Arrhenius and Gleason revisited: new hybrid models resolve an old controversy

Aim Studies have typically employed species–area relationships (SARs) from sample areas to fit either the power relationship or the logarithmic (exponential) relationship. However, the plots from empirical data often fall between these models. This article proposes two complementary and hybrid models as solutions to the controversy regarding which model best fits sample‐area SARs. Methods The two models are and , where S_A is number of species in an area, A, where z, b, c₁ and c₂ are predetermined parameters found by calculation, and where d and n are parameters to be fitted. The number of parameters is reduced from six to two by fixing the model at either end of the scale window of the data set, a step that is justified by the condition that the error or the bias, or both, in the first and the last data points is negligible. The new hybrid models as well as the power model and the logarithmic model are fitted to 10 data sets. Results The two proposed models fit well not only to Arrhenius’ and Gleason’s data sets, but also to the other six data sets. They also provide a good fit to data sets that follow a sigmoid (or triphasic) shape in log–log space and to data sets that do not fall between the power model and the logarithmic model. The log‐transformation of the dependent variable, S, does not affect the curve fit appreciably, although it enhances the performance of the new models somewhat. Main conclusions Sample‐area SARs have previously been shown to be convex upward, convex downward (concave), sigmoid and inverted sigmoid in log–log space. The new hybrid models describe successfully data sets with all these curve shapes, and should therefore produce good fits also to what are termed triphasic SARs.     

Effects of sampling protocol on the shapes of species richness curves

Aim Scheiner (Journal of Biogeography, 2009, 36 , 2005–2008) criticized several issues regarding the typology and analysis of species richness curves that were brought forward by Dengler (Journal of Biogeography, 2009, 36 , 728–744). In order to test these two sets of views in greater detail, we used a simulation model of ecological communities to demonstrate the effects of different sampling schemes on the shapes of species richness curves and their extrapolation capability. Methods We simulated five random communities with 100 species on a 64 × 64 grid using random fields. Then we sampled species–area relationships (SARs, contiguous plots) as well as species–sampling relationships (SSRs, non‐contiguous plots) from these communities, both for the full extent and the central quarter of the grid. Finally, we fitted different functions (power, quadratic power, logarithmic, Michaelis–Menten, Lomolino) to the obtained data and assessed their goodness‐of‐fit (Akaike weights) and their extrapolation capability (deviation of the predicted value from the true value). Results We found that power functions gave the best fit for SARs, while for SSRs saturation functions performed better. Curves constructed from data of 32² grid cells gave reasonable extrapolations for 64² grid cells for SARs, irrespective of whether samples were gathered from the full extent or the centre only. By contrast, SSRs worked well for extrapolation only in the latter case. Main conclusions SARs and SSRs have fundamentally different curve shapes. Both sampling strategies can be used for extrapolation of species richness to a target area, but only SARs allow for extrapolation to a larger area than that sampled. These results confirm a fundamental difference between SARs and area‐based SSRs and thus support their typological differentiation.     

Shapes and functions of species–area curves: a review of possible models

Aim This paper reviews possible candidate models that may be used in theoretical modelling and empirical studies of species–area relationships (SARs). The SAR is an important and well‐proven tool in ecology. The power and the exponential functions are by far the models that are best known and most frequently applied to species–area data, but they might not be the most appropriate. Recent work indicates that the shape of species–area curves in arithmetic space is often not convex but sigmoid and also has an upper asymptote. Methods Characteristics of six convex and eight sigmoid models are discussed and interpretations of different parameters summarized. The convex models include the power, exponential, Monod, negative exponential, asymptotic regression and rational functions, and the sigmoid models include the logistic, Gompertz, extreme value, Morgan–Mercer–Flodin, Hill, Michaelis–Menten, Lomolino and Chapman–Richards functions plus the cumulative Weibull and beta‐P distributions. Conclusions There are two main types of species–area curves: sample curves that are inherently convex and isolate curves, which are sigmoid. Both types may have an upper asymptote. A few have attempted to fit convex asymptotic and/or sigmoid models to species–area data instead of the power or exponential models. Some of these or other models reviewed in this paper should be useful, especially if species–area models are to be based more on biological processes and patterns in nature than mere curve fitting. The negative exponential function is an example of a convex model and the cumulative Weibull distribution an example of a sigmoid model that should prove useful. A location parameter may be added to these two and some of the other models to simulate absolute minimum area requirements.     

