Global climate change and its impact on disease embedded in ...

Global climate change and its impact on disease embedded in ecological communities Philippe A. Rossignol, Jennifer Orme-Zavaleta, and Annette M. Rossignol

ABSTRACT

AUTHORS Philippe A. Rossignol Department of Fisheries and Wildlife, Nash Hall 104, Oregon State University Corvallis, Oregon 97331; [email protected] Philippe Rossignol is professor in the Department of Fisheries and Wildlife at Oregon State Univesity, where he has been since 1988. He received his Ph.D. from the Faculty of Medicine at the University of Toronto in 1978. He was then a postdoctoral fellow at the Harvard School of Public Health. His main interests are in public health entomology, specifically malariology, and theoretical community ecology.

We present the techniques of qualitative analysis of complex communities and discuss the impact of climate change as a press perturbation. In particular, we focus on the difficult problem of disease and parasites embedded in animal communities, notably zoonotic diseases. Climate change can potentially affect population densities of hosts and vectors, as well as their life expectancy. Recent advances may provide insight in predicting change in risk of zoonotic disease following climate shifts. We conclude that the impact of change on ecological communities can be profound but subtle, complex, and ambiguous, even under basic mathematical assumptions about the structure of a community when in equilibrium.

Jennifer Orme-Zavaleta Office of Research and Development, National Health and Environmental Effects Research Laboratory, Western Ecology Division, U.S. Environmental Protection Agency, 200 SW 35th St., Corvallis, Oregon 97333; [email protected]

INTRODUCTION

Annette M. Rossignol Department of Public Health, Waldo Hall, Oregon State University, Corvallis, Oregon 97331; [email protected]

A scientific consensus is approaching on global climate change and its anthropogenic causes (see Dore et al., 2003). Health, ecological, and agricultural impacts are being documented (see Martens, 1998; Burns et al., 2003; Cotton, 2003; O’Reilly et al., 2003; Tan and Shibasaki, 2003; Verburg et al., 2003). It is probably too early yet to evaluate the full impact of global climate change, but it is possible to formulate scenarios on its effects. It is important, therefore, to be able to formulate hypotheses about the impact of change and ideally to predict the potential consequential scenarios for life on Earth and the future health of humankind. One difficulty in predicting possible scenarios arises from the complexity of the natural communities in which pathogens and parasites are embedded. This complexity, namely, in number of species, links (interactions), and dynamics, presents several difficult conceptual challenges. These problems arise because the number of potential links

Copyright #2006. The American Association of Petroleum Geologists/Division of Environmental Geosciences. All rights reserved. DOI:10.1306/eg.11160404032

Environmental Geosciences, v. 13, no. 1 (MArch 2006), pp. 55 – 63

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Jennifer Orme Zavaleta is associate director for science at the Environmental Protection Agency’s Western Ecology Division in Corvallis, Oregon. She has worked for the EPA since 1981. She obtained her Ph.D. from the Department of Fisheries and Wildlife at Oregon State University (OSU) in 2003 and is a courtesy Professor at OSU. Her main research interests are in risk assessment and integration of health and ecological risk assessment.

Annette Rossignol is professor of epidemiology in the Department of Public Health at Oregon State University, where she has been teaching since 1988. She received her D.Sc. from the Harvard School of Public Health in 1981. Her main interests are in environmental epidemiology , international health, and ethics. She is the author of a text entitled Principles and Practice of Epidemiology: An Engaged Approach, to be published by McGraw-Hill company in December 2005.

ACKNOWLEDGEMENTS The authors thank members of the Loop Group in the Department of Fisheries and Wildlife, Oregon State University, and specifically Hiram Li for his constructive comments. The information in this article has been funded in part by the U.S. Environmental Protection Agency. It has been subjected to the agency’s peer review process and has been approved for publication. The conclusions and opinions are solely those of the authors and are not necessarily the views of the agency.

