2004 Progress Report: Model Choice Stochasticity, and Ecological Complexity

EPA Grant Number: R829402C005
Subproject: this is subproject number 005 , established and managed by the Center Director under grant R829402
(EPA does not fund or establish subprojects; EPA awards and manages the overall grant for this center).

Center: Center for Integrating Statistical and Environmental Science
Center Director: Stein, Michael
Title: Model Choice Stochasticity, and Ecological Complexity
Investigators: Dwyer, Greg , Coram, Marc , Pfister, Cathy , Wang, Mei , Wootton, Timothy
Current Investigators: Dwyer, Greg , Coram, Marc , Elderd, Bret , Forester, James , Pfister, Cathy , Wang, Mei , Wootton, Timothy
Institution: University of Chicago
EPA Project Officer: Packard, Benjamin H
Project Period: March 12, 2002 through March 11, 2007
Project Period Covered by this Report: March 12, 2004 through March 11, 2005
RFA: Environmental Statistics Center (2001) RFA Text |  Recipients Lists
Research Category: Environmental Statistics , Ecological Indicators/Assessment/Restoration , Health , Ecosystems , Air


Mathematical models are of increasing importance in helping ecologists and environmental scientists to understand human-influenced ecosystems. Historically, most ecological models have been deterministic and thus were of limited usefulness in environmental management. Work in the ecology group has therefore focused on incorporating stochasticity into ecological models and on using models to quantify uncertainty in management decisions. For concreteness, each project focuses on particular organisms, but the models and methods that we are developing are of wide applicability.

Our projects fall into two categories—population growth and species interactions. The population growth project is in collaboration with the U.S. Environmental Protection Agency (EPA) scientists Drs. Diane Nacci and Jason Grear at the Atlantic Ecology Division of the National Health and Environmental Effects Research Laboratory (NHEERL). Drs. Anne Fairbrother and Jennifer Orme-Zavaleta of NHEERL’s Western Ecology Division have expressed interest in the model structure results from the species interactions project. In particular, they are interested in how the models of the epidemic dynamics produced by the group can be applied to other diseases of concern (i.e., West Nile Virus and Chronic Wasting Disease). During the next year, we expect to build upon these collaborations with EPA. In addition to these collaborations, we will continue to submit articles to journals (see Publications), and our expectation is that the articles that are currently in review or in revision will be published.

Progress Summary:

Population Growth

In the population growth project, we are attempting to assess the impact of variation among individuals on population growth rates, by extending simple models and assessing their general usefulness. First, principal investigators (PIs) Cathy Pfister and Mei Wang have been adapting population matrix models to account for variation among individuals. Matrix models are widely used in ecology and environmental management and are often helpful in quantifying the effects of pollutants on population growth rates. Such models, however, typically lump individuals into a small number of age or state groups, which may bias estimates of population growth. Pfister and Wang (2005) have therefore developed a method of incorporating variability in growth rates among individuals into matrix population models. The usefulness of this method is that it allows for a major complication without greatly complicating the underlying model; in particular, the increase in the number of parameters that must be estimated is quite modest. Moreover, Mei Wang and a colleague have developed an algorithm that can be applied to this and related models to exhaustively categorize the different possible impacts that a given change in a life-history parameter can have on population growth rates (Sun and Wang, 2007).

An important issue in the use of matrix models, including in Pfister and Wang’s work, is the trade-off between the higher realism afforded by more complex models and the smaller amount of data needed to parameterize less complex models. Indeed, one of the reasons why ages or stages in matrix models are often lumped together is that there are not enough data available to calculate survival rates or fecundities for more than a few age or stage classes. Models with lumped classes, known as partial life cycle (PLC) models, may thus be useful when data are lacking, even though their predictions are less accurate. To assess the relative importance of systematic error versus higher data requirements in such models, Bret Elderd, postdoctoral research associate with the Ecology Group, is working with Jason Grear of EPA to explore the usefulness of PLC models (Grear and Elderd, in preparation). PLC models are more likely to be useful if survivorship changes little over the course of an organism’s lifetime. Elderd and Grear’s approach is therefore to consider how rapidly the error in PLC predictions increases as survivorship changes more strongly with age. The overall goal of this analysis will be to determine if and when these simple models fail to properly predict population growth.

Elderd has also been working on expanding the usefulness of matrix population models for analyzing experimental data. Matrix models often figure in ecological experiments in the form of Life-Table Response Experiment (LTRE) analyses. LTRE analyses allow investigators to quantify the impact of experimental treatments on population growth, in order to better understand how changes in life-history parameters, due for example to anthropogenic change, are likely to affect population growth. Most LTRE studies, however, use only single-factorial experiments and deterministic models. Elderd and Doak (2006) have extended their method to include nested multiple-factorial experiments and have shown how the method can incorporate the use of stochastic models.

Species Interactions

The work being carried out in the population-growth projects essentially assumes that interactions among species are weak. For many organisms, this assumption is a useful simplification, but for just as many or more, it is not. A second focus of our group thus lies in the development and application of models of species interactions, whether those interactions are between pairs or instead involve complex networks of many species. Our approach involves the incorporation of stochastic elements into species-interaction models in order to use data to estimate the likelihood of each model. In turn, the likelihood of each model will be used in a model-selection criterion in order to determine which model is the best. Our immediate goal is thus to develop methods by which ecologists and environmental managers can determine how complex a model they need to describe a given set of ecological interactions. Such assessments are almost always necessary, because typically we do not know which model is most appropriate. Model-selection criteria allow the investigator to rigorously identify the model that does the best job of describing the data and is thus the most appropriate description of the process. For example, in applications, researchers may have to choose between a model that includes differential effects of pollution on species interactions and one that does not. Our longer term goal is thus to provide models that can help us to predict how anthropogenic perturbations may influence communities and ecosystems.

Results to Date

Population Growth. As we have described, by incorporating variability among individuals into matrix population models, Pfister and Wang (2005) have developed a method of accounting for autocorrelation in growth among individuals. That is, in real populations, some individuals consistently grow more rapidly while others grow more slowly, but such effects are typically ignored in matrix models (see Section above on Population Growth). An account of the mathematical details of this method can be found in a Center for Integrating Statistical and Environmental Science (CISES) technical report by Sun and Wang (2007). Drs. Nacci and Grear of EPA’s Atlantic Ecology Division recognized that the method could be used to examine the impact of pollution on the population demography of the common loon (Gavia immer), which is an indicator species for environmental health in the Northeast and Midwest. Their interest in using the method prompted a site visit to CISES by Dr. Grear. Dr. Grear gave a seminar to the whole CISES group and had a number of meetings with CISES personnel, including Cathy Pfister, Mei Wang, and Bret Elderd. Not only is this site visit likely to lead to the use of Pfister and Wang’s (2005) methodology by Nacci’s group, but it also has led to a collaboration on another intriguing aspect of population demography. Specifically, Elderd and Grear have been collaborating on a project that investigates the importance of stochasticity in correctly forecasting population growth given the central assumptions of PLC (See Section above on Population Growth). This collaboration has resulted in additional discussions via conference calls and at the Ecological Society of America’s (ESA) annual meeting. Elderd has also made progress on developing methodologies for determining the impacts of nested experimental treatments on population growth by extending LTRE analysis to include nested components and a stochastic factor (Elderd and Doak, 2006). Each of these analyses represents a substantial contribution within the study of population ecology, and our hope is that each will find application in environmental management problems, including those examining environmental stressors and ecosystem health.

Species Interactions. Work in the species interactions project has focused mostly on disease dynamics. First, Bret Elderd, Vanja Dukic, and Greg Dwyer have developed a Bayesian framework for forecasting a disease’s rate of spread. The basis of this approach is to first construct prior distributions on each model parameter, using literature data on observations of infected individuals. Elderd, et al. (2006) next used maximum likelihood to fit the model to epidemic data and finally combined the likelihood and the priors to calculate the posterior distribution of each parameter. This method allows researchers to combine data from multiple sources to produce an overall prediction of the intensity of future epidemics. In combination with the Bayesian framework, the model itself has been used by Elderd, et al. (2006) to assess the effectiveness of different public health interventions for reducing the severity of epidemics of introduced diseases, notably due to bio-terrorist smallpox attacks.

Beyond its usefulness in public health, this approach can be of widespread usefulness in environmental management, because it allows a rigorous assessment of how uncertainty in a model’s parameter values is translated into uncertainty in the model’s predictions. One obstacle to direct application to wildlife diseases, however, is that the model does not allow for stochastic fluctuations in transmission rates. To address this problem, Elderd and Dwyer, in collaboration with Marc Coram of the Statistics Department, have developed a stochastic epidemic model, as well as a method of fitting this model to epidemic data (Dwyer, et al., in preparation). The group first incorporated the effects of Gaussian white noise on transmission to represent the effects of random fluctuations due to weather. To fit the model to data, they then generated many realizations of the model, calculated the likelihood of each realization under the assumption of Gaussian measurement errors, and maximized the average likelihood across realizations. A difficulty with this procedure, however, is that the likelihood is dominated by a small number of good-fitting realizations. To get around this problem, Coram has constructed an importance-sampling scheme, which permits a reduction in the variance of the likelihoods by preferentially sampling parameter values from regions of parameter space that provide better fits to the data. Drs. Fairbrother and Orme-Zavaleta of EPA’s Western Ecology Division have expressed interest in the applicability of these models for other diseases of concern.

Tim Wootton’s research has been focusing on community dynamics and species interactions. Currently, he is using a long-term data set on intertidal communities to investigate an emerging ecological theory about community assemblage (i.e., Hubbell’s neutral theory). This study will test for the effects of variation in species and their interactions on community composition.

Future Activities:

Population Growth

Future work in the population growth group will continue to explore how variation among individuals is quantitatively described and how it relates to extinction risk and viability in populations. In particular, Pfister and Wang have come to realize that continued progress is likely to require an accounting of the relative roles of genetic versus environmental factors on the performance of individuals and the viability of populations. Again using data from coastal marine populations, they will continue to develop general models that seek to predict the trajectories of these populations with the least complexity necessary. Elderd and Grear will extend their initial work on PLC methods beyond a strictly deterministic analysis to include stochastic effects. This analysis will be conducted by constructing a variety of models of differing levels of complexity, such that more complex models allow for stronger effects of age or stage on survival or fecundity. Once the models are constructed, they will use the most detailed model to generate stochastic realizations of population growth. Next, they will fit each model to the simulated data and compare the ability of different models to reproduce the data.

Species Interactions

The species interaction project is currently assessing the usefulness of Coram’s importance sampling technique. An important future direction for this work is to extend the host pathogen model in two ways. First, a basic feature of the model is that it assumes that the transmission rate is drawn from a Gaussian distribution with some mean and variance. Because the underlying assumption is that this stochasticity is due to weather effects on host behavior or disease transmission rates, they will modify the model to instead assume that the transmission rate is a function of weather covariates such as temperature or rainfall, plus a stochastic term. The group then plans to fit the model to Dwyer’s data on gypsy moth pathogens, and eventually to data on chronic wasting disease in white-tailed deer, as suggested by Drs. Fairbrother and Orme-Zavaleta of EPA’s Western Ecology Division.

Second, over the longer term, the group plans to extend the model to consider multiple pathogens, as disease spread in wildlife populations often involves more than one disease. Additionally, Wootton and Elderd are attempting to use a data set of species interactions along with path analysis tools and Markov-chain analysis to examine species interactions within a community setting. By quantifying the strength of species interactions, those interactions that impact the community to the greatest extent can be identified. It is changes in these interaction strengths that may be the first signs of ecosystem stress.