The species–area relationship in centipedes (Myriapoda: Chilopoda): a comparison between Mediterranean island groups

The present study article examines the shapes of centipede species–area relationships (SARs) in the Mediterranean islands, compares the results of the linear form of the power model between archipelagos, discusses biological significance of the power model parameters with other taxa on the Aegean archipelago, and tests for a significant small‐island effect (SIE). We used 11 models to test the SARs and we compared the quality‐of‐fit of all candidate models. The power function ranked first and Z‐values was in the range 0.106–0.334. We assessed the presence of SIEs by fitting both a continuous and discontinuous breakpoint regression model. The continuous breakpoint regression functions never performed much better than the closest discontinuous model as a predictor of centipede species richness. We suggest that the relatively low Z‐values in our data partly reflect better dispersal abilities in centipedes than in other soil invertebrate taxa. Longer periods of isolation and more recent island formation may explain the somewhat lower constant c in the western Mediterranean islands compared to the Aegean islands. Higher breakpoint values in the western Mediterranean may also be a result of larger distance to the mainland and longer separation times. Despite the differences in the geological history and the idiosyncratic features of the main island groups considered, the overall results are quite similar and this could be assigned to the ability of centipedes to disperse across isolation barriers. © 2011 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, 105 , 146–159.     

吉林蛟河42 hm2针阔混交林样地植物种-面积关系

 种-面积关系是生态学中的基本问题, 其构建方式对种-面积关系的影响以及最优种-面积模型的选择仍然存在争议。该文利用吉林蛟河42 hm²针阔混交林样地数据, 分别采用巢式样方法和随机样方法建立对数模型、幂函数模型和逻辑斯蒂克模型, 并通过赤池信息量准则(AIC)检验种-面积模型优度。结果表明, 种-面积关系受到取样方法的影响, 随机样方法的拟合效果优于巢式样方法。采用随机样方法构建的幂指数模型(AIC = 89.11)和逻辑斯蒂克模型(AIC = 71.21)优于对数模型(AIC = 113.81)。根据AIC值可知, 随机样方法构建的逻辑斯蒂克模型是拟合42 hm²针阔混交林样地种-面积关系的最优模型。该研究表明: 在分析种-面积关系时不仅应考虑尺度效应, 还需注意生境变化及群落演替的影响。     

Species–area relationship and a tentative interpretation of the function coefficients in an ecosystem simulation

In this paper, we identified the best species–area relationship (SAR) models from amongst 28 different models gathered from the literature, using an artificial predator–prey simulation (EcoSim), along with investigating how sampling approaches and sampling scales affect SARs. Further, we attempted to determine a plausible interpretation of SAR model coefficients for the best performing SAR models. This is the most extensive quantitatively based investigation of the species–area relationship so far undertaken in the literature.We gathered 28 different models from the literature and fitted them to sampling data from EcoSim using non-linear regression and ΔAICc as the goodness-of-fit criterion. Afterwards, we proposed a machine-learning approach to find plausible relationships between the models’ coefficients and the spatial information that likely affect SARs, as a basis for extracting rules that provide an interpretation of SAR coefficients.We found the power function family to be a reasonable choice and in particular the Plotkin function based on ΔAICc ranking. The Plotkin function was consistently in the top three in terms of the best ranked SAR functions. Furthermore, the simple power function was the best-ranked model in nested sampling amongst models with two coefficients. We found that the Plotkin, quadratic power, Morgan–Mercer–Floid and the generalized cumulative Weibull functions are the best ranked models for small, intermediate, large, and very large scales, respectively, in nested sampling, while Plotkin (in small to intermediate scales) and Chapman–Richards (in large to very large scales) are the best ranked functions in random sampling. Finally, based on rule extractions using machine-learning techniques we were able to find interpretations of the coefficients for the simple and extended power functions. For instance, function coefficients corresponded to sampling scale size, patch number, fractal dimension, average patch size, and spatial complexity.Our main conclusions are that SAR models are highly dependent on sampling scale and sampling approach and that the shape of the best ranked SAR model is convex without an asymptote for smaller scales (small, intermediate) and it is sigmoid for larger scales (large and very large). For some of the SAR model coefficients, there are clear correlations with spatial information, thereby providing an interpretation of these coefficients. In addition, the slope z measuring the rate of species increase for SAR models in the power function family was found to be directly proportional to beta diversity, which confirms the view that beta diversity and SAR models are to some extent both measures of species richness.     

