Population Dynamics: New Approaches and Synthesis

By Naomi Cappuccino and Peter W. Price

An understanding of the dynamics of populations is critically important to ecologists, evolutionary biologists, wildlife managers, foresters, and many other biologists. This edited treatise brings together the latest research on how populations fluctuate in size, the factors that drive these changes, and the theories explaining how populations are regulated. The book also includes specific chapters dealing with insects of economic importance.

• Language: English
• Publisher: Elsevier Science
• Release date: Sep 1, 1995
• ISBN: 9780080539256

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PART I

INTRODUCTION

Chapter 1

Novel Approaches to the Study of Population Dynamics

Naomi Cappuccino

I. Population Dynamics: A Brief Historical Review of the Major Concepts

The study of population dynamics is an old discipline with roots that antedate the modern science of ecology. The relative stability of most natural populations, as well as the occasional wild fluctuations of a few, has interested philosophers and natural historians since the beginning of historical time (Edgerton, 1973). In 1798, the Rev. T. R. Malthus launched the modern study of the control of population growth, noting that animal populations increase geometrically up to the limit set by their resources. Although the study of population dynamics has come a long way since Malthus, explaining the stability and persistence of populations remains one of the most difficult challenges confronting twentieth-century ecologists.

Population dynamics is a subject with a history of provoking heated debate among ecologists. Throughout the twentieth century, interest in the debate has ebbed and flowed, the exact focus has shifted, and the terminology has evolved. Early authors were interested in whether biotic factors (Howard and Fiske, 1911) or climatic factors (Uvarov, 1931) controlled populations. Nicholson (1933) was the first to insist on the important point that in order for populations to be regulated (controlled or balanced in his terms), they must be subject to factors that act in a density-dependent manner (he preferred the term density-governing). Density-dependent factors impinge more severely upon the population when its density is high, and provide a mechanism by which the population returns to its equilibrium density following a perturbation. For the past four decades, the debate has mainly centered on the issue of density-dependent regulation, how frequent or strong it is in nature (Milne, 1958; Dempster, 1958; Strong, 1984), and whether it is indeed necessary to explain the persistence of populations (Andrewartha and Birch, 1954; Den Boer, 1968).

Early approaches to the question of regulation were mostly phenomenological: analyses of long-term abundance data (e.g., Morris, 1963) and short-term, more detailed measures of the mortality factors (k-factors) impinging upon populations (Varley and Gradwell, 1960). Although this descriptive approach was quantitative and quite rigorous, the tests designed to detect density dependence from time-series data and from k-factor analysis were often unreliable (Murdoch and Walde, 1989). Although recent advances in time-series analysis have greatly improved this technique as a tool for understanding regulation (Turchin, Chapter 2, this volume), the earlier problems with this approach probably contributed to the feeling among many ecologists that population regulation was an intractable topic for study.

At the same time that ecologists were wearying of the population regulation debate, emphasis in ecology was shifting to areas of inquiry with more appeal and apparent explanatory power. The 1970s saw the rise of community ecology, spurred on by the exciting ideas of Robert MacArthur (Kareiva, 1989). Evolutionary ecology, especially the study of life-history theory, and topics in behavioral ecology such as optimal foraging theory also flourished during this decade. The field of plant–herbivore interactions got off the ground in the early 1970s (Feeny, 1970; Root, 1973), initiating two decades of research that would lay the groundwork for an improved understanding of insect population dynamics. Yet it was not until the mid-1980s that population ecology would regain its place at the center of ecological interest (Stiling, 1994).

Throughout the late 1980s and into the current decade, the study of population dynamics has been steadily gaining momentum. General interest in a field is often stimulated by the consolidation of ideas that have been in the air, and the consequent formation of a new paradigm. In population ecology, a great deal of excitement has been provided by ideas that fall under the rubric nonequilibrium theory. Although over the decades authors have questioned both the equilibrial notion of local density-dependent regulation as well as the usefulness of the very concept of equilibrium (e.g., Andrewartha and Birch, 1954; Den Boer, 1968), the assumption of a balance of nature has nevertheless dominated ecological thinking (Edgerton, 1973). Within the past decade, however, nonequilibrial ideas have come into vogue, especially in North America (Koetsier et al., 1990; Murdoch, 1994).

