Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 110
7 Understanding Communities and Ecosystems An ecological community is an assemblage of populations of differ- ent species (plants, animals, fungi, microbes, etc.) at a given place and time. The living organisms of a community cannot be separated from their physical and chemical environment, and the combination of a community and an environment is referred to as an ecosystem. Although a commu- nity is often characterized by a dominant feature—as is, for example, a desert community or an oak savanna community—its species composi- tion has a significant random component. Community ecology is concerned with explaining patterns of diver- sity, the distribution and abundance of species within the context of these assemblages, and the underlying processes. The field of community ecol- ogy has developed rapidly over the last few decades, driven by the need to understand the consequences of anthropogenic impacts on the func- tioning of ecological communities. Our understanding of how communities assemble has changed over time (Kingsland, 1991). It has ranged from regarding an ecological com- munity as a random assemblage (Gleason, 1926) to thinking of it as a “complex organism” (Clements, 1936). In the beginning of community ecology, questions focused on community structure, population dynam- ics, and, in the case of plant communities, on succession (Grinnell, 1917; Clements et al., 1929). The abiotic (nonliving) environment was assigned a minor role until Lindeman’s seminal paper (1942) on the trophic-dy- namic aspects of ecology, which established the ecosystem as the funda- mental unit. Mathematics has played a vital role in framing community ecology 110

OCR for page 110
111 UNDERSTANDING COMMUNITIES AND ECOSYSTEMS concepts. Deterministic models (systems of differential or difference equa- tions) dominated theoretical advances for much of the history of the field, and they continue to be the single most important choice of modeling framework for analytical models. In the 1920s and 1930s, two key con- cepts were formalized using deterministic models: competition and pre- dation. Mathematical models greatly enhanced our understanding of both processes. Competition has been identified as an important process of eco- logical communities ever since Darwin proposed it as the chief mecha- nism in the evolution of species (Darwin, 1859). The competition models by Lotka (1932) and Volterra (1926), formulated as systems of differential equations, provide a theoretical framework for the dynamic interactions within a trophic level.1 This framework was further developed by Elton (1927, 1933) using the concept of a niche, which he defined as “the status of an animal in its community.” He linked this concept to competition in order to explain how multiple species can persist within a community. A mathematical formulation of the niche concept was finally given by Hutchinson (1957), who defined a niche as a subset of an n-dimensional hypervolume. This concept is still useful today. While the models of Lotka and Volterra describe phenomena, they lack mechanisms for competition. Tilman’s (1982) resource competition model led the way from phenom- enological to mechanistic competition models. Like the Lotka-Volterra models, mechanistic competition models are also based on systems of dif- ferential equations and continue to form the conceptual basis for under- standing competition among multiple species. Predation is by definition a process that occurs between trophic lev- els. Lotka (1925) and Volterra (1926) were the first to provide a math- ematical formulation of this process, again using systems of differential equations. Differential equations model continuous time dynamics and are thus well suited for populations with overlapping generations. How- ever, this does not always hold for biological situations. For instance, the seasonal dynamics of a host and an associated parasitoid2 are better de- scribed by discrete time models. To include this aspect of biological real- ism into models, Nicholson and Bailey (Nicholson, 1933; Nicholson and Bailey, 1935) promoted systems of difference equations to describe preda- tion models. Difference equations are now commonly employed to model interactions among species with nonoverlapping generations. In the 1950s and 1960s the focus shifted toward understanding the 1A trophic level is a stratum of the food chain consisting of species the same number of steps from the primary source of nutrition. 2A parasitoid is an insect that lays its eggs in, on, or near a host and whose offspring consume the host as they develop.

OCR for page 110
112 MATHEMATICS AND 21ST CENTURY BIOLOGY relationship between the diversity of an ecological community and its sta- bility (Real and Levin, 1991). Using qualitative arguments, Odum (1953), MacArthur (1955), and Elton (1958) concluded that diversity and stability were positively correlated. Despite the absence of carefully designed ex- periments and mathematical models to corroborate this claim, it remained unchallenged until May (1972, 1974) and others investigated models of randomly assembled communities whose dynamics were described by systems of differential equations, similar to those of Lotka and Volterra. These theoretical studies led to the opposite conclusion: Stability and di- versity were negatively correlated. The conclusion was based on rigorous mathematics, though it lacked the synergy and validation that come from combining theoretical and empirical work. It became widely accepted by community ecologists but was questioned by ecosystem ecologists (Patten, 1975; McNaughton, 1977; Loreau et al., 2002). The diversity-stability de- bate was revived in the 1990s, when carefully designed experiments and mathematical models that directly addressed the variability of species abundances questioned the negative correlation between stability and di- versity (see Box 7.1). Models that try to address basic principles or processes and that focus on ideas rather than on specific biological systems have played a large role in ecology. These will be referred to as conceptual models. Many of the conceptual models of community ecology are framed as systems of differential or difference equations. This framework carries an implicit assumption of spatial homogeneity, but of course it is known that spatial movements and dispersal of individuals and spatial interactions among individuals can lead to spatial heterogeneity. Spatial movement was first included in ecological models in the 1950s with Skellam’s (1951) work on the spread of muskrats. The equations were identical to those developed by Fisher (1937) to describe the spread of a novel allele. Both partial differ- ential equations and integro-differential equations are commonly em- ployed now to model movement and dispersal (Okubo, 1980; Holmes et al., 1994; Okubo and Levin, 2001). They are also used to investigate the effects of spatially dependent factors on the dynamics of multispecies communities. This has led to the insight that biotic interactions alone can generate spatial patterns. Stochastic models are rarely employed in theoretical ecological stud- ies owing to the difficulties in analyzing them, even though both environ- mental and demographic stochasticity play an important role in the dy- namics of ecological communities. Demographic stochasticity refers to randomness that is inherent in demographic processes, such as birth or death. It is of particular importance when populations are small. Environ- mental stochasticity—for instance, unexplained variation in precipitation or temperature that may affect fecundity or the survival of species—can

