STABILITY AND PERSISTANCE OF POPULATIONS AND ASSEMBLAGES:

THEORY AND LABORATORY AND FIELD STUDIES

A Departmental Thesis Presented to the Faculty

of

California State University, Hayward

In Partial Fulfillment

of the Requirements for the Degree

Master of Science in Biological Science

By

Burke A. Strobel

.Ttme, 1996 Aknowledgments

I would like to thank the Moss Landing Marine Laboratories' librarians Sheila

Baldridge and Joan Parker, whose efforts made many important papers available to me. The use ofJay Hager's and Brett Strobel's computers for text and graphics respectively was invaluable to the project. Discussions with Christopher Moyer solidified many of the ideas on disturbances presented. I also thank my committee members for their patience and insights, and, most of all, my wife Katie for her support, understanding, and even courage.

111 Table of Contents

Acknowledgments iii

Table of Contents IV

List of Figures vi

I. Introduction 1

IT. Stability Theory: Populations 13

Introduction 13

Carrying capacity 17

Mechanisms 25

Density dependence 29

Detection of density dependence 32

Assumptions 35

Ill. Stability Theory: Paired Interactions 37

Interspecific competition 37

Amensalism 46

IV v

Predation 47

Mutualism 66

N. Stability ofMultispecies Interactions 70

Indirect effects 70

V. Stability of Assemblages 76

Considerations 76

Diversity 78

Alternative stable states 81

VI. Conclusion 84

Bibliography 89 List of Figures

Figure 1. Gompertz curve of population growth 18

Figure 2. Logistic curve of population growth 19

Figure 3. Holling type I predator functional response 49

Figure 4. Holling type I1 predator functional response 50

Figure 5. Holling type ill predator functional response 51

Figure 6. Lotka-Volterra predator/prey trajectory 53

Figure 7. Predator/prey trajectory

(prey: self-limiting; predators: type I1 functional response; d=O.Ol) 57

Figure 8. Predator/prey trajectory

(prey: self-limiting; predators: type I1 functional response; d=0.03) 58

Figure 9. Predator/prey trajectory

(prey: self-limiting; predators; type I1 functional response, self limiting) 62

vi As many populations persist for long periods, yet remain finite, there can be little doubt that they experience some fonn of density-dependent limitation (Pollard et a!.

1987).

I. Introduction

Stability of populations and assemblages is a widely assumed trait given the continued presence offamiliar species and associations of species. Holling (1973) identifies two ecological views of natural systems as qualitative and quantitative. The qualitative view concerns itself with presence or absence of populations within an arbitrarily defined system. The continued presence of a species' population within a system is termed persistence. The quantitative view concerns itself with the degree of constancy of numbers of individuals in the species' populations in the system. This concern focuses on two questions: are the observed numbers of individuals constant or fluctuating, and why are they that way, or, stated another way, how do the numbers of individuals respond to forces that could potentially alter them? The first question is a straightforward matter of unbiased observation. Numerical constancy, however, does not, in itself, provide an answer to the second question. The absence of fluctuation can be due to the lack of perturbing forces or the presence of mechanisms that compensate for them. The change in numbers with time is dictated by the strength offorces potentially altering them and the population response to these forces, whether through

1 2 resisting them or recovering from them. Ecologists are divided over which of these predominates in determining population dynamics ( eg. Andrewartha and Birch 1954,

Nicholson 1933). It is the population's response to disturbances, when they occur, that defines its stability. References to stability infer the existence of a size that a given population would have in the total absence of disturbance. This is the equilibrium, where the population's per capita rate of increase equals its per capita rate of decrease.

The questions whether or not and under what circumstances populations and assemblages are or are not stable or persistant are central to the science of ecology, which attempts to account for organisms' distributions and abundances. As the human species continually occupies new resources and exploits populations of organisms, it becomes increasingly important to consider how other species in an impacted system will respond. They may remain unaltered, increase in number or biomass, decrease, or go extinct. Their numbers could become more or less variable, all to humans' eventual benefit or detriment. Despite the importance of understanding populations' and assemblages' persistance and stability, researchers' conclusions, indeed the entire issue remain vague. Lack of aggreement on definitions for specific terms, nebulous concepts, and preconcieved notions all seem to have added to this lack of clarity regarding ecological persistance and stability.

Most of the contributions to stability theory have been by theoreticians. To a lesser extent the predictions made by theory have been investigated in laboratory and 3 field studies, although such predictions have often, without being first demonstrated, formed the basis of investigations of field data (see 'detection of density dependence' below). To some extent field and laboratory studies have in turn guided the development of theory. It is the objective of this review to explore what ecology understands about the dynamics of persistance and stability in populations and assemblages. It will attempt to draw together and give cohesion to the scattered body of theory with the ultimate goal of evaluating the usefulness of the concept of population and assemblage stability.

The existence of forces capable of disturbing populations, in the form of environmental variation and catastrophes, is obvious and it is possibly as a result of this observation that the stability qualities of populations and assemblages have traditionally been invoked to explain persistence of populations. Egerton (1973) traces the related concept of the "balance of nature" back to Greek philosophy. The existence offossils of unknown organisms in the geological record first introduced the possibility of extinction to human thought in the 19th century (Egerton 1973). The assumption of stability continued into the 20th century and became central to plant ecology, where it was incorporated into Clement's view of the plant assemblage as a superorganism

(Mcintosh 1982). One of the distinguishing traits of such a superorganism was homeostasis, or self regulation that maintains an equilibrium of its parts and processes. 4

Although H. A Gleason advanced an opposing view of populations undergoing separate dynamics and more or less coincidentally co-occuring, his "individualistic concept" was not fully appreciated until the 1950's by such researchers as Whittaker

(1956), who examined plant species' distributions along environmental gradients. The assumption of homeostasis in "mature" assemblages of organisms continued ( eg.

Margalef 1963, 1974) and many researchers assume that ecological systems are best described by the stability principles of engineers (eg. Berryman 1991).

The mathematical formulations ofLotka (1925) and Volterra (1926) stimulated interest in describing situations when interacting populations might be expected to be stable or proceed to extinction. Similar modeling of laboratory insect populations by

Nicholson (1933), in conjunction with Lotka-Volterra theory, gave rise to the field of theoretical ecology. Much of theoretical ecology examines models of populations assumed to react in specific ways to their environment. Stability of these models is determined by looking for equilibria. Some researchers have opposed the extrapolation of findings in this body of theory to real systems. They suggest that environmental variation renders the entire equilibrium concept next to useless. Such criticisms, appearing as early as the 1930's, were acknowledged by Nicholson (1933) and countered by the concept of an equilibrium being the point approached but often never attained by a population as it varies under different environmental conditions. Similar arguments are echoed in more recent debates (Dennis and Taper 1994, Berryman 5

1991). Clearly the continuing controversy revolves around the issue of the predominance of the mean equilibrium, given environmental "noise", or the variance about the mean in determining population and assemblage dynamics. The opinions are likely to be as diverse as are the positions along the continuum between the two extremes occupied by real populations and assemblages.

A myriad of ecological terms have been used in reference to stability.

