A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove

The spread of invasive species is a long studied subject that garners much interest in the ecological research community. Historically the phenomenon has been approached using a purely deterministic mathematical framework (usually involving differential equations of some form). These methods, while scientifically meaningful, are generally highly simplified and fail to account for uncertainty in the data and process, of which our knowledge could not possibly exist without error. We propose a hierarchical Bayesian model for population spread that accommodates data sources with errors, dependence structures between population dynamics parameters, and takes into account prior scientific understanding via non-linear relationships between model parameters and space-time response variables. We model the process (i.e., the bird population in this case) as a Poisson response with spatially varying diffusion coefficients as well as a logistic population growth term using a common reaction-diffusion equation that realistically mimics the ecological process. We focus the application on the ongoing invasion of the Eurasian Collared-Dove.     

A Bayesian hierarchical framework for calibrating aquatic biogeochemical models

Model practitioners increasingly place emphasis on rigorous quantitative error analysis in aquatic biogeochemical models and the existing initiatives range from the development of alternative metrics for goodness of fit, to data assimilation into operational models, to parameter estimation techniques. However, the treatment of error in many of these efforts is arguably selective and/or ad hoc. A Bayesian hierarchical framework enables the development of robust probabilistic analysis of error and uncertainty in model predictions by explicitly accommodating measurement error, parameter uncertainty, and model structure imperfection. This paper presents a Bayesian hierarchical formulation for simultaneously calibrating aquatic biogeochemical models at multiple systems (or sites of the same system) with differences in their trophic conditions, prior precisions of model parameters, available information, measurement error or inter-annual variability. Our statistical formulation also explicitly considers the uncertainty in model inputs (model parameters, initial conditions), the analytical/sampling error associated with the field data, and the discrepancy between model structure and the natural system dynamics (e.g., missing key ecological processes, erroneous formulations, misspecified forcing functions). The comparison between observations and posterior predictive monthly distributions indicates that the plankton models calibrated under the Bayesian hierarchical scheme provided accurate system representations for all the scenarios examined. Our results also suggest that the Bayesian hierarchical approach allows overcoming problems of insufficient local data by “borrowing strength” from well-studied sites and this feature will be highly relevant to conservation practices of regions with a high number of freshwater resources for which complete data could never be practically collected. Finally, we discuss the prospect of extending this framework to spatially explicit biogeochemical models (e.g., more effectively connect inshore with offshore areas) along with the benefits for environmental management, such as the optimization of the sampling design of monitoring programs and the alignment with the policy practice of adaptive management.     

Lifting a veil on diversity: a Bayesian approach to fitting relative-abundance models.

Bayesian methods incorporate prior knowledge into a statistical analysis. This prior knowledge is usually restricted to assumptions regarding the form of probability distributions of the parameters of interest, leaving their values to be determined mainly through the data. Here we show how a Bayesian approach can be applied to the problem of drawing inference regarding species abundance distributions and comparing diversity indices between sites. The classic log series and the lognormal models of relative- abundance distribution are apparently quite different in form. The first is a sampling distribution while the other is a model of abundance of the underlying population. Bayesian methods help unite these two models in a common framework. Markov chain Monte Carlo simulation can be used to fit both distributions as small hierarchical models with shared common assumptions. Sampling error can be assumed to follow a Poisson distribution. Species not found in a sample, but suspected to be present in the region or community of interest, can be given zero abundance. This not only simplifies the process of model fitting, but also provides a convenient way of calculating confidence intervals for diversity indices. The method is especially useful when a comparison of species diversity between sites with different sample sizes is the key motivation behind the research. We illustrate the potential of the approach using data on fruit-feeding butterflies in southern Mexico. We conclude that, once all assumptions have been made transparent, a single data set may provide support for the belief that diversity is negatively affected by anthropogenic forest disturbance. Bayesian methods help to apply theory regarding the distribution of abundance in ecological communities to applied conservation.     