OCR for page 110
113 UNDERSTANDING COMMUNITIES AND ECOSYSTEMS have significant effects on communities, as illustrated by the work of Chesson and Warner (Chesson and Warner, 1981; Chesson, 1994), who introduced a general modeling framework to address the role of environ- mental stochasticity in species coexistence. This work demonstrated the importance of nonlinear, species-specific responses to the environment that can resonate into future generations. Demographic stochasticity has also been incorporated into individual- based spatial models where interactions among small groups of individu- als are important. The study of these models was initiated by Spitzer (1970) in the United States and Dobrushin (1971) in the Soviet Union. The models are spatially explicit Markov processes, called interacting particle systems. These models were originally developed for problems in statisti- cal physics, but it soon became clear that local interactions are important in other fields as well, including community ecology. The study of inter- acting particle systems and their discrete-time analogs, discrete-time cel- lular automata, has greatly advanced our understanding of the role of space and local interactions in the dynamics of ecological communities. This remains a very active area of research (Durrett and Levin, 1994; Neuhauser, 2001). Interacting particle systems or cellular automata are easy to formu- late, so much so that there are now numerous theoretical ecology papers that base the analysis of spatially explicit models solely on simulations. Their mathematical analysis, however, is a highly nontrivial matter. Re- sults from simulation studies can be quite misleading, because the be- havior of a finite system can differ from that of the related infinite system (Neuhauser and Pacala, 1999), and the results may not be robust with respect to the choice of local interactions (Anderson and Neuhauser, 2002). Dynamics, in particular in two spatial dimensions, may also be slow enough so that it takes a long time for the system to accurately re- flect long-term behavior. For instance, the voter model (Clifford and Sudbury, 1973; Holley and Liggett, 1975) and the multitype contact pro- cess (Neuhauser, 1992; Neuhauser and Pacala, 1999) in two spatial di- mensions exhibit clustering of like community members, with clusters growing indefinitely. Computer simulations have led researchers to be- lieve that it is possible for competing species to coexist in such systems, yet rigorous mathematical analysis shows eventual exclusion of all but one type in arbitrarily large regions. This demonstrates the need for rig- orous mathematical analysis. Some analytical methods for dealing with local spatial interactions and/or stochasticity have been developed, such as metapopulation mod- els and the moment approximation. Metapopulations are spatially implicit models (Levins, 1969; Hanski, 1999). They are formulated as systems of differential equations and track the dynamics of populations on a finite or

OCR for page 110
114 MATHEMATICS AND 21ST CENTURY BIOLOGY BOX 7.1 The Productivity-Stability-Diversity Debate The relationship between productivity, stability, and diversity has been of long-standing interest, from both a purely academic point of view and a management perspective, where it has become pressing to understand the consequences of the large-scale diversity loss caused by anthropogenic disturbances. The following illustrates how increasingly more sophisticated mathematical models in combination with carefully designed experiments expand our understanding of important processes. The past 50 years have seen a lively debate on whether diversity re- sults in more stable and more productive ecosystems or whether the oppo- site is true. The arguments in favor of a positive correlation between stabil- ity and diversity in the 1950s were based on superficial comparisons between species-poor agricultural systems and species-rich tropical sys- tems. The opposite conclusion, reached in the 1970s, was based on rigor- ous mathematical analysis of the equilibrium behavior of multispecies models. Early on in the discussions there was confusion, partly because different groups of researchers used different definitions of stability. The multiple definitions of stability were clarified by Pimm (1984), but the de- bate is still unresolved. Loss of biodiversity can affect ecosystem processes such as nutrient cycling and energy flow. It is thus not surprising that ecosystem ecologists increasingly joined the debate on the role of biodiversity. This coming to- gether of community ecology and ecosystem ecology since the beginning of the 1990s has helped refocus and expand the debate. A series of short- term and longer-term experiments were conducted to understand the role of biodiversity on ecosystem processes (Lawton et al., 1993; Naeem et al., 1994; Tilman and Downing, 1994). Theoretical studies soon followed. infinite number of patches. Dispersal among the patches is assumed to be on a complete graph; that is, all patches are equally accessible from any other patch. Moment approximations, commonly employed in statistical physics, have proved to be useful in community ecology for studying spa- tial clustering (Bolker and Pacala, 1997). The connections among the four major modeling frameworks (ordinary differential equation, partial differential equation, integro- differential equation, and interacting particle system) are well established (Durrett and Levin, 1994). As the interaction neighborhood in an inter- acting particle system increases either through an increase in movement relative to demographic processes or an increase in dispersal, a partial differential equation in the former case and an integro-differential equa- tion in the latter case become good approximations; removing, in addi-