(interactions) between variables in a system increases exponentially with the number of variables, with the result that the number of possible effects from changes in even one variable in the system can be astronomical (Dambacher and Rossignol, 2001). With respect to climate change, yet another challenge is identifying the effects, such as the affected variables and the types of input or perturbation, in the system arising from the climate change. Finally, theoretical knowledge, specifically the mathematics required to predict the impact of perturbations on ecologic, community, and human health systems, still is developing and somewhat lagging. Based on parsimonious assumptions on new mathematical developments and using available practical tools of qualitative analysis, we discuss the potential impact of climate change on ecological communities and human health, focusing on zoonoses. The impact and consequences of direct input into epidemiological parameters, for example, a change in the duration of parasite development in malaria, have been analyzed in detail (Martens, 1998). If the input is to a distant community member, however, the effect on parasite turnover (taken here as mortality rate or as the inverse of life expectancy) requires community-level understanding, and as noted previously, it is fair to say that systemwide, community modeling still is in its infancy in epidemiology and public health. We present a mathematically based analysis of the behavior of perturbed communities and provide illustrations when possible. From our analysis, we conclude that the impact of change on ecological communities can be profound but subtle, complex, and ambiguous, even under basic mathematical assumptions about the structure of a community when in equilibrium.

Climate Change and Health Climate change is a topic that inspires considerable debate among the scientific and political communities. At issue here is not whether climate changes, but instead, the consequences of climate change on human and ecosystem health and the function of human activity in influencing climate (McMichael et al., 1996). Climate, per se, is generally defined as the average weather over a period of time for a particular geographic region (IPCC, 1996). Climate is influenced by natural interactions among the atmosphere, water (oceans and freshwaters), land mass, and land cover. Increasing evidence is present that human activity can 56

Climate Change and Its Impact on Disease

affect climate through changes to atmospheric composition, habitat alteration, and water quantity, quality, and salinity (NRC, 2001). Similarly, human health is influenced by climate through air and water quality, food productivity, temperature ranges, extreme weather events, and disease potential (McMichael et al., 1996; Patz and Balbus, 1996; Patz et al., 1996; Watson and McMichael, 2001). The cyclical relationship among human activity, climate change, and human health presents an unconventional challenge in risk evaluation. Instead of the typical approach of assessing health risks resulting from, for example, a single chemical exposure, complex scenarios are needed to forecast health outcomes resulting from the direct and indirect effects of climate change on human health (McMichael, 1997; NRC, 2001). Several models have been used to forecast climate scenarios. The two most noted models are the Hadley model developed by the Hadley Centre for Climate Prediction and Research of the United Kingdom and the Canadian model developed by the Canadian Climate Centre (USEPA, 2000; NRC, 2001). The primary difference between these models is the forecast of temperature increase by the year 2100, with the Canadian model predicting a more rapid increase in temperature accompanied by a decrease in precipitation in the future (USEPA, 2000). In its assessment of climate impacts to the Mid-Atlantic region of the United States, the U.S. Environmental Protection Agency determined that there was no basis to suggest that either model or simulation was more accurate than the other. Both scenarios were thought to be plausible given the projected changes in atmospheric composition (USEPA, 2000). Furthermore, neither the Hadley nor the Canadian model accounts well for year-to-year fluctuations in climate, such as El Nin˜o events. Our analysis of the effects of climate change on human health takes both long- and short-term changes into account. Another widely used modeling approach to ecological systems is the population matrix analysis, essentially a life-table approach, which has been widely applied in ecology (Caswell, 2001); its focus is on a single population, an approach that has been criticized for its reductionism (Kareiva, 1994). Alternatively, system analysis considers a community’s flows of energy and provides a thermodynamic analysis of a system (Eisen, 1988). Food-web analysis and modeling, basically a subset of system ecology, wherein biomass transfer is analyzed, has made significant contributions, particularly in the study of fisheries (Pauly et al., 2000). All approaches, including our own, provide narrow,

incomplete but complementary perspectives on system dynamics that ultimately provide a valuable synthesis (Levins, 1966).