Population dynamics and control of invasive Spartina alterniflora: inference and forecasting under uncertainty

Managing invaded ecosystems entails making decisions about control strategies in the face of scientific uncertainty and ecological stochasticity. Statistical tools such as model selection and Bayesian decision analysis can guide decision-making by estimating probabilities of outcomes under alternative management scenarios, but these tools have seldom been applied in invasion ecology. We illustrate the use of model selection and Bayesian methods in a case study of smooth cordgrass (Spartina alterniflora) invading Willapa Bay, Washington. To address uncertainty in model structure, we quantified the weight of evidence for two previously proposed hypotheses, that S. alterniflora recruitment varies with climatic conditions (represented by sea surface temperature) and that recruitment is subject to an Allee effect due to pollen limitation. By fitting models to time series data, we found strong support for climate effects, with higher per capita seedling production in warmer years, but no evidence for an Allee effect based on either the total area invaded or the mean distance between neighboring clones. We used the best-supported model to compare alternative control strategies, incorporating uncertainty in parameter estimates and population dynamics. For a fixed annual removal effort, the probability of eradication in 10 years was highest, and final invaded area lowest, if removals targeted the smallest clones rather than the largest or randomly selected clones. The relationship between removal effort and probability of eradication was highly nonlinear, with a sharp threshold separating -0% and -100% probability of success, and this threshold was 95% lower in simulations beginning early rather than late in the invasion. This advantage of a rapid response strategy is due to density-dependent population growth, which produces alternative stable equilibria depending on the initial invasion size when control begins. Our approach could be applied to a wide range of invasive species management problems where appropriate data are available.     

Sensitivity of ecological models to their climate drivers: statistical ensembles for forcing.

Global and regional numerical models for terrestrial ecosystem dynamics require fine spatial resolution and temporally complete historical climate fields as input variables. However, because climate observations are unevenly spaced and have incomplete records, such fields need to be estimated. In addition, uncertainty in these fields associated with their estimation are rarely assessed. Ecological models are usually driven with a geostatistical model's mean estimate (kriging) of these fields without accounting for this uncertainty, much less evaluating such errors in terms of their propagation in ecological simulations. We introduce a Bayesian statistical framework to model climate observations to create spatially uniform and temporally complete fields, taking into account correlation in time and space, spatial heterogeneity, lack of normality, and uncertainty about all these factors. A key benefit of the Bayesian model is that it generates uncertainty measures for the generated fields. To demonstrate this method, we reconstruct historical monthly precipitation fields (a driver for ecological models) on a fine resolution grid for a climatically heterogeneous region in the western United States. The main goal of this work is to evaluate the sensitivity of ecological models to the uncertainty associated with prediction of their climate drivers. To assess their numerical sensitivity to predicted input variables, we generate a set of ecological model simulations run using an ensemble of different versions of the reconstructed fields. We construct such an ensemble by sampling from the posterior predictive distribution of the climate field. We demonstrate that the estimated prediction error of the climate field can be very high. We evaluate the importance of such errors in ecological model experiments using an ensemble of historical precipitation time series in simulations of grassland biogeochemical dynamics with an ecological numerical model, Century. We show how uncertainty in predicted precipitation fields is propagated into ecological model results and that this propagation had different modes. Depending on output variable, the response of model dynamics to uncertainty in inputs ranged from uncertainty in outputs that matched that of inputs to those that were muted or that were biased, as well as uncertainty that was persistent in time after input errors dropped.     

Modelling the dynamics of coral reef macroalgae using a Bayesian belief network approach

Macroalgae are a major benthic component of coral reefs and their dynamics influence the resilience of coral reefs to disturbance. However, the relative importance of physical and ecological processes in driving macroalgal dynamics is poorly understood. Here we develop a Bayesian belief network (BBN) model to integrate many of these processes and predict the growth of coral reef macroalgae. Bayesian belief networks use probabilistic relationships rather than deterministic rules to quantify the cause and effect assumptions. The model was developed using both new empirical data and quantified relationships elicited from previous studies. We demonstrate the efficacy of the BBN to predict the dynamics of a common Caribbean macroalgal genus Dictyota. Predictions of the model have an average accuracy of 55% (implying that 55% of the predicted categories of Dictyota cover were assigned to the correct class). Sensitivity analysis suggested that macroalgal dynamics were primarily driven by top–down processes of grazing rather than bottom–up nutrification. BBNs provide a useful framework for modelling complex systems, identifying gaps in our scientific understanding and communicating the complexities of the associated uncertainties in an explicit manner to stakeholders. We anticipate that accuracies will improve as new data are added to the model.     