Which function describes the species–area relationship best? A review and empirical evaluation

Aim The aims of this study are to resolve terminological confusion around different types of species–area relationships (SARs) and their delimitation from species sampling relationships (SSRs), to provide a comprehensive overview of models and analytical methods for SARs, to evaluate these theoretically and empirically, and to suggest a more consistent approach for the treatment of species–area data. Location Curonian Spit in north‐west Russia and archipelagos world‐wide. Methods First, I review various typologies for SARs and SSRs as well as mathematical models, fitting procedures and goodness‐of‐fit measures applied to SARs. This results in a list of 23 function types, which are applicable both for untransformed (S) and for log‐transformed (log S) species richness. Then, example data sets for nested plots in continuous vegetation (n = 14) and islands (n = 6) are fitted to a selection of 12 function types (linear, power, logarithmic, saturation, sigmoid) both for S and for log S. The suitability of these models is assessed with Akaike’s information criterion for S and log S, and with a newly proposed metric that addresses extrapolation capability. Results SARs, which provide species numbers for different areas and have no upper asymptote, must be distinguished from SSRs, which approach the species richness of one single area asymptotically. Among SARs, nested plots in continuous ecosystems, non‐nested plots in continuous ecosystems, and isolates can be distinguished. For the SARs of the empirical data sets, the normal and quadratic power functions as well as two of the sigmoid functions (Lomolino, cumulative beta‐P) generally performed well. The normal power function (fitted for S) was particularly suitable for predicting richness values over ten‐fold increases in area. Linear, logarithmic, convex saturation and logistic functions generally were inappropriate. However, the two sigmoid models produced unstable results with arbitrary parameter estimates, and the quadratic power function resulted in decreasing richness values for large areas. Main conclusions Based on theoretical considerations and empirical results, I suggest that the power law should be used to describe and compare any type of SAR while at the same time testing whether the exponent z changes with spatial scale. In addition, one should be aware that power‐law parameters are significantly influenced by methodology.     

A Symmetric Extended Logistic Model with Applications to Experimental Toxicity Data

The fit of the logit and probit models for quantal response data can be improved by embedding these classical models within a richer parametric family indexed by one or two shape parameters. In this paper, a symmetric extended logistic model indexed by a shape parameter λ is discussed with application to dose response curves. The usual maximum likelihood method is employed to estimate the parameters of the model. The need to include the shape parameter λ is illustrated by analyzing a set of real experimental data and comparing the fit of the extended logistic model to those obtained by the standard logit and probit models.     

The importance of samples and isolates for species–area relationships

Species–area relationships (SARs) are a key tool for understanding patterns of species diversity. A framework for the interpretation of SARs and their prediction under different landscape configurations remains elusive, however. This article addresses one of these configurations: how species' minimum-area requirements affect the shape of island or other isolate Species–area curves. We distinguish between two classes of SARs: sample-area curves, compiled entirely within larger contiguous areas, and isolate curves, compiled between isolated areas. We develop this conceptual and graphic model in order to illuminate landscape-scale diversity patterns, to discuss how various landscape and species characteristics affect outcomes, and to investigate the dynamics of local extinction under conditions of habitat fragmentation. Minimum-area effects on actual islands and other isolates predictably cause Species–area curves either to be sigmoid in arithmetic space or to be lowered for smaller areas. In order to illustrate the inherent shape of isolate curves, this study fits convex and sigmoid regression models to empirical isolate (island) data sets that cover the small scales expected to include inflection points.     