A MATHEMATICALLY BASED ANALYSIS OF PERTURBED ECOLOGIC COMMUNITIES Our analysis begins with an explanation of the general methodology we used to evaluate the effects of climate change on ecologic systems, covering basic terminology, underlying assumptions of the mathematical model, and the model components and inputs. We then focus specifically on climate change as a perturbation into an ecologic system. Assumptions By perturbed communities, we mean ecologic systems, including those affecting human health, wherein parameters of birth or death have been changed, and stable equilibrium levels are potentially affected. We will assume that system dynamics are nonlinear, but that interactions (or links) between variables are linearizable (partial derivatives are linear). This assumption is a debated generalization. Although there is no doubt that interactions between organisms, including between human beings and other variables, can be extremely complex (nonlinearizable, so-called ‘‘functional’’ responses), it has been shown, nevertheless, that our assumption is sufficiently predictive at a practical level (Hulot et al., 2000; Schmitz, 2002). We have demonstrated also that community structure, the overall assemblage of interactions between variables, is of overriding concern in system dynamics, and that qualitative predictions of changes in natural systems resulting from our analytic approach are correct, useful, and accurate (Dambacher et al., 2002, 2003a, b). Another assumption of our analysis is that of local stability. Stability is a controversial concept in ecology and has led to considerable confusion. Grimm and Wissel (1997) document more than 100 definitions of local stability in the ecological literature. Strictly speaking, stability is the ability of a community to recover from a local perturbation. Liberally, stability is equated with persistence (presence of over time) (Connell and Sousa, 1983; Dambacher et al., 2002), a definition that generally is accepted. In addition, virtually any ecologically meaningful definition of stability or persistence ultimately will rely on similar mathematical criteria (Logofet, 1993).

Model Components and Input Community models consist of variables, typically populations of a species or of similar species (a guild or trophic level), but which also can be more anthropogenic variables, such as urbanization or an economic variable. Variables interact through links (or are linked through interactions) that describe the density-dependent impact that a variable has on the magnitude of another variable such as an organism’s birth rate or death rate. Links can be absent, positive, negative, and single or paired. For example, a typical predator-prey relationship would consist of two links of opposite sign pointing in the opposite direction (Figure 1). In addition to links, self-effects may exist; self-effects represent density-dependent interactions of a variable on itself. A self-effect might arise, for example, from logistic (intraspecific regulatory) processes or reliance on any component with overall negative feedback; thus, a selfeffect link can represent another complex but stable system with which the variable interacts. A community can be represented as a so-called ‘‘community or Jacobian matrix’’ (Berlow et al., 2004) (Figure 1). Elements of the matrix can be represented qualitatively as + 1, 1, or 0 (positive, negative, or no effect) (Quirk and Ruppert, 1965) or quantitatively. A system is a term generally reserved for the structure describing energy or mass flows in a community (Eisen, 1988). A food web is a special case and is both a system describing transfer of mass and a trophic community. A community is broader in that nontrophic interactions are included. Two general types of perturbation can impact a system. The first is a pulse, wherein a significant change occurs in the magnitude of a system variable, but the change, instead of being permanent, is transitory, and the variable later returns to its prepulse magnitude. In pulsed systems, the term ‘‘resistance’’ is commonly used to refer to a system’s degree of change and ‘‘resilience’’ to its speed of return. The second is a press, in which a permanent change occurs in the magnitude of a parameter (a positive press, for example, might be a permanent increase or decrease in an organism’s birth rate and/or death rate). Both types of perturbations have been studied analytically (Bender et al., 1984); generally, the effects of press perturbations are considerably better understood. The following discussion generally examines the effect of climate change acting as a press. Given an incomplete knowledge of a community, the presence or absence of stability, nevertheless, can be determined from the so-called Routh-Hurwitz criteria (May, 1974). In essence, the criteria assess, Rossignol et al.

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Predicting Change The qualitative effects on all variables in Figure 1 of a positive press perturbation to a variable is read down a column and, thus, down the first column for a press to variable N2, with a predicted increase in variables N2 and N3 and a decrease in N1. Qualitative prediction matrix þ þ þ þ þ

Figure 1. Signed digraph of a community consisting of a resource (N1), a consumer (N2), and a top predator (N3). The qualitative matrix below tabulates the links with the digraph. Each element of the matrix can be labeled aij, where i refers to the row, and j refers to the column. Thus, a11 refers to the effect to 1 from 1 and is negative, a12 refers to the effect to 1 from 2 and is negative, and so on. from qualitative relationships, whether a system is capable of recovery following a perturbation. These criteria recently were revised to remove the redundancy present in Hurwitz’s initial 1895 formulation (Dambacher et al., 2003b). From the community matrix, it is possible to predict the impact of a press perturbation on the density (or abundance) of a variable from the inverse of the community matrix. Predicting change in a variable’s turnover rate also is possible (Puccia and Levins, 1985, 1991), and a recent study will provide an algorithm to predict changes in variable turnover (Orme-Zavaleta and Rossignol, 2004; Dambacher et al., 2005). Calculations of stability conditions and of predictions have been simplified with the use of a computer interface (available on request) and a published Maple program (Dambacher et al., 2002). 58


