–Volterra model, we show that the global densities and densities of pairs (or spatial covariances) establish a bi-power law at the stationary state and also in their transient approach to this state. This relationship implies that the dynamics of global densities can be written simply as a function of those densities alone without invoking pairs (or higher order moments). The exponents of the bi-power law for the predation rate exhibit a remarkable robustness to changes in model parameters. Evidence is presented for a connection of our findings to the existence of a critical phase transition in the dynamics of the spatial system. We discuss the application of similar modified mean-field equations to other ecological systems for which similar transitions have been described, both in models and empirical data.     

A theory for ecological survey methods to map individual distributions

Spatially explicit approaches are widely recommended for ecosystem management. The quality of the data, such as presence/absence or habitat maps, affects the management actions recommended and is, therefore, key to management success. However, available data are often biased and incomplete. Previous studies have advanced ways to resolve data bias and missing data, but questions remain about how we design ecological surveys to develop a dataset through field surveys. Ecological surveys may have multiple spatial scales, including the spatial extent of the target ecosystem (observation window), the resolution for mapping individual distributions (mapping unit), and the survey area within each mapping unit (sampling unit). We developed an ecological survey method for mapping individual distributions by applying spatially explicit stochastic models. We used spatial point processes to describe individual spatial placements using either random or clustering processes. We then designed ecological surveys with different spatial scales and individual detectability. We found that the choice of mapping unit affected the presence mapped fraction, and the fraction of the total individuals covered by the presence mapped patches. Tradeoffs were found between these quantities and the map resolution, associated with equivalent asymptotic behaviors for both metrics at sufficiently small and large mapping unit scales. Our approach enabled consideration of the effect of multiple spatial scales in surveys, and estimation of the survey outcomes such as the presence mapped fraction and the number of individuals situated in the presence detected units. The developed theory may facilitate management decision-making and inform the design of monitoring and data gathering.     

基于空间分析的保护生物学研究

 保护生物学家和生态学家早就认识到只有准确地辨识保护对象的空间位置、 范围、 及其相邻的关系(例如边缘)和连接度, 以及依存的地形和气候等生境条件, 才能发现生物种群和生境在空间的扩散与收缩、 增长与灭绝的动态, 揭示分布的格局, 从而系统、 全面地了解保护对象和生境的存在状态、 破碎程度和变化趋势, 进行有效的自然保护。 得益于新兴的空间分析技术, 保护生物学自20世纪90年代以来取得了很大的进步。基于空间分析的保护生物学研究是最近10年左右大力发展的新保护生物学的重要基础。 该文结合作者的研究工作,综述了基于空间分析的保护生物学项目, 探讨了保护生物学发展历史、 主要研究方法与应用、 以及今后的可能发展趋势。 在生物多样性的丰度和分布的空间解绎部分,通过综述世界保护监测中心的图解全球生物多样性的工作, 如国家尺度的生物多样性水平、 植物多样性的分布中心和维管束植物科的多样性等的空间分布 ,介绍了 Dobson等图示美国主要濒危植物、 鸟类、 鱼类和软体动物等4个主要类群在县(County) 为基本空间单位上分布的空间格局, 讨论了生物多样性空间解绎的意义。在第二部分用世界资源研究所的全球森林监测(Global forest watch)项目, 美国的国家保护缺失区分析(GAP analysis)项目, 美国林务局的无路自然区域(Roadless area)保护项目和加拿大自然审计(Nature audit)项目, 以及北美和东亚生物多样性空间分布的比较分析和生物入侵的空间分析等具体实例来说明生物多样性空间分布变化比较分析方法的应用。 过去20年来, 面向空间格局的生态学和保护生物学研究得到了快速的发展, 特别是空间格局的描述、 由地统计演变而成的空间统计、 地理信息系统、 基于个体(或栅格)的空间解绎模拟模型、 基于斑块(Patch)的种群理论及其发展(如复合种群理论, 源 汇模型等)等。在第三部分, 以美国森林破碎度空间格局分析和美国太平洋西北演替后期森林的空间格局分析为例, 介绍了空间格局分析在保护生物学中的应用。 同时介绍了澳大利亚保护生态学家Lindenmayer 和美国著名景观生态学家Franklin 2002年提出的模板(Matrix)保护理论,把保护的眼光不局限在面积不多而且分散的保护区中,应注意景观模板和保护区相邻的原生区域的综合保护, 这样将大大扩展保护的范围, 并且平衡保护与发展的关系。最后, 介绍了在保护生物学中已有一定应用的空间模型和模拟, 包括了空间解绎模型(Spatial explicit model)、 基于过程(Process-based)的空间模拟模型、 面向代理(Agent-based)的空间适应模拟模型(SWAM)以及与此有关的动态全球植被模型(DGVM)。 通过上面的讨论和综述, 预测一个新的保护生物学的分支: 空间保护生物学, 已经逐渐成熟问世, 这门基于现代信息技术和空间技术的新学科已经而且还将为全球生物多样性的研究和保育作出重大的贡献。     

