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This chapter examines the dynamics of consumer–resource interaction, one of the fundamental building blocks of food webs. In particular, it considers how consumer–resource systems that are nonexcitable and excitable respond to changes in interaction strength. The chapter begins with a discussion of two classes of interaction-strength metrics: the first focuses on instantaneous rates of change in one species with respect to another species; the second follows the longer-term influence of the removal of (or change in) one species on the density of another focal species. Continuous consumer–resource models are then described, after which two underlying mechanisms that are behind the stabilization of consumer–resource interactions are analyzed. The chapter concludes with a review of microcosm experiments and empirical data that show consistency with the proposed consumer–resource theory.

This chapter examines the basic assumptions of classic food web theory. It first considers the classic whole-community approach, which assumes that any specific matrix represents a sample from a “statistical universe” of interaction strengths for a given set of n species. It then describes some matrix approaches to see if context-dependent techniques can be applied to matrix theory, along with the simple graphical techniques of Gershgorin discs employed as an intuitive approach to eigenvalues. It argues that there are some rather intriguing “gravitational-like” properties of Gershgorin discs for some important biologically motivated matrices. The chapter proceeds by discussing some classic whole-matrix results that highlight the connections between the stability of lower-dimensional modules and whole food webs. Finally, it shows how the ideas derived from classic whole-system matrix approaches generally agree with the results of modular theory.

The chapter summarizes the history of recent scientific understanding of food webs, particularly shallow water food webs. The chapter outlines a history of key papers ranging from Elton to the most recent synthetic studies of meta-analysis of multiple food webs in many ecosystems. Changes in general organizing principles over time and conflicts with recent understanding are emphasized including an outline of the considerable progress over time and hope for an emerging discipline that takes advantage of emerging multidisciplinary tools and collaborations.

This chapter extends the consumer–resource theory to include simple but common three-species modules behind the construction of whole food webs, with particular emphasis on food chains and omnivory. It first considers some common simple modular food web structures and whether the dynamics of subsystems can be seen using the framework laid out in previous chapters. Specifically, it asks when common food web structure increases or weakens the relative interaction strengths and/or when a food web structure modifies flux between consumers and resources in a density-dependent manner such that the food web tends to increase flux rates in some situations and decrease the coupling in other situations. The chapter also explores how stage structure can influence food chain stability before concluding with a review of empirical evidence on the dynamical implications of omnivory for food webs.

This chapter considers four-species modules and the role of generalism (effectively a three-species module with a consumer feeding on two resources). It first examines how generalists affect the dynamics of food webs by focusing on a set of modules that contrast generalist consumer dynamics relative to the specialist case. It then discusses organismal trade-offs that play a role in governing the diamond food web module and the intraguild predation module, arguing that such tradeoffs influence the flux of matter, the organization of interaction strengths, and ultimately the stability of communities. The chapter also reviews empirical evidence showing that apparent competition and the diamond module with and without intraguild predation are ubiquitous, and that weak interactions in simple modules seem to promote less variable population dynamics.

This chapter examines the influence of biological lags on consumer–resource dynamics, with particular emphasis on how consumer–resource cycles, or the lack thereof, interact with population level dynamical phenomena. It first considers discrete consumer–resource interactions before discussing the dynamics of stage-structured consumer–resource interactions. It then explains how stage structure promotes the possibility of alternative stable states and changes consumer–resource interaction strength. It also shows how a change in population structure affects food web interactions and/or the strengths of food webs. Finally, it reviews empirical results that show how stage structure and food web interaction influence ecological stability. The chapter argues that weak and inherently stable consumer–resource interactions can mute a potentially unstable population level phenomenon, and that a dynamically decoupled stable stage class can strongly stabilize other stages and the consumer–resource interaction.

This chapter describes ecology of the Galapagos rocky reef system and the important role of biogeographic position on biodiversity, the El Niño cycle, and the history of resource extraction on the current state of the ecosystem. The chapter presents a model of the energetic pathways in the ecosystem and its predictions for fisheries yields and the role of key species. The history of exploitation is outlined as well as the role of the current marine protected areas to develop sustainable management system.