Do microbial processes regulate the stability of a coral atoll's enclosed pelagic ecosystem?

Complex marine ecosystems contain multiple feedback cycles that can cause unexpected responses to perturbations. To better predict these responses, complicated models are increasingly being developed to enable the study of feedback cycles. However, the sparseness of ecological data often limits the direct empirical parameterization of all model parameters. Here we use a Bayesian inverse analysis approach to synthesize empirical data and ecological theory derived from published studies of a coral atoll's enclosed pelagic ecosystem (Takapoto Atoll, French Polynesia). We then use the estimates of flux magnitudes to parameterize probabilistic compartment models with two forms of heterotrophic consumption: (1) “bottom-up” donor-controlled heterotrophic consumption and (2) “top-down” mass-action heterotrophic consumption. We explore how the flux magnitudes affect the ecosystem's stability properties of resilience, reactivity, and resistance under both assumptions for heterotrophic consumption. The models suggest that the microbial uptake of dissolved organic carbon (DOC) regulates the long term rate of return to steady state following a temporary or pulse perturbation (resilience), and the cycling of carbon between abiotic pools and heterotrophic compartments regulates the short-term response (reactivity). In the bottom-up process model, the sensitivity of steady state masses following a sustained or press perturbation (resistance) is highest for the DOC pool following a sustained change to the microbial uptake rate of DOC. Further, a change in the microbial uptake of DOC propagates through the ecosystem and affects the steady state values of zooplankton. The analysis suggests that the food web is highly dependent on the recycling between the abiotic and biotic carbon pools, particularly as mediated by the microbial consumption of DOC, and this recycling determines how the ecosystem responds to perturbations.     

Linking fate model in freshwater and PBPK model to assess human internal dosimetry of B(a)P associated with drinking water

In the present study, we demonstrate an integrated modeling approach for predicting internal tissue concentrations of chemicals by coupling a multimedia environmental model and a generic physiologically based pharmacokinetic (PBPK) model. A case study was designed for a region situated on the Seine river watershed, downstream of the Paris megacity, and for benzo(a)pyrene emitted from industrial zones in the region. In this case study, these two models are linked only by water intake from riverine system for the multimedia model into human body for the PBPK model. The limited monitoring data sets of B(a)P concentrations in bottom sediment and in raw river water, obtained at the downstream of Paris, were used to re-construct long-term daily concentrations of B(a)P in river water. The re-construction of long-term series of B(a)P level played a key role for the intermediate model calibration (conducted in multimedia model) and thus for improving model input to PBPK model. In order to take into account the parametric uncertainty in the model inputs, some input parameters relevant for the multimedia model were given by probability density functions (PDFs); some generic PDFs were updated with site-specific measurements by a Bayesian approach. The results of this study showed that the multimedia model fits well with actual annual measurements in sediments over one decade. No accumulation of B(a)P in the organs was observed. In conclusion, this case study demonstrated the feasibility of a full-chain assessment combining multimedia environmental predictions and PBPK modeling, including uncertainty and sensitivity analyses.     

A geostatistical approach for mapping thematic classification accuracy and evaluating the impact of inaccurate spatial data on ecological model predictions

Spatial information in the form of geographical information system coverages and remotely sensed imagery is increasingly used in ecological modeling. Examples include maps of land cover type from which ecologically relevant properties, such as biomass or leaf area index, are derived. Spatial information, however, is not error-free: acquisition and processing errors, as well as the complexity of the physical processes involved, make remotely sensed data imperfect measurements of ecological attributes. It is therefore important to first assess the accuracy of the spatial information being used and then evaluate the impact of such inaccurate information on ecological model predictions. In this paper, the role of geostatistics for mapping thematic classification accuracy through integration of abundant image-derived (soft) and sparse higher accuracy (hard) class labels is presented. Such assessment leads to local indices of map quality, which can be used for guiding additional ground surveys. Stochastic simulation is proposed for generating multiple alternative realizations (maps) of the spatial distribution of the higher accuracy class labels over the study area. All simulated realizations are consistent with the available pieces of information (hard and soft labels) up to their validated level of accuracy. The simulated alternative class label representations can be used for assessing joint spatial accuracy, i.e., classification accuracy regarding entire spatial features read from the thematic map. Such realizations can also serve as input parameters to spatially explicit ecological models; the resulting distribution of ecological responses provides a model of uncertainty regarding the ecological model prediction. A case study illustrates the generation of alternative land cover maps for a Landsat Thematic Mapper (TM) subscene, and the subsequent construction of local map quality indices. Simulated land cover maps are then input into a biogeochemical model for assessing uncertainty regarding net primary production (NPP).     