OCR for page 110
115 UNDERSTANDING COMMUNITIES AND ECOSYSTEMS Systems of differential equations still dominated theoretical investigations, but there was an increased focus on ecosystem processes (e.g., Loreau, 1998). A different class of mathematical models found their way into the debate. Instead of deterministic systems of differential equations, where stability is based on eigenvalue properties, stochastic models were intro- duced that allowed keeping track of variability of both individual species and the entire community (e.g., Lehman and Tilman, 2000). Much of the empirical and theoretical work includes only primary producers and disregards trophic links (but, see Ives et al., 2000). The po- tential importance of this link has been pointed out by Paine (2002). The experiments ignored belowground processes. Wardle and van der Putten (2002) point out the lack of evidence for a diversity-productivity relation- ship in decomposer systems. The role of symbiotic organisms also warrants further study (van der Heijden and Cornelissen, 2002). The role of biodiversity in belowground processes has only recently received attention (Freckman et al., 1997; special issue of BioScience, February 1999). Theoretical work will need to be closely linked to experimental work. To guide experiments, it needs to focus on quantities that are measurable in field experiments. To have predictive power, models need to be param- eterized by experimental data. Future theoretical investigations will need to include the complex interactions among different trophic levels, belowground processes, the evolutionary potential of the organisms, envi- ronmental fluctuations, and spatial structure. They will also need to ad- dress nonequilibrium behavior. There will be an increased need for long- term data in different ecosystems. The current experiments indicate that the dominant process can change over time (Fargione et al., 2004), and it will be important to provide ways to statistically test for such changes (e.g., Loreau and Hector, 2001). tion, spatial heterogeneities results in an ordinary differential equation. Looking at this another way, if one needs to include the effects of fluctua- tions, correlations, and spatial heterogeneities, the simple framework of ordinary differential equations no longer suffices. Instead, the much more complicated framework of interacting particle systems (or similar pro- cesses) must be understood. The past 30 years of research in this area have considerably improved our understanding, but much work remains, because the properties of more complex multispecies assemblages em- bedded in ever-changing environments are only beginning to be re- vealed. Analytical models will be increasingly complemented by complex simulation models that attempt to incorporate nonlinearities, nonequi- librium behavior, genetic composition, space, demographic, and envi-

OCR for page 110
116 MATHEMATICS AND 21ST CENTURY BIOLOGY ronmental stochasticity. Even though (or because) computers have greatly expanded our ability to study large and complex systems, there remains a need for analytical methods. Many of the complex systems have large numbers of parameters that make exhaustive simulations nearly impossible. Developing mathematically tractable approximations of a complex simulation model can yield valuable insights into the be- havior of complex models. Ecological interactions are often complex and nonlinear and involve multiple species. Multiple stable states are a hallmark of such systems, which can lead to catastrophic changes under disturbances (Scheffer and Carpenter, 2003, and references therein). Mathematical modeling has yielded significant insights into dynamic consequences of the presence of multiple stable states. Modeling has also been applied to the recovery of systems that have undergone environmental degradation. It is often diffi- cult to restore the original system, and it has been conjectured that this is because the system has reached a different equilibrium state (or, more generally, is in a different domain of attraction). The importance of studying transient dynamics was pointed out by Hastings (2004). Most ecological interactions are probably far away from equilibrium. Large-scale anthropogenic perturbations, such as land-use change or nitrogen addition, are additional processes that result in nonequilibrium situations. Some of the mathematical theory has been de- veloped, in particular when different timescales are involved. Most field experiments are studied over only short timescales, even if the dynamics are slow, thus probably describing dynamics that are not in equilibrium. COMPUTATION Multispecies interactions across trophic levels, including ecosystem processes, provide statistical and modeling challenges for community ecologists. The statistical analysis of large data sets that often cannot sim- ply be analyzed using standard statistical software packages requires model development and computational methods to estimate parameters and test hypotheses. The theoretical study of large, complex systems re- sults in models that are often analytically intractable. Computational ad- vances have made possible the study of these models, which are currently framed as systems of differential equations. Increasingly though, a spatial component and stochastic factors are included, and both equilibrium and nonequilibrium dynamics are investigated. Few tools are currently avail- able to deal with these frameworks when applied to large systems. Inference in community ecology frequently deals with multiple com- peting hypotheses. Model selection as a way to distinguish between hy- potheses provides alternatives to traditional hypothesis testing (see

OCR for page 110
117 UNDERSTANDING COMMUNITIES AND ECOSYSTEMS Johnson and Omland, 2004, for a review). The idea here is to formulate two or more models with different embedded hypotheses, compare them with data, and analyze the goodness of fit to reveal which of the hypoth- eses appear to be borne out by the data. This framework was initially developed over 30 years ago (Akaike, 1973) but is only now receiving attention in ecology. It provides a way to quantify the relative support for competing hypotheses based on data. Further development of this useful tool will probably impact both experimental design and statistical analy- sis in ecology. Assessment of uncertainty remains a key challenge in ecological mod- eling (Brewer and Gross, 2003). Few models include a stochastic compo- nent, so they are not set up to provide a distribution of results from mul- tiple runs. In addition, different modeling approaches can yield different predictions even if the same scenarios are modeled, reflecting uncertainty in our knowledge of the underlying processes. Averaging over different models has recently been suggested as a way to increase the robustness of results (Koster et al., 2004). However, there is no general theory at this point that lends credence to such ad hoc methods. Predictive models of ecosystems also increasingly include economic and social components. For instance, the goal of a recent National Center for Ecological Analysis and Synthesis workshop, “Global Biodiversity Sce- narios” (Chapin et al., 2001), was to combine vegetation and climate mod- els with economic and social scenarios to predict the effects of human impact on major biomes. The management of natural ecosystems relies increasingly on sophis- ticated models. Spatial heterogeneity and demographic and environmen- tal stochasticity are often key driving factors. Spatial control, a mathemati- cally sophisticated and computationally intensive tool, appears to be a promising methodology (Hof and Bevers, 1998, 2002). FUTURE DIRECTIONS Interactions at the community level are influenced by and influence all other levels of organization, from genes to ecosystems, including abi- otic conditions such as temperature, precipitation, and nutrient availabil- ity. For a full understanding of processes at the community level, integra- tion across disciplines, scales, and levels of organization will be needed. The following exemplify this integration and highlight some of the math- ematical developments that need to occur in order to accomplish this inte- gration. First come the processes discussed earlier that shape ecological communities: competition and predation. Ecology has traditionally been divided into community ecology and ecosystem ecology. Community ecology focuses on population dynamics