This chapter explains the use of modular or motif-based theory to interpret the dynamics of whole food webs. According to Robert Holt, modules are “as motifs with muscles.” Holt's modular theory focuses on the implications of the strength of the interactions on the dynamics and persistence of these units. In this book, the term “module” means all motifs that include interaction strength, whereas the term “motif” represents all possible subsystem connections, including the trivial one-node/species case to the n-node/species cases. Part 2 considers the dynamics of important ecological modules or motifs such as populations, consumer–resource interactions, food chains, and omnivory, while Part 3 uses the logic attained from this modular or motif-based theory in order to elucidate the dynamics of whole food webs. The book argues that ecologists must make a concerted effort to understand how coupling different modules ultimately modifies flux within each individual module.

This chapter examines some of the potential empirical signatures of instability in complex adaptive food webs. It first considers the role of adaptive behavior on food web topology, ecosystem size, and interaction strength before discussing the implications of this behavior for ecosystem dynamics and stability. It then analyzes the results of empirical investigations of Canadian Shield lake trout food webs and how human influences and ecosystems coupled in space may drive biomass pyramids, potentially leading to species loss. It also explores the tendency of subsidies, through human impacts, to homogenize natural ecosytems and concludes by assessing some of the changing conditions that are being driven by humans and how these may change ecosystems.

This chapter examines how nutrient recycling and decomposition affect the dynamics and stability of food webs. It first reviews some of the existing theory on detritus and food web dynamics before discussing the basics of a model that takes into account grazing food webs and whole ecosystems. It then describes the N-R-D (nutrient pool, resource, detritus) submodule as well as the full N-C-R-D (nutrient pool, consumer, resource, detritus) model. It also explores how detritus may act to distribute nutrients by considering a model that begets nonequilibrium dynamics. It shows that detritus tends to stabilize consumer–resource interactions relative to the purely community module (no recycling) because the detritus tends to fall out of phase with the resource–nutrient interaction. The addition of a consumer–resource incteraction to the N-R-D module, even in a closed system, eventually can drive overshoot dynamics and destabilization by increased production, coupling, or interaction strength.

This chapter explores ecological networks and their properties. Ecological networks summarize the many potential interactions between species within a community by representing species as nodes in the network and using links between nodes to represent the interactions between species. The earliest and best-studied ecological networks are food webs that describe who eats whom within a community (trophic links). Most food webs contain a few strong and many weak links between species; the preponderance of weak links promotes food web stability. Body size is a key trait in determining the pattern and strength of trophic interactions in food webs. Mutualistic networks describe the positive interactions between species in a community, where patterns of species associations may be characterized as either “nested” or “modular”. Nestedness may increase stability in mutualistic networks. A major challenge to future research is to incorporate multiple types of species interactions into the same ecological network.

This book explores how interaction strength affects the dynamics of food webs. It aims to conceptually synthesize our current understanding of one of the big questions in ecology and evolution: What governs ecological stability? The book discusses the consequences of human impacts for the intricate, detailed spatial and temporal structure that underlies most pristine ecosystems. It asserts that ecologists never saw the balance of nature as a perfect equilibrium process. This chapter defines stability and examines the role whole systems play in governing stable ecosystem function. It also considers the stability problem by presenting examples that illustrate how humans cause ecological instability and ecosystem collapse. It concludes with an overview of the book's proposed theory about food webs and ecosystems that can help elucidate the ways that perturbations (such as human impact) ought to influence the sustainability of ecosystems.

Invasive alien species cause impacts partly through the changes they impose on biotic interactions which lead to altered ecosystem processes in recipient systems. This chapter reviews the types of direct and mediated (indirect) biotic interactions established by alien species. It also discusses possible ways of quantifying the strength and identifying the drivers of interactions. We present models and approaches that allow us to investigate the effects of biotic interactions on the spatial distribution and dynamics of species. Interaction fidelity and promiscuity are explored, as are the concepts of interaction switching and rewiring which are often experienced by species in their novel environments. The chapter ends with an outline of the potential eco-evolutionary forces experienced by interacting native and alien species, and a discussion of how such novel biological contexts could shape both current and future biotic interactions.

Uncovering the fundamental properties of ecological stability is a central question in theoretical biology since its inception at the turn of the century. Here, motivated by simple modular theory (e.g., population models to few species models), we review the role of interactions strength and lags on dynamics and stability. Specifically, we argue that modular theory consistently finds that lags combined with high growth rates or strong interaction strengths underly all forms of instability in ecological models. To fully explore this relationship, we first need to understand the role of both explicit lags—using lagged versions of classical models, such as the lagged logistic population model, as well as the more subtle role of implicit lags that arise in all biological models of growth. Given this, and the realization that nature is replete with lags (e.g., age structure, stage structure, predator-prey, reproductive lags, recycling lags), it becomes important to understand how lags, both implicit and explicit, interact. With an eye towards correcting the frequently overlooked role of lags on stability we review existing mathematical examples that argue lags can combine to drive instability (lag excitation) or inhibit the expression of instability by cancelling each other out effectively (lag cancellation). We suggest that further understanding the role of lags and how they interact within whole webs and ecosystems remains an important research area for the future.