þ þ

A large number of factors that introduce increased complexity can be added to the basic assumptions of a community model. Examples of these factors include nonlinearizable interactions (McCann et al., 1998), host switching (Kondoh, 2003), stochastic inputs and responses (Brown and Rothery, 1993), and meta-community dynamics (Hanski, 2001). We suggest, however, that the parsimonious assumptions and components of a community model that we use, the basis of a substantial part of modern community ecology, have held up well and are sufficient for qualitative analysis. In addition, our approach conforms with the approach suggested by Puccia and Levins (1985, 1991), in the sense that when confronted with complex input, such as climate change, into a multifaceted and multivariable community system, progress in understanding the system can be made by ‘‘sacrificing precision for generality and realism’’ (Puccia and Levins, 1985, p. 10). Levins’ suggestion that such tradeoffs in hypothesis stating sometimes are needed in the quest to understand, predict, and modify nature was the subject of a classic discussion (Levins, 1966).

CLIMATE AS A PERTURBATION A perturbation can have multiple effects, namely, on density, turnover, harmonic oscillations, transient nonharmonic oscillations, and variance.

Change in Density A scenario is that climate change acts as a positive or negative press on a single variable in a particular community that can be depicted as a simple straight

chain (Figure 1). The desiccation effect of climate warming (such as from aridification) on plants could lead to a so-called ‘‘bottom-up’’ impact on the food chain relying on it, each component of which would be negatively impacted. Intuitively, as resources are lost in plants, all herbivores and their predators will suffer. A top-down input, however, to, for example, a top predator would lead to alternating decreases and increases in equilibrium levels down the food chain (Dambacher and Rossignol, 2001). This effect occurs because as a predator declines, its herbivore prey benefits, which, in turn, reduces plants. These effects are well understood (Hairston et al., 1960). The effects of an input to more than one or to all variables in a community, however, become ambiguous and most easily missed observationally and confused experimentally. If climate warming negatively impacts all self-regulated variables in a predator-prey chain, as illustrated in Figure 1, an observed effect may be obvious only on top predators or their parasites (N3) (Voigt et al., 2003; Arkoosh et al., 2004). If an input was to have alternatively positive and negative effects on different variables, only an analysis of system structure would allow the detection of change with any degree of confidence. The danger is that the ambiguity arising from a qualitative analysis of great complexity will cause many predictions to be unreliable in direction and magnitude (Dormoy and Collet, 1988). Recent theoretical developments allow us to identify which qualitative predictions are reliable (Dambacher et al., 2003a, b, 2005). As an example of the results of perturbations to both the top and bottom variables in a trophic chain that have been interpreted as evidence of decoupling (Bodini, 1998), a decoupled system behaves as if several interacting variables in the community no longer are linked. Thus, no observable effects exist from the perturbations. Alternatively, it may be that the effects of this double press are substantial but cancel each other out at some trophic levels and, thus, are theoretically ambiguous (Dambacher et al., 2002). We suggest that the last interpretation is the more parsimonious. As a consequence of this phenomenon, the impact of climate change may present difficult problems in documenting. This difficulty arises because climate change is likely to impact many if not all variables in a community, sometimes in opposite directions. The theoretical ambiguity that arises from such a situation or from any type of input (Wooton, 2002) will make an effect difficult to confirm and differentiate from random effects.

Change in Turnover To compound further field observations and interpretations, even when no change in density is predicted or observed, changes to a variable following a press somewhere in the community, nevertheless, can be profound. If input from a press to a variable leads to no change in density because of compensating effects in the community, changes in turnover, nevertheless, can be drastic. A population with an increased birth rate and no change in density will have a drastically shortened age pyramid and lower life expectancy. Example Using the model illustrated in Figure 1 and based on an analysis in Puccia and Levins (1985, p. 53ff), a negative (reversing the above) press perturbation to the system arising from an increase in the death rate of the affected variable would yield the following qualitative changes in life expectancy. Let us assume that the middle variable (N2) represents a vector, with the bottom (N1) being its host and the top (N3) being a predator. It can be seen that vector life expectancy would be increased in the first case, with a concomitant increase in transmission potential, although vector population density would have decreased (from prediction matrix in Figure 1). Community effects can have complex consequences predictable only from analysis of the inverse of the community matrix. Life expectancy matrix þ 0

þ þ þ

Change in life expectancy is of potential importance to human health because vector-borne diseases are determined principally by turnover. For example, mosquito density is the most intuitive aspect of malaria transmission but, probably, the least important in determining the malaria incidence rate. Ronald Ross demonstrated that malaria transmission is counterintuitive because it is the mosquitoes’ life expectancy, and not the number of female mosquitoes, that has the greatest effect on malaria transmission (Bailey, 1982); the theoretical impact of climate change on malaria has been studied by Martens (1998), who concluded that the risk of malaria is likely to increase in temperate areas as vector life expectancy and extrinsic Rossignol et al.