OCR for page 110
118 MATHEMATICS AND 21ST CENTURY BIOLOGY and the interplay between the biotic and abiotic environment. Ecosystem ecology deals with fluxes of nutrients and energy. Models in community ecology describe the dynamics of biomass or individuals, whereas mod- els in ecosystem ecology describe fluxes of matter and energy among func- tional units. The past 15 years have increasingly witnessed research at the interface of the two ecologies (Naeem et al., 2002). Research that addressed the diversity-stability-productivity debate illustrates this emerging syn- thesis of the two fields (Box 7.1). Research in this area will probably see greater integration across spatial scales and across levels of organization. Food web studies are another example where integration across fields, scales, and levels of organization is occurring. Food webs are complex networks of interacting groups of species. A community ecology approach focuses on particular species and attempts to understand their interac- tions as described by competition, predation, or facilitation. A classic study by Paine (1966) illustrates this approach: Recognizing that detailed bookkeeping of the calorie consumption of the members of a food web could explain food web structure and, ultimately, the diversity of a local community, Paine manipulated food webs through removal (or addition) experiments so as to assess the importance of each link. As one of its most significant conclusions, the study demonstrated a drastic decrease in di- versity after removal of the starfish Pisaster B. glandula from an intertidal community, thus identifying predation as an important process for main- taining diversity. Paine and Levin (1981) introduced a disturbance model that modeled the dynamics of gaps left behind by a predator and their subsequent recolonization. The model was parameterized by field data and yielded predictions that compared well with observations. An ecosystem approach to food webs disregards species identities and instead focuses on functional groups, such as autotrophs, detritus, het- erotrophs, and nutrient pools. This approach leads to compartment mod- els that track the flux of matter and energy among the compartments. This flux is typically described by systems of differential equations. Reiners (1986) proposed a theoretical framework for ecosystem dy- namics that included both energy and nutrient considerations, calling it ecological stoichiometry. The recent book by Sterner and Elser (2002) on ecological stoichiometry provides a synthesis of processes at the cellular level to ecosystem levels based on such stoichiometry and the resulting nutrient demands of the biota. Food web models that combine both ap- proaches are still in their early stages but have already yielded interesting insights into the importance of food quality in addition to food quantity (Loladze et al., 2000). These new models combine classical community ecology models with insights from nutrient dynamics. They are largely phenomenological but will likely become more mechanistic as our under- standing of these processes across all levels of organization increases.

OCR for page 110
119 UNDERSTANDING COMMUNITIES AND ECOSYSTEMS Additional insights into food web structure can be gained by compar- ing large food webs across different ecosystems. Such comparison has revealed structural commonalities, and it has been proposed that com- mon mechanisms are responsible for network structure (Dunne et al., 2004). Recently, Brose et al. (2004) attempted to unify the relationships between species richness and spatial scaling and between species richness and trophic interactions to extend the spatial scale at which food web theory applies. Another area of activity that requires sophisticated modeling, math- ematical analysis, and statistical tools is epidemiology or, more generally, host-pathogen systems. The increased attention to disease dynamics stems from the global threat of emerging and reemerging diseases, such as avian flu, West Nile virus, or SARS. Modeling often involves much more than simple disease dynamics as embodied in the standard models of Kermack and McKendrick (1927). Human behavior, socioeconomic factors, and spa- tiotemporal dynamics play a significant role and must be taken into ac- count to adequately capture the dynamics. Increasingly, researchers are studying diseases not only from a public health perspective but also with respect to how they interact with the ecological environment. Known as disease ecology, this emerging field is highly interdisciplinary, drawing from epidemiology and ecology. Complex dynamics stemming from multispecies interactions complicate the analysis and make predictions difficult. Progress in this area will require collaborations among epidemi- ologists, ecologists, statisticians, and mathematicians. Microbial communities will increasingly be the focus of community and ecosystem ecology studies. They provide the opportunity for true integration across levels of organization, similar to the integration in physical systems that resulted in a description of macroscopic phenom- ena based on microscopic processes. Molecular biology techniques are beginning to reveal the diversity of microbes. Large-scale genome analy- sis is needed to assess the metabolic capacity of microbes, because pro- teins will need to be identified and their functions understood to reveal the metabolic pathways. Ecological studies will reveal the activity of path- ways as a function of the biotic and abiotic environment; this is necessary to link the metabolic potential of microbes to community-level processes. To accomplish this integration, statistical analysis of genomic data based on evolutionary models will need to be linked to physiological models and, finally, to community-level models. Development of such models will require close collaboration between experimentalists and theoreti- cians. The importance of microbial studies is discussed in Box 7.2. To illustrate the need for integration between the fields of evolution and ecology in the context of community ecology, the committee revisits a theme discussed in Chapter 6. Community ecologists largely view eco-

