Grantee Research Project Results
Final Report: Framework for Predicting the Effects of Environmental Change on Populations
EPA Grant Number: R823588Title: Framework for Predicting the Effects of Environmental Change on Populations
Investigators: Nisbet, Roger M. , Murdoch, William W.
Institution: University of California - Santa Barbara
EPA Project Officer: Hahn, Intaek
Project Period: August 15, 1995 through August 14, 1998
Project Amount: $360,037
RFA: Exploratory Research - Environmental Biology (1995) RFA Text | Recipients Lists
Research Category: Biology/Life Sciences , Human Health , Aquatic Ecosystems
Objective:
This research was part of a larger program that aimed to produce mathematical models that can translate the effects of environmental stress on individual aquatic organisms to the dynamics of populations. Specifically, we aimed to develop testable, individual-based models capable of predicting population response to environmental insults.
We focused on an ecologically important genus, Daphnia, for which sufficient empirical data are available, or can be obtained from well-chosen experiments, to permit powerful tests of models. We had a subcontract with Dr. E. McCauley (University of Calgary) who, with previous EPA support, conducted experiments on the response of individual Daphnia pulex to varied environmental conditions (temperature and pH). The experiments funded under this grant aimed to:
- Determine Daphnia population dynamics, when faced with contrasting environments and food schedules; and
- Follow cohorts of individuals within a population, with the aim of estimating fecundity, growth rates and mortality rates for individuals within a population.
The primary objectives of our modeling were to:
- Develop a general model of individual growth and reproduction, and to test its applicability at the low food levels encountered in natural environments;
- Develop a model relating Daphnia mortality to energetics;
- Extend the individual models to cover new environments; and
- Determine whether the results of the population experiments (described above) were predictable using individual-based models parameterized using previous data on individuals grown in isolation.
Modeling Objectives 1 through 3 represented continuation of previous work. Modeling Objective 4 was new to the present grant and, to our knowledge, is the first rigorous test of a fundamental premise of individual-based modeling. As such, it offers EPA an objective evaluation of the practical utility of an increasingly used approach to predicting the population level consequences of environmental change.
Background. Although the research conducted with support from this grant focused on Daphnia, our aim was the development of general theory. Thus, we based our models on general mechanisms, but focused our testing effort on laboratory data on individuals and populations of this particular genus, with which we have extensive modeling experience. Models that survive such detailed scrutiny and rigorous testing have some credibility when used in field situations where testing is much more difficult.
Highlights of our research on Daphnia over the past 15 years include the following:
- A demonstration of common dynamic patterns in a large number of Daphnia populations in laboratory, microcosm, and field systems (Murdoch and McCauley, 1985; McCauley and Murdoch, 1987).
- A meta-analysis that accounted for data on growth and reproduction in D. pulex using an empirically-based DEB model (McCauley, et al., 1990; Gurney, et al., 1990).
- Experimental studies relating to the "paradox of enrichment" (Rosezweig, 1971), a theoretical prediction that large amplitude population cycles should occur in highly enriched systems. McCauley and Murdoch (1990) demonstrated the absence of such cycles in enriched mesocosms. McCauley, et al. (1999), recently demonstrated that these cycles may occur in carefully prepared microcosms with accelerated nutrient recycling and no inedible algae, but also found evidence of multiple attractors.
- A detailed theoretical study of five hypothesized mechanisms that might be responsible for the observed stability of most Daphnia populations, and rejection of four of these (Murdoch, et al., 1998).
- A model of carbon flow and phosphorus recycling that showed sequestering of phosphorus by Daphnia is unlikely to be the sole cause of observed population stability (Nisbet, et al., 1991).
- A simple, carbon-based, food-web model that identified two mechanisms whereby the presence of inedible algae might stabilize a Daphnia-algae system (Kretzschmar, et al., 1993).
- A demonstration of the ability of simple individual-based models to predict biomass (or carbon) densities and the outcome of competition between Daphnia and Bosmina in laboratory cultures, and also the density of edible algae in the field (Nisbet, et al., 1997).
- A demonstration of the ability of physiologically-based, stage-structured models to predict laboratory population dynamics (McCauley, et al., 1996).