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incubation period of the parasite are affected. Furthermore, time delays in parasite development in the vector and host, ranging from 2 weeks to many years, and in the hosts’ immune response complicate the understanding of human malaria epidemiology (Paul et al., 2003). Climate as a Source of Harmonic Oscillatory Behavior Input to a community can have other ramifications. A press to a variable can result in a change in eigenvalues. Eigenvalues are the coefficients of the characteristic polynomial of a matrix; intuitively, they are simply the feedback cycles (loops) of a system. Eigenvalues scale the response duration of variables such that the inverse of the real component of the largest eigenvalue is proportional to the community’s recovery time (resilience), the time it takes the community to return to its equilibrium level. Longer recovery time is regarded, ecologically speaking, as a form of instability (Pimm and Lawton, 1978). Furthermore, eigenvalues commonly have an imaginary component to their solution. A press can result in changes in the selfeffects of variables, in their pairwise interactions, or in the number of variables in the system (from invasion or extinction). Changes in the self-effects of variables will affect recovery time, whereas changes in variables’ pairwise interactions or in the number of variables that are linked will impact oscillatory behavior following a perturbation (Jorgensen et al., 2000). Transient Behavior Even in the absence of harmonic oscillations during recovery, an increase in the variance of eigenvalues can result in an increased likelihood of transient behavior, namely, of initial movement of a variable in a direction that is opposite to the direction toward equilibrium (G. Thompson and P. A. Rossignol, unpublished data). For heuristic purposes, let us assume, for example, a two-variable system with real negative eigenvalues, wherein recovery will be monotonic, that is, without harmonic oscillations. On a phase plane, a point (other than exactly on the eigenvectors) near the major eigenvector will move to the equilibrium point more or less parallel to the major eigenvector until it enters the area between the isoclines that encompass the minor eigenvector. This initial movement will not necessarily be in the direction of the equilibrium point. Only once it approaches the minor eigenvector will recovery be monotonic (Figure 2). 60


Figure 2. Behavior of a variable moving towards equilibrium (N*). The recovery can be monotonic (dashed), can have a transient movement, and can then be monotonic (solid) or be harmonically oscillatory (dotted). Displacement away from system equilibrium, as with a climate-induced shift in equilibrium, may cause unexpected nonharmonic behavior even in the simplest communities. Paradoxically, an input ultimately causing a decrease in the prevalence of a human parasite could be accompanied by an initial transient increase, causing the disease to be perceived as spreading. Climate as a Source of Variance Another effect of climate change may not so much be a change in mean temperatures as in the variances of regional temperatures, leading to larger variances in the strengths of interactions between community variables but not in their mean values. Strictly speaking, the community therefore is not pressed. Again, this apparently subtle effect can have profound impact on communities. A qualitative technique called ‘‘time averaging’’ (Puccia and Levins, 1985, 1991) represents one analytic tool to assess the impact of increased variances. Time averaging derives covariance between variables in a constantly fluctuating but bounded (limited) community. Time averaging, however, has remained poorly studied and poorly understood. We suggest that it is a field deserving of further study. In a community with nonlinearizable interactions, the impact of variance, independent of perturbations, is to introduce previously unavailable niches for new species of organisms (Levins, 1979). In short, increased variance favors invasions and introductions of new species into a community and, conversely, decreased variance leads to losses in biodiversity. This

phenomenon is of great concern and also merits considerable further theoretical investigation because changes in temperature variance as well as in mean temperature likely will affect the consequences of climate change on ecologic communities and the potential for changes in the incidence of human disease.

