Uncertainty analysis of transient population dynamics

Two types of demographic analyses, perturbation analysis and uncertainty analysis, can be conducted to gain insights about matrix population models and guide population management. Perturbation analysis studies how the perturbation of demographic parameters (survival, growth, and reproduction parameters) may affect the population projection, while uncertainty analysis evaluates how much uncertainty there is in population dynamic predictions and where the uncertainty comes from. Previously, both perturbation analysis and uncertainty analysis were conducted on the long-term population growth rate. However, the population may not reach its equilibrium state, especially when there is management by harvesting or hunting. Recently, there has been an increased interest in short-term transient dynamics, which can differ from asymptotic long-term dynamics. There are currently techniques to conduct perturbation analyses of short-term transient dynamics, but no techniques have been proposed for uncertainty analysis of such dynamics. In this study, we introduced an uncertainty analysis technique, the general Fourier Amplitude Sensitivity Test (FAST), to study uncertainties in transient population dynamics. The general FAST is able to identify the amount of uncertainty in transient dynamics and contributions by different demographic parameters. We applied the general FAST to a mountain goat (Oreamnos americanus) matrix population model to give a clear illustration of how uncertainty analysis can be conducted for transient dynamics arising from matrix population models.     

A hierarchical analysis of terrestrial ecosystem model Biome-BGC: Equilibrium analysis and model calibration

The increasing complexity of ecosystem models represents a major difficulty in tuning model parameters and analyzing simulated results. To address this problem, this study develops a hierarchical scheme that simplifies the Biome-BGC model into three functionally cascaded tiers and analyzes them sequentially. The first-tier model focuses on leaf-level ecophysiological processes; it simulates evapotranspiration and photosynthesis with prescribed leaf area index (LAI). The restriction on LAI is then lifted in the following two model tiers, which analyze how carbon and nitrogen is cycled at the whole-plant level (the second tier) and in all litter/soil pools (the third tier) to dynamically support the prescribed canopy. In particular, this study analyzes the steady state of these two model tiers by a set of equilibrium equations that are derived from Biome-BGC algorithms and are based on the principle of mass balance. Instead of spinning-up the model for thousands of climate years, these equations are able to estimate carbon/nitrogen stocks and fluxes of the target (steady-state) ecosystem directly from the results obtained by the first-tier model. The model hierarchy is examined with model experiments at four AmeriFlux sites. The results indicate that the proposed scheme can effectively calibrate Biome-BGC to simulate observed fluxes of evapotranspiration and photosynthesis; and the carbon/nitrogen stocks estimated by the equilibrium analysis approach are highly consistent with the results of model simulations. Therefore, the scheme developed in this study may serve as a practical guide to calibrate/analyze Biome-BGC; it also provides an efficient way to solve the problem of model spin-up, especially for applications over large regions. The same methodology may help analyze other similar ecosystem models as well.     

Automatic sensitivity analysis of a finite volume model for two-dimensional shallow water flows

Given a numerical model for solving two-dimensional shallow water equations, we are interested in the robustness of the simulation by identifying the rate of change of the water depths and discharges with respect to a change in the bottom friction coefficients. Such a sensitivity analysis can be carried out by computing the corresponding derivatives. Automatic differentiation (AD) is an efficient numerical method, free of approximation errors, to evaluate derivatives of the objective function specified by the computer program, Rubar20 for example. In this paper AD software tool Tapenade is used to compute forward derivatives. Numerical tests were done to show the robustness of the model and to demonstrate the efficiency of these AD-derivatives.     

Behavioral games involving a clever prey avoiding a clever predator: An individual-based model of dusky dolphins and killer whales