Predicting a ‘tree change’ in Australia's tropical savannas: Combining different types of models to understand complex ecosystem behaviour

In this study, key ecological modelling limitations of a process-based simulation model and a Bayesian network were reduced by combining the two approaches. We demonstrate the combined modelling approach with a case study investigating increases in woody vegetation density in northern Australia's tropical savannas. We found that by utilising the strengths of a simulation model and a Bayesian network we could both forecast future change in woody vegetation density and diagnose the reasons for current vegetation states. The local conditions of climate, soil characteristics and the starting population of trees were found to be more important in explaining the likelihood of change in woody vegetation density compared to management practices such as grazing pressure and fire regimes. We conclude that combining the strengths of a process and BN model allowed us to produce a simple model that utilised the ability of the process model to simulate ecosystem processes in detail and over long time periods, and the ability of the BN to capture uncertainty in ecosystem response and to conduct scenario, sensitivity and diagnostic analysis. The overall result was a model that has the potential to provide land managers with a better understanding of the behaviour of a complex ecosystem than simply utilising either modelling approach in isolation.     

Sensitivity analysis of the two dimensional application of the Generic Ecological Model (GEM) to algal bloom prediction in the North Sea

Harmful algae can cause damage to co-existing organisms, tourism and farmers. Accurate predictions of algal future composition and abundance as well as when and where algal blooms may occur could help early warning and mitigating. The Generic Ecological Model is an instrument that can be applied to any water system (fresh, transitional or coastal) to calculate the primary production, chlorophyll-a concentration and phytoplankton species composition. It consists of physical, chemical and ecological model components which are coupled together to build one generic and flexible modelling tool. In this paper the model has been analyzed to assess sensitivity of the simulated chlorophyll-a concentration to a subset of ecologically significant input factors. Only a small number of approaches could be considered as suitable for several reasons including the model complexity, engagement of numerous interacting parameters and relatively long time of a single simulation. Thus, sensitivity analysis has been carried out with the use of the Morris method and later enriched by the computation of the correlation ratios of the selected parameters on the model response at more than a few locations in the modelled area. The obtained results are in agreement with expert knowledge of the ecological processes in the North Sea and correspond well with local characteristics.     

Circuit-theory applications to connectivity science and conservation

Conservation practitioners have long recognized ecological connectivity as a global priority for preserving biodiversity and ecosystem function. In the early years of conservation science, ecologists extended principles of island biogeography to assess connectivity based on source patch proximity and other metrics derived from binary maps of habitat. From 2006 to 2008, the late Brad McRae introduced circuit theory as an alternative approach to model gene flow and the dispersal or movement routes of organisms. He posited concepts and metrics from electrical circuit theory as a robust way to quantify movement across multiple possible paths in a landscape, not just a single least-cost path or corridor. Circuit theory offers many theoretical, conceptual, and practical linkages to conservation science. We reviewed 459 recent studies citing circuit theory or the open-source software Circuitscape. We focused on applications of circuit theory to the science and practice of connectivity conservation, including topics in landscape and population genetics, movement and dispersal paths of organisms, anthropogenic barriers to connectivity, fire behavior, water flow, and ecosystem services. Circuit theory is likely to have an effect on conservation science and practitioners through improved insights into landscape dynamics, animal movement, and habitat-use studies and through the development of new software tools for data analysis and visualization. The influence of circuit theory on conservation comes from the theoretical basis and elegance of the approach and the powerful collaborations and active user community that have emerged. Circuit theory provides a springboard for ecological understanding and will remain an important conservation tool for researchers and practitioners around the globe.     

Assessment of uncertainties in expert knowledge, illustrated in fuzzy rule-based models

The coherence between different aspects in the environmental system leads to a demand for comprehensive models of this system to explore the effects of different management alternatives. Fuzzy logic has been suggested as a means to extend the application domain of environmental modelling from physical relations to expert knowledge. In such applications the expert describes the system in terms of fuzzy variables and inference rules. The result of the fuzzy reasoning process is a numerical output value. In such a model, as in any other, the model context, structure, technical aspects, parameters and inputs may contribute uncertainties to the model output. Analysis of these contributions in a simplified model for agriculture suitability shows how important information about the accuracy of the expert knowledge in relation to the other uncertainties can be provided. A method for the extensive assessment of uncertainties in compositional fuzzy rule-based models is proposed, combining the evaluation of model structure, input and parameter uncertainties. In an example model, each of these three appear to have the potential to dominate aggregated uncertainty, supporting the relevance of an ample uncertainty approach.     

A Bayesian approach to retransformation bias in transformed regression

Ecological data analysis often involves fitting linear or nonlinear equations to data after transforming either the response variable, the right side of the equation, or both, so that the standard suite of regression assumptions are more closely met. However, inference is usually done in the natural metric and it is well known that retransforming back to the original metric provides a biased estimator for the mean of the response variable. For the normal linear model, fit under a log-transformation, correction factors are available to reduce this bias, but these factors may not be generally applicable to all model forms or other transformations. We demonstrate that this problem is handled in a straightforward manner using a Bayesian approach, which is general for linear and nonlinear models and other transformations and model error structures. The Bayesian framework provides a predictive distribution for the response variable so that inference can be made at the mean, or over the entire distribution to incorporate the predictive uncertainty.     

A dynamic simulation model to support reduction in illegal trade within legal wildlife markets

Sustainable wildlife trade is critical for biodiversity conservation, livelihoods, and food security. Regulatory frameworks are needed to secure these diverse benefits of sustainable wildlife trade. However, regulations limiting trade can backfire, sparking illegal trade if demand is not met by legal trade alone. Assessing how regulations affect wildlife market participants’ incentives is key to controlling illegal trade. Although much research has assessed how incentives at both the harvester and consumer ends of markets are affected by regulations, little has been done to understand the incentives of traders (i.e., intermediaries). We built a dynamic simulation model to support reduction in illegal wildlife trade within legal markets by focusing on incentives traders face to trade legal or illegal products. We used an Approximate Bayesian Computation approach to infer illegal trading dynamics and parameters that might be unknown (e.g., price of illegal products). We showcased the utility of the approach with a small-scale fishery case study in Chile, where we disentangled within-year dynamics of legal and illegal trading and found that the majority (∼77%) of traded fish is illegal. We utilized the model to assess the effect of policy interventions to improve the fishery's sustainability and explore the trade-offs between ecological, economic, and social goals. Scenario simulations showed that even significant increases (over 200%) in parameters proxying for policy interventions enabled only moderate improvements in ecological and social sustainability of the fishery at substantial economic cost. These results expose how unbalanced trader incentives are toward trading illegal over legal products in this fishery. Our model provides a novel tool for promoting sustainable wildlife trade in data-limited settings, which explicitly considers traders as critical players in wildlife markets. Sustainable wildlife trade requires incentivizing legal over illegal wildlife trade and consideration of the social, ecological, and economic impacts of interventions.     

Emerging,diverging and converging paradigms on sustainable development

Sustainable development has been used in various contexts by theoreticians and practitioners from a number of disciplines. This review explores some of these contexts from basic and applied ecological sciences, social sciences and philosophical works. It is concluded that there is a need to develop a theoretical paradigm that helps to explain the reasons underlying human resource use—a fundamental question that has been ignored in many other studies. A theoretical, evolutionary approach and several premises derived from this approach are offered. The theoretical framework suggests that societal scale, social structure, interrelatedness of individuals, and reciprocal relations between individuals may all be important in determining the types of management programmes that promote sustainable resource use by humans. Examples of solutions to various sustainability issues at different scales, based on different kinds of incentive structures, are also presented.     