Populations and assemblages have been shown to respond to disturbances in various ways that have been claimed to indicate stability. The most fundamental distinction is between persistence and stability as described above. These two population responses have been confused in the past (Connell and Sousa 1983). The tendency for certain populations to remain relatively constant through time as seen in time series data has been referred to as stability (eg. Matthews eta!. 1988). This has also variously been referred to as constancy (Orians 1974, Whittaker 1974), conservatism (mentioned in

Connell and Sousa 1983), endurance (Margalef 1969), and stabilization (Reddingius and Den Boer 1987). For the sake of clarity I will use the term constancy. Consistent census numbers does not equal stability because disturbing forces may by lacking

(Connell and Sousa 1983, above). It should be kept in mind, however, that some potential disturbances always exist, if only the change from day to night and there are forces that no assemblage could resist, such as lava flows. What is important is that the 6 numerical or other response of the population to potentially disturbing forces it is likely to face at some place and time in its habitat be quantified. The ability of a population to remain at or within its equilibrium distribution when faced with a potentially disturbing force has been called inertia (Orians 1974, Murdoch 1970), persistence

(Margalef 1969), and resistance (Connell and Sousa 1983). If a population completely resists a potential external disturbance, it cannot be said to be disturbed. The speed of return to the equilibrium or the equilibrium distribution has been referred to as elasticity

(Orians 1974, Connell and Sousa 1983), resilience (Holling 1973), and has been discussed for populations under the term 'return time' (May et al. 1974). The distance from equilibrium that a population or assemblage can be disturbed and still return has been called amplitude (Orians 1974, Connell and Sousa 1983). Holling's (1973) use of the term resilience also includes amplitude. In this paper I will explain terms as I use them.

Mention of disturbance is unavoidable when discussing stability. Sousa (1984) contrasts the traditional definition of disturbances as "uncommon, irregular events that cause abrupt structural changes in natural communities and move them away from static, near equilibrium conditions" with a new definition, where a disturbance is "a discrete, punctuated killing, displacement, or damaging of one or more individuals (or colonies) that directly or indirectly creates an opportunity for new individuals (or 7 colonies) to become established." The latter definition above is, of course, an assemblage-oriented view and would be of little use in the consideration of single populations. Reference to the puctuated nature of disturbance is key to definitions of disturbance. This separates disturbances from environmental trends. Sousa discusses the "regime" of disturbance under the discriptors areal extent, or size of disturbed area, magnitude, frequency, and predictability.

Perhaps the greatest difficulty in defining disturbance comes from its perception as a movement in some population or assemblage characteristic from equilibrium. The ecological meaning of equilibrium is difficult to conceive (see above) due to the many kinds of variability (Strong 1986). Even in the undoubtedly simplistic view of equilibrium as a distribution of values (Dennis and Taper 1994, see below), it is impossible to say if an impinging factor has moved the quantity of the interesting variable in the direction it was already moving within that distribution or away from it.

Another difficulty with defining disturbance is its relativity to the population(s) under study. The result is the nebulose division between Sousa's (1984) two components of disturbance magnitude, intensity and severity, as well as the inclusion of turnover rate or rotation period, clearly an assemblage characteristic, as a discriptor of a disturbance's regime. Intensity is the strength of the disturbing force. Severity is the effect on the assemblage. The former term says little about the latter and seems to be of most ecological use when comparing two populations' responses to the same force 8

I would suggest that disturbance be defined as a punctuated, proportional alteration in some population or assemblage characteristic of interest within the spatially defined unit under consideration that is significant, ie. outside some level of probability of occurance, given present environmental conditions and omitting the discrete event. A force causes rather than is the disturbance. A definition of disturbance must allow disturbances in different populations or assemblages to be comparable if it is to be of more than qualitative use. For instance, a disturbing force that kills a certain number of individuals might be more detrimental to a smaller population than a larger population. If the disturbance is defined as being 'the death of one third of the population', however, then questions such as 'how will the same disturbance affect a large and a small population differently?' can be asked. Under this definition certain experimental treatments would not be considered disturbances, eg. the addition of grazers or the maintaining of increased nutrient levels (McNaughton

1977). These would be alterations in the environment of the study organisms. Despite this, the above definition seems most compatible with stability theory as discussed in this paper.

This definition raises the questions of frequency, severity as used by Sousa

(1984), and scale. With increased frequency, severity would be expected to be reduced as organisms that remained would be those best adapted to withstand that type of disturbance. One single moderately sized wave, frequent on an exposed rocky shore, 9

might wreak havoc on a protected Salicomia salt marsh but hardly affects aMytilus bed. With increasing frequency of waves, a salt marsh would give way to other, more

exposed shore assemblages, provided there were a supply of propagules, and so

severity would decrease. Severity would be expected to decrease, generally speaking, with an increase in scale. A landslide might be disastrous to a hillside plant assemblage, but, when a range of hills is considered, landslides might be frequent enough to actually be maintaining the observed species diversity, and it surly would do little to cause a

large alteration in the range's overall diversity. A further illustration of this point is

Sousa's (1984) inclusion of predators as agents of biological disturbance. On a very

local scale mobile predators may be fairly infrequent and cause large changes in

populations when they appear (Roughgarden 1986). Sousa (1984), however, discusses

predation in terms of rates, which are larger scale phenomena. A finite rate of change

in population size is the ratio of one time period's population size to the size of the

previous time period's population. It provides a percentage of survivors or a

percentage of mortalities. Predation rates on large scales, in fact, are a major

component of prey mortality rates, that help determine equilibria. The distinction of

"small scale" from "large scale" is arbitrary and in this case occurs at the smallest spatial

extent within which prey mortality due to predation would be expected to be identical

to that found in adjacent prey patches. One individual does not have a mortality rate.

The scale at which predation can be said to occur at a more or less consistent rate is 10 considered here a "large scale". The terms disturbance and perturbation are synonymous but I will use the term disturbance.

Connell and Sousa (1983) point out that, for reasons discussed above, the spatial scale of the stability analysis must be defined. Because scale affects other processes such as the influence of migration relative to birth and death rates (see below) and the continuity between local populations and assemblages (Whittaker 1956,

Underwood 1986) this paper assumes a spatially defined approach to the issue of stability, where study boundaries are precisely defined, regardless of the degree of arbitrariness. Even after the spatial scale of a study has been established, consideration oflarger scales such as the region surrounding the spatially defined study unit is often vital to understanding the processes observed (Ricklefs 1987). Extrapolation to other assemblages occupying similar -sized areas requires an understanding of regional effects. The extreme of this regional effect is dealt with under the heading of

"supply-side ecology" (eg. Roughgarden 1986) where population and assemblage dynamics are largely determined by recruitment into the study system.

The temporal scale of the study is also important. Assemblages can be observed to show constancy in the sense ofOrians (1974) simply because the individuals in it are long-lived relative to the time span of the observation (Frank 1968).

Connell and Sousa (1983) suggest that observations span a time equal to or greater 11 than the longest-lived individual in the assemblage. An alternative is to make observations for a time span equal to ten or more generations of the species or some species in the assemblage that has a large impact on the assemblage ( eg. Paine et a!.