Ecological concepts often have hazy definitions (Peters, 1991) and the buzzword nonequilibrium dynamics is no exception. Purely nonequilibrial populations are not subject to density-dependent factors that return their densities toward an equilibrium; they fluctuate randomly and eventually reach extreme densities (Nisbet and Gurney, 1982). Of course, populations cannot continue to increase infinitely; all populations have an upper limit imposed by essential resources such as food or space. Eventually, a purely nonequilibrial local population will randomly walk its way to extinction.

While randomly walking populations are unambiguously nonequilibrial, the term nonequilibrium dynamics is most often used to refer to other types of dynamics in which local populations do not trend toward a point equilibrium. This includes notions such as density-vagueness (Strong, 1984) and stochastic boundedness (Chesson, 1978). In these cases, populations experience density dependence only at density extremes (ceilings and floors). However, because the fluctuations of such populations are bounded, these forms of dynamics are consistent with the notion of regulation. In this sense they are misplaced in the category nonequilibrium (Murdoch, 1994).

The other form of nonequilibrium dynamics that has attracted a great deal of attention is the concept of metapopulations dynamics (Gilpin and Hanski, 1991). Building on ideas present since Nicholson (1954) and Birch (1958) and formalized by Den Boer (1968) and Levins (1969), metapopulation theory attempts to explain the persistence of metapopulations (ensembles of subpopulations) as a consequence of spatially heterogeneous local dynamics coupled with dispersal between subpopulations. Individual subpopulations are not assumed to be regulated, and indeed they may go extinct. Because subpopulations fluctuate asynchronously, however, extinctions and outbreaks do not occur at the level of the entire metapopulation. Dispersal allows for the recolonization of vacant habitat islands following local extinction. There is no explicit assumption of local density dependence in metapopulation models, hence the inclusion under the rubric nonequilibrial. However, a metapopulation composed of purely nonequilibrial (i.e., randomly walking) subpopulations will also randomly walk to extinction (Chesson, 1981).

II. Traditional and Novel Approaches to the Study of Population Dynamics

A. Observations: The Search for Patterns in Time Series

1. The Search for Evidence of Regulation

The study of population dynamics has been approached from a variety of angles. As already mentioned, the traditional approach has been observational, based on the analysis of time-series data. Although earlier analyses were often unable to distinguish reliably between random-walk dynamics and regulation in simulated time series, as Turchin points out (Chapter 2, this volume), newer analytical techniques and longer runs of data have improved this situation. Evidence for regulation is now commonly found in time series that are sufficiently long. Abundant evidence for regulation is also revealed by calculating the cumulative variance of natural populations (Murdoch and Walde, 1989). A regulated population, whose fluctuations are bounded, has a cumulative temporal variance that levels off after a certain length of time.

Using the criterion for regulation that a population must be bounded (Murdoch, 1994) or have a stationary probability distribution (Turchin, Chapter 2, this volume), the definition of regulation becomes quite broad and includes complex dynamical behaviors such as cycles and chaos (Murdoch, 1994). The only dynamical patterns that do not fit these definitions are those of populations undergoing random walks or increasing oscillations, or those with a trending equilibrium. Thus the question is no longer is it regulated? but "how is it regulated?"

A good example of two organisms that may both be considered regulated but that exhibit radically different dynamics is provided by the spruce budworm (Choristoneura fumiferana) and the tiger swallowtail butterfly (Papilio glaucus). The spruce budworm is a particularly destructive pest in the boreal forest of North America. Its population fluctuations are bounded, however, since their cumulative variance has been shown to level off with time (Murdoch and Walde, 1989). The upper limit to spruce budworm density is very close to that imposed by the abundance of its food plant, balsam fir. On the lower end, it is barely detectable in the habitat. If we consider the spruce budworm to be regulated, we need to specify how its regulation is different from that of the tiger swallowtail, whose larvae are quite rare relative to their food resources. We must distinguish between the many forms that regulation can take: simple local regulation, metapopulation regulation, complex dynamics. To help field ecologists in this endeavor, several chapters of this book may serve as guides. Harrison and Cappuccino (Chapter 7) address the problem of testing for local regulation via density-perturbation experiments. Walde (Chapter 9) and Roland and Taylor (Chapter 10) provide examples of how one might assess the effect of metapopulation structure on herbivore–natural enemy systems. Dwyer (Chapter 11) discusses the mechanisms resulting in complex dynamics in model systems; the sort of dynamics that Myers and Rothman (Chapter 12) and Rossiter (Chapter 13) seek to understand in populations of forest Lepidoptera.

2. The Comparative Approach

Another use of time-series data involves the comparative approach. This approach often sidesteps the question of regulation, concentrating instead on the magnitude of fluctuations, typically measured as the standard deviation of the log-transformed abundance. Once the temporal variability of a large number of species has been calculated, one may look for characteristics that are associated with variability. Theory often suggests species characteristics that might be correlated with a tendency to fluctuate greatly. For instance, Southwood (1981) has suggested that r-selected species—species that are smaller and have short generation times and high reproductive rates—should be more likely to erupt. The characteristics associated with r-selection have proven to be only weak or occasional predictors of temporal variability (Nothnagle and Schultz, 1987; Gaston and Lawton, 1988a; Morse et al., 1988; Root and Cappuccino, 1992). Contradictory predictions have been made regarding the relative variability of specialists versus generalists. Watt (1964) suggested that generalists, which have less trouble finding food resources in the habitat, would more efficiently take advantage of occasional good growth conditions and outbreak. On the other hand, MacArthur (1955) predicted that specialists would be more variable because they are more susceptible to population crashes brought on by the vagaries of weather acting upon their single food resource. Analyses by Redfearn and Pimm (1988) and Gaston and Lawton (1988b) revealed weak support for Watt’s hypothesis; however, Root and Cappuccino (1992) found no relationship between specialization and variability.

One ecological characteristic, the tendency to aggregate, has been shown to be consistently associated with high temporal variability. Outbreak insects often feed in groups or lay their eggs in masses or otherwise aggregate during some stage of their life cycle (Watt, 1964; Cappuccino, 1987; Hanski, 1987; Nothnagle and Schultz, 1987; Rhoades, 1985; Hunter, 1991; Root and Cappuccino, 1992). Capitalizing on this well-documented pattern, as well as other life-history characteristics consistently associated with high temporal variability, Hunter (Chapter 3, this volume) investigates population dynamics from an evolutionary perspective. In one of the rare studies to address population dynamics with phylogenetic analyses, she assesses the number of independent origins of characters associated with eruptive dynamics.

Although the link between spatial patterns and temporal dynamics has now been well documented, the causal link between clumping and outbreaking remains theoretical (Hanski, 1987). In Chapter 4, Cappuccino, Damman, and Dubuc discuss experimental manipulations of spatial pattern that may allow us to understand the mechanisms driving the increased temporal variability of clumped species.

Although the comparative method is typically used to reveal ecological and life-history characteristics associated with outbreak dynamics, it can also be used to determine whether sources of mortality or the prevalence of density dependence differ among outbreak and nonoutbreak species. This is the approach taken in Chapter 5 by Auerbach, Connor, and Mopper. Though they find neither differences in life-history characteristics nor differences in dominant mortality sources between leaf miners with outbreak or nonoutbreak dynamics, they document, as do several other authors of this volume, the importance of bottom-up effects and horizontal effects, such as interference competition and cannibalism, on leaf-miner dynamics. Competition and premature leaf abscission often impose density-dependent mortality on latent and eruptive miner species alike, and are thus likely candidates in the regulation of miner density.

The comparative method is a powerful approach to understanding population dynamics, especially when it is combined with rigorous phylogenetic analysis or experimentation on mechanisms. To take advantage of this approach, however, we need census data on nonoutbreak species. Though many insect pests are well studied, much of the information on the dynamics, ecological characteristics, and mortality sources of nonoutbreak species is anecdotal at best. We would make more rapid progress toward understanding the outbreaks of pests if every laboratory working on a pest species also had a team working on the comparative ecology of related nonpest species or other species using the same host plant.

B. Mechanisms: The Factors Driving Population Change

1. Three Less Commonly Studied Factors Influencing Population Change

Since the 1970s when field experimentation in ecology became popular, studies of mechanisms, the factors that influence population change, became increasingly prevalent. Perhaps as a result of the seminal paper of Hairston et al. (1960), in which they argued that herbivorous insects are controlled by their enemies, the emphasis in insect population ecology has mostly been on predation and parasitism as mechanisms driving population change. Models of herbivore–natural enemy dynamics focused on the behavioral details of predation and parasitism (Hassell, 1976); thus experimental studies mostly concentrated on mechanistic details such as the functional response of enemies when presented with different densities of prey or hosts (e.g., Waage, 1979).

Other mechanisms have received much less attention from experimentalists, even though their potential importance has been long recognized by theoreticians. Three less commonly studied mechanisms—dispersal, disease, and maternal effects—are given special consideration in this volume.

Dispersal has been seen as a central mechanism in population models (mathematical and conceptual), at least since Nicholson (1954) recognized that local host–parasitoid systems may go extinct, and that vacant habitat can be recolonized by dispersal from other patches. Despite the central role of dispersal in modern population models, this mechanism is one of the most poorly studied (Taylor, 1990). In Chapter 6, Denno and Peterson review studies of dispersal in a variety of insect taxa and argue that, for many species, dispersal is the single most important mechanism influencing population size and stability. Dispersal is also key to understanding population dynamics in fragmented landscapes, of the kind discussed by Hanski and Kuussaari (Chapter 8), Walde (Chapter 9), and Roland and Taylor (Chapter 10).

Despite the appealing theoretical work of Anderson and May (1980), suggesting that pathogens are responsible for the cyclic dynamics of forest lepidopterans, pathogens have traditionally taken a back seat to predators and parasites in both theory and empirical work. As much as insect ecologists tend to shun the study of pathogens, insect pathologists typically ignore population ecology and concentrate instead on the physiology of the insect–pathogen interaction (Ignoffo, 1978). Slowly, however, the number of experimental ecologists addressing pathogen–host dynamics is growing. Roland and Taylor (Chapter 10), Dwyer (Chapter 11), and Myers and Rothman (Chapter 12) describe experiments on pathogen spread and host susceptibility in forest pests with complex dynamics.

With the exception of the earlier studies of Chitty (1960) and Wellington (1960), population ecologists have been concerned primarily with numbers of individuals and have typically glossed over variation in the quality of individuals in a population. However, newer theory has underscored the importance of variability in the quality of individuals within a population (Nisbet et al., 1989). One mechanism responsible for potentially important differences in the quality of individuals within a population is the maternal effect. Although maternal effects have been well known to ecological geneticists, who strive to eliminate such nongenetical effects from their experimental designs, we are only beginning to appreciate their role in population dynamics. In Chapter 13, MaryCarol Rossiter describes how maternal effects can generate time lags and contribute to the complex dynamics of gypsy moths. She then surveys the literature for the importance of maternal effects in other insect taxa and discusses the circumstances under which maternal effects are likely to be important.

2. Linking Mechanisms to Dynamics through Models

The link between mechanisms and the dynamical patterns we see in nature is often made through mathematical models. However, the gulf between theory and empiricism (Kareiva, 1989) has often resulted in only a weak coordination between models and field studies. A prime example of both the importance of models and the confusion they often engender among empiricists is the vast literature on herbivore–parasitoid interactions in which spatial patterns in parasitism are central to the stability of the models (e.g., Hassell, 1985; Chesson and Murdoch, 1986; Pacala et al., 1990). Countless empirical studies, many of which are inconclusive, have been performed to determine the spatial density dependence in parasitism rates (reviewed by Stiling, 1987; Walde and Murdoch, 1988). Most of these studies involve pest species (Kareiva, 1990), and thus are looking for stabilizing mechanisms in the populations least likely to show them. Furthermore, only recently have the inner workings of host–parasitoid models been laid bare in terms that are accessible to empiricists. In a clear review of the theory, Taylor (1993) explains how spatial patterns such as inversely density-dependent parasitism can lead, counterintuitively, to stability in host–parasitoid dynamics.

A superb example of how theory and fieldwork can, and should, be combined to understand population dynamics is provided by the work of Bill Murdoch and his colleagues (Reeve and Murdoch, 1985; Murdoch and Stewart-Oaten, 1989; Murdoch et al., 1987, 1989; Murdoch, 1994). In a series of studies on a particularly stable system, California redscale Aonidiella aurantii and its parasitoid Aphytis melinus, they tested and rejected eight mechanisms that had the potential to stabilize the system, including spatial heterogeneity in attack rates. Most ecologists are not quite so effective at coupling ground-breaking theory with elegant experimentation. Nevertheless, to avoid performing experiments that only weakly pertain to theory, we need to bridge the gap between theoreticians and empiricists. Several chapters in this book, including those by Dwyer (Chapter 11), Hanski and Kuussaari (Chapter 8), and Belovsky and Joern (Chapter 18), represent major steps in bringing mathematical theory and empiricism closer together.

C. The Varied Processes of Population Regulation: Testing Old Theories and Building New Ones

The question of regulation has always been central to the study of population dynamics. Regulation is the process that translates mechanisms such as predation or dispersal into the long-term dynamical patterns of temporal constancy or persistence that are so often observed in nature. Although most studies testing for regulation have involved analysis of observational data, the most effective way to test for local regulation is to perform convergence experiments (Murdoch, 1970) in which density is manipulated to see if populations return to their previous levels. Perturbed populations of herbivorous ladybirds (Ohgushi and Sawada, 1985; Ohgushi, Chapter 15, this volume), tephritid flies (Cappuccino, 1992), and tussock moths (Harrison, 1994) have all shown tendencies to return to preperturbation densities. In Chapter 7, Harrison and Cappuccino review the literature on perturbation experiments and discuss ways in which the experimental approach to regulation can be improved.

Populations may also be stabilized as a result of metapopulation dynamics (Gilpin and Hanski, 1991). In Chapter 8, Hanski and Kuussaari explain metapopulation theory and provide evidence that many butterfly populations, while locally unstable, are indeed regulated at the level of the metapopulation. Most evidence for metapopulation regulation comes from the observations that are consistent with this theory: local populations fluctuate asynchronously, sometimes go extinct, and are later recolonized by dispersers from other subpopulations. An ideal way to test metapopulation theory would be either to manipulate dispersal, perhaps by putting up large barriers to movement (Reeve, 1990), or to alter the degree of fragmentation of the habitat. The large scale at which such experiments would have to be done makes them all but impossible in most systems, but may be possible for very small organisms such as phytophagous mites. In studying the predator–prey metapopulation dynamics of mites in apple orchards, Walde (Chapter 9) has performed one of the first field manipulations of metapopulation structure. Roland and Taylor (Chapter 10) take advantage of experiments performed by lumber companies in northern Alberta. They show how the longer duration of forest tent caterpillar outbreaks in more highly fragmented forest can be explained by both lower disease incidence and lack of parasitoid dispersal to habitat isolates.

D. Case Studies: Building and Testing Theory in Well-Studied Systems

The best ecology is often accomplished through a mix of observation, mechanistic studies, and process-oriented studies (Feinsinger and Tiebout, 1991). With the development of long-term, spatially explicit data sets, several researchers are in the position to address population stability using a combination of observation and experimentation. This book also features work on a variety of well-studied systems, including seed-feeders (Solbreck, Chapter 14), herbivorous coccinellids (Ohgushi, Chapter 15), tenthredinid sawflies (Price, Craig, and Roininen, Chapter 16), bark beetles (Reeve, Ayres, and Lorio, Chapter 17), and rangeland grasshoppers (Belovsky and Joern, Chapter 18).

These contributions tie together several of the themes addressed in earlier chapters. In Chapter 15, Ohgushi emphasizes the importance of oviposition physiology and behavior in the remarkably stable populations of the herbivorous lady beetle. His chapter, as well as those by Solbreck and Price et al., underscores the prevalence of bottom-up and lateral or horizontal factors, also detected in the reviews by Auerbach et al. (Chapter 5) and Harrison and Cappuccino (Chapter 7). The herbivores described in these three chapters all track closely the carrying capacity set by the host plant, which may, however, be quite variable in space and time, as shown by Solbreck in Chapter 14.

The southern pine beetle populations examined by Reeve et al. in Chapter 17 exhibit the sort of complex dynamics described by both Turchin (Chapter 2) and Dwyer (Chapter 11). The delayed numerical response of a predatory clerid beetle with a long generation time is probably responsible for the cyclic dynamics. Rangeland grasshoppers can also reach outbreak densities, but do not do so in all locations, as described by Belovsky and Joern (Chapter 18). Their models, tested with a series of large-scale density manipulations, suggest that grasshopper densities may be attracted to two domains: one at high densities set by the interaction with the host plant, the other at lower densities imposed by enemies.

III. The Role of Herbivorous Insects in the Study of Population Dynamics

All of the chapter authors primarily study herbivorous insects. Herbivorous insects have been central to the question of population regulation and stability from the early debates on density dependence (Nicholson, 1958; Andrewartha and Birch, 1954; Birch, 1958) up through the modern discussions of the prevalence of different sorts of regulation in nature. Despite the central place that insects and insect ecologists have occupied in the development and testing of population dynamical theory, the approaches we take apply quite well to other types of organisms. Many of us have indeed addressed the question of regulation in other taxa, both in this book (Chapter 10) and in other publications (Myers and Krebs, 1971; Hanski et al., 1991; Turchin, 1993).

IV. Population Dynamics: The New (Pluralist) Synthesis

One of the most refreshing aspects of the renaissance of population dynamics is the absence of polarizing debate. Early participants in the debates about density dependence and density independence were often arguing at cross-purposes (Sinclair, 1989), since density-dependent and density-independent factors are not mutually exclusive. Neither does the presence of agents capable of regulating local populations preclude the existence of metapopulation processes in the same or other species. The new synthesis in population dynamics is thus pluralist in nature, reflecting the diversity of the natural world and the range of ways in which populations can be regulated. A wide variety of subdisciplines in ecology have greatly contributed to our current understanding of population dynamics; in the concluding chapter of this volume, Price and Hunter discuss how this, too, contributes to the new pluralist synthesis.

Theory in population ecology has never been scarce. The time has come for empiricists to devise clever tests of these theories, to help refine theory by opening channels of communication with theorists, and to participate in the building of empirically based theory. This is the exciting challenge of population ecology as we enter the next century. We hope that the chapters presented in this volume will provide a guide to ways in which population dynamical questions may be addressed, as well as a stimulus to the creativity of the next generation of population ecologists.

Acknowledgments

I thank Peter Price, Mike Singer, and L. Ramakrishnan for helpful comments on this manuscript.

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PART II

OBSERVATION AND COMPARATIVE APPROACHES

Chapter 2

Population Regulation: Old Arguments and a New Synthesis

Peter Turchin

I. Introduction: The Never-Ending Debate

Population regulation is one of the central organizing themes in ecology (Dennis and Taper, 1994; Murdoch, 1994). Yet, ever since the concept of population regulation by density-dependent mechanisms was formulated by Nicholson (1933), regulation has been the subject of an acrimonious debate (or actually a number of debates about its various aspects and implications), which continues to this very day.

In 1949, Charles Elton wrote: It is becoming increasingly understood by population ecologists that the control of populations, i.e., ultimate upper and lower limits set to increase, is brought about by density-dependent factors (p. 19). Subsequent history showed that this statement was somewhat premature. In a very influential book, Andrewartha and Birch argued that density-dependent factors are not a general theory because, as we have seen … they do not describe any substantial body of empirical facts (Andrewartha and Birch, 1954, p. 649). Andrewartha and Birch then proposed an alternative theory of population limitation by density-independent factors. The arguments of Andrewartha and Birch set the stage for the ensuing controversies, which could be grouped around two general themes, one logical and the other empirical: (1) What does population regulation mean, and are density-dependent factors a necessary condition for regulation? (2) Can we detect density-dependent regulation in real populations, and if yes, with what frequency does it operate?

Here are some highlights from the debate, with quotes showing how vitriolic the exchange occasionally became. Milne (1957) criticized Nicholson’s theory as mistaken, because (1) it is based on a false assumption, namely, that enemy action is perfectly density-dependent … and (2) it asserts that this density-dependent action is responsible for natural control of decrease as well as increase, which is ridiculous. Milne advanced his own theory of imperfect density dependence, which is largely forgotten now (Berryman, 1992). Similar concepts are density vagueness of Strong (1986) and regulation by ceilings and floors of Dempster (1983), according to which a population can fluctuate in a largely density-independent manner for most of the time, until it approaches either a lower or an upper extreme. Milne, Dempster, and Strong felt that they were departing greatly from the density-dependence school of thought, but they were basically proposing that density dependence involved nonlinearities, and that there could be a great deal of noise—hardly controversial, in retrospect. In a more significant departure from Nicholson’s theory, Andrewartha and Birch felt that density dependence was not at all necessary to prevent outbreaks of organisms. This idea was later developed by Den Boer (1968), who suggested that density-independent fluctuations in natural populations can become stabilized by stochastic processes (i.e., by chance) via a mechanism that he called spreading of risk. These proposals have now been rejected on logical grounds, and it is generally accepted that population regulation cannot occur in the absence of density dependence (Murdoch and Walde, 1989; Hanski, 1990; Godfray and Hassell, 1992). But what is regulation? Wolda (1989) wrote a paper with a rather plaintive title, The equilibrium concept and density dependence tests: What does it all mean?, where he concluded that the concept of an equilibrium was fundamentally impractical and unusable in the analysis of field data. Berryman (1991) countered with a paper entitled Stabilization or regulation: What it all means!, arguing that, on the contrary, equilibrium has a well-defined meaning and can be estimated by an appropriate analysis of