Scattering experiments provide the main source of information on the properties of elementary particles. Here we present the theory of scattering in both classical and non-relativistic quantum physics. We introduce the basic notions of cross section and of range and strength of interactions. We work out some illustrative examples. The concept of resonant scattering, central to almost all applications in particle physics, is explained in the simple case of potential scattering, where we derive the Breit–Wigner formula. This framework of non-relativistic potential scattering turns out to be very convenient for introducing several other important concepts and results, such as the optical theorem, the partial wave amplitudes and the corresponding phase shifts and scattering lengths. The special cases of scattering at low energies, and that in the Born approximation, are studied. We also offer a first glance at the problem of the infrared divergences for the case of Coulomb scattering.

How the diversity of an ecological community affects its stability is an old and important question (Forbes, 1887; Elton, 1927; Nicholson, 1933). The science of ecology grew out of the study of natural history in the nineteenth century, when nature was viewed as wondrous, mysterious, complex, and largely in balance (even if murderous to experience from an individual’s point of view; Forbes, 1887). Whereas our current scientific view is more textured and guarded, the ‘balance of nature’ still permeates the popular press. Some vestiges also remain in the scientific literature. Over the last 100 years, conclusions about the relationship between ecological diversity and stability have varied wildly (May, 2001; Ives, 2005). The goal of this chapter is to show that these wildly varying conclusions are due largely to wildly varying definitions of both stability and diversity. To do this, I will take two tacks, one for stability and the other for diversity. For stability, I will give an abbreviated history of the changing definitions of stability, merging both empirical and theoretical studies. I make no pretence of being comprehensive, but will instead pick highlights that show how the definition of stability often changes from one study to the next. For diversity, I will present a theoretical model to illustrate how different ‘diversity effects’ on stability can be parsed out. This model shows in a concrete way how any theoretical study (and, for that matter, empirical study) necessarily makes a long list of assumptions to derive any conclusion about diversity and stability. The multiple definitions of stability, and the multiple roles of diversity, argue against any general relationship between stability and diversity. In the final section of the chapter, I will argue that understanding the relationship between diversity and stability requires the integration of theory and experiment. Theory is needed to define in unambiguous terms the meanings of stability and diversity. Experiments are needed to ground theory in reality. Unfortunately, rarely is this done. To present an abbreviated history of the changing definitions of stability, I will discuss theoretical and empirical studies side by side.

In all areas of ecology, from studies of individual organisms through populations to communities and ecosystems, there have been huge empirical and theoretical advances over the past several decades. Our guess—a testable hypothesis—is that the worldwide research community of ecologists has grown by roughly an order-of-magnitude since the 1960s, as is of course true for other areas of the life sciences. One consequence is that it is harder to put together a book like the present one. And we find it especially hard when we compare this chapter on community patterns with the corresponding chapter in the second edition of Theoretical Ecology. For the chapters on single populations, for example, there has been growth both in understanding the nonlinear dynamical phenomena that can arise, along with a host of well-designed field and laboratory experiments which illustrate these processes. The narrative, however, retains a unifying central thread, and much of the task of overview and compression lies in choosing good examples from an increasing panoply of choice. For communities, on the other hand, we find so many different yet intersecting areas of growth, many of which have recently produced booklength collections of papers, that the task of choosing which topics to emphasize and which to elide is invidious. The result is necessarily quirky. Without further apology, here is an outline. One broad area of community ecology deals with models for the dynamical behaviour of collections of many interacting species—either within a single trophic level or more generally—essentially as a scale-up of models for single and pairwise- interacting populations. This was the subject of the preceding chapter. Here, we begin by emphasizing the importance of work which views communities from, as it were, a plumber’s perspective, looking at patterns of flow of energy or nutrients or other material. But we then move on quickly to other topics. These include: the network structure of food-webs (connectance, interaction strengths, etc.); what determines species’ richness (niche versus null models); relative abundance of species (observed patterns and suggested causes); succession and disturbance; species–area relations; and scaling laws (with suggested connections among some such laws).