HUMAN DISEASES EMBEDDED IN ECOLOGICAL COMMUNITIES Transmission of human diseases that arise from the natural environment, of which zoonotic diseases are the most important, can be categorized into two broad categories: diseases that are caused by direct transmission and diseases that are caused by vector-borne transmission. In the case of a directly transmitted zoonosis, such as rabies, risk can be directly evaluated from an element in the inverse of the community matrix. For example, over the long term, if a fox population increases, presumably, the number of contacts between foxes and human beings or their pets will increase, along with rabies incidence among humans. Transient behavior, described above, however, could modulate risk in the short term by causing a temporary drop in human rabies before the final monotonic increase. This counterintuitive behavior could lead to a false conclusion if only the immediate short-term behavior is examined.

Vector-Borne Zoonoses Many of the mathematical considerations discussed here are needed to predict changes in the more complex diseases, namely, the vector-borne zoonoses. Not only are vectors and parasites embedded in complex natural communities, but both turnover and relative abundances are crucial in predicting the effects of climate change. We suggest that predicting changes in the risks to humans from zoonoses is not possible without an understanding and analysis of the overall relevant target community (Orme-Zavaleta and Rossignol, 2004). The combination of change in density and turnover has important consequences for diseases embedded in animal communities, particularly vector-borne zoonotic diseases, such as Lyme borreliosis and West Nile encephalitis. A vector must contact hosts at least twice to acquire and then transmit a pathogen or parasite; the vector must be abundant enough relative to the host,

live long enough on average to survive between these two blood meals, and bite suitable hosts, or the pathogen cycle will be interrupted. These considerations lead to complex mathematical relationships among species, namely, the host, vector, and pathogen. These relationships have been elaborated in powerful mathematical models, particularly for malaria (Bailey, 1982). Realistically, each system variable interacts with others in a complex natural community, and the impact of a perturbation on a variable linked to a disease cycle but not directly involved in it may profoundly impact on the risk of human disease. Let us consider the example above, wherein N1 is a host, N2 is a vector, and N3 is a beneficial agent of biological control. If climate change was to decrease the birth rate of the vector, it would result in a decrease in its abundance (from the negative of the qualitative prediction matrix in the text above; Input is to the third column, and effects on the vector are the third element down), which would at first seem beneficial. However, it also would cause an increase in vector life expectancy (from the negative of the life expectancy matrix in the text above, third element down in the third column). The consequence is of dubious benefit at best and possibly deleterious at worst.

CONCLUSION Global climate change scenarios will be complex and incorporate factors ranging from psychology, economics, climatology, epidemiology, ecology, etc. We have focused on a single but worrisome aspect, namely, risk of zoonotic disease arising from perturbed natural ecosystems. We suggest that the effect of climate change may be a paradoxical decrease in biodiversity but an increase in zoonotic disease. The first would occur because of loss of niches and degradation of the environment and is a relatively straightforward Eltonian conclusion (Li et al., 1999). The second would arise as a consequence both of increased variance in Malthusian parameters, which could facilitate invasion and spread, and of density increase of pest species and pathogens. In combination, changes in temperature would modify turnover rates and, thus, the incubation period of parasites. Taken together, reduced speciosity will focus interactions between the remaining species, and the rapidly flowing embedded pathogens will present a significantly increased risk. Subsystems with embedded pathogens will increase in relative number and in intensity. Our theoretical and practical ability to Rossignol et al.

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predict and control these events may be lagging and may require immediate attention. Global climate change will be a perturbation that affects all organisms of a community, and the effects of climate change will be paradoxical and difficult to gauge. We suggest that community analysis provides a tool to predict and interpret these disturbing and pervasive anthropogenic effects.