OCR for page 110
120 MATHEMATICS AND 21ST CENTURY BIOLOGY BOX 7.2 Microbial Ecology Microbes are microscopic organisms that are not visible with the na- ked eye. They were discovered by Antony van Leeuwenhoek (1632-1723). Prokaryotic microbes (bacteria) are the oldest organisms on earth. The fos- sil record indicates that they evolved more than 3.8 billion years ago. Eu- karyotic microbes, such as fungi and protozoa, appear to have evolved at least a billion years later. Microbes with their unrivaled metabolic capacity play an important role in biogeochemical cycles. Since human activities have profoundly al- tered virtually every biogeochemical cycle, it is important to understand the roles of microbes in these cycles. Advances in molecular biology, in particular in genomics, have greatly expanded our ability to study naturally occurring microbes that have eluded us thus far owing to the difficulties in culturing them. For instance, Zehr et al. (2001) recently demonstrated that many unicellular microbes in the oxygenated region of the sea have nif genes, indicating that oceanic nitrogen fixation might be much higher than previously thought. Microbes provide opportunities for integration across all levels of or- ganization, from genes to ecosystems (Stahl and Tiedjen, 2002). Venter et al. (2004), using shotgun sequencing of microbes in the ocean, have given us a static glimpse into the enormous diversity of largely unknown organ- isms that are responsible for biogeochemical cycles. Their study demon- strated the feasibility of large, community-level genomics analysis to assess diversity. It is a long way from the assessment of microbial diversity to understanding the function of microbes in ecological communities. It will require integration of genomic, proteomic, and metabolomic data with community-level models. New modeling and statistical approaches will need to be developed to deal with these very large and complex systems. System theoretical approaches are currently being championed as the key to unraveling the metabolic capacity of microbes and their role in community dynamics. An integrative approach has been suggested (Wolkenhauer et al., 2004). The complex interactions are often described by block diagrams and a network, ultimately represented through differen- tial equations, which are the mainstay of control engineers for dealing with processes. A standard equilibrium analysis of such large systems is often not satisfying, because the systems are so complex. Modularization of net- works has been suggested to understand these large complex systems (Saez- Rodriguez et al., 2004). In addition, transient dynamics might dominate much of naturally occurring communities. It is important to realize that revealing metabolic capacity alone will not be sufficient. Environmental conditions affect the expression of meta- bolic pathways (Dauner and Sauer, 2001; Dauner et al., 2001). It is thus necessary to experimentally understand metabolic activities as a function of environmental conditions in order to predict community dynamics. This will require close collaboration between experimentalists and theoreticians.

OCR for page 110
121 UNDERSTANDING COMMUNITIES AND ECOSYSTEMS logical communities as genetically homogeneous (but, see Ford, 1964). Over the last 10 years, an increasing number of studies have demonstrated the importance of including evolutionary processes in studies of ecologi- cal communities. For instance, invasive species or the assembly of novel communities can alter ecological interactions and impose strong selection on all members of a community (Reznick et al., 1997, 2001; Davis and Shaw, 2001). Evolution within a predator-prey system has been studied, for instance, by Shertzer et al. (2002) and Yoshida et al. (2003), who com- bined theoretical and empirical studies to demonstrate that the evolution within such a system (an algal prey and its rotifer predator) can shape population dynamics. The empirical system showed oscillations in quali- tative agreement with theoretical studies. However, there was quantita- tive disagreement: Both the cycle period and the phase between predator and prey differed from theoretical predictions. Shertzer et al. (2002) sug- gested a new model that incorporated evolution of the algal prey and demonstrated that rapid evolution of the prey could explain the observed pattern. Yoshida et al. (2003) confirmed this model experimentally by growing the algal prey with and without its predator. Their study showed that resistance to the predator was a heritable trait and that there was a trade-off between resistance and competitive ability. It has been suggested that this trade-off and predation contribute to the maintenance of genetic diversity. (See Johnson and Agrawal, 2003, for a summary of these stud- ies.) These studies demonstrate the importance of allowing genetic varia- tion and incorporating it into ecological models. The study of these com- plex interactions is in its early stages. Only a combination of empirical and theoretical studies will yield much-needed insights. The distribution and abundance of each species is a function of the whole community composition and the genetic composition of each indi- vidual in the context of the community. When ecological interactions and genetic composition of populations reciprocally affect each other, both factors need to be considered. Antonovics (1992) proposed a new frame- work, “community genetics” (a term suggested by J.J. Collins at Arizona State University), which is a synthesis of evolutionary genetics and com- munity ecology and focuses on the role of genetic variation in determin- ing community structure (Luck et al., 2003; Neuhauser et al., 2003; Whitham et al., 2003). Models that incorporate both ecological and genetic factors quickly become quite complex because they must track not only the dynamics of the species but also the genetic composition of the indi- viduals. These models often also include a spatial component, adding to their complexity. A community genetics perspective seems to be particularly useful when dealing with strong selection in a community context. As argued in Neuhauser et al. (2003), this is particularly likely to occur during transient dynamics following large-scale perturbations, such as habitat reduction

OCR for page 110
122 MATHEMATICS AND 21ST CENTURY BIOLOGY or expansion. Habitat reduction due to land-use changes has been occur- ring at an unprecedented rate. The concomitant loss of genetic diversity can accelerate extinction. Habitat expansion can be observed in both agri- cultural and natural systems, for instance through the introduction of a novel organism such as a genetically modified organism or an exotic spe- cies invasion. The final example illustrates the need for integration at the global scale. The effects of human activities on global climate were for the first time illustrated by Keeling et al. (1976) when they published data from Mauna Loa in Hawaii showing a clear increase in atmospheric carbon dioxide over many decades. It became clear that, in order to assess changes in the global carbon cycle, global measurements were needed. Satellite data that became available in the 1980s made it possible to estimate net primary production from remote sensing data. Satellites now capture a continuous stream of spectral data at resolutions at and below the 1-kilo- meter scale. For instance, the NASA Earth Observing System Terra satel- lite uses the Moderate Resolution Imaging Spectroradiometer (MODIS) to measure the spectral reflectance of terrestrial vegetation. This data set is used to produce a weekly data set of primary production of the entire vegetated surface, a critical quantity for assessing carbon dynamics. Understanding carbon and nutrient cycles at global and regional scales is a very active area of ecology that integrates across community ecology and ecosystem ecology. As an example, predicting an increase in temperature as a function of an increase in carbon dioxide at the spatial scale of the whole earth was already accomplished by Arrhenius (1896). It has proved much more difficult to make predictions at regional scales, which requires linking vegetation models to global circulation models. To parameterize such models, estimates of primary production at a regional scale are needed. This will require advances in retrieving accurate esti- mates based on spectral information, relating those estimates to measure- ment of the actual state on the ground in a region, and incorporating the data into ecosystem process models. This field provides clear opportuni- ties for linking computational models to observational data. REFERENCES Akaike, H. 1973. Information theory as an extension of the maximum likelihood principle. Pp. 267-281 in Second International Symposium on Information Theory. B.N. Petrov and F. Csaki, eds. Budapest: Akademia Kiado. Anderson, K., and C. Neuhauser. 2002. Patterns in spatial simulations—Are they real? Ecol. Model. 155: 19-30. Antonovics, J. 1992. Toward community genetics. Pp. 426-449 in Plant Resistance to Herbi- vores and Pathogens: Ecology, Evolution, and Genetics. Chicago, Ill.: University of Chicago Press.

OCR for page 110
123 UNDERSTANDING COMMUNITIES AND ECOSYSTEMS Arrhenius, S. 1896. On the influence of carbonic acid in the air upon temperature on the ground. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 41: 237-275. Bolker, B.M., and S.W. Pacala. 1997. Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol. 52: 179-197. Brewer, C.A., and L.J. Gross. 2003. Training ecologists to think with uncertainty in mind. Ecology 84: 1412-1414. Brose, U., A. Ostling, K. Harrison, and N.D. Martinez. 2004. Unified spatial scaling of species and their trophic interactions. Nature 428: 167-171. Chapin, F.S., O.E. Sala, and E. Huber-Sannwald, eds. 2001. Global Biodiversity in a Changing Environment: Scenarios for the 21st Century. New York, N.Y.: Springer-Verlag. Chesson, P. 1994. Multispecies competition in variable environments. Theor. Popul. Biol. 45: 227-276. Chesson, P.L., and R.R. Warner. 1981. Environmental variability promotes coexistence in lottery competitive systems. Am. Nat. 117: 923-943. Clements, F.E. 1936. Nature and structure of the climax. J. Ecol. 24: 252-284. Clements, F.E., J.E. Weaver, and H.C. Hanson. 1929. Plant Competition: An Analysis of Com- munity Functions. Washington, D.C.: Carnegie Institution. Clifford, P., and A. Sudbury. 1973. A model for spatial conflict. Biometrika 60: 581-588. Darwin, C. 1859. The Origin of Species. Reprinted in 1985. London, England: Penguin Books. Dauner, M., and U. Sauer. 2001. Stoichiometric growth model of riboflavin-producing Bacil- lus subtilis. Biotechnol. Bioeng. 76: 132-143. Dauner, M., T. Storni, and U. Sauer. 2001. Bacillus subtilis metabolism and energetics in carbon-limited and excess-carbon chemostat culture. J. Bacteriol. 183: 7308-7317. Davis, M.B., and R.G. Shaw. 2001. Range shifts and adaptive responses to quaternary cli- mate change. Science 292: 673-679. Dobrushin, R.L. 1971. Markov processes with a large number of locally interacting compo- nents: Existence of a limit process and its ergodicity. Problemy Peredachi Informatsii 7: 149-164. Dunne, J.A., R.J. Williams, and N.D. Martinez. 2004. Network structure and robustness of marine food webs. Marine Ecol. Progress Ser. 273: 291-302. Durrett, R., and S.A. Levin. 1994. The importance of being discrete (and spatial). Theor. Popul. Biol. 46: 363-394. Elton, C. 1927. Animal Ecology. London, England: Sedgwick and Jackson. Elton, C. 1933. The Ecology of Animals. London, England: Methuen. Elton, C.S. 1958. The Ecology of Invasion by Animals and Plants. London, England: Methuen. Fargione, J., R. Dybzinski, C. Clark, J. Hille Ris Lambers, S. Harpole, M. Loreau, and D. Tilman. 2004. From selection to complementarity: Temporal trends in a long-term biodiversity experiment. 88th Ecological Society of America Annual Meeting, Savannah, Georgia, August 3-8, 2003. Washington, D.C.: Ecological Society of America. Fisher, R.A. 1937. The wave of advance of advantageous genes. Ann. Eugen. 7: 353-369. Ford, E.B. 1964. Ecological Genetics. London, England: Methuen. Freckman, D.W., T.H. Blackburn, L. Brussaard, P. Hutchings, M.A. Palmer, and P.V.R. Snelgrove. 1997. Linking biodiversity and ecosystem functioning of soils and sediments. AMBIO 26: 556-562. Gleason, H.A. 1926. The individualistic concept of the plant association. Bull. Torrey Botani- cal Club 53: 7-26. Grinnell, J. 1917. The niche-relationship of the Californian thrasher. Auk 34: 427-433. Hanski, I. 1999. Metapopulation Ecology. Oxford, U.K.: Oxford University Press. Hastings, A. 2004. Transients: The key to long-term ecological understanding? Trends Ecol. Evol. 19: 39-45.

OCR for page 110
124 MATHEMATICS AND 21ST CENTURY BIOLOGY Hof, J.G., and M. Bevers. 1998. Spatial Optimization in Ecological Applications. New York, N.Y.: Columbia University Press. Hof, J.G., and M. Bevers. 2002. Spatial Optimization for Managed Ecosystems. New York, N.Y.: Columbia University Press. Holley, R., and T.M. Liggett. 1975. Ergodic theorems for weakly interacting particle systems and the voter model. Ann. Probab. 3: 643-663. Holmes, E.E., M.A. Lewis, J.E. Banks, and R.R. Veit. 1994. Partial differential equations in ecology: Spatial interactions and population dynamics. Ecology 75: 17-29. Hutchinson, G.E. 1957. Population studies: Animal ecology and demography. Pp. 415-427 in Cold Spring Harbor Symposia on Quantitative Biology. Vol. 22. Woodbury, N.Y.: CSHL Press. Ives, A.R., J.L. Klug, and K.Gross. 2000. Stability and species richness in complex communi- ties. Ecol. Lett. 3: 399-411. Johnson, J.B., and K.S. Omland. 2004. Model selection in ecology and evolution. Trends Ecol. Evol. 19: 101-108. Johnson, M.T.J., and A.A. Agrawal. 2003. The ecological play of predator-prey dynamics in an evolutionary theatre. Trends Ecol. Evol. 18: 549-551. Keeling, C.D., R.B. Bacastow, A.E. Bainbridge, C.A. Ekdahl, P.R. Guenther, L.S. Waterman, and J.F.S. Chin. 1976. Atmospheric carbon dioxide variations at Mauna Loa Observa- tory, Hawaii. Tellus 28: 538-551. Kermack, W.O., and A.G. McKendrick. 1927. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. London Ser. A 115: 700-721. Kingsland, S.E. 1991. Defining ecology as a science. Pp. 1-13 in Foundations of Ecology: Classic Papers with Commentaries. L.A. Real and J.H. Brown, eds. Chicago, Ill.: University of Chicago Press. Koster, R.D., P.A. Dirmeyer, Z. Guo, G. Bonan, E. Chan, P. Cox, C.T. Gordon, S. Kanae, E. Kowalczyk, D. Lawrence, P. Liu, C.-H. Lu, S. Malyshev, B. McAvaney, K. Mitchell, D. Mocko, T. Oki, K. Oleson, A. Pitman, Y.C. Sud, C.M. Taylor, D. Verseghy, R. Vasic, Y. Xue, and T. Yamada. 2004. Regions of strong coupling between soil moisture and pre- cipitation. Science 305: 1138-1140. Lawton, J.H., S. Naeem, R.M. Woodfin, V.K. Brown, A. Gange, H.J.C. Godfray, P.A. Heads, S. Lawler, D. Magda, C.D. Thomas, L.J. Thompson, and S. Young. 1993. The Ecotron: A controlled environmental facility for the investigation of population and ecosystem processes. Phil. Trans. Roy. Soc. Lond. B 341: 181-194. Lehman, C.L., and D. Tilman. 2000. Biodiversity, stability, and productivity in competitive communities. Am. Nat. 156: 534-552. Levins, R. 1969. Some demographic and genetic consequences of environmental heterogene- ity for biological control. Bull. Entomol. Soc. Am. 15: 237-240. Lindeman, R.L. 1942. The trophic-dynamic aspects of ecology. Ecology 23: 399-418. Loladze, I., Y. Kuang, and J.J. Elser. 2000. Stoichiometry in producer-grazer systems: Linking energy flow and element cycling. Bull. Math. Biol. 62: 1137-1162. Loreau, M. 1998. Biodiversity and ecosystem functioning: A mechanistic model. Proc. Nat. Acad. Sci. U.S.A. 95: 5632-5636. Loreau, M. 2000. Biodiversity and ecosystem functioning: Recent theoretical advances. Oikos 91: 3-17. Loreau, M., A. Downing, M. Emmerson, A. Gonzales, J. Hughes, P. Inchausti, J. Joshi, J. Norberg, and O. Sala. 2002. A new look at the relationship between diversity and sta- bility. Chapter 7 in Biodiversity and Ecosystem Functioning: Synthesis and Perspectives. M. Loreau, S. Naeem, and P. Inchausti, eds. Oxford, U.K.: Oxford University Press. Loreau, M., and A. Hector. 2001. Partitioning selection and complementarity in biodiversity experiments. Science 412: 72-76.

OCR for page 110
125 UNDERSTANDING COMMUNITIES AND ECOSYSTEMS Lotka, A. 1925. Elements of Physical Biology. Baltimore, Md.: Williams & Wilkins Co.. Lotka, A.J. 1932. The growth of mixed populations: Two species competing for a common food supply. J. Wash. Acad. Sci. 22: 461-469. Luck, G.W., G.C. Daily, and P.R. Ehrlich. 2003. Population diversity and ecosystem services. Trends Ecol. Evol. 18: 331-336. MacArthur, R.H. 1955. Fluctuations of animal populations and a measure of community stability. Ecology 36: 533-536. May, R.M. 1972. Will large and complex systems be stable? Nature 238: 413-414. May, R.M. 1974. Stability and Complexity in Model Ecosystems. Princeton, N.J.: Princeton Uni- versity Press. McNaughton, S.J. 1977. Diversity and stability of ecological communities: A comment on the role of empiricism in ecology. Am. Nat. 111: 515-525. Naeem, S., M. Loreau, and P. Inchausti. 2002. Biodiversity and ecosystem functioning: The emergence of a synthetic ecological framework. Pp. 3-11 in Biodiversity and Ecosystem Functioning: Synthesis and Perspectives. M. Loreau, S. Naeem, and P. Inchausti, eds. Ox- ford, U.K.: Oxford University Press. Naeem, S., L.J. Thompson, S.P. Lawler, J.H. Lawton, and R.M. Woodfin. 1994. Declining biodiversity can alter the performance of ecosystems. Nature 368: 734-737. Neuhauser, C. 1992. Ergodic theorems for the multitype contact process. Probab. Theory Rel. Fields 91: 467-506. Neuhauser, C. 2001. Mathematical challenges in spatial ecology. Notices of the American Math- ematical Society 48: 1304-1314. Neuhauser, C., D.A. Andow, G.E. Heimpel, G. May, R.G. Shaw, and S. Wagenius. 2003. Community genetics: Expanding the synthesis of ecology and genetics. Ecology 84: 545-558. Neuhauser, C., and S. Pacala. 1999. An explicitly spatial version of the Lotka-Volterra model with interspecific competition. Ann. Appl. Probab. 9: 1226-1259. Nicholson, A.J. 1933. The balance of animal populations. J. Anim. Ecol. 2: 132-178. Nicholson, A.J., and V.A. Bailey. 1935. The balance of animal populations, Part I. Proc. Zool. Soc. London 3: 551-598. Odum, E.P. 1953. Fundamentals of Ecology. Philadelphia, Pa.: Saunders. Okubo, A. 1980. Diffusion and Ecological Problems: Mathematical Models. Biomathematics, Vol. 10. New York, N.Y.: Springer. Okubo, A., and S.A. Levin. 2001. Diffusion and Ecological Problems: Modern Perspectives. New York, N.Y.: Springer. Paine, R.T. 1966. Food web complexity and species diversity. Am. Nat. 100: 65-75. Paine, R.T. 2002. Trophic control of production in a rocky intertidal community. Science 296: 736-739. Paine, R.T. , and S.A. Levin. 1981. Intertidal landscapes: Distribution and dynamics of pat- tern. Ecol. Monogr. 51: 145-178. Patten, B.C. 1975. Ecosystem linearization: An evolutionary design problem. Am. Nat. 109: 529-539. Pimm, S.L. 1984. The complexity and stability of ecosystems. Nature 307: 321-326. Real, L.A., and S.A. Levin. 1991. The role of theory in the rise of modern ecology. Pp. 177-191 in Foundations of Ecology: Classic Papers with Commentaries. L.A. Real and J.H. Brown, eds. Chicago, Ill.: University of Chicago Press. Reiners, W.A. 1986. Complementary models for ecosystems. Am. Nat. 127: 59-73. Reznick, D.N., M.J. Butler, and F.H. Rodd. 2001. Life-history evolution in guppies. VII. The comparative ecology of high- and low-predation environments. Am. Nat. 157: 126-140. Reznick, D.N., F.H. Shaw, F.H. Rodd, and R.G. Shaw. 1997. Evaluation of the rate of evolu- tion in natural populations of guppies (Poecilia reiculata). Science 275: 1934-1937.

OCR for page 110
126 MATHEMATICS AND 21ST CENTURY BIOLOGY Saez-Rodriguez, J., A. Kremling, H. Conzelmann, K. Bettenbrock, and E.D. Gilles. 2004. Modular analysis of signal transduction networks. IEEE Control Systems Magazine 24(4): 35-52. Scheffer, M., and S.R. Carpenter. 2003. Catastrophic regime shifts in ecosystems: Linking theory to observation. Trends Ecol. Evol. 18: 648-656. Shertzer, K.W., S.P. Ellner, G.F. Fussmann, and N.G. Hairston Jr. 2002. Predator-prey cycles in an aquatic microcosm: Testing hypotheses of mechanism. J. Anim. Ecol. 71: 802-815. Skellam, J.G. 1951. Random dispersal in theoretical populations. Biometrika 38: 196-218. Spitzer, F. 1970. Interaction of Markov processes. Adv. Math. 5: 246-290. Stahl, D.A., and J.M. Tiedje. 2002. Microbial Ecology and Genomics: A Crossroads of Opportunity. American Academy of Microbiology Critical Issues Colloquia Report. Washington, D.C.: American Society for Microbiology. Sterner, R.W., and J.J. Elser. 2002. Ecological Stoichiometr. Princeton, N.J.: Princeton Univer- sity Press. Tilman, D. 1982. Resource Competition and Community Structure. Princeton, N.J.: Princeton University Press. Tilman, D., and J.A. Downing. 1994. Biodiversity and stability in grasslands. Nature 367: 363-365. van der Heijden, M.G.A., and J.H.C. Cornelissen. 2002. The critical role of plant-microbe interactions on biodiversity and ecosystem functioning: Arbuscular mycorrhizal asso- ciations as an example. Pp. 181-194 in Biodiversity and Ecosystem Functioning: Synthesis and Perspectives. M. Loreau, S. Naeem, and P. Inchausti, eds. Oxford, U.K.: Oxford Uni- versity Press. Venter, J.C., K. Remington, J.F. Heidelberg, A.L. Halpern, D. Rusch, J.A. Eisen, D. Wu, I. Paulsen, K.E. Nelson, W. Nelson, D.E. Fouts, S. Levy, A.H. Knap, M.W. Lomas, K. Nealson, O. White, J. Peterson, J. Hoffman, R. Parsons, H. Baden-Tillson, C. Pfannkoch, Y.H. Rogers, and H.O. Smith. 2004. Environmental genome shotgun sequencing of the Sargasso Sea. Science 304: 66-74. Volterra, V. 1926. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI 2: 31-113. Wardle, D.A., and W.H. van der Putten. 2002. Biodiversity, ecosystem functioning and above-ground-below-ground linkages. Pp. 155-168 in Biodiversity and Ecosystem Func- tioning, Synthesis and Perspectives, M. Loreau, S. Naeem, and P. Inchausti, eds. New York, N.Y.: Oxford University Press. Whitham, T.G., W.P. Young, G.D. Martinsen, C.A. Gehring, J.A. Schweitzer, S.M. Shuster, G.M. Wimp, D.G. Fischer, J.K. Bailey, R.L. Lindroth, S. Woolbright, and C.R. Kuske. 2003. Community and ecosystem genetics: A consequence of the extended phenotype. Ecology 84: 559-573. Wolkenhauer, O., B.K. Ghosh, and K.-H. Cho. 2004. Control and coordination in biochemical networks. IEEE Control Systems Magazine 24(4): 30-34. Yoshida, T., L.E. Jones, S.P. Ellner, G.F. Fussmann, and N.G. Hairston Jr. 2003. Rapid evolu- tion drives ecological dynamics in a predator-prey system. Nature 424: 303-306. Zehr, J.P., J.B. Waterbury, P.J. Turner, J.P. Montoya, E. Omoregie, G.F. Steward, A. Hansen, and D.M. Karl. 2001. New nitrogen-fixing unicellular cyanobacteria discovered in the North Pacific subtropical gyre. Nature 412: 635-638.