Final 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 , 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
RFA: Environmental Statistics Center (2001) RFA Text |  Recipients Lists
Research Category: Environmental Statistics , Ecological Indicators/Assessment/Restoration , Health , Ecosystems , Air

Objective:

To understand human-influenced systems, ecologists have increasingly relied upon mathematical models to describe processes ranging from population growth to ecosystem dynamics. Historically, most ecological models have been deterministic, and thus were of limited usefulness in environmental management. By focusing on incorporating stochasticity into ecological models, the ecology group developed methods for quantifying uncertainty and for examining how uncertainty in outcomes changes management decisions. For concreteness, each sub-project focused on particular organisms, but the models and methods that we developed are of wide applicability.

Our projects fell into two categories; population growth and species interactions. The population growth project was in collaboration with EPA scientist Dr. Jason Grear at the Atlantic Ecology Division of the National Health and Environmental Effects Research Laboratory (NHEERL). The paper based on this collaboration has led to a manuscript that is in revision at Oikos (Grear and Elderd, 2008). In addition to these collaborations, we have published extensively in a wide-variety of journals (see Publications).

Summary/Accomplishments (Outputs/Outcomes):

1.1 Models of population growth

In the population-growth project, we assessed the impact of variation among individuals on population growth rates, by extending simple models and assessing their general usefulness. First, PI’s Cathy Pfister and Mei Wang adapted 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 stage groups, which may bias estimates of population growth. Pfister and Wang (2005) 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 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 tradeoff 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 is 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, a former postdoctoral research associate with the Ecology Group has been working with Jason Grear of the EPA to explore the usefulness of PLC models (Grear and Elderd 2008). PLC models are more likely to be useful if survivorship changes little over the course of an organism’s lifetime. If survivorship varies significantly over multiple age classes, the PLC approach can drastically overestimate or underestimate population growth rate, whether the model in question is deterministic or stochastic. To correct for this error in population growth rate, Grear and Elderd proposed a simple solution based on the shape of the organism’s survivorship curve. The resulting approach can provide reasonable estimates of population growth even when data are quite limited.

Elderd also expanded the usefulness of matrix population models for analyzing experimental data, through a collaboration with Dr. Dan Doak at the University of California, Santa Cruz. 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) extended this method to include nested multiple-factorial experiments, and showed how the method can incorporate stochastic models.

A further goal of collaboration between ecologists and statisticians is assessing the limitations of existing data sets and making recommendations on how they can be improved. Datasets on natural populations can have constraints and limitations based on the cost of data collection and the logistical hurdles in the field. We assessed the biases imposed by survey designs for coastal fish populations and made recommendations for how to remedy these biases. Many ecological margins (e.g. rivers, coasts, forests) are ’curvy’ and require special attention and correction to sampling biases. In collaboration with the Alaska Department of Fish and Game, we developed a transect sampling method for more accurate animal estimation, even in the absence of detailed maps (Ma, et al. submitted). The methodology is particularly relevant to any state or federal agency that has a mandate to sample widely distributed species over poorly described geographies.

1.2 Species interactions

The work being carried out in the population-growth projects essentially assumed that interactions between 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 lay 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 involved 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 was used in a model-selection criterion, in order to determine which model is the best. Our immediate goal was 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 was thus to provide models that can help us to predict how anthropogenic perturbations may influence communities and ecosystems.

Work in the species interactions project focused on (1) disease dynamics and (2) intertidal community dynamics. Bret Elderd, Vanja Dukic, and Greg Dwyer developed a Bayesian framework for forecasting a disease’s rate of spread (Elderd, et al. 2006). The basis of this approach is to first construct prior distributions on each model parameter, using literature data on observations of infected individuals. Second, one uses maximum likelihood to fit the model to epidemic data, and finally one combines the likelihood and the priors to calculate the joint posterior distribution of the parameters. This method allows researchers to combine data from multiple sources to predict future epidemics. Elderd et al. (2006) used this approach to assess the effectiveness of different public health interventions for reducing the severity of epidemics of introduced diseases, notably due to bioterrorist smallpox attacks. Note that Elderd will be beginning a tenure-track faculty position at Louisiana State University in the fall of 2008.

Beyond 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, have developed a stochastic epidemic model, as well as a method of fitting this model to epidemic data (Dwyer et al., in preparation). In early work, the group assumed simply that the effects of stochasticity on transmission were due only to Gaussian white noise, representing the effects of random fluctuations due to weather. In nature, however, such fluctuations are likely to be autocorrelated, as weather fronts move in and out of the region in which the epidemic is proceeding. The group has therefore extended the model so that transmission on any given day is an autoregressive process of order 1. A useful feature of the resulting model is that it combines classical autoregression with mechanistic differential equation models, thereby unifying two very different approaches to disease spread.

To fit the model to data, the group next 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 a Monte-Carlo integration 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. This approach was also used in an effort to estimate dispersal rates from genetic data, an effort spear-headed by Greg Dwyer’s Ph.D. student Pauline Fujita. Fujita is now a staff scientist at the University of California at Santa Cruz Genome Browser project.