Sampling-Design Effects on Properties of Species-Area Relationships – A Case Study from Estonian Dry Grassland Communities

Despite widespread use of species-area relationships (SARs), dispute remains over the most representative SAR model. Using data of small-scale SARs of Estonian dry grassland communities, we address three questions: (1) Which model describes these SARs best when known artifacts are excluded? (2) How do deviating sampling procedures (marginal instead of central position of the smaller plots in relation to the largest plot; single values instead of average values; randomly located subplots instead of nested subplots) influence the properties of the SARs? (3) Are those effects likely to bias the selection of the best model? Our general dataset consisted of 16 series of nested-plots (1 cm²–100 m², any-part system), each of which comprised five series of subplots located in the four corners and the centre of the 100-m² plot. Data for the three pairs of compared sampling designs were generated from this dataset by subsampling. Five function types (power, quadratic power, logarithmic, Michaelis-Menten, Lomolino) were fitted with non-linear regression. In some of the communities, we found extremely high species densities (including bryophytes and lichens), namely up to eight species in 1 cm² and up to 140 species in 100 m², which appear to be the highest documented values on these scales. For SARs constructed from nested-plot average-value data, the regular power function generally was the best model, closely followed by the quadratic power function, while the logarithmic and Michaelis-Menten functions performed poorly throughout. However, the relative fit of the latter two models increased significantly relative to the respective best model when the single-value or random-sampling method was applied, however, the power function normally remained far superior. These results confirm the hypothesis that both single-value and random-sampling approaches cause artifacts by increasing stochasticity in the data, which can lead to the selection of inappropriate models.     

Species‐area relationships of lemurs in a fragmented landscape in Madagascar

We used species‐area relationships (SARs) to investigate the effects of habitat loss on lemur biogeography. We measured species richness via visual surveys on line transects within 42 fragments of dry deciduous forest at the Ambanjabe field site in Ankarafantsika National Park, Madagascar. We measured human disturbance and habitat characteristics within 38 of the 42 fragments. We measured the distance between each fragment and the nearest settlement, continuous forest, and nearest neighboring fragment. We fit 10 candidate SAR models to the data using nonlinear least squares regression and compared them using Akaike's Information Criterion (AIC). To determine how habitat characteristics, as well as area, influenced species richness, we ran a hierarchical partitioning procedure to select which variables to include in generalized additive models (GAMs) and compared them using AIC. Contrary to expectations, we found that lemurs form convex SARs, without a “small island effect”, and with the power model being the most likely SAR model. Although we found that four variables (area, survey effort, and total human disturbance, and mean tree height) independently contributed greater than 10% of the variation in lemur species richness, only area was included in the most likely model. We suggest that the power model was the most likely SAR model and our inability to detect a “small island effect” are the result of Microcebus spp. being edge tolerant and capable of dispersing through matrix, scale issues in the study design, and low γ‐diversity in the landscape. However, more study is needed to determine what role human disturbance plays in influencing species richness in lemurs.     

Island biogeography of the Mediterranean sea: the species–area relationship for terrestrial isopods

Aim We looked at the biogeographical patterns of Oniscidean fauna from the small islands of the Mediterranean Sea in order to investigate the species–area relationship and to test for area‐range effects. Location The Mediterranean Sea. Methods We compiled from the literature a data set of 176 species of Oniscidea (terrestrial isopods) distributed over 124 Mediterranean islands. Jaccard's index was used as input for a UPGMA cluster analysis. The species–area relationship was investigated by applying linear, semi‐logarithmic, logarithmic and sigmoid models. We also investigated a possible ‘small island effect’ (SIE) by performing breakpoint regression. We used a cumulative and a sliding‐window approach to evaluate scale‐dependent area‐range effects on the log S/log A regression parameters. Results Based on similarity indexes, results indicated that small islands of the Mediterranean Sea can be divided into two major groups: eastern and western. In general, islands from eastern archipelagos were linked together at similarity values higher than those observed for western Mediterranean islands. This is consistent with a more even distribution of species in the eastern Mediterranean islands. Separate archipelagos in the western Mediterranean could be discriminated, with the exception of islets, which tended to group together at the lowest similarity values regardless of the archipelago to which they belong. Islets were characterized by a few common species with large ranges. The species–area logarithmic model did not always provide the best fit. Most continental archipelagos showed very similar intercepts, higher than the intercept for the Canary island oceanic archipelago. Sigmoid regression returned convex curves. Evidence for a SIE was found, whereas area‐range effects that are dependent on larger scale analyses were not unambiguously supported. Main conclusions The Oniscidea fauna from small islands of the Mediterranean Sea is highly structured, with major and minor geographical patterns being identifiable. Some but not all of the biogeographical complexity can be explained by interpreting the different shapes of species–area curves. Despite its flexibility, the sigmoid model tested did not always provide the best fit. Moreover, when the model did provide a good fit the curves looked convex, not sigmoid. We found evidence for a SIE, and minor support for scale‐dependent area‐range effects.     