REFERENCES CITED Arkoosh, M. R., L. Johnson, P. A. Rossignol, and T. K. Collier, 2004, Predicting press perturbations on salmon communities: Implications for monitoring: Canadian Journal of Fisheries and Aquatic Sciences, v. 61, p. 1166 – 1175. Bailey, N. T. J., 1982, The biomathematics of malaria: London, Charles Griffin and Co., 194 p. Bender, E. A., T. J. Case, and M. E. Gilpin, 1984, Perturbation experiments in community ecology: Theory and practice: Ecology, v. 65, p. 1 – 13. Berlow, E. L., et al., 2004, Interaction strengths in food webs: Issues and opportunites: Journal of Animal Ecology, v. 73, p. 585 – 598. Bodini, A., 1998, Representing ecosystem structure through signed digraphs; model reconstruction, qualitative predictions and management: The case of a freshwater ecosystem: Oikos, v. 83, p. 93 – 106. Brown, D., and P. Rothery, 1993, Models in biology: Mathematics, statistics and computing: Chichester, United Kingdom, John Wiley & Sons Publishing, 688 p. Burns, C. E., K. M. Johnson, and O. J. Schmitz, 2003, Global climate change and mammalian species diversity in U.S. national parks: Proceedings of the National Academy of Science U.S.A., v. 100, p. 11,474 – 11,477. Caswell, H., 2001, Matrix population models, 3d ed.: Massachusetts, Sinauer Associates Publisher, 722 p. Connell, J. H., and W. P. Sousa, 1983, On the evidence needed to judge ecological stability or persistence: American Naturalist, v. 121, p. 789 – 8245. Cotton, P. A., 2003, Avian migration phenology and global climate change: Proceedings of the National Academy of Science, v. 100, p. 12,219 – 12,222. Dambacher, J. M., and P. A. Rossignol, 2001, The golden rule of complementary feedback: Association for Computing Machinery Special Interest Group in Symbolic and Algebraic Manipulation Bulletin, v. 35, p. 1 – 9. Dambacher, J. M., H. W. Li, and P. A. Rossignol, 2002, Relevance of community structure in assessing indeterminacy of ecological predictions: Ecology v. 83, p. 1372 – 1385. Dambacher, J. M., H. W. Li, and P. A. Rossignol, 2003a, Qualitative predictions in model ecosystems: Ecological Modelling, v. 161, p. 79 – 93. Dambacher, J. M., H.-K. Luh, H. W. Li, and P. A. Rossignol, 2003b, Qualitative stability and ambiguity in model ecosystems: American Naturalist, v. 161, p. 876 – 888. Dambacher, J. M., R. Levins, and P. A. Rossignol, 2005, Life expectancy change in perturbed communities: Derivation and qualitative analysis: Mathematical Biosciences, v. 197, p. 1 – 14. Dore, J. E., R. Lukas, D. W. Sandler, and D. M. Karl, 2003, Climate-driven changes to the atmosphere CO2 sink in the subtropical North Pacific Ocean: Nature v. 424, p. 754 – 757. Dormoy, J.-L., and J. Collet, 1988, New methods in qualitative

62


calculus: http://www.dormoy.org/JLuc/PaperSOR88.pdf (accessed 2004). Eisen, M., 1988, Mathematical methods and models in the biological sciences — Nonlinear and multidimensional theory: New Jersey, Prentice Hall Publishing, 323 p. Grimm, V., and C. Wissel, 1997, Babel, or the ecological stability discussions: An inventory and analysis of terminology and a guide to avoiding confusion: Oecologia, v. 109, p. 323 – 334. Hairston, N. G., F. E. Smith, and L. B. Slobodkin, 1960, Community structure, population control, and competition: American Naturalist, v. 94, p. 421 – 425. Hanski, I., 2001, Spatially realistic theory of metapopulation ecology: Naturwissenschaften, v. 88, p. 372 – 381. Hulot, F. L., G. Lacroix, F. Lescher-Moutoue´, and M. Loreau, 2000, Functional diversity governs ecosystem response to nutrient enrichment: Nature, v. 405, p. 340 – 344. Intergovernmental Panel on Climate Change (IPCC), 1996, Climate change 1995, in J. T. Houghton, L. G. Meira Filho, B. A. Callander, N. Harris, A. Kattenberg, and K. Maskell, eds., The science of climate change: Cambridge, Cambridge University Press, p. 55 – 64. Jorgensen, J., A. M. Rossignol, C. J. Puccia, R. Levins, and P. A. Rossignol, 2000, On the variance of eigenvalues of the community matrix: Derivation and appraisal: Ecology, v. 81, p. 2928 – 2931. Kareiva, P., 1994, Higher order interactions as a foil to reductionist ecology: Ecology, v. 75, p. 1527 – 1528. Kondoh, M., 2003, Foraging adaptation and the relationship between food-web complexity and stability: Science, v. 299, p. 1388 – 1391. Levins, R., 1966, The strategy of model building in population biology: American Scientist, v. 54, p. 421 – 431. Levins, R., 1979, Coexistence in a variable environment: American Naturalist, v. 114, p. 765 – 783. Li, H. W., P. A. Rossignol, and G. Castillo, 1999, Risk analysis of species introduction: Insights from qualitative modeling, in R. Claudi and J. H. Leach, eds., Nonindigenous freshwater organisms: Vectors, biology, and impacts: Boca Raton, Florida, Lewis Publishers, chapter 30, p. 431 – 447. Logofet, D. O., 1993, Matrices and graphs: Stability problems in mathematical ecology: Boca Raton, Florida, CRC Press, 308 p. Martens, P., 1998, Health and climate change: London, Earthscan Publishing, 176 p. May, R. M., 1974, Stability and complexity in model ecosystems, 2d ed.: New Jersey, Princeton University Press, 265 p. McCann, K., A. Hastings, and G. R. Huxel, 1998, Weak trophic interactions and the balance of nature: Nature, v. 395, p. 794 – 798. McMichael, A. J., 1997, Global environmental change and human health: Impact assessment, population vulnerability, and research priorities: Ecosystem Health, v. 3, p. 200 – 210. McMichael, A. J., A. Haines, R. Sloof, and S. Kovats, eds., 1996, Climate change and human health: Geneva, World Health Organization, 296 p. National Research Council (NRC), 2001, Climate change science: An analysis of some key questions: Washington, D.C., National Academy Press, p. 1 – 24. O’Reilly, C. M., S. R. Alin, P.-D. Plinster, A. S. Cohen, and B. A. McKee, 2003, Climate change decreases aquatic ecosystem productivity of Lake Tanganyika, Africa: Nature, v. 424, p. 766 – 768. Orme-Zavaleta, J., and P. A. Rossignol, 2004, Community level analysis of risk of vector-borne disease: Transactions of the Royal Society of Tropical Medicine and Hygiene, v. 98, p. 610 – 618. Patz, J., and J. M. Balbus, 1996, Methods for assessing public health vulnerability to global climate change: Climate Research, v. 6, p. 11 – 125.