Items 4, 6, and 7 were in part supported by this grant. However, a notable omission in the above list is any evaluation of whether we can predict the details of population dynamics of Daphnia using an empirically-based model of growth and reproduction of individual organisms, coupled with a similar empirically-based model of mortality. This was the primary focus of the research reported here.
Modeling Individual Growth and Reproduction in Different Environments. During the tenure of the grant, we greatly improved our ability to formulate, fit, and test models of growth and reproduction in individual organisms. We use dynamic energy budget (DEB) models, in which differential equations describe the rates at which individual organisms assimilate and utilize energy from food for maintenance, growth, reproduction, and development. These rates depend on the state of the organism (age, size, sex, nutritional status, etc.) and the state of its environment (food density, temperature, etc.). Solutions of the model equations represent the life history of individual organisms in a potentially variable environment. Testing a model involves solving an inverse problem where we determine the values for unknown parameters and/or functions that cause a partially specified model to best fit (in the sense of minimizing some objective function) the data on growth and fecundity at different food levels. For this work, we used a software package ddefit (Wood, 1999; code available from http://www.ruwpa.st-and.ac.uk/simon.html).
The unknown parameters in our models represent such quantities as maintenance rate or a half-saturation constant for feeding. The unknown functions characterize the size dependence of feeding rate and of the proportion of energy allocated to reproduction. There is independent data on some of these, and we use consistency of fitted parameters and functions with this independent information as a criterion for model validation.
- High and Medium Food Environments
Our data on Daphnia growth and reproduction at high and medium levels of food are well described by a simple DEB model that assumes there is a single "currency" (carbon) for both biomass and energy. The carbon mass of somatic tissue in a female animal is related to some measure of its length, L, by an allometric relation. The animal assimilates carbon from food at a rate A and respires carbon for basal maintenance at a rate M. The net production, P = A - M, represents surplus carbon that can be assigned to "growth" (i.e., production of new somatic tissue) or "reproduction" (production of eggs). Energy reserves are not considered explicitly.
Assimilation rate depends multiplicatively on food density and on length, with the food dependence represented as a type-2 functional response (see Chapter 4 of Gurney and Nisbet 1998 for more discussion of this assumption). Maintenance rate is assumed proportional to carbon weight, and the partitioning of net production between growth and reproduction is assumed to depend only on size.
Figure 1 (below) shows that this simple model gives an excellent fit to data on the growth and fecundity of isolated individuals from a single clone of D. pulex grown in algal food at 4 carbon densities (see below). The fitted maintenance rate, the half saturation constant in the functional response, and the fitted function characterizing the length dependence of assimilation rate are all consistent with independent data in the literature.
- Growth and Reproduction at Very Low Food Levels
The model of Section 1 does not correctly predict physiological performance at very low food levels, such as those regularly experienced in the field. Daphnia pulex can grow and even produce a few eggs in small-volume (25 mL) transfer cultures under conditions where the model predicts no growth. This finding is consistent with previous work of Lynch (1989) who studied a wider range of food densities, but in smaller containers and whose experiments were modeled by Andersen (1997); growth at medium and high food was consistent with simple energetic considerations, but individuals performed better than predicted when faced with low food.
To explore this phenomenon in more detail, we used data from a pervious study of individual D. pulex grown in small (25 mL) containers with 2 day transfers, and with food concentrations ranging from 0.05 to 0.8 mg C/L (McCauley, et al., 1990). A key diagnostic for the validity of a simple net production model with no explicit representation of reserves is a plot of cumulative reproduction against size; the simple model predicts data collapse on to a single curve. This data collapse did occur with high and medium food (the data shown in Figure 1), but not in the low-food, long-transfer-interval cultures. We showed that the low food results can be explained by a variant of the net production model in which the allocation rules are modified to account for brief periods of starvation. Further details of this work are reported in Noonburg, et al. (1998).
- Changed Environments
We completed an extensive series of experiments on Daphnia growth and reproduction at one temperature (23?C) and three different pH values (8.3, 7.3, and 6.3). The range in pH represents substantial environmental stress, because our clone came from a hard water system with pH 8.0-8.3.
Analysis of these data on growth and reproduction in different environments is in progress. However, we have completed a preliminary study comparing the data at 20 and 23?C with pH 7.3. Changing temperature from 20 to 23?C has remarkably little effect on individual growth and reproduction, but substantially reduces longevity. One immediate result from the experiments at different pH is that no animals were reproductively viable at pH 6.3.