Faced with an intermittent but potent threat, animals exhibit behavior that allows them to balance foraging needs and avoid predators and over time, these behaviors can become hard-wired adaptations with both species trying to maximize their own fitness. In systems where both predator and prey share similar sensory modalities and cognitive abilities, such as with marine mammals, the dynamic nature of predator-prey interactions is poorly understood. The costs and benefits of these anti-predator adaptations need to be evaluated and quantified based on the dynamic engagement of predator and prey. Many theoretic models have addressed the complexity of predator-prey relationships, but few have translated into testable mechanistic models. In this study, we developed a spatially-explicit, geo-referenced, individual-based model of a prototypical adult dusky dolphin off Kaikoura, New Zealand facing a more powerful, yet infrequent predator, the killer whale. We were interested in two primary objectives, (1) to capture the varying behavioral game between a clever prey and clever predator based on our current understanding of the Kaikoura system, (2) to compare evolutionary costs vs. benefits (foraging time and number of predator encounters) for an adult non-maternal dusky dolphin at various levels of killer whale-avoidance behaviors and no avoidance rules. We conducted Monte Carlo simulations to address model performance and parametric uncertainty. Mantel tests revealed an 88% correlation (426 × 426 distance matrix, km²) between observed field sightings of dusky dolphins with model generated sightings for non-maternal adult dusky dolphin groups. Simulation results indicated that dusky dolphins incur a 2.7% loss in feeding time by evolving the anti-predator behavior of moving to and from the feeding grounds. Further, each evolutionary strategy we explored resulted in dolphins incurring an additional loss of foraging time. At low killer whale densities (appearing less than once every 3 days), each evolutionary strategy simulated converged towards the evolutionary cost of foraging, that is, the loss in foraging time approached the 2.7% loss experienced by evolving near shore-offshore movement behavior. However, the highest level of killer whale presence resulted in 38% decreases in foraging time. The biological significance of these losses potentially incurred by a dusky dolphin is dependent on various factors from dolphin group foraging behavior and individual energy needs to dolphin prey availability and behavior.     

A global Bayesian sensitivity analysis of the 1d SimSphere soil–vegetation–atmospheric transfer (SVAT) model using Gaussian model emulation

Sensitivity analysis consists of an integral and important validatory check of a computer simulation model before the code is used in performing any kind of analysis operation. The present paper demonstrates the use of a relatively new method and tool for conducting global sensitivity analysis (GSA) for environmental models, providing simultaneously the first GSA study of the widely used 1d soil–vegetation–atmospheric transfer (SVAT) model named SimSphere. A software platform called the Gaussian emulation machine for sensitivity analysis (GEM SA), which has been developed for performing a GSA via Bayesian theory, is applied to SimSphere model in order to identify the most responsive model inputs to the simulation of key model outputs, detect their interactions and derive absolute sensitivity measures concerning the model structure. This study is also very timely in that, use of this particular SVAT model is currently being considered to be used in a scheme being developed for the operational retrieval of the soil surface moisture content by National Polar-orbiting Operational Environmental Satellite System (NPOESS), in a series of satellite platforms that are due to be launched in the next 12 years starting from 2016.The employed GSA method was found capable of identifying the most responsive SimSphere inputs and also of capturing their key interactions for each of the simulated target quantities on which the GSA was conducted. The most sensitive model inputs were the topography parameters (slope, aspect) as well as the fractional vegetation cover and soil surface moisture availability. The implications of these findings for the future use of SimSphere are discussed.     

Estimating the critical phosphorus loading of shallow lakes with the ecosystem model PCLake: Sensitivity, calibration and uncertainty

There is a vast body of knowledge that eutrophication of lakes may cause algal blooms. Among lakes, shallow lakes are peculiar systems in that they typically can be in one of two contrasting (equilibrium) states that are self-stabilizing: a ‘clear’ state with submerged macrophytes or a ‘turbid’ state dominated by phytoplankton. Eutrophication may cause a switch from the clear to the turbid state, if the P loading exceeds a critical value. The ecological processes governing this switch are covered by the ecosystem model PCLake, a dynamic model of nutrient cycling and the biota in shallow lakes. Here we present an extensive analysis of the model, using a three-step procedure. (1) A sensitivity analysis revealed the key parameters for the model output. (2) These parameters were calibrated on the combined data on total phosphorus, chlorophyll-a, macrophytes cover and Secchi depth in over 40 lakes. This was done by a Bayesian procedure, giving a weight to each parameter setting based on its likelihood. (3) These weights were used for an uncertainty analysis, applied to the switchpoints (critical phosphorus loading levels) calculated by the model. The model was most sensitive to changes in water depth, P and N loading, retention time and lake size as external input factors, and to zooplankton growth rate, settling rates and maximum growth rates of phytoplankton and macrophytes as process parameters. The results for the ‘best run’ showed an acceptable agreement between model and data and classified nearly all lakes to which the model was applied correctly as either ‘clear’ (macrophyte-dominated) or ‘turbid’ (phytoplankton-dominated). The critical loading levels for a standard lake showed about a factor two uncertainty due to the variation in the posterior parameter distribution. This study calculates in one coherent analysis uncertainties in critical phosphorus loading, a parameter that is of great importance to water quality managers.     