Journal Articles on this Report : 4 Displayed | Download in RIS Format

Other subproject views: All 14 publications 13 publications in selected types All 12 journal articles
Other center views: All 115 publications 69 publications in selected types All 47 journal articles
Type Citation Sub Project Document Sources
Journal Article Elderd BD, Doak DF. Comparing the direct and community-mediated effects of disturbance on plant population dynamics:flooding, herbivory, and Mimulus guttatus. Journal of Ecology 2006;94(3):656-669. R829402 (Final)
R829402C005 (2004)
R829402C005 (2006)
R829402C005 (Final)
  • Full-text: InterScience Full Text
  • Abstract: Blackwell Synergy Abstract
  • Other: InterScience PDF
  • Journal Article Elderd BD, Dukic VM, Dwyer G. Uncertainty in predictions of disease spread and public health responses to bioterrorism and emerging diseases. Proceedings of the National Academy of Sciences of the United States of America 2006;103(42):15693-15697. R829402 (Final)
    R829402C005 (2004)
    R829402C005 (2006)
    R829402C005 (Final)
  • Full-text from PubMed
  • Abstract from PubMed
  • Associated PubMed link
  • Full-text: PNAS full text
  • Abstract: PNAS abstract
  • Other: PNAS PDF
  • Journal Article Pfister CA, Wang M. Beyond size: matrix projection models for populations where size is an incomplete descriptor. Ecology 2005;86(10):2673-2683. R829402 (Final)
    R829402C005 (2004)
    R829402C005 (2006)
    R829402C005 (Final)
  • Abstract: ESA-Abstract
  • Other: University of Chicago-PDF
  • Journal Article Sun L, Wang M. An algorithm for a decomposition of weighted digraphs: with applications to life cycle analysis in ecology. Journal of Mathematical Biology 2007;54(2):199-226. R829402 (Final)
    R829402C005 (2004)
    R829402C005 (2006)
    R829402C005 (Final)
  • Abstract from PubMed
  • Abstract: SpringerLink-Abstract
  • Other: U Chicago PDF
  • Supplemental Keywords:

    RFA, Scientific Discipline, Health, Economic, Social, & Behavioral Science Research Program, PHYSICAL ASPECTS, Air, Geographic Area, Ecosystem Protection/Environmental Exposure & Risk, Applied Math & Statistics, particulate matter, Health Risk Assessment, Ecosystem/Assessment/Indicators, Ecological Effects - Environmental Exposure & Risk, Monitoring/Modeling, Risk Assessments, Environmental Monitoring, Ecological Effects - Human Health, Physical Processes, Ecological Risk Assessment, Environmental Statistics, Environmental Engineering, Engineering, Chemistry, & Physics, Ecological Indicators, EPA Region, particulates, risk assessment, ecological effects, monitoring, health risk analysis, watersheds, emissions monitoring, ecological health, ozone , particulate, stratospheric ozone, ozone, sediment transport, computer models, exposure, air pollution, chemical transport modeling, chemical transport, trend monitoring, statistical models, human exposure, ecological risk, water, ecosystem health, environmental indicators, PM, ecological models, chemical transport models, Region 5, data models, air quality, statistical methodology, human health risk, stochastic models

    Relevant Websites:

    http://www.stat.uchicago.edu/~cises/ Exit

    Progress and Final Reports:

    Original Abstract
  • 2002 Progress Report
  • 2003
  • 2005
  • Final Report

  • Main Center Abstract and Reports:

    R829402    Center for Integrating Statistical and Environmental Science

    Subprojects under this Center: (EPA does not fund or establish subprojects; EPA awards and manages the overall grant for this center).
    R829402C001 Detection of a Recovery in Stratospheric and Total Ozone
    R829402C002 Integrating Numerical Models and Monitoring Data
    R829402C003 Air Quality and Reported Asthma Incidence in Illinois
    R829402C004 Quasi-Experimental Evidence on How Airborne Particulates Affect Human Health
    R829402C005 Model Choice Stochasticity, and Ecological Complexity
    R829402C006 Statistical Approaches to Detection and Downscaling of Climate Variability and Change