sars: an R package for fitting,evaluating and comparing species–area relationship models

The species–area relationship (SAR) constitutes one of the most general ecological patterns globally. A number of different SAR models have been proposed. Recent work has shown that no single model universally provides the best fit to empirical SAR datasets: multiple models may be of practical and theoretical interest. However, there are no software packages available that a) allow users to fit the full range of published SAR models, or b) provide functions to undertake a range of additional SAR‐related analyses. To address these needs, we have developed the R package ‘sars’ that provides a wide variety of SAR‐related functionality. The package provides functions to: a) fit 20 SAR models using non‐linear and linear regression, b) calculate multi‐model averaged curves using various information criteria, and c) generate confidence intervals using bootstrapping. Plotting functions allow users to depict and scrutinize the fits of individual models and multi‐model averaged curves. The package also provides additional SAR functionality, including functions to fit, plot and evaluate the random placement model using a species–sites abundance matrix, and to fit the general dynamic model of oceanic island biogeography. The ‘sars’ R package will aid future SAR research by providing a comprehensive set of simple to use tools that enable in‐depth exploration of SARs and SAR‐related patterns. The package has been designed to allow other researchers to add new functions and models in the future and thus the package represents a resource for future SAR work that can be built on and expanded by workers in the field.     

Choosing the right sigmoid growth function using the unified‐models approach

Growth of the young is an important part of the life history in birds. However, modelling methods have paid little attention to the choice of regression model used to describe its pattern. The aim of this study was to evaluate whether a single sigmoid model with an upper asymptote could describe avian growth adequately. We compared unified versions of five growth models of the Richards family (the four‐parameter U‐Richards and the three‐parameter U‐logistic, U‐Gompertz, U‐Bertalanffy and U4‐models) for three traits (body mass, tarsus‐length and wing‐length) for 50 passerine species, including species with varied morphologies and life histories. The U‐family models exhibit a unified set of parameters for all models. The four‐parameter U‐Richards model proved a good choice for fitting growth curves to various traits – its extra d‐parameter allows for a flexible placement of the inflection point. Which of the three‐parameter U‐models was the best performing varied greatly between species and between traits, as each three‐parameter model had a different fixed relative inflection value (fraction of the upper asymptote), implying a different growth pattern. Fixing the asymptotes to averages for adult trait value generally shifted the model preference towards one with lower relative inflection values. Our results illustrate an overlooked difficulty in the analysis of organismal growth, namely, that a single traditional three‐parameter model does not suit all growth data. This is mostly due to differences in inflection placement. Moreover, some biometric traits require more attention when estimating growth rates and other growth‐curve characteristics. We recommend fitting either several three‐parameter models from the U‐family, where the parameters are comparable between models, or only the U‐Richards model.     

Evolutionary model selection with a genetic algorithm: a case study using stem RNA

The choice of a probabilistic model to describe sequence evolution can and should be justified. Underfitting the data through the use of overly simplistic models may miss out on interesting phenomena and lead to incorrect inferences. Overfitting the data with models that are too complex may ascribe biological meaning to statistical artifacts and result in falsely significant findings. We describe a likelihood-based approach for evolutionary model selection. The procedure employs a genetic algorithm (GA) to quickly explore a combinatorially large set of all possible time-reversible Markov models with a fixed number of substitution rates. When applied to stem RNA data subject to well-understood evolutionary forces, the models found by the GA 1) capture the expected overall rate patterns a priori; 2) fit the data better than the best available models based on a priori assumptions, suggesting subtle substitution patterns not previously recognized; 3) cannot be rejected in favor of the general reversible model, implying that the evolution of stem RNA sequences can be explained well with only a few substitution rate parameters; and 4) perform well on simulated data, both in terms of goodness of fit and the ability to estimate evolutionary rates. We also investigate the utility of several distance measures for comparing and contrasting inferred evolutionary models. Using widely available small computer clusters, our approach allows, for the first time, to evaluate the performance of existing RNA evolutionary models by comparing them with a large pool of candidate models and to validate common modeling assumptions. In addition, the new method provides the foundation for rigorous selection and comparison of substitution models for other types of sequence data.     