A population model for the peach fruit moth, Carposina sasakii Matsumura (Lepidoptera: Carposinidae), in a Korean orchard system

A population model for the peach fruit moth, Carposina sasakii Matsumura, was constructed to understand the population dynamics of this pest species and to develop an effective management strategy for various orchard (apple, peach, apple + peach) systems. The model was structured by the five developmental stages of C. sasakii: egg, larva, pupa, larval-cocoon (overwintering larva), and adult. The model consisted of a series of component models: (1) a bimodal spring adult emergence model, (2) an adult oviposition model, (3) stage emergence models of eggs, larvae, and pupae, (4) a larval survival rate model in fruits, (5) a larval-cocoon formation model, and (6) an insecticide effect model. Simulations using the model described the typical patterns of C. sasakii adult abundance in various orchard systems well, and was specific to the composition of host plants: three adult abundance peaks (first peak, mid-season peak, and last peak) a year with decreased peaks after the first peak in monoculture orchards of late apple, two adult peaks a year with a much higher last peak in monoculture orchards of early peach, and three adult peaks a year with much higher later peaks in mixed orchards of late apple and early peach. The average deviation between model outputs and actual records for first and second adult peak dates was 2.8 and 3.9 d, respectively, in simulations without an insecticide effect. The deviation decreased when insecticide effects were incorporated into the model. We also performed a sensitivity analysis of our model, and suggest possible applications of the model.     

Implications of different population model structures for management of threatened plants

Population viability analysis (PVA) is a reliable tool for ranking management options for a range of species despite parameter uncertainty. No one has yet investigated whether this holds true for model uncertainty for species with complex life histories and for responses to multiple threats. We tested whether a range of model structures yielded similar rankings of management and threat scenarios for 2 plant species with complex postfire responses. We examined 2 contrasting species from different plant functional types: an obligate seeding shrub and a facultative resprouting shrub. We exposed each to altered fire regimes and an additional, species‐specific threat. Long‐term demographic data sets were used to construct an individual‐based model (IBM), a complex stage‐based model, and a simple matrix model that subsumes all life stages into 2 or 3 stages. Agreement across models was good under some scenarios and poor under others. Results from the simple and complex matrix models were more similar to each other than to the IBM. Results were robust across models when dominant threats are considered but were less so for smaller effects. Robustness also broke down as the scenarios deviated from baseline conditions, likely the result of a number of factors related to the complexity of the species’ life history and how it was represented in a model. Although PVA can be an invaluable tool for integrating data and understanding species’ responses to threats and management strategies, this is best achieved in the context of decision support for adaptive management alongside multiple lines of evidence and expert critique of model construction and output.     

Sensitivity of species-distribution models to error,bias, and model design: An application to resource selection functions for woodland caribou

Models that predict distribution are now widely used to understand the patterns and processes of plant and animal occurrence as well as to guide conservation and management of rare or threatened species. Application of these methods has led to corresponding studies evaluating the sensitivity of model performance to requisite data and other factors that may lead to imprecise or false inferences. We expand upon these works by providing a relative measure of the sensitivity of model parameters and prediction to common sources of error, bias, and variability. We used a one-at-a-time sample design and GPS location data for woodland caribou (Rangifer tarandus caribou) to assess one common species-distribution model: a resource selection function. Our measures of sensitivity included change in coefficient values, prediction success, and the area of mapped habitats following the systematic introduction of geographic error and bias in occurrence data, thematic misclassification of resource maps, and variation in model design. Results suggested that error, bias and model variation have a large impact on the direct interpretation of coefficients. Prediction success and definition of important habitats were less responsive to the perturbations we introduced to the baseline model. Model coefficients, prediction success, and area of ranked habitats were most sensitive to positional error in species locations followed by sampling bias, misclassification of resources, and variation in model design. We recommend that researchers report, and practitioners consider, levels of error and bias introduced to predictive species-distribution models. Formal sensitivity and uncertainty analyses are the most effective means for evaluating and focusing improvements on input data and considering the range of values possible from imperfect models.     