The simple net production model described in Section 1 is applicable to the 23?C data. Indeed, the data collapse on the plot of cumulative reproduction against size is, if anything, tighter than at 20?C (see Figure 2).
- Modeling Growth and Reproduction in Fluctuating Environments
The scope of simple DEB models like that described in Section 1 is very limited. A model that represents either an organism that feeds in distinct "meals," or any organism that experiences periods of food shortage, must incorporate some representation of energy reserves. This includes any model capable of describing the batch culture conditions used in most standardized toxicity tests.
There are two families of deterministic DEB model that take account of reserve dynamics, distinguished by the assumed priorities for utilization of energy when food is scarce (see Chapter 4 of Gurney and Nisbet, 1998). Assimilation models (Kooijman, 1986, 1993) assume that a specified fraction of assimilated food is assigned to reproduction, while net production models (Ross and Nisbet, 1990; Nisbet, Ross, and Brooks, 1996; Lika and Nisbet, 1999) subtract off basal maintenance before assigning some fraction to reproduction. We have fitted several DEB models, including modified versions of the Kooijman and Lika-Nisbet models, to the data in Figure 1, and obtained excellent fits. The modified Lika-Nisbet model alone is consistent with the analyses of low food dynamics (Section 3 above).
Modeling Mortality for Individuals in Isolation. We followed the daily growth, reproduction, and molting of a large number of individual D. pulex reared in isolation in different food environments from birth until death. Two contrasting approaches were used to analyze these data. We first performed a standard analysis of age specific survival rates, obtaining estimates of daily hazard rate. Results are shown below in Figure 3.
Figure 3. Estimates of daily mortality rate for Daphnia pulex grown in isolation at four food levels. Points shown are 5-day moving averages.
Our second set of analyses aimed to relate mortality to individual energetics. We recast individual growth and reproduction in relation to instar, and, for each instar, calculated the amount of energy allocated to growth and reproduction (i.e., the "net production" described above). Thus, for each individual we had a complete "instar by instar" history of its energy allocation. Because we know the instar at which the individual died, we can relate the death of an individual to its age, length, and energy allocation in the previous instar(s). We used logistic regression models to determine how the probability of dying during any particular instar is related to the size, age, and performance of individuals in previous instars. We found significantly different relationships between net production and probability of dying for juveniles, young adults, and old adults, a finding consistent with the results of our previous hazard rate analysis. However, the logistic regression analysis yields explicit relationships between an individual's net production and the probability of dying.
The principal findings were:
- Juvenile mortality is food dependent, being high at the lowest and the highest food treatments.
- For each food level, mortality is lowest in the first 20 days of adult life.
- The "low mortality" phase coincides with the increasing portion of the fecundity curve.
- Within the low mortality phase, increasing food density from 0.1 a 0.4 mg C/L decreases the death rate.
- In later life, the rate of increase of mortality increases with food level.
Daphnia Population Dynamics in Fluctuating Environments. Natural environments are constantly fluctuating, and any predictions of population dynamics on the basis of individual performance should, in principle, recognize this. The problem for modelers is how much detail to include in models that incorporate environmental variability. To investigate this, we studied replicated Daphnia populations in three contrasting transfer culture treatments. In all treatments, non-growing, algal food was supplied at the same average rate of 1.5 mg C/L/day. Treatments differed in the time interval between transfers?1, 2, and 4 days, respectively. Daphnia in transfer culture eat most of the food supplied within a few hours of each transfer; thus, the experimental design forces individuals within a population to live in alternating periods of glut and famine, the extremity of the fluctuations being determined by the transfer interval. As Daphnia at 20?C molt every 2-3 days, the design also allows us to compare situations where all animals experience at least one period of ample food per molt with situations where the periods of starvation exceed the inter-molt time.
Analysis of the results is still in progress. However, work to date establishes the following patterns in populations at equilibrium in response to increasing transfer interval:
- Total biomass decreases.
- Adult biomass decreases.
- Juvenile biomass is unchanged between 1- and 2-day transfers, but is higher for 4-day transfers.