1985). Connell and Sousa (1983) also suggest that observing recruits replacing adults for each adult or estimating the number ofjuveniles and establishing their separate probabilities of survival to adulthood. This has the problem of survival probabilities changing over time. A potentially shorter-term experimental technique suggested by

Underwood (1986) and used by others ( eg. Stimson and Black 1975, Farrell 1988) is to disturb populations or assemblages and observe their behavior relative to undisturbed control plots. If treatments are levels of disturbance, then convergence between them is evidence for recovery or elasticity sensu Connell and Sousa (1983). Constancy can only be demonstrated by long-term observations or their equivalent.

The following overview of ecological stability theory is structured around levels of organization. The fundamental level of organization considered here is the population. A population is defined as a group of individuals of one species living in an area (Begon eta!. 1990). Although virtually no species occur naturally alone or in pairs, it is my opinion upon reviewing the literature that an incremental consideration of levels of organization is important to understanding naturally occurring, complex 12 interactions. Even on the level of the assemblage certain strong interactions between two or a few species can sometimes determine the dynamics of the whole (Paine 1980).

The final level of organization considered is that of the assemblage as a whole. In the literature, "communities" have been assumed to possess qualities that could not be predicted by summing their parts. Communities have been supposed to consist of tightly coevolved organisms that form a system characterized by homeostasis and having distinct boundaries (see Underwood 1986). Underwood (1986) reviews some of the key difficulties with the community concept. Communities are supposed to consist of all species in an area. The actual choice of study species is often arbitrary. It is my intent to remain impartial by referring to collectives of populations in a unit area, whether they actively interact or not, simply as assemblages. The degree to which individuals remain in the unit area under study is, in part, a function of the spatial scale used. Oksanen et al. (1981) briefly discuss the occasional occurrence of nomadic species in a unit area in terms of their model of trophic interactions. As spatial units under study are often too small to support a resident population of certain species, which nonetheless may have a large effect on the dynamics of other resident organisms, the consideration of different spatial scales can be important. The alternative is to describe the dynamics of interacting populations within a unit area at a discrete time

(eg. population sizes quantified once per year at the same time of year) and represent nomadic species as their average presence. Depending on the interval between population measures, this could potentially lead to population sizes less than one. II. Stability Theory: Populations

Introduction

Before stability in population size is discussed it is useful to provide a mathematical and conceptual foundation for the discussion. Den Boer and Reddingius

(1989) distinguish between populations that are "stabilized" (remaining within relatively narrow upper and lower limits) and populations that are "regulated" (governed by density-dependent factors leading to stabilization). The difference between stabilization and regulation is the latter causes the former, but the former need not be caused by the latter. Boundedness is a term synonymous with stabilization as used here, differing only in that the narrowness of the bounds relative to the average population size is generally deemphasized (den Boer 1991, Murdoch 1994). Constancy (Orians 1974, see above) differs only in that it does not incorporate a range of natural variation.

Murdoch (1994) defines regulation as being the result of density-dependent feedback processes that can lead to stabilization, implying that density-dependent feedback can occur without boundedness being the result. The fundamental equation for a population without overlap of generations is

equation (1) where Xi is the natural logarithm of population size (N) at time i and ei is an independent random variable. A population described by equation 1, a random walk

13 14 population, will fluctuate randomly over time with increasing variance until it inevitably goes extinct, ie. it is unbounded. Den Boer (1991) points out, however, that simulated random walk populations can persist for several centuries. The addition of a drift parameter, r, to equation 1, corresponding to the intrinsic, instantaneous per head rate of change ofthe population or any other drift-e.g. caused by a change in resource abundance, Pollard et al. (1987)

equation (2) introduces a potential buffer against extinction. When r is positive the population more than replaces itself in the absence of random fluctuations. The term r is a log quantity.

The addition of a further parameter, b, multiplied either by Xi or Ni, into the equation describes a population with a drift and density dependence.

Xi+ r=xi+bxi+r+ei

equation (3a) or

Xi+r=xi+bNi+r+ei.

equation (3b)

Equation 3a is a discrete time, stochastic Gompertz model and equation 3b is a discrete time, stochastic logistic model (Dennis and Taper 1994, see figures 1 and 2). The use of discrete time (difference) equations such as those above for modeling populations 15 with complete overlap in generations is considered inappropriate (May 1981) because processes that determine population growth happen continuously whereas discrete equations introduce a time lag equal to the interval between censuses, yet such use of these equations occurs frequently in the literature (eg.Vickery and Nudds 1984,

Turchin and Taylor 1992, Dennis and Taper 1994). There seems to be little recourse, however, as field populations are generally quantified only at certain discrete times.

The continuous (differential) Gompertz equation,

dN/dt=-bNin(N/K),

equation ( 4a) and the continuous logistic equation,

dN/dt=rN(1-N/K),

equation (4b) where K is the equilibrium population size (N*) attained by each as described below, behave in an analogous manner to equations 3a and 3b without stochastic elements when a time lag is added representing generation time, or the time between i and i+ 1

(May 1974). Time lags are discussed further below. The dynamics of difference equations are categorized by May (1974). Their analysis consist of generalizing the model to

equation (5) 16 where j{N;) is the net growth rate of the population expressed as a function ofN. The slope of the density dependence at the equilibrium point (bin Hassel et al. 1975) characterizes the dynamics of the population described by the given equation with the given parameters.

b=-[N(dfldN)]*

equation (6)

The symbol * denotes the equilibrium value of the term it follows. The mathematical population is locally stable when O2, diverging oscillations. When the return time (1/r) is longer than the delay in density dependence (time lag) then the return to the equilibrium is smooth.

When the reverse is true, the return, if it occurs, is oscillatory (May 1974).

An environmental factor "whose action affects survival and reproduction" has been defined as being density dependent if its intensity is correlated with population density (Murray 1982). Density dependence, then, would be a correlation of a population's rate of change with its density. This definition allows for positive (inverse) and negative density dependence. The definition cited by Pollard et al. (1987-from

Huffaker and Messenger) of density dependent action as "... the actions of repressive environmental factors, collectively or singly, which intensity as the population density 17 increases and relax as this density falls ... " is limited to negative density dependence.

For this reason the former definition seems appropriate.

Carrying capacity

Equations 3a and 3b describe theoretical populations whose densities in the absence of ei return smoothly to an equilibrium population size or density (x*= r/-b and x*=ln(r/-b) respectively). Theoretical equilibrium is signified asK. Inclusion of ei results in a replacing of the above point equilibriums with probability distributions of long-term population sizes, among which r/-b or ln(r/-b) are merely "return points"

(Dennis and Taper 1994). When a single population is considered, with all environmental factors constant (ie. mortality included in the term 'r' and food resources replacing themselves at a rate equal to consumption) and 'r' positive, this equilibrium is generally considered to be at carrying capacity (Odum 1953 in Dhondt 1988).