Patz, J., P. R. Epstein, T. A. Burke, and J. M. Balbus, 1996, Global climate change and emerging infectious diseases: Journal of the American Medical Association, v. 275, p. 217 – 223. Paul, R. E. L., F. Ariey, and V. Robert, 2003, The evolutionary ecology of Plasmodium: Ecology Letters, v. 6, p. 866 – 880. Pauly, D., V. Christensen, and C. Walters, 2000, Ecopath, Ecosim, and Ecospace as tools for evaluating ecosystem impact of fisheries: International Council for the Exploration of the Sea Journal of Marine Science, v. 57, p. 697 – 706. Pimm, S. L., and J. H. Lawton, 1978, On feeding on more than one trophic level: Nature, v. 275, p. 542 – 544. Puccia, C. J., and R. Levins, 1985, Qualitative modeling of complex systems: Cambridge, Harvard University Press, 259 p. Puccia, C. J., and R. Levins, 1991, Qualitative modeling in ecology: Loop analysis, signed digraphs, and time averaging, in P. Fishwick and P. A. Luker, eds., Qualitative simulation modeling and analysis: New York, Springer-Verlag Publishers, chapter 6, p. 119 – 143. Quirk, J., and R. Ruppert, 1965, Qualitative economics and the stability of equilibrium: Review of Economic Studies, v. 32, p. 311 – 326. Schmitz, O. J., 2002, Linearity in the aggregate effects of mul-

tiple predators in a food web: Ecology Letters, v. 5, p. 168 – 172. Tan, G., and R. Shibasaki, 2003, Global estimation of crop productivity and the impacts of global warming by GIS and EPIC integration: Ecological Modelling, v. 168, p. 357 – 370. U.S. Environmental Protection Agency (USEPA), 2000, Preparing for a changing climate: The potential consequences of climate variability and change — Mid-Atlantic overview: http://www .essc.psu.edu/mara/results/overview_report/index.html (accessed 2004). Verburg, P. R., E. Hecky, and H. King, 2003, Ecological consequences of a century of warming in Lake Tanganyika: Science, v. 301, p. 505 – 507. Voigt, W., et al., 2003, Trophic levels are differentially sensitive to climate: Ecology, v. 84, p. 2444 – 2453. Watson, R. T., and A. J. McMichael, 2001, Global climate change — The latest assessment: Does global warming warrant a health warning?: Global Change and Human Health, v. 2, p. 64 – 75. Wooton, J. T., 2002, Indirect effects in complex ecosystems: Recent progress and future challenges: Journal of Sea Research, v. 48, p. 157 – 172.

Rossignol et al.

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