Competing regression models for longitudinal data

The choice of an appropriate family of linear models for the analysis of longitudinal data is often a matter of concern for practitioners. To attenuate such difficulties, we discuss some issues that emerge when analyzing this type of data via a practical example involving pretest–posttest longitudinal data. In particular, we consider log‐normal linear mixed models (LNLMM), generalized linear mixed models (GLMM), and models based on generalized estimating equations (GEE). We show how some special features of the data, like a nonconstant coefficient of variation, may be handled in the three approaches and evaluate their performance with respect to the magnitude of standard errors of interpretable and comparable parameters. We also show how different diagnostic tools may be employed to identify outliers and comment on available software. We conclude by noting that the results are similar, but that GEE‐based models may be preferable when the goal is to compare the marginal expected responses.     

Determination of the unknown age at first capture of western rock lobsters (Panulirus cygnus) by random effects model

We propose a method for fitting growth curves to multiple recapture data of lobsters when the age at first capture is unknown. The von Bertalanffy growth curve is used to model the growth. To account for individual variability, the unknown age in logarithmic scale of a lobster at first capture, the individual asymptotic size, and the individual growth coefficient of its carapace length are modeled as random effects with a trivariate normal distribution. Unlike previously suggested models, the present model permits correlation between the growth coefficient and the age at first capture and can be fitted readily using existing software. The error structures between consecutive recaptures of a lobster are assumed to be a first-order autoregressive process with unequally spaced time points. A comparison between this model and the Fabens growth equation is given. The proposed method is a flexible method and can be applied to fit different growth equations when the age at first capture is unknown.     

Sampling design and the shape of species–area curves on the regional scale

Species–area relationships (SARs) are one of the fundamental patterns in ecology. However, how the way they were constructed influences resulting SAR shapes has gained astonishingly little attention. We use data of the distribution atlas of Polish butterflies to compare SARs constructed from four different designs: adding up species numbers of independent areas (species accumulation curves using contiguous and non-contiguous areas), using a nested design, and comparing species numbers of independent areas of different sizes. It appeared that the way of constructing SARs influences the outcome. We attribute this influence to the pronounced faunal dissimilarities of more distant areas (spatial species turnover). The nested design resulted in significantly higher slopes and lower intercepts of power function SARs than the other designs. SARs from all four sampling designs showed a pronounced downward curvature on small spatial scales. Only the nested design predicted species densities correctly. The implications of these results for the use of SARs in bioconservation are discussed.     

Bird species numbers in an archipelago of reeds at Lake Velence, Hungary

1 Bird species numbers were studied on 109 reed islands at Lake Velence, Hungary, in the 1993 and 1994 breeding seasons. The aim was to describe and account for the abundance and distribution patterns of the bird species. 2 It was expected that an exponential model would fit the calculated species–area curves. However, for the 1993 data, both the power function (LogS ~ LogArea) and the exponential (S ~ LogArea) models did so, while the power function, exponential and linear (S ~ A) models fitted the curves for the 1994 data. 3 The results showed that the pattern was not random: a collection of small islands held more species than a few large islands with the same total area. 4 The relative species richness of small islands is a result of the preference of most common passerine bird species for the edges of reed islands. Most individuals were found in the first 5 m of the reedbed, and no edge avoidance was detected on a local spatial scale. Large, rarer species (e.g. Great White Egret), however, were found to be dependent on large reed islands. 5 Comparison of results with two other studies on bird communities of reed islands revealed that the type of landscape matrix (e.g. deep water, shallow water or agricultural lands) among reed patches significantly influences bird communities. Deep water was dominated by grebes and coot, shallow water by reed‐nesting passerines, and farmed areas by reed‐ and bush‐nesting passerines.