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.     

Comparison of sensitivity analysis techniques: A case study with the rice model WARM

The considerable complexity often included in biophysical models leads to the need of specifying a large number of parameters and inputs, which are available with various levels of uncertainty. Also, models may behave counter-intuitively, particularly when there are nonlinearities in multiple input-output relationships. Quantitative knowledge of the sensitivity of models to changes in their parameters is hence a prerequisite for operational use of models. This can be achieved using sensitivity analysis (SA) via methods which differ for specific characteristics, including computational resources required to perform the analysis. Running SA on biophysical models across several contexts requires flexible and computationally efficient SA approaches, which must be able to account also for possible interactions among parameters. A number of SA experiments were performed on a crop model for the simulation of rice growth (Water Accounting Rice Model, WARM) in Northern Italy. SAs were carried out using the Morris method, three regression-based methods (Latin hypercube sampling, random and Quasi-Random, LpTau), and two methods based on variance decomposition: Extended Fourier Amplitude Sensitivity Test (E-FAST) and Sobol’, with the latter adopted as benchmark. Aboveground biomass at physiological maturity was selected as reference output to facilitate the comparison of alternative SA methods. Rankings of crop parameters (from the most to the least relevant) were generated according to sensitivity experiments using different SA methods and alternate parameterizations for each method, and calculating the top-down coefficient of concordance (TDCC) as measure of agreement between rankings. With few exceptions, significant TDCC values were obtained both for different parameterizations within each method and for the comparison of each method to the Sobol’ one. The substantial stability observed in the rankings seem to indicate that, for a crop model of average complexity such as WARM, resource intensive SA methods could not be needed to identify most relevant parameters. In fact, the simplest among the SA methods used (i.e., Morris method) produced results comparable to those obtained by methods more computationally expensive.     

Identifying key life-traits for the dynamics and gene flow in a weedy crop relative: Sensitivity analysis of the GeneSys simulation model for weed beet (Beta vulgaris ssp. vulgaris)

The benefits of genetically modified herbicide-tolerant (GMHT) sugar beet (Beta vulgaris) varieties stem from their presumed ability to improve weed control and reduce its cost, particularly targeting weed beet, a harmful annual weedy form of the genus Beta (i.e. B. vulgaris ssp. vulgaris) frequent in sugar beet fields. As weed beet is totally interfertile with sugar beet, it is thus likely to inherit the herbicide-tolerance transgene through pollen-mediated gene flow. Hence, the foreseeable advent of HT weed beet populations is a serious threat to the sustainability of GM sugar beet cropping systems. For studying and quantifying the long-term effects of cropping system components (crop succession and cultivation techniques) on weed beet population dynamics and gene flow, we developed a biophysical process-based model called GeneSys-Beet in a previous study. In the present paper, the model was employed to identify and rank the weed life-traits as function of their effect on weed beet densities and genotypes, using a global sensitivity analysis to model parameters. Monte Carlo simulations with simultaneous randomization of all life-trait parameters were carried out in three cropping systems contrasting for their risk for infestation by HT weed beets. Simulated weed plants and bolters (i.e. beet plants with flowering and seed-producing stems) were then analysed with regression models as a function of model parameters to rank processes and life-traits and quantify their effects. Key parameters were those determining the timing and success of growth, development, seed maturation and the physiological end of seed production. Timing parameters were usually more important than success parameters, showing for instance that optimal timing of weed management operations is more important than its exact efficacy. The ranking of life-traits though depended on the cropping system and, to a lesser extent, on the target variable (i.e. GM weeds vs. total weed population). For instance, post-emergence parameters were crucial in rotations with frequent sugar beet crops whereas pre-emergence parameters were most important when sugar beet was rare. In the rotations with frequent sugar beet and insufficient weed control, interactions between traits were small, indicating diverse populations with contrasted traits could prosper. Conversely, when sugar beet was rare and weed control optimal, traits had little impact individually, indicating that a small number of optimal combinations of traits would be successful. Based on the analysis of sugar beet parameters and genetic traits, advice for the future selection of sugar beet varieties was also given. In climatic conditions similar to those used here, the priority should be given to limiting the presence of hybrid seeds in seed lots rather than decreasing varietal sensitivity to vernalization.