The varied use of the term carrying capacity is followed through history by

Dhondt (1988). The carrying capacity of an arbitrarily defined environmental unit is the maximum equilibrium population or biomass of a species that unit could indefinitely support, given constant environmental conditions. In his discussion of the equilibrium concept Wolda (1989) suggests that the environmental conditions so considered be the total of those over the course of one year being repeated year after year until the population stabilizes. Narrower time frames would not include the full range variation 18

100 -,------~ 80 ------60 ------z 40 ------20 ------

0 ~-+--+--+--+-~--~~~~--~~--r-~ 0 100 200 300 400 500 600 Time

Figure 1. Gompertz curve of population growth. Drawn using difference equation 3

(r=.05, b=-0.01) 19

500 ,------~~------, 400 ' 300 2 200 100

0 +-~~-r--~--r-~---r--+-~--~~ 0 100 200 300 400 500 Time

Figure 2. Logistic curve of population growth. Drawn using difference equation 4

(r=.05, b=-.0001). 20 in environmental conditions necessary for the persistence of most any population.

Edwards and Fowle (1955 in Dhondt 1988) "regard carrying capacity as represented by the maximum number of animals of given species and quality that can in a given ecosystem survive through the least favorable environmental conditions occurring within a stated time interval ... usually one year." This definition considers the carrying capacity of an ecosystem, the bounds of which would generally be difficult to define.

As spatial is at least as important as temporal delineation, it seems more logical to consider some spatially defined unit. Considering periods longer than one year would be "counterproductive"(Wolda 1989). An alternative interval might be generation time.

Using generation time would allow for a consideration of historical events which might affect a population's recruitment or mortality response to a given year's environmental conditions. A population, however, would respond to the environmental conditions of the year at hand, granted in a way determined by the individuals' preexisting quality. A possible illustration of this temporal aspect of carrying capacity is a study of game animal populations in a fenced reserve over 25 years (Ben-Shahar 1993). A decline in numbers of zebra and wildebeest was attributed to animals not being allowed to undergo annual migrations to other habitats to fulfill their full nutritional requirements.

Populations eventually stabilized at a level well below that of surrounding areas.

Steward and Bjornn (1989) found that, in the summer, the salmonid carrying capacity of pools was determined by the supply offood and in winter by the amount of cover. 21

Carrying capacity of animals is generally considered to be set by food supply (Keller

1983, Crete 1987), and will be discussed in this respect unless otherwise qualified, but can be set by any limiting resource, eg. space (Agami and Reddy 1990), nest boxes

(Kluger 1951 in den Boer 1991), refuges etc. Though Caughley (1979 in Dhondt

1988) equated carrying capacity (as a rate of consumption) with the rate of production,

Giles (1978) stated that an area must supply "... all of the energetic and physiological requirements ... " of a population. The value of the carrying capacity is widely assumed to affect population growth, even if carrying capacity is never attained.

Carrying capacity can be estimated by determining the resource utilization requirements per individual and dividing it into the availability of the resource within the defined environmental unit (Crete 1989, Salvanes et al. 1992), or by measuring the population or density after enough time has elapsed for both population and resource to stabilize (Crete 1987 in Crete 1989), provided it is known that all other environmental factors are constant and inter. :eci :·~ competition is lacking or insignificant. It can be determined by manipulating density and the supply of the known limiting resource to look for convergence in population sizes at levels corresponding to availability of the resource (Eisenberg 1966). Estimates of carrying capacity are more common for lower than for higher trophic levels (Salvanes et al. 1992).

Edwards and Fowle defined the units of carrying capacity as numbers of individuals (1955). Giles defined carrying capacity in terms of biomass (1978). I think 22 that the units of carrying capacity should be defined in terms of the limiting resource.

When refuges are critical and limiting, a population might be limited in terms of numbers if refuges are defended, or possibly volume if they are not. If prey are limiting in biomass, then their production can support a limited predator biomass (Grant 1986 in Levitan 1989). Since a maintained population is at issue, an equilibrium consumer population requires a constant rate of supply of prey, not a particular standing biomass.

Populations are often modeled in terms of numbers of individuals, however age classes within a population differ in biomass and resource requirements (Gutierrez 1992).

Some populations, eg. sea urchins (Levitan 1989), can, in fact, regulate somatic size and therefore biomass, without changing numerically. If space is limiting, then carrying capacity should represent spatial units and individuals should be spatially defined, not in terms of biomass as in Agami and Reddy (1992). Once space has been fully utilized, plants can often continue to increase in biomass by growing upwards (Agami and

Reddy 1992). A given amount of space equates to a certain biomass of plants only if it can be shown that no other resource ( eg. nutrients) can influence the final biomass once all space has been occupied. As space is not consumed during population maintenance, spatial carrying capacity does not correspond to a rate. Defining K in terms other than that of the limiting resource separates the concept and the model it is included in from any biological meaning. 23

The logistic equation:

dN/dt=rN(1-N/K.)

equation ( 4b) where N is the number of individuals, t is time, a is the instantaneous per capita rate of change, and K is the equilibrium population or carrying capacity sensu Odum (1953), was designed to describe a population's approach to a constant equilibrium so is analogous to equation 3b without stochastic elements. It is a ratio dependent equation, meaning the approach to equilibrium (dN/dt-70) is set by a ratio of consumer demand to resource supply. Logistic self-limitation is often used to describe the response of single populations and prey populations to carrying capacity (Berryman 1992). In predator/prey equations the Latka- Volterra prey dependent response cNP or some prey-dependent functional response cf(N)P, where c is a coefficient converting N to an instantaneous per capita rate of production of predators (P), are most often used to describe predator responses to prey numbers. Per capita predator reproduction is thus considered a linear function of prey acquisition, which, in tum, is a function solely of prey quantity. There is no maximum instantaneous per capita rate of increase.

Additionally only one prey density will lead to an equilibrium population of predators and that population can be of any size (Berryman 1992, Rosenzweig and MacArthur

1963). Such equations do not infer a carrying capacity that dictates an equilibrium proportional to resource supply. The logic behind the equilibrium population of 24 consumers being proportional to resource supply is that, through respiration, individuals consume resources as a rate. This, in turn, must be maintained by a rate of resource production as is implied in ratio dependence (see also Gutierrez, 1992, for a conceptually equivalent, though biologically more explicit model). Ratio dependence will be further discussed under predation. Latka-Volterra predator/prey models diverged fundamentally from logistic single population models in that they assume that predators do not have to "share" their resource in instantaneous time, whereas prey populations are widely modeled as being logistically self-limiting. A large body of ecological literature on stability of populations integrated in assemblages is based on the assumption of prey-dependence at higher trophic levels and ratio dependence at the lowest trophic level (Berryman, 1992).

What is the supposed ecological meaning of the density dependent term b? This is most easily answered by rewriting equations 3a and 3b without the ei term to give the linear functions

ln(Ni+ dN)=-bxi-r

equation (7a) and

ln(Ni+dNi)=-bNi-r

equation (7b) 25

The term ln(N;+1/N;) represents mortality, which increases linearly (slope=-b) with population size or the logarithm of population size, depending on the equation used, beyond K (x-intercept). Below K mortality becomes negative, corresponding to reproduction or immigration. The y-intercept, then, is the maximum instantaneous per capita rate of increase. Nonlinear theory inserts an exponent Q over x; or N; in the density dependent term to describe curvatures in the density dependence (Berryman

1991).

