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.     

Future distribution modelling: A stitch in time is not enough

The last two decades have seen an increasing number of studies assessing the impact of climate change upon biodiversity. A central assumption underpinning research into the potential future habitat of terrestrial biota is that species are presently in equilibrium with their environments and that quantitative climate models adequately represent the distribution of species. Recently, many alarming predictions have emerged concerning the extinction and redistribution of species. Here, we show that even large-scale models of the climatic niche dimensions of species are temporally variable. Distributional models were developed for Salix (willow) species occurring in the province of Ontario, Canada, using three historical climate data sets. Although historical data very accurately represented the distributions of willows, the inherent variability within the models of species based on different periods greatly influenced the direction and magnitude of projected distributional change. We expose a fundamental uncertainty with respect to predicting the responses of species to climate change.     

On the latent state estimation of nonlinear population dynamics using Bayesian and non-Bayesian state-space models

Nonlinear state-space models have been increasingly applied to study population dynamics and data assimilation in environmental sciences. State-space models can account for process error and measurement error simultaneously to correct for the bias in the estimates of system state and model parameters. However, few studies have compared the performance of different nonlinear state-space models for reconstructing the state of population dynamics from noisy time series. This study compared the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and Bayesian nonlinear state-space models (BNSSM) through simulations. Synthetic population time series were generated using the theta logistic model with known parameters, and normally distributed process and measurement errors were introduced using the Monte Carlo simulations. At higher levels of nonlinearity, the UKF and BNSSM had lower root mean square error (RMSE) than the EKF. The BNSSM performed reliably across all levels of nonlinearity, whereas increased levels of nonlinearity resulted in higher RMSE of the EKF. The Metropolis–Hastings algorithm within the Gibbs algorithm was used to fit the theta logistic model to synthetic time series to estimate model parameters. The estimated posterior distribution of the parameter θ indicated that the 95% credible intervals included the true values of θ (=0.5 and 1.5), but did not include 1.0 and 0.0. Future studies need to incorporate the adaptive Metropolis algorithm to estimate unknown model parameters for broad applications of Bayesian nonlinear state-space models in ecological studies.     

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.     

Reconstructing historical marine ecosystems using food web models: Northern British Columbia from Pre-European contact to present

Mass-balance trophic models (Ecopath with Ecosim) are developed for the marine ecosystem of northern British Columbia (BC) for the historical periods 1750, 1900, 1950 and 2000 AD. Time series data are compiled for catch, fishing mortality and biomass using fisheries statistics and literature values. Using the assembled dataset, dynamics of the 1950-based simulations are fitted to agree with observations over 50 years to 2000 through the manipulation of trophic flow parameters and the addition of climate factors: a primary production anomaly and herring recruitment anomaly. The predicted climate anomalies reflect documented environmental series, most strongly sea surface temperature and the Pacific Decadal Oscillation index. The best-fit predator–prey interaction parameters indicate mixed trophic control of the ecosystem. Trophic flow parameters from the fitted 1950 model are transferred to the other historical periods assuming stationarity in density-dependent foraging tactics. The 1900 model exhibited an improved fit to data using this approach, which suggests that the pattern of trophic control may have remained constant over much of the last century. The 1950 model is driven forward 50 years using climate and historical fishing drivers. The resulting ecosystem is compared to the 2000 model, and the dynamics of these models are compared in a predictive forecast to 2050. The models suggest similar restoration trajectories after a hypothetical release from fishing.     

New approaches to modelling fish-habitat relationships

Ecologists often develop models that describe the relationship between faunal communities and their habitat. Coral reef fishes have been the focus of numerous such studies, which have used a wide range of statistical tools to answer an equally wide range of questions. Here, we apply a series of both conventional statistical techniques (linear and generalized additive regression models) and novel machine-learning techniques (the support vector machine and three ensemble techniques used with regression trees) to predict fish species richness, biomass, and diversity from a range of habitat variables. We compare the techniques in terms of their predictive performance, and we compare a subset of the models in terms of the influence each habitat variable has for the predictions. Prediction errors are estimated by cross-validation, and variable importance is assessed using permutations of individual variable values. For predictions of species richness and diversity the tree-based models generally and the random forest model specifically are superior (produce the lowest errors). These model types are all able to model both nonlinear and interaction effects. The linear model, unable to model either effect type, performs the worst (produces the highest errors). For predictions of biomass, the generalized additive model is superior, and the support vector machine performs the worst. Depth range, the difference between maximum and minimum water depth at a given site, is identified as the most important variable in the majority of models predicting the three fish community variables. However, variable importance is highly dependent upon model type, which leads to questions regarding the interpretation of variable importance and its proper use as an indicator of causality. The representation of ecological relationships by tree-based ensemble learners will improve predictive performance, and provide a new avenue for exploring ecological relationships, both statistical and causal.     

Multivariate Bayesian analysis of atmosphere–ocean general circulation models

Numerical experiments based on atmosphere–ocean general circulation models (AOGCMs) are one of the primary tools in deriving projections for future climate change. Although each AOGCM has the same underlying partial differential equations modeling large scale effects, they have different small scale parameterizations and different discretizations to solve the equations, resulting in different climate projections. This motivates climate projections synthesized from results of several AOGCMs’ output. We combine present day observations, present day and future climate projections in a single highdimensional hierarchical Bayes model. The challenging aspect is the modeling of the spatial processes on the sphere, the number of parameters and the amount of data involved. We pursue a Bayesian hierarchical model that separates the spatial response into a large scale climate change signal and an isotropic process representing small scale variability among AOGCMs. Samples from the posterior distributions are obtained with computer-intensive MCMC simulations. The novelty of our approach is that we use gridded, high resolution data covering the entire sphere within a spatial hierarchical framework. The primary data source is provided by the Coupled Model Intercomparison Project (CMIP) and consists of 9 AOGCMs on a 2.8 by 2.8 degree grid under several different emission scenarios. In this article we consider mean seasonal surface temperature and precipitation as climate variables. Extensions to our model are also discussed.     

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.     

Uncertainty analyses for calibrating a soil carbon balance model to agricultural field trial data in Sweden and Kenya

How do additional data of the same and/or different type contribute to reducing model parameter and predictive uncertainties? Most modeling applications of soil organic carbon (SOC) time series in agricultural field trial datasets have been conducted without accounting for model parameter uncertainty. There have been recent advances with Monte Carlo-based uncertainty analyses in the field of hydrological modeling that are applicable, relevant and potentially valuable in modeling the dynamics of SOC. Here we employed a Monte Carlo method with threshold screening known as Generalized Likelihood Uncertainty Estimation (GLUE) to calibrate the Introductory Carbon Balance Model (ICBM) to long-term field trail data from Ultuna, Sweden and Machang’a, Kenya. Calibration results are presented in terms of parameter distributions and credibility bands on time series simulations for a number of case studies. Using these methods, we demonstrate that widely uncertain model parameters, as well as strong covariance between inert pool size and rate constant parameters, exist when root mean square simulation errors were within uncertainties in input estimations and data observations. We show that even rough estimates of the inert pool (perhaps from chemical analysis) can be quite valuable to reduce uncertainties in model parameters. In fact, such estimates were more effective at reducing parameter and predictive uncertainty than an additional 16 years time series data at Ultuna. We also demonstrate an effective method to jointly, simultaneously and in principle more robustly calibrate model parameters to multiple datasets across different climatic regions within an uncertainty framework. These methods and approaches should have benefits for use with other SOC models and datasets as well.     

Ensemble Bayesian model averaging using Markov Chain Monte Carlo sampling

Bayesian model averaging (BMA) has recently been proposed as a statistical method to calibrate forecast ensembles from numerical weather models. Successful implementation of BMA however, requires accurate estimates of the weights and variances of the individual competing models in the ensemble. In their seminal paper (Raftery et al. Mon Weather Rev 133:1155–1174, 2005) has recommended the Expectation–Maximization (EM) algorithm for BMA model training, even though global convergence of this algorithm cannot be guaranteed. In this paper, we compare the performance of the EM algorithm and the recently developed DiffeRential Evolution Adaptive Metropolis (DREAM) Markov Chain Monte Carlo (MCMC) algorithm for estimating the BMA weights and variances. Simulation experiments using 48-hour ensemble data of surface temperature and multi-model streamflow forecasts show that both methods produce similar results, and that their performance is unaffected by the length of the training data set. However, MCMC simulation with DREAM is capable of efficiently handling a wide variety of BMA predictive distributions, and provides useful information about the uncertainty associated with the estimated BMA weights and variances.     

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.     

Using global sensitivity analysis of demographic models for ecological impact assessment

Population viability analysis (PVA) is widely used to assess population‐level impacts of environmental changes on species. When combined with sensitivity analysis, PVA yields insights into the effects of parameter and model structure uncertainty. This helps researchers prioritize efforts for further data collection so that model improvements are efficient and helps managers prioritize conservation and management actions. Usually, sensitivity is analyzed by varying one input parameter at a time and observing the influence that variation has over model outcomes. This approach does not account for interactions among parameters. Global sensitivity analysis (GSA) overcomes this limitation by varying several model inputs simultaneously. Then, regression techniques allow measuring the importance of input‐parameter uncertainties. In many conservation applications, the goal of demographic modeling is to assess how different scenarios of impact or management cause changes in a population. This is challenging because the uncertainty of input‐parameter values can be confounded with the effect of impacts and management actions. We developed a GSA method that separates model outcome uncertainty resulting from parameter uncertainty from that resulting from projected ecological impacts or simulated management actions, effectively separating the 2 main questions that sensitivity analysis asks. We applied this method to assess the effects of predicted sea‐level rise on Snowy Plover (Charadrius nivosus). A relatively small number of replicate models (approximately 100) resulted in consistent measures of variable importance when not trying to separate the effects of ecological impacts from parameter uncertainty. However, many more replicate models (approximately 500) were required to separate these effects. These differences are important to consider when using demographic models to estimate ecological impacts of management actions.     

Quantifying parameter uncertainty in a coral reef model using Metropolis-Coupled Markov Chain Monte Carlo

Coral reefs are threatened ecosystems, so it is important to have predictive models of their dynamics. Most current models of coral reefs fall into two categories. The first is simple heuristic models which provide an abstract understanding of the possible behaviour of reefs in general, but do not describe real reefs. The second is complex simulations whose parameters are obtained from a range of sources such as literature estimates. We cannot estimate the parameters of these models from a single data set, and we have little idea of the uncertainty in their predictions.We have developed a compromise between these two extremes, which is complex enough to describe real reef data, but simple enough that we can estimate parameters for a specific reef from a time series. In previous work, we fitted this model to a long-term data set from Heron Island, Australia, using maximum likelihood methods. To evaluate predictions from this model, we need estimates of the uncertainty in our parameters. Here, we obtain such estimates using Bayesian Metropolis-Coupled Markov Chain Monte Carlo. We do this for versions of the model in which corals are aggregated into a single state variable (the three-state model), and in which corals are separated into four state variables (the six-state model), in order to determine the appropriate level of aggregation. We also estimate the posterior distribution of predicted trajectories in each case.In both cases, the fitted trajectories were close to the observed data, but we had doubts about the biological plausibility of some parameter estimates. We suggest that informative prior distributions incorporating expert knowledge may resolve this problem. In the six-state model, the posterior distribution of state frequencies after 40 years contained two divergent community types, one dominated by free space and soft corals, and one dominated by acroporid, pocilloporid, and massive corals. The three-state model predicts only a single community type. We conclude that the three-state model hides too much biological heterogeneity, but we need more data if we are to obtain reliable predictions from the six-state model. It is likely that there will be similarly large, but currently unevaluated, uncertainty in the predictions of other coral reef models, many of which are much more complex and harder to fit to real data.     

Sensitivity of atmospheric dispersion simulations by HYSPLIT to the meteorological predictions from a meso-scale model

Mesoscale transport and dispersion of air pollutants from a few major point sources in the Mississippi Gulf coastal region is calculated using a coupled modeling system consisting of the atmospheric dynamical model WRF and the lagrangian particle model HYSPLIT. The sensitivity of the dispersion model results to the meteorological fields is studied by conducting an ensemble of simulations using the WRF model for the same dispersion case. Several parameterization schemes for the physical processes of boundary layer turbulence and land surface temperature/moisture prediction in WRF are used in various combinations to produce different meteorological members which are then used for dispersion simulation. The uncertainty in the simulated concentration probabilities to the meteorological model configurations and the ensemble mean are presented. The parameters used for determining the uncertainties include the wind fields, temperature, area of concentration and the levels of concentration. The results indicate that dispersion model results are influenced by the choices made in respect of the planetary boundary layer and land surface schemes in the mesoscale model to produce the meteorological forecast thereby leading to certain amount of uncertainty in the resultant concentrations. Results show that the specific choices made about the atmospheric model configuration can significantly after the simulated concentrations.     

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).     

Spatial uncertainty analysis of population models

This paper describes an approach for conducting spatial uncertainty analysis of spatial population models, and illustrates the ecological consequences of spatial uncertainty for landscapes with different properties. Spatial population models typically simulate birth, death, and migration on an input map that describes habitat. Typically, only a single “reference” map is available, but we can imagine that a collection of other, slightly different, maps could be drawn to represent a particular species’ habitat. As a first approximation, our approach assumes that spatial uncertainty (i.e., the variation among values assigned to a location by such a collection of maps) is constrained by characteristics of the reference map, regardless of how the map was produced. Our approach produces lower levels of uncertainty than alternative methods used in landscape ecology because we condition our alternative landscapes on local properties of the reference map. Simulated spatial uncertainty was higher near the borders of patches. Consequently, average uncertainty was highest for reference maps with equal proportions of suitable and unsuitable habitat, and no spatial autocorrelation. We used two population viability models to evaluate the ecological consequences of spatial uncertainty for landscapes with different properties. Spatial uncertainty produced larger variation among predictions of a spatially explicit model than those of a spatially implicit model. Spatially explicit model predictions of final female population size varied most among landscapes with enough clustered habitat to allow persistence. In contrast, predictions of population growth rate varied most among landscapes with only enough clustered habitat to support a small population, i.e., near a spatially mediated extinction threshold. We conclude that spatial uncertainty has the greatest effect on persistence when the amount and arrangement of suitable habitat are such that habitat capacity is near the minimum required for persistence.     

Rapid assessment of management options for promoting stock rebuilding in data-poor species under climate change

The development of species recovery plans requires considering likely outcomes of different management interventions, but the complicating effects of climate change are rarely evaluated. We examined how qualitative network models (QNMs) can be deployed to support decision making when data, time, and funding limitations restrict use of more demanding quantitative methods. We used QNMs to evaluate management interventions intended to promote the rebuilding of a collapsed stock of blue king crab (Paralithodes platypus) (BKC) around the Pribilof Islands (eastern Bering Sea) to determine how their potential efficacy may change under climate change. Based on stakeholder input and a literature review, we constructed a QNM that described the life cycle of BKC, key ecological interactions, potential climate-change impacts, relative interaction strengths, and uncertainty in terms of interaction strengths and link presence. We performed sensitivity analyses to identify key sources of prediction uncertainty. Under a scenario of no climate change, predicted increases in BKC were reliable only when stock enhancement was implemented in a BKC hatchery-program scenario. However, when climate change was accounted for, the intervention could not counteract its adverse impacts, which had an overall negative effect on BKC. The remaining management scenarios related to changes in fishing effort on BKC predators. For those scenarios, BKC outcomes were unreliable, but climate change further decreased the probability of observing recovery. Including information on relative interaction strengths increased the likelihood of predicting positive outcomes for BKC approximately 5–50% under the management scenarios. The largest gains in prediction precision will be made by reducing uncertainty associated with ecological interactions between adult BKC and red king crab (Paralithodes camtschaticus). Qualitative network models are useful options when data are limited, but they remain underutilized in conservation.     

Predicting extinctions as a result of climate change

Widespread extinction is a predicted ecological consequence of global warming. Extinction risk under climate change scenarios is a function of distribution breadth. Focusing on trees and birds of the eastern United States, we used joint climate and environment models to examine fit and climate change vulnerability as a function of distribution breadth. We found that extinction vulnerability increases with decreasing distribution size. We also found that model fit decreases with decreasing distribution size, resulting in high prediction uncertainty among narrowly distributed species. High prediction uncertainty creates a conservation dilemma in that excluding these species under-predicts extinction risk and favors mistaken inaction on global warming. By contrast, including narrow endemics results in over-predicting extinction risk and promotes mistaken inaction on behalf of individual species prematurely considered doomed to extinction.     

Parameter uncertainties in the modelling of vegetation dynamics—Effects on tree community structure and ecosystem functioning in European forest biomes

Dynamic vegetation models are useful tools for analysing terrestrial ecosystem processes and their interactions with climate through variations in carbon and water exchange. Long-term changes in structure and composition (vegetation dynamics) caused by altered competitive strength between plant functional types (PFTs) are attracting increasing attention as controls on ecosystem functioning and potential feedbacks to climate. Imperfect process knowledge and limited observational data restrict the possibility to parameterise these processes adequately and potentially contribute to uncertainty in model results. This study addresses uncertainty among parameters scaling vegetation dynamic processes in a process-based ecosystem model, LPJ-GUESS, designed for regional-scale studies, with the objective to assess the extent to which this uncertainty propagates to additional uncertainty in the tree community structure (in terms of the tree functional types present and their relative abundance) and thus to ecosystem functioning (carbon storage and fluxes). The results clearly indicate that the uncertainties in parameterisation can lead to a shift in competitive balance, most strikingly among deciduous tree PFTs, with dominance of either shade-tolerant or shade-intolerant PFTs being possible, depending on the choice of plausible parameter values. Despite this uncertainty, our results indicate that the resulting effect on ecosystem functioning is low. Since the vegetation dynamics in LPJ-GUESS are representative for the more complex Earth system models now being applied within ecosystem and climate research, we assume that our findings will be of general relevance. We suggest that, in terms of carbon storage and fluxes, the heavier parameterisation requirement of the processes involved does not widen the overall uncertainty in model predictions.