Decomposing the heterogeneity of species distributions into multiple scales: a hierarchical framework for large‐scale count surveys

We introduce a novel spatially explicit framework for decomposing species distributions into multiple scales from count data. These kinds of data are usually positively skewed, have non‐normal distributions and are spatially autocorrelated. To analyse such data, we propose a hierarchical model that takes into account the observation process and explicitly deals with spatial autocorrelation. The latent variable is the product of a positive trend representing the non‐constant mean of the species distribution and of a stationary positive spatial field representing the variance of the spatial density of the species distribution. Then, the different scales of emergent structures of the distribution of the population in space are modelled from the latent density of the species distribution using multi‐scale variogram models. Multi‐scale kriging is used to map the spatial patterns previously identified by the multi‐scale models. We show how our framework yields robust and precise estimates of the relevant scales both for spatial count data simulated from well‐defined models, and in a real case‐study based on seabird count data (the common guillemot Uria aalge) provided by large‐scale aerial surveys of the Bay of Biscay (France) performed over a winter. Our stochastic simulation study provides guidelines on the expected uncertainties of the scales estimates. Our results indicate that the spatial structure of the common guillemot can be modelled as a three‐level hierarchical system composed of a very broad‐scale pattern (～ 200 km) with a stable location over time that might be environmentally controlled, a broad‐scale pattern (～ 50 km) with a variable shape and location, that might be related to shifts in prey distribution, and a fine‐scale pattern (～ 10 km) with a rather stable shape and location, that might be controlled by behavioural processes. Our framework enables the development of robust, scale‐dependent hypotheses regarding the potential ecological processes that control species distributions.     

The ecology of information: an overview on the ecological significance of making informed decisions

Information is characterized as the reduction of uncertainty and by a change in the state of a receiving organism. Thus, organisms can acquire information about their environment that reduces uncertainty and increases their likelihood of choosing a best‐matching strategy. We define the ecology of information as the study of how organisms acquire and use information in decision‐making and its significance for populations, communities, landscapes and ecosystems. As a whole, it encompasses the reception and processing of information, decision‐making, and the ecological consequences of making informed decisions. The first two stages constitute the domains of, e.g. sensory ecology and behavioral ecology. The exploration of the consequences of information use at larger spatial and temporal scales in ecology has lagged behind these other disciplines. In our overview we characterize information, discuss statistical decision theory as a quantitative framework to analyze information and decision‐making, and discuss potential ecological ramifications. Rather than attempt a cursory review of the enormity of the scope of information we highlight information use in development, breeding habitat selection, and interceptive eavesdropping on alarm calls. Through these topics we discuss specific examples of ecological information use and the emerging ecological consequences. We emphasize recurring themes: information is collected from multiple sources, over varying temporal and spatial scales, and in many cases links heterospecifics to one another. We conclude by breaking from specific ecological contexts to explore implications of information as a central organizing principle, including: information webs, information as a component of the niche concept, and information as an ecosystem process. With information having such an enormous reach in ecology we further cast a spotlight on the potential harmful effects of anthropogenic noise and info‐disruption.     

Spatial heterogeneity in habitat selection: Nest site selection by greater prairie-chickens

Ecological relationships of animals and their environments are known to vary spatially and temporally across scales. However, common approaches for evaluating resource selection by animals assume that the processes of habitat selection are stationary across space. The assumption that habitat selection is spatially homogeneous may lead to biased inference and ineffective management. We present the first application of geographically weighted logistic regression to habitat selection by a wildlife species. As a case study, we examined nest site selection by greater prairie-chickens at 3 sites with different ecological conditions in Kansas to assess whether the relative importance of habitat features varied across space. We found that 1) nest sites were associated with habitat conditions at multiple spatial scales, 2) habitat associations across spatial scales were correlated, and 3) the influence of habitat conditions on nest site selection was spatially explicit. Post hoc analyses revealed that much of the spatial variability in habitat selection processes was explained at a regional scale. Moreover, habitat features at local spatial scales were more strongly associated with nest site selection in unfragmented grasslands managed intensively for cattle production than they were in fragmented grasslands within a matrix of farmland. Female prairie-chickens exhibited spatial variability in nest site selection at multiple spatial scales, suggesting plasticity in habitat selection behavior. Our results highlight the importance of accounting for spatial heterogeneity when evaluating the ecological effects of habitat components. © 2013 The Wildlife Society.     

Using hierarchical sampling to understand scales of spatial variation in early coral recruitment

Ecological patterns are created by processes acting over multiple spatial and temporal scales. By combining spatially explicit sampling with variance components models, the relative importance of spatial scale to overall variability can be determined. We used a spatially structured experimental design in the Mombasa Marine National Park in Kenya to quantify variation in coral recruitment across four spatial scales (~1–1,000 m) and to generate hypotheses about processes affecting recruitment and potential sources of post-settlement mortality during early life history. For the dominant recruiting corals (Pocillopora spp.), variation in recruitment on surfaces protected from fish grazing was greatest at the largest spatial scale examined (1,000 m). We hypothesize that recruitment on protected surfaces varies mainly with larval delivery due to different lagoonal circulation and water flow between sites. Conversely, variation on surfaces exposed to fishes was greatest at the smallest spatial scale (1 m). We hypothesize that recruitment on exposed surfaces mainly reflects local differences in the scale and intensity of fish grazing, which may obscure larval delivery patterns. Spatial variation in recruitment can affect many ecological processes and factors, including growth, survival to maturity, the distribution of habitat, and variation in species interaction strengths. This study demonstrates how spatially explicit sampling, followed by variance components modeling to partition variance across scales, can help to identify potential drivers of patterns at each relevant scale.     

The effect of static and dynamic spatially structured disturbances on a locally dispersing population

Previous models of locally dispersing populations have shown that in the presence of spatially structured fixed habitat heterogeneity, increasing local spatial autocorrelation in habitat generally has a beneficial effect on such populations, increasing equilibrium population density. It has also been shown that with large-scale disturbance events which simultaneously affect contiguous blocks of sites, increasing spatial autocorrelation in the disturbances has a harmful effect, decreasing equilibrium population density. Here, spatial population models are developed which include both of these spatially structured exogenous influences, to determine how they interact with each other and with the endogenously generated spatial structure produced by the population dynamics. The models show that when habitat is fragmented and disturbance occurs at large spatial scales, the population cannot persist no matter how large its birth rate, an effect not seen in previous simpler models of this type. The behavior of the model is also explored when the local autocorrelation of habitat heterogeneity and disturbance events are equal, i.e. the two effects occur at the same spatial scale. When this scale parameter is very small, habitat fragmentation prevents the population from persisting because sites attempting to reproduce will drop most of their offspring on unsuitable sites; when the parameter is very large, large-scale disturbance events drive the population to extinction. Population levels reach their maximum at intermediate values of the scale parameter, and the critical values in the model show that the population will persist most easily at these intermediate scales of spatial influences. The models are investigated via spatially explicit stochastic simulations, traditional (infinite-dispersal) and improved (local-dispersal) mean-field approximations, and pair approximations.     

The effect of dispersal and neighbourhood in games of cooperation

The prisoner's dilemma (PD) and the snowdrift (SD) games are paradigmatic tools to investigate the origin of cooperation. Whereas spatial structure (e.g. nonrandom spatial distribution of strategies) present in the spatially explicit models facilitates the emergence of cooperation in the PD game, recent investigations have suggested that spatial structure can be unfavourable for cooperation in the SD game. The frequency of cooperators in a spatially explicit SD game can be lower than it would be in an infinitely large well-mixed population. However, the source of this effect cannot be identified with certainty as spatially explicit games differ from well-mixed games in two aspects: (i) they introduce spatial correlations, (ii) and limited neighbourhood. Here we extend earlier investigations to identify the source of this effect, and thus accordingly we study a spatially explicit version of the PD and SD games with varying degrees of dispersal and neighbourhood size. It was found that dispersal favours selfish individuals in both games. We calculated the frequency of cooperators at strong dispersal limit, which in concordance with the numerical results shows that it is the short range of interactions (i.e. limited neighbourhood) and not spatial correlations that decreases the frequency of cooperators in spatially explicit models of populations. Our results demonstrate that spatial correlations are always beneficial to cooperators in both the PD and SD games. We explain the opposite effect of dispersal and neighbourhood structure, and discuss the relevance of distinguishing the two effects in general.