- Fecundity (eggs per adult per day) has no statistically significant change between 1- and 2-day transfers, and increases sharply (2-3 fold) for 4-day transfers.
We interpret these results as suggesting that the primary effect of long transfer intervals is an increase in juvenile mortality. This is consistent with our observations on the food dependence of individual animals and has implications for the interpretation of field data where measures of fecundity (e.g., egg ratios) are frequently used to infer mortality rates.
We have completed analogous experiments at a different temperature (23?C) and at the same three pH levels as were used in our study of individuals. The analysis of these experiments is still in progress; however, one potentially important finding is persistence of a population at the lowest pH where individuals were unable to grow to reproductive maturity.
Comparison of Vital Rates in Individuals and Populations. The central tenet of individual based population models (IBMs) is that a population is simply a collection of individuals that grow, develop, reproduce, and die at rates determined by their own physiological state and by their environment. There is a large body of data on growth of individual Daphnia in isolation?including the results of countless standardized toxicity tests that can be fitting using DEB models similar to those discussed in Section 3 (Kooijman and Bedaux, 1996). These data, and similar information for other organisms, will have considerable "added value" if it can be used to parameterize an IBM.
The "ideal" way to perform experiments to estimate rates of demographic processes within a population is to use individual markings. Unfortunately, no nonintrusive technique has been developed that can be applied to Daphnia individuals that molt every 2 to 3 days. Thus, no marks known to us can be sustained long enough to provide data for estimating demographic and growth rates. Our alternative was the artificial assembling of age-structured populations with readily identifiable cohorts. Cohorts of individuals were raised to one of four sizes: neonates of length 0.7?0.1 mm, immature adults (or "adolescents") of 1.0?0.1 mm., "small adults" of 1.4?0.1 mm, and "large adults" of 2.0?0.1 mm. Individuals from these cohorts were used to create a population with the same mix of size classes as the populations with 2-day transfers described in Section 5. The evolution of these multi-modal "equilibrium" size distributions was used to estimate fecundity and rates of growth and mortality in populations near equilibrium.
Analysis of the results is still in progress. The most important finding was that age at first reproduction was different in populations. Individuals in isolation were typically first observed with eggs at lengths of around 1.4 mm, but in populations, this threshold was increased to 1.8 mm.
The mortality rates were compared with those predicted from the studies of individuals in isolation with the following results:
- Mortality rates for juveniles are predicted to be much greater than the corresponding values for small adults. Measured values for the cohorts were 0.05 and 0.01 day-1 in agreement with the prediction.
- Mortality rates were predicted to be greater for small adults than for large adults. Measured values were 0.01 and 0.02 day-1 in agreement with the prediction.
- Mortality rate for juveniles was predicted to be 0.05 day-1, in exact agreement with the measured value.
- Mortality rate for small adults was predicted to be 0.02 day-1, approximately twice the value measured in the cohorts. This is the size class with lowest mortality rates and both values are inferred from a small number of deaths; thus, the difference may not be highly significant statistically. However, taken together with the change in the size threshold for reproduction, it points to the possibility that individuals in this size range are postponing reproduction and using the energy saved both to promote growth and reduce mortality.
Once mortality rates in the population are known, the juvenile development time can be inferred from theory we developed to describe the demography of food-limited populations at equilibrium (Gurney, et al., 1996; Nisbet, et al., 1997). If NA, NJ denote adults and juvenile populations, mJ is the juvenile mortality rate, and b is the fecundity, then the juvenile stage duration, TJ, is given by:
If "adults" and "juveniles" are defined assuming a maturation size of 1.8 mm, then this formula yields TJ = 29-32 days.
The juvenile development time also can be estimated from the cohort data. If juvenile growth is exponential, and we again assume that the size of maturity is 1.8 mm, then from measured growth rates in the cohort experiments, we predict that TJ = 27 days. We have not yet analyzed data from all the replicates, and hence do not have confidence intervals around this value; however, it is safe to remark that it is close to the predicted value from equilibrium demography theory.
In summary, Daphnia in populations initiated reproduction at a larger size than individuals in isolation. Mortality and growth rates were broadly consistent with those of isolated individuals and with the observed equilibrium demography.
Conclusions:
The most important findings from this research are the following:
- Simple DEB models of individuals fit data on individual growth and reproduction, except in situations where organisms experience sustained long intervals with low food.
- Mortality rates for individuals in isolation can be related to individual energy budgets.
- Juveniles in populations grow and die at rates consistent with measured individual physiology. Adults may delay committing energy to reproduction; however, once the size at first reproduction is known, the demography of populations at equilibrium can be predicted from data on individuals.
- Fluctuating environments may significantly influence demography. Populations experiencing prolonged periods of starvation have higher fecundity and juvenile mortality.
We end this report with a comment on the importance of our work beyond the narrow problem of understanding the effects of environmental stress of the population dynamics of Daphnia. Daphnia are of course an important genus in many temperate freshwater bodies, so our work has some immediate applicability. However, developing truly general theory is our long-term objective; indeed without some generality, theory and models have little to contribute to ecology, and even less to ecological applications, as we can never study experimentally all combinations of organisms and stressors that are of concern. The challenge is achieving generality without sacrificing the security that comes from working with testable (and tested) models. There is no a priori way of determining the level of generality to which we can legitimately aspire.
We are optimistic that the work described above can contribute to general theory for four reasons. First, the processes included in our models (feeding, assimilation, respiration, and reproduction) are shared by many other organisms, and must constrain their physiology. Second, although the only environmental changes considered in the present work involved temperature and pH, other forms of environmental stress (e.g., some toxic chemicals) are known to influence energy budgets by modifying these same rate processes. Third, our Daphnia work opens the possibility of developing models of very general ecological phenomena (response to variable environments, short-term adaptations). Finally, the population phenomenon that originally motivated our work, the apparent stability of a consumer-resource with an "equilibrium" resource level well below carrying capacity, is common in nature. We do not believe in "universal" explanations?ecology is not physics?but the key to modeling is knowing what to leave out. Understanding what is important (and therefore what is less important) in one system should provide guidance to workers in others.
References:
Andersen T. Pelagic nutrient cycles: herbivores as sources and sinks. Springer-Verlag, Berlin, NY, 1997.
Gurney WSC, McCauley E, Nisbet RM, Murdoch WW. The physiological ecology of Daphnia: a dynamic model of growth and reproduction. Ecology 1990;71:716-732.
Gurney WSC, Nisbet RM. Ecological dynamics. Oxford University Press, New York, NY, 1998.
Kooijman SALM. Population dynamics on basis of energy budgets. The dynamics of physiologically structured populations. Metz JAJ, Diekmann O, eds. Springer-Verlag, Berlin, NY, 1986, pp. 266-297.
Kooijman SALM. Dynamic energy budgets in biological systems: theory and applications in ecotoxicology. Cambridge University Press, New York, NY, 1993.
Kooijman SALM, Bedaux JJM. The analysis of aquatic toxicity data. VU University Press, Amsterdam, 1996.
Kretzschmar M, Nisbet RM, McCauley E. A predator-prey model for zooplankton grazing on competing algal populations. Theoretical Population Biology 1993;44:32-66.
Lika L, Nisbet RM. A dynamic energy budget model based on partitioning of net production. Journal of Mathematical Biology 1999 (submitted for publication).
Lynch M. The life history consequences of resource depression in Daphnia pulex. Ecology 1989;70:246-256.
McCauley E, Murdoch WW. Cyclic and stable populations: plankton as paradigm. The American Naturalist 1987;129:97-121.
McCauley E, Murdoch WW. Predator-prey dynamics in environments rich and poor in nutrients. Nature 1990;343:455-457.
McCauley E, Murdoch WW, Nisbet RM. Growth, reproduction, and mortality of Daphnia pulex: life at low food. Functional Ecology 1990;4:505-514.
McCauley E, Murdoch WW, Nisbet RM, Gurney WSC. The physiological ecology of Daphnia: development of a model of growth and reproduction. Ecology 1990;71:703-715.
McCauley E, Nisbet RM, de Roos AM, Murdoch WW, Gurney WSC. Structured population models of herbivorous zooplankton. Ecological Monographs 1996;66(4):479-501.
McCauley E, Nisbet RM, Murdoch WM, de Roos AM, Gurney WSC. Large-amplitude cycles of Daphnia and its algal prey in enriched environments. Nature 1999;402:653-656.
Murdoch WW, McCauley E. Three distinct types of dynamic behavior shown by a single planktonic system. Nature 1985;316:628-630.
Murdoch WW, Nisbet RM, McCauley E, de Roos AM, Gurney WSC. Plankton abundance and dynamics across nutrient levels: Tests of hypotheses. Ecology 1998;79(4):1339-1356.
Nisbet RM, McCauley E, de Roos AM, Murdoch WW, Gurney WSC. Population dynamics and element recycling in an aquatic plant-herbivore system. Theoretical Population Biology 1991;40:125-147.
Nisbet RM, Mccauley E, Gurney WSC, Murdoch WW, deRoos AM. Simple representations of biomass dynamics in structured populations. In: Othmer HG, Adler FR, Lewis MA, Dallon JC, eds. Case Studies in Mathematical Modeling—Ecology, Physiology, and Cell Biology. Upper Saddle River, NJ: Prentice Hall 1997, pp. 61-79.
Nisbet RM, Muller EB, Brooks AJ, Hosseini P. Models relating individual and population response to contaminants. Environmental Modeling and Assessment 1997;2:7-12.
Nisbet RM, Ross AH, Brooks AJ. Empirically-based dynamic energy budget models: theory and an application to ecotoxicology. Nonlinear World 1996;3:85-106.
Noonburg EG, Nisbet RM, McCauley E, Gurney WSC, Murdoch WW, de Roos AM. Experimental testing of dynamic energy budget models. Functional Ecology 1998;12(2):211-222.
Rosenzweig ML. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 1997;171:385-387.
Ross AH, Nisbet RM. Dynamic models of growth and reproduction of the mussel Mytilus edulis L. Functional Ecology 1990;4:777-487.
Wood SN. Partially specified population models: modelling, data fitting and inference with incomplete ecological information. Ecology 1999 (submitted for publication).
Journal Articles on this Report : 6 Displayed | Download in RIS Format
Other project views: | All 12 publications | 11 publications in selected types | All 8 journal articles |
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De Roos AM, McCauley E, Nisbet RM, Gurney WSC, Murdoch WW. What individual life histories can (and cannot) tell about population dynamics. Aquatic Ecology 1997;31(1):37-45. |
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Gurney WSC, Middleton DAJ, Nisbet RM, McCauley E, Murdoch WW, DeRoos A. Individual energetics and the equilibrium demography of structured populations. Theoretical Population Biology 1996;49(3):344-368. |
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McCauley E, Nisbet RM, De Roos AM, Murdoch WW, Gurney WSC. Structured population models of herbivorous zooplankton. Ecological Monographs 1996;66(4):479-501. |
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McCauley E, Nisbet RM, Murdoch WW, de Roos AM, Gurney WSC. Large-amplitude cycles of Daphnia and its algal prey in enriched environments. Nature 1999;402(6762):653-656. |
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Muller EB, Nisbet RM. Survival and production in variable resource environments. Bulletin of Mathematical Biology 2000;62(6):1163-1189. |
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Nisbet RM, Diehl S, Wilson WG, Cooper SD, Donalson DD, Kratz K. Primary-productivity gradients and short-term population dynamics in open systems. Ecological Monographs 1997;67(4):535-553. |
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Supplemental Keywords:
modeling, population response, population dynamics, environmental change effects, ecology, Daphnia., RFA, Scientific Discipline, Ecosystem Protection/Environmental Exposure & Risk, Ecology, exploratory research environmental biology, Ecosystem/Assessment/Indicators, Chemical Mixtures - Environmental Exposure & Risk, Ecosystem Protection, Chemistry, Monitoring/Modeling, Ecological Effects - Environmental Exposure & Risk, Ecological Effects - Human Health, Biology, Ecological Indicators, ecological exposure, anthropogenic stress, environmental monitoring, ecological modeling, exposure, population level response, ecological impacts, predictive model, ecosystem health, population models, predicting environmental change, environmental impactProgress and Final Reports:
Original AbstractThe perspectives, information and conclusions conveyed in research project abstracts, progress reports, final reports, journal abstracts and journal publications convey the viewpoints of the principal investigator and may not represent the views and policies of ORD and EPA. Conclusions drawn by the principal investigators have not been reviewed by the Agency.