Tim Wootton’s research focuses on community dynamics and species interactions. He used 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 tested for the effects of variation in species and their interactions on community composition (Wootton 2005). As such, it was the first empirical analysis of the neutral theory of community assembly, and it demonstrated the need for understanding species interactions in describing community dynamics. Wootton was also active in research regarding Lyme disease dynamics and rates of human infection (Tsao et al. 2004). This work showed that vaccinating mice, a primary reservoir for this disease, reduces disease incidence among mice. Similar ideas can also be readily applied to other diseases of concern where no human vaccine exists.

James Forester worked with Tim Wootton, Greg Dwyer and Marc Coram to identify direct and indirect species interactions in intertidal communities using observational time series of fixed quadrats (Forester et al., in preparation). These data are particularly challenging to work with because they are percent cover (therefore bounded between zero and 100) and exhibit variability that changes as a function of the density of the dominant mussel species (Mytilus californianus). This density-dependent stochasticity is of biological interest because it represents variability in immigration at low density and increased susceptibility to wave disturbances at higher density. They have modeled this system as a series of linked, non-linear betabinomial mixture models that describe the system dynamics and identify how species interactions change through time. Forester will be a post-doctoral fellow with Paul Moorcroft’s lab at Harvard University in the fall of 2008.

Contributions to Understanding of Environmental Problems

The ecology group attempted to understand the mechanisms determining the number and distribution of living organisms in the environment. Understanding these mechanisms is crucial for understanding how human impacts on the environment will contribute to the risk that species go extinct, or at least suffer a severe reduction in numbers. In particular, one would like to be able to predict how a given human impact will affect the population of a given species. Because of this interest in prediction, it is crucial to be able to construct mathematical models of how populations will grow or decline in response to environmental perturbations. For example, low doses of mercury are common in the environment, and clearly have at least low-level toxic effects on individual loons. Predicting how these low-level toxic effects are translated into reductions in population growth, however, is a difficult task. The best way to make such predictions is by using mathematical models of loon population growth.

In constructing ecological models, a key problem is that the number of different factors that can affect a population is very large. Clearly, models that are too simple will give inaccurate predictions because they neglect key mechanisms. A less obvious problem is that, if we attempt to construct a model that is almost as complicated as nature itself, the uncertainty in the model’s parameters will be so large that the model predictions will be unreliable. The crucial task is therefore to develop methods of constructing models that take into account as much complexity as can be reliably supported by the data. The ecology group managed to do this for several important classes of organisms, across a range of habitats.


Journal Articles on this Report : 11 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 Dwyer G, Morris WF. Resource-dependent dispersal and the speed of biological invasions. The American Naturalist 2006;167(2):165-176. R829402 (Final)
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  • Abstract from PubMed
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  • Other: Mendeley abstract
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  • 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)
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  • Full-text: InterScience Full Text
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  • Abstract: Blackwell Synergy Abstract
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  • 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)
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  • Abstract: PNAS abstract
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  • Journal Article Grear JS, Elderd BD. Bias in population growth rate estimation: sparse data, partial life cycle analysis and Jensen's inequality. Oikos 2008;117(10):1587-1593. R829402 (Final)
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  • Abstract: Wiley
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  • Journal Article Ma L, Stein ML, Wang M, Shelton AO, Pfister CA, Wilder KJ. A method for unbiased estimation of population abundance along curvy margins. Environmetrics 2011;22(3):330-339. R829402C005 (Final)
  • Abstract: Wiley
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  • Other: Universoty of Chicag-PDF
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  • Journal Article Novak M, Wootton JT. Estimating nonlinear interaction strengths: an observation-based method for species-rich food webs. Ecology 2008;89(8):2083-2089. R829402 (Final)
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  • Abstract from PubMed
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  • 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)
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  • 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)
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  • Journal Article Tsao JI, Wootton JT, Bunikis J, Luna MG, Fish D, Barbour AG. An ecological approach to preventing human infection: vaccinating wild mouse reservoirs intervenes in the Lyme disease cycle. Proceedings of the National Academy of Sciences of the United States of America 2004;101(52):18159-18164. R829402 (Final)
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  • Journal Article Wooton JT, Pfister CA, Forester JD. Dynamical patterns and ecological impacts of declining ocean pH in a high-resolution multi-year dataset. Proceedings of the National Academy of Sciences of the United States of America 2008;105(48):18848-18853. R829402 (Final)
    R829402C005 (Final)
  • Full-text from PubMed
  • Abstract from PubMed
  • Associated PubMed link
  • Abstract: PNAS abstract
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  • Other: PNAS PDF
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  • Journal Article Wootton JT. Field parameterization and experimental test of the neutral theory of biodiversity. Nature 2005;433(7023):309-312. R829402 (Final)
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  • Other: UNIMI-PDF
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  • 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, ecological effects, monitoring, particulates, risk assessment, health risk analysis, ecological health, particulate, watersheds, stratospheric ozone, ozone , emissions monitoring, computer models, exposure, ozone, sediment transport, air pollution, trend monitoring, chemical transport, chemical transport modeling, human exposure, statistical models, ecological risk, ecosystem health, environmental indicators, PM, water, data models, Region 5, chemical transport models, ecological models, air quality, human health risk, statistical methods, stochastic models

    Progress and Final Reports:

    Original Abstract
  • 2002 Progress Report
  • 2003
  • 2004 Progress Report
  • 2005

  • 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