Testing hypotheses in evolutionary ecology with imperfect detection: capture-recapture structural equation modeling

Studying evolutionary mechanisms in natural populations often requires testing multifactorial scenarios of causality involving direct and indirect relationships among individual and environmental variables. It is also essential to account for the imperfect detection of individuals to provide unbiased demographic parameter estimates. To cope with these issues, we developed a new approach combining structural equation models with capture-recapture models (CR-SEM) that allows the investigation of competing hypotheses about individual and environmental variability observed in demographic parameters. We employ Markov chain Monte Carlo sampling in a Bayesian framework to (1) estimate model parameters, (2) implement a model selection procedure to evaluate competing hypotheses about causal mechanisms, and (3) assess the fit of models to data using posterior predictive checks. We illustrate the value of our approach using two case studies on wild bird populations. We first show that CR-SEM can be useful to quantify the action of selection on a set of phenotypic traits with an analysis of selection gradients on morphological traits in Common Blackbirds (Turdus merula). In a second case study on Blue Tits (Cyanistes caeruleus), we illustrate the use of CR-SEM to study evolutionary trade-offs in the wild, while accounting for varying environmental conditions.     

Using the expert model PERPEST to translate measured and predicted pesticide exposure data into ecological risks

An important topic in the registration of pesticides and the interpretation of monitoring data is the estimation of the consequences of a certain concentration of a pesticide for the ecology of aquatic ecosystems. Solving these problems requires predictions of the expected response of the ecosystem to chemical stress. Up until now, a dominant approach to come up with such a prediction is the use of simulation models or safety factors. The disadvantage of the use of safety factors is a crude method that does not provide any insight into the concentration–response relationships at the ecosystem level. On the other hand, simulation models also have serious drawbacks like that they are often very complex, lack transparency, their implementation is expensive and there may be a compilation of errors, due to uncertainties in parameters and processes. In this paper, we present the expert model prediction of the ecological risks of pesticides (PERPEST) that overcomes these problems. It predicts the effects of a given concentration of a pesticide based on the outcome of already performed experiments using experimental ecosystems. This has the great advantage that the outcome is more realistic. The paper especially discusses how this model can be used to translate measured and predicted concentrations of pesticides into ecological risks, by taking data on measured and predicted concentrations of atrazine as an example. It is argued that this model can be of great use to evaluate the outcome of chemical monitoring programmes (e.g. performed in the light of the Water Framework Directive) and can even be used to evaluate the effects of mixtures.     

Ecological applications of multilevel analysis of variance

A Bayesian representation of the analysis of variance by A. Gelman is introduced with ecological examples. These examples demonstrate typical situations encountered in ecological studies. Compared to conventional methods, the multilevel approach is more flexible in model formulation, easier to set up, and easier to present. Because the emphasis is on estimation, multilevel models are more informative than the results from a significance test. The improved capacity is largely due to the changed computation methods. In our examples, we show that (1) the multilevel model is able to discern a treatment effect that is smaller than the conventional approach can detect, (2) the graphical presentation associated with the multilevel method is more informative, and (3) the multilevel model can incorporate all sources of uncertainty to accurately describe the true relationship between the outcome and potential predictors.     

Estimation methods for nonlinear state-space models in ecology

The use of nonlinear state-space models for analyzing ecological systems is increasing. A wide range of estimation methods for such models are available to ecologists, however it is not always clear, which is the appropriate method to choose. To this end, three approaches to estimation in the theta logistic model for population dynamics were benchmarked by Wang (2007). Similarly, we examine and compare the estimation performance of three alternative methods using simulated data. The first approach is to partition the state-space into a finite number of states and formulate the problem as a hidden Markov model (HMM). The second method uses the mixed effects modeling and fast numerical integration framework of the AD Model Builder (ADMB) open-source software. The third alternative is to use the popular Bayesian framework of BUGS. The study showed that state and parameter estimation performance for all three methods was largely identical, however with BUGS providing overall wider credible intervals for parameters than HMM and ADMB confidence intervals.