Mechanisms

Ail density dependence occurs by definition in the population growth rate, it can mechanistically occur in any or all of the instantaneous rates which sum to give the total instantaneous rate of change. What rates determine population growth depends on the definition of population used. The growth rate of the entire population of a species can be set only by mortality and reproduction (Varley et al. 1973). Many questions in ecology, however, deal with a spatially defined area, into and out of which movements of individuals may occur. Additions to a population due to movement are called immigration, an instantaneous rate added to the birth rate. Subtractions due to movement are termed emigration, an instantaneous rate added to mortality. Emigration has generally been thought to result in individuals with little chance of survival, and lumped into the term "gross mortality," but this may not necessarily be the case 26

(Lidicker 1975). Surviving emigrants might influence long-term population stability by affecting the rate of immigration in later years as considered in metapopulation dynamics. This has not, to my knowledge, been discussed in such terms in the literature, however.

The theory is mathematically straightforward, but how well does it describe reality? Does density dependence occur in a population's vital rates and, does it then lead to constancy, fulfilling Reddingius and den Boer's (1989) definition of regulation?

One school of thought argues that density dependence does not need to be proven. It is a logical prerequisite for the assumed persistence of populations (Royama 1977,

Berryman 1991). Another school holds that density dependence leading to regulation must be demonstrated in nature before it is assumed (Wolda, 1991). The assumption of populations persisting has itself been questioned (den Boer 1991). As the former logical deduction does not necessarily require regulation in the sense of narrow limits, the two groups in part argue past one another. "Density vagueness" (Strong 1986) has been advanced as a compromising concept allowing for density independence at all but very high and possibly very low densities.

Birth rate is considered by some to be the most important rate affected by density dependence (Branch 1974). Fecundity in animals can increase with food supply in two ways, as illustrated by a comparison of two marine snails (Spight and Emlen

1976). One breeds many times a year. Increased food supply is directly translated into 27 egg production at the expense of growth. The other breeds once a year. Increased consumption is allocated to growth. In both clutch size is directly proportional to body size, so both show increased fecundity. Both the former (Eisenberg 1966) and latter

(Branch 1974, Sutherland 1970) processes have been demonstrated in other aquatic gastropods. As has been mentioned, density dependence is expected to occur in the rate of addition of biomass to the population. This rate includes somatic growth and reproduction. Often a certain amount of somatic growth must occur before reproduction is possible. Forrester (1990) found evidence of density dependence in the growth rates ofjuvenile coral reef fish. High densities slowed growth, delaying maturation and thus fecundity. Recruitment in a tropical lizard population was positively correlated with previous per capita food intake and negatively correlated with density (Andrews 1991 ).

Laboratory studies have revealed a range of density dependent responses in mortality that correspond to -b values between 0 and infinity (Varley et al. 1973). A value of -b between 0 and 1 is described as being some degree of 'contest' alone in intraspecific competition (Hassel1974). Contest competition, introduced by Nicholson

(1954), describes density dependence whereby the surviving individuals obtain all the resources they require and the unsuccessful do not. The logistic equation describes a contest situation. Examples of contest might be competition for refuges, nest sites, or territories. A value of -b greater than 1 indicates decreasing contest and increasing 28

'scramble' (Hassel 1975). Through scramble competition all individuals in a population equally share a limiting resource and all are adversely affected. In its most extreme case there is 100% survival when initial N is below carrying capacity and 100% mortality when initial N is above it. Density dependent mortality can occur through predation and resource limitation. Both a functional response on the part of predators

(see Berryman 1989) and behavioral responses of prey (Sih 1979) can potentially lead to density dependent mortality. Resource limitation would occur when a population was above carrying capacity, either due to oscillating population dynamics (high r and/or b), time lags, or downward stochastic shifts in carrying capacity. Mortality caused artificially increased densities of sea urchins in enclosures to converge on

surrounding densities (Keller 1983). In some instances the limiting resource might be refuge from predation ( eg. Errington 193 8), in which the two factors are difficult to

separate.

For the present discussion and its criteria of a spatially defined unit the terms

dispersal and migration are used synonymously. The concept that animals migrate from

high to low density was supported by laboratory studies with crayfish (Bovbjerg 1959).

Field studies have found evidence of density dependent dispersal in freshwater bivalves

(Kat 1982) and juvenile mantids (Hurd and Eisenberg 1984). Lidicker (1975)

distinguishes between saturation and presaturation dispersal. Saturation dispersal

occurs in animals as an act of desperation in overcrowded conditions and involves 29 individuals with little chance of survival. It is a process that would be expected to act on populations only at or near K and so would not be adequately described by a parameter such as b. Presaturation dispersal, however, occurs far below equilibrium and so may help stabilize mortality. The study of migration in freshwater bivalves (Kat

1982) gives some evidence of density dependent immigration. It is unlikely that migration into a spatially defined unit would be affected by density as strongly as emigration. The rate of initial immigration would depend on the density of surrounding areas were dispursal density dependent. Whether or not immigrants subsequently emigrate, then, might depend on carrying capacity. A linear relationship in immigration versus density might only be expected in a species with a great deal of constant movement, where the length of intermittent stay times were a function oflocal resources.

Density dependence

Density dependence as defined by Murray (1982) and determination of population size seem to part ways over the nature of the carrying capacity and how populations respond to it. Environmental factors such as the number of refuges or nest sites are not correlated with density, but can determine carrying capacity. The population, then, is not limited by density dependent factors (Andrewartha and Birch

1954, Murray 1982). The term 'b' would not be expected to describe the relation of 30 population size to density because b is linear over the range of a population's densities.

In the above example b would only be expected to hold above a critical density or

"ceiling" (Strong 1986), suggesting the population description:

equ. 2 when x;

equ. 3a or 3b when x;>x:c

What iffood is the limiting resource? Food, as mentioned above, is supplied at a rate.

An increase in density in the absence of rigid territories would lower the per capita supply offood, making it a true density dependent factor sensu Murray (1982).

Equation 3a or 3b might well describe a food-limited population as the weaker individuals succumbed to predation or starvation before the carrying capacity had quite been attained.

The use of the term density dependence suggests the existence of the antonym

'density independence'. Andrewartha and Birch (1954) argue against this dichotomy.

Density independent factors are expected to affect proportions of populations, regardless of density, and are commonly equated with abiotic influences such as weather or random disturbances. Susceptibility to extreme weather conditions, however, varies by individual. Therefore small populations should react more variously than large ones. Susceptibility to disturbance might depend on the ratio of refuges from that disturbance to population size. Neither factor can be considered strictly density independent. 31

The distinction between density dependent and independent population dynamics might be defined by whether a populations dynamics are better described by equations 1 or 2 than by 3a or 3b (eg. Dennis and Taper 1994, Pollard et al1987,

Vickery and Nudds 1984). Den Boer (1991) found no significant difference in the increase in log range of census data for 62 beetle species from simulated random walks.

DenBoer and Reddingius (1989) found only one ofl2 sets of insect census data to vary significantly from random walks using both the test ofPollard et al. (1987) and

Reddingius' (1971) log range test. Vickery and Nudds (1984) found that, of 12 sets of duck census data that were of reliable length, half failed to show significant density dependence. An analysis of92 serial censuses of bird populations (Murdoch 1994) using several tests found at most over 50% to be density dependent using asymptotic or asymptoting cumulative variance. In contrast, random walk populations indicated density dependence in 64% of the cases. In Murdoch's review (1994) populations showed little interannual variability, so the difficulty may be one of detection.

Many populations have shown evidence of density dependence using various tests (see below) and manipulations (Gould et al., 1990, Stimson and Black 1975,

Eisenberg 1966). The latter have been cited as strong evidence of regulation

(Berryman 1991 ). However, manipulations have generally been conducted for less than one year. Rapid convergence of density or resource altered plots with control plots is evidence of density dependence, but density dependence need not lead to regulation 32

(Wolda 1991). Tropical lizard populations have shown density dependent fecundity,

yet are highly unstable (Andrews 1991).

Stability, therefore, appears to be absent or undetectable in many populations,

but what of persistence? Strict random walk populations increase in variance until a

lower threshold is crossed that leads to extinction (but see den Boer 1991). Strong

(1986) distinguishes between bounded and unbounded random walks. The former

walks walk randomly at intermediate densities until sharp density dependence occurs at

"ceilings" and "floors". Ceilings ultimately would be set by resource limitation and

floors might be set by refugia or background immigration.

Detection of density dependence

Analysis oftime series data is a common method researchers use to test for

density dependence in natural populations. Most tests for density dependence involve

looking for a statistic of a parameter predicted by the above theory to occur in time

series data and applying some measure of significance. An obvious statistic is (b+ 1), or

B (Pollard eta!. 1987), that should result from a regression ofx; (abscissa) against Xi+ I

(ordinate) as proposed by Morris (1959). A significantly correlated slope less than one

should indicate negative density dependence. This test has been found to indicate

density dependence even in simulated random walk populations (Maelzer 1970) and to

be unreliable (St. Amant 1970). This stems from the observation that, given a body of 33 data, an unusually high value is likely to be followed by a lower value and a very low value by a higher one. The effect is increased with increased variance in the data and decreased number of data points (Maelzer 1970). Regressing Xi+! against x1 and taking the reciprocal of the slope in addition to Morris' regression as being density dependent only when both are <1 (Varley and Gradwell1963) has been criticized as being too conservative (low power) (Slade 1977). Other techniques have attempted to estimate

Busing the slope of the major or principle axis of a bivariate scatterplot with successive log population values as ordinate and abscissa (Slade 1976 in Slade 1977) and the major axis when variates are standardized (Ricker 1975 in Slade 1977). The effectiveness of these estimations has been tested using a Gompertz model (Slade

1977).

An alternative method of testing for density dependence is to use computers to generate distributions of test statistics drawn from simulated random walk populations

(Bulmer 1975, Vickery and Nudds 1984, den Boer 1991). Such distributions provide a rejection region for the testing of the null hypothesis of density independence in the observed data. Related tests randomize the serial order of observed coefficients of reproduction (R;=x;wx1) to determine the likelihood of obtaining the observed product moment correlation coefficient ofR values on x values (Pollard et al1987) or other statistic, eg. log range (Reddingius and den Boer 1989). Dennis and Taper (1994) proposed a density dependence test using a likelihood ratio statistic based on the· 34 logistic model, (equ. 3b) and Pollard et al. (1987) suggested a modification ofVickery and Nudds' (1984) simulation procedure that amounts to an identical likelihood ratio test based on a Gompertz model, (equ 3a).

In theory the power of such tests relies partly on which model, 3a or 3b, better describes the growth of the actual population. With additional complexity (added parameters), models can be made to describe data sets more closely. It is possibly as an inverse result of this that density dependence tests based on more complex and therefore more flexible models seem to detect density dependence more frequently ( eg.

Dennis and Taper 1994 over Pollard et al. 1987) or of more complex nature (eg.

Turchin and Taylor 1992 over Hassel et al. 1976). Before a significant positive result is concluded to prove density dependence, I would suggest that a data set be tested using the model established by the first data set. Tests for density dependence that rely on a model have the disadvantage of ignoring the uniqueness of the habitat ( eg. presence of refuges) and the biology of the organism as well as assuming that populations spend most of their time recovering from disturbance to an equilibrium value or range rather than being disturbed.

The incorporation of time lags into single species analysis seems logical if not necessary in many systems. Theoretical discussions require a time lag for any form of density dependent population dynamics other than exponential damping. A lag of generation time is intrinsic in discrete time models (May et al. 1975). Time lags can 35 arise as a result of interspecific interactions and population structure (Royama 1981).

Time lags are significant in the dynamics of many insect populations (Turchin 1990) and have been implicated as a driving force in vole cycles (Hornfeldt 1994). Extraction of time lag information from time series data has been attempted using the regression of the rate of increase (R) on lagged population density (Berryman 1991). This shares the difficulties of other regression methods (Walda 1991). Time lags have also been investigated by plotting the autocorrelation coefficient (r) values against the time lag between the regressed population sizes (Turchin 1990, Hornfeldt 1994, Turchin and

Taylor 1994). When time lags are due to interspecific interactions it seems reasonable to assume that the strength of a lagged effect is positively related to the strength of coupling in interactions, eg. degree of specialization of predators on the subject prey population.

Assumptions

There are many assumed sources of variation in the dynamics of single populations. Strong (1986) divides types of variance into two categories, contingent and residual (inherent). Contingent variance is the result of a fluctuating environment with which the population parameters covary deterministically. An example might be an increase in the exposure of a rocky intertidal zone to water, either lower tidal height, increased wave exposure, or parial shade, leading to higher microalgal productivity, 36 increased carrying capacity for grazers and therefore higher r. Contingent variance can also occur in the slope of b. For example, lower temperatures might decrease metabolic rates in fish without affecting r. This might decrease b and increase K simultaneously. Residual variance is what remains after contingent variance has been explained. It includes individual, or demographic, variability that decreases with increased population size or density (Pollard et al. 1987), and random events such as disturbance, that might be expected to decrease with an increase in the spatial scale of a study site (see discussion of disturbance). Variability that is contingent on a small scale can become residual on a larger scale (Strong 1986).

An implicit assumption in considering the dynamics of single populations is that the variation in parameters is due to abiotic factors and that the influence of other species is either consistently described by some function of the study population's density, or can be reduced to a time lag in density dependence. This is the case in many laboratory studies but is highly unlikely in natural populations. Ill. Stability Theory: Paired Interactions

Interspecific competition

The stability and even persistence of a population in a unit area can be greatly

influenced by the presence of other species. Generally speaking, if the influence of a

second species on the first is via a decrease in the availability of resources they share,

interspecific competition is taking place. This influence can be embodied in a change in

abundance, fecundity or survivorship in both populations. How can the persistence of

both populations in the same unit area be explained and, under what circumstances

might their populations be stable?

Competition is generally modeled with differential equations. The logistic

equation

dN/dt=rN[(K-N)/K]

equation (8)

can be altered to account for a competing species population by adding the

competitor's density multiplied by a coefficient (a) converting it into equivalents of the

first species,

dN1/dt=r!NJ[(K.J-{N 1+a12Nz} )/K1]

dNz/dt=rzNz[(Kz-{Nz+azJNi})/Kz].

equations (9)

37 38

This is the Latka-Volterra formulation of competition. By setting dN/dt=O and representing each species as a linear function of the other, the theoretical outcome of the competitive interaction can be determined. Competition exclusion of species 2 by species 1 occurs when K1>K2/a21 and K2~1/a 12 . A hypothetical illustration might be a mussel species (N1) and a kelp species (N2) competing for space. The existence of one in a plot reduces the potential density of the other. If 50 mussels fit in a plot (K1=50) and the area occupied by one kelp holdfast is the same as that occupied by 5 mussels, ie. a 12=5, then K2=10 if space on the primary substrate is the only source of intraspecific competition in seaweeds, (a21=1/5). Then K1=K2/a21 and K2=K1/a12. If kelp canopies shade one another, however, and K2=5, then a 21=1/10, K2K2/a21 and K2>K1/a:12.

Using the above simple illustration, if whiplash by kelp clears an area equal to 13 mussels (a:12=B) and mussels are able to clear substrate around them of seaweed so that a21=1/7, then the conditions for an unstable equilibrium are met and whichever species gains the upper hand excludes the other. A stable equilibrium according to

Latka-Volterra theory only occurs when K1

Yodzis (1986) distinguishes between competition for space and consumptive competition in a way that seems to indicate that they are fundamentally different. As can be concluded from my examples, I do not see why space should not be viewed as directly analogous to any other limiting resource as viewed through the framework of

Lotka-Volterra theory, as 'resource' is only implied in the term K and so is either consumable and provided at a rate or is merely used and thus a standing quantity. I fail to see why "niche control" could not apply to a spatially defined unit where space is the limiting resource to the inhabitants. Not all units of space are equally usable by all sessile organisms. Y odzis himself outlines his scheme for the categorization of competition using Lotka-Volterra terms. Note that in the above examples the occupation of space by any one species would not be expected to follow a logistic curve as is described by the Latka-Volterra equations. The logistic curve assumes that the organisms within the unit of study produce the observed recruits and so r is logarithmic. In an open, space-limited system settlement is constant or varies according to some external factor or factors. The curve would most likely have a slope equal to the rate of supply of propagules at the origin and decrease linearly or nonlinearly to an asymptote at K. The interactions between species would be expected to be similar to that described above, as their outcome depends only on their respective

K's and a's. 40

To summarize the above theory, stable coexistence between two species occurs when, under a given set of environmental conditions, the effects of intraspecific competition outweigh those of interspecific competition in both species. This mathematical formulation forms the framework for many proposed mechanisms leading to persistence in stable or unstable form. These mechanisms can be generalized as forms of refuges from competition for inferior species. Refuges in the above general sense lead to an increase in the term K/a. through decreasing the average realized a. value by releasing the inferior species from some of the competitive effects of the superior one. As categorized for this review, these refuges include spatial refuges, partitioned resources, and probability refuges.

Spatial refuges could include a certain portion of the area of interest and thus the resources it contains, being usable by the competitively inferior but not the superior species. This would have the effect of simultaneously increasing K of the inferior species and decreasing the net competitive effect of the superior on the inferior species.

According to theory, in a deterministic world, the addition of refuges for inferior competitors should automatically lead to coexistence of competitors. A similar type of refuge would be areas where the competitive imbalance between two species was reversed, leading to coexistence on a larger scale. A temporal version of such refuges is easily imagined. Though the K's and a.'s of a model predict an eventual outcome, provided they remain constant, the time required for an equilibrium to occur is 41 determined by the r's. If, within this time, another resource becomes more limiting, then alternate dynamics will dominate. Provided the inferior species prevails under these alternate circumstances and shifts in dynamics occur too often for one species to exclude the other, persistence, though not stability, is provided for.

These two refuge types differ in the predicted size of the equilibrium populations of the two species, assuming the resources are continuously distributed amongst the individuals of both species. This assumption means that, either movement of individuals of both species is high relative to the study area, so individuals potentially encounter all resources unless excluded somehow, resources are relatively uniformly distributed across the study area as are individuals of both species, so that they are encountered, though variously utilized by both species, or resources are highly mobile and so are encountered by both species. It is important to distinguish between refuges where the limiting resources are continuous with those utilized by the dominant competitor and refuges where there is no exchange of resources or individuals with non-refuge areas when assessing the total impact of two species on each other within a unit area. The latter case is equivalent to two separate populations of each species and

the predicted result is the attainment ofK for each species in the areas where it is

competitively superior (Schoener 1974). In the former case, the inferior competitor's

population as maintained by refuges from exclusion can still reduce the quantity of

common resources available to the superior competitor, and reduce its population size 42 below its K. The mechanism inferred by Savolainen and Vepsalainen (1989) to explain the coexistence of certain of several study ant populations corresponds to a spatial refuge from interspecific competition at increasing distance from colonies of competitively superior territorial ants.

Were space the resource most limiting the numbers of both species, then spatial refuges would be a form of resource partitioning. Resource partitioning means that species in the same assemblage differ in the way they utilize resources. The result of resource partitioning is that one or both species have a refuge from interspecific competition with the other relative to the partitioned resource. This translates to stable coexistence in a deterministic environment provided that the inferior competitor's resource utilization is partially nonoverlapping with the superior competitor's, and that the limiting resource is either the resource partitioned or is indirectly partitioned as a result. If the supply of food in the form of insects limited lizard populations, for example, partitioning of perch heights between competing lizard species would not lead to coexistence if insects moved freely and frequently between those heights.

The Lotka-Volterra equations, though widely used, have been criticized as lacking specificity regarding the mechanism of resource utilization and, thus, competition. An alternative and more explicit model proposed by MacArthur (1968) and discussed in terms of competition (Abrams 1980, Schoener 1974a, 1974b) is the two level consumer-resource model 43

equation (10)

equation (11) describing the resource as a separate population, where R is the number of consumers, i, that can be produced per unit energy, anc is the per-resource-unit consumption rate of resource k by i, bnc is the number of units of energy extracted by i from a unit of k, and

Ci is the cost of maintenance and replacement of an individual of i. The resource, k, is logistically selflimiting in the absence of consumers, where rk and Kk are respectively the intrinsic per capita rate of growth and carrying capacity of k. The resource acquisition term of the competitor follows the behavior of randomly interacting particles without a maximum rate of acquisition per individual competitor. In the case of resources entering a system at a fixed rate, such as detritus production or invertebrate drift in streams a different model has been proposed for the resource level ofmodelll,

equation (12) where Sk is the quantity of k entering the system per unit time.

Equations lla, b, and 12 form the basis of many attempts to characterize the effect of resource partitioning on the competition coefficient, a, relating the effect of an 44 individual of one species on the growth rate of a competitor's population to the effect of a competitor on the growth rate of its own population. When equation 11 b is solved for equilibrium (dFJdt=O) and the resulting Fk is inserted in equation 11a at equilibrium, a linear equation relating one competitor to the other results, in which

equation (13)

(see Schoener 1974}. When it is assumed the term bK/r is the same for all resources, the a's are equivalent to frequencies of utilization of habitats and k represents all resources in a habitat, and the total consumption of all resources by one species equals the total consumption of all resources by the other, then the a's can be replaced with p's, the relative utilization of habitats by competitor i. The resulting equation

equation (14) forms the core ofMacArthur and Levin's (1967) theory oflimiting similarity. Due to the above assumptions, equation 14 is unrealistic (Abrams 1980, 1983). When the provision of resource k is better described by equation 12, then the competition coefficient that results is simply equal to the ratio of a's (aij=ajJa;k). The nature of the competition coefficient is expected to differ from resource to resource (Abrams 1980) 45 and depending on the nature of competition (Schoener 1974). Roughgarden (1986) discusses evidence that resource partitioning explains the coexistence of competing lizard populations on Carribbean islands. As mentioned before, a. represents only the ratio of one-to- one inter- to intraspecific competition and does not reveal either the intensity of competition (Abrams 1980) or the predicted population-level outcome of a competitive interaction (Perrson 1985).

Probability refuges can result when competitors show a large degree of intraspecific aggregation and localized resource use (Abrams 1984). This would result in a decrease in the realized ratio of inter- to intraspecific competition. An example would be Atkinson and Shorrocks' (1981) simulation of the effects of naturally occurring degrees of aggregation offruitflies (1984) competing in ephemeral resource patches on their coexistence. Spatial variation caused by random disturbances is a second form of probability refuge. When patches of resources are made randomly available to two competitors and their distribution among patches is also random, some patches will lack interspecific competitors and global coexistence will be possible

(Slatkin 1974). In an open system where the supply ofpropagules is limited, coexistance between competitors can result when recruitment and mortality of the dominant competitor balance at a low population size. This condition is analogous to a decreased Koominont without the competitive interaction being affected. 46

A mensa/ism

Amensalism is another significant interaction between two species includes any

of a potentially large variety of interactions that reduce the growth rate of one

population without having any effect on the other. Probably one of the most common

forms appears as a special case of extreme asymmetrical competition, where one of the

competition coefficients (a) equals zero. It seems likely that many species are able to

assume control of a resource at a rate unaffected by the presence or absence of certain

other species already using that resource, especially when only two species are

considered. The expansion of oak woodlands is probably little impeded by dandelions,

although the latter might have difficulty growing in the shade of the former. In the

context of the above competition theory, dandelions could only stably coexist in the

presence of oak trees if oaks created a canopy that allowed some light through (were

unable to completely dominate the resource even at their own K =>

K,,k

inadvertently destroying another in the pursuit of a third prey item, which might be

modeled as a predator effect in the prey equation (see below) without a complimentary

term in the predator's growth rate. Such an interaction would not be expected affect

the stability of either population. except to decrease the growth rate and realized K of

the affected population and so possibly influence the outcome of the affected species'

interaction with a third species. Indirect effects will be further discussed below. 47

Predation

The nature of the potential stability of two interacting populations is significantly altered if one population's growth rate increases with increased density or size of the other and the growth rate of the second population decreases with increased density or size of the first. This describes the interaction of a prey species with one of its predator species. The Latka- Volterra equations intended to describe the change in population size of interacting predator and prey species are:

dN/dt=aN-bNP

equation (15)

dP/dt=cbNP-dP,

equation (16) where N represents the prey population, P represents the predator population, and a, b, c, and d are constants representing the prey intrinsic instantaneous rate of growth, the per-N mortality rate given P, the per-P conversion rate ofN to new predators, and the per capita predator mortality rate respectively. Because most predators can only

consume a limited number of prey per unit time it becomes apparent that the interaction term bNP must be replaced with some functional response, b(N)P, to realistically

describe actual interactions. Holling (1959) divides possible functional responses into

three types. A type I response is identical to bNP at relatively low densities but upon

reaching a saturation point abruptly assumes a constant maximum value (figure 3). 48

Filter feeders might be expected to show a type I response as the quantity of prey items they consume is determined at lower prey densities by the cross-sectional area of their filtering apparatus. An example of a type II response is the Holling "disc" equation, equivalent to the Michaelis-Menten equation of interacting enzymes and substrates

(Berryman 1992, Real1977):

b(N)=mN/(w+N)

equation (17) where m is the maximum attack rate and w is the prey density at which the attack rate is half maximal. The attack rate steadily decreases as N increases and m is approached asymptotically (figure 4). Thompson (1975) demonstrated type II functional responses in predatory damselfly nymphs that varied in parameter values for different predator and prey sizes. A type III functional response rises sigmoidally to an asymptotic maximum attack rate (figure 5). Such a response occurs when, at relatively low prey densities, an increase in prey density leads to an increase in predator search efficiency or a decrease in a predator's handling time of prey.

The outcome ofLotka-Volterra modeled interactions can be ascertained using graphical methods similar to those described for competitive interactions (Rosenzweig and MacArthur 1963). When the change in either population is set to zero, then an equilibrium of the species at hand becomes a function of the density of the other species. The nature of the two modeled populations' joint deterministic return to their 49

2 -. -- . ----. --.. ------j- - ~ ~ (.) ------co :::: co

prey density

Figure 3. A Holling type I predator functional response. 50

·------:----·------' '

.::s::. ------() co -co

prey density

Figure 4. A Holling type II predator functional response. 51

Q) ------co...... ::.:: ------() ro ~

prey density

Figure 5. A Holling type III predator functional response. 52 mutual equilibrium depends on both of their equilibrium functions of the other, or isoclines, which in turn are dictated by the functional response chosen for the predator and thus the prey's mortality and the function that describes the prey's rate of increase in the absence of predators. Predator's are generally assumed to have a constant background mortality rate which includes the per capita cost of maintenance (Arditi and Ginzburg 1989) in the form of individuals not born. Berryman (1992) outlines the effect of various functional responses on the shapes of the resulting isoclines and the system's return to equilibrium. Lotka-Volterra predator-prey systems exhibit neutral stability. They cycle indefinitely if undisturbed and assume a new cycle when disturbed

(see figure 6). In a real environment this is equivalent to unstable. Prey self-limitation introduces stability in the form of damped oscillations.

Each of the functional responses described above are those of individual predators given a range of prey densities. Theoretically, stability can also be enhanced if a predator species' instantaneous consumption rate decreases with an increase in predator density. Traditional predator-prey models assume that there is no additional affect of predator density on the resulting value, ie. they are prey dependent. Prey dependent models have a predator isocline that is perpendicular to the prey axis, ie. one density of prey leads to an equilibrium population of predators, regardless ofthe predator population size, as set by the intersection with the prey isocline. Arditi and

Akvakaya (1990) analyzed 15 laboratory data sets and found at least partial ratio 53

200

Predator

..-.. 150 ..._..a.. ,_(/) .....0 100 co Prey '"0 ,_(]) a.. 50

0 0 50 100 150 200 250 Prey (N)

Figure 6. Predator/prey graph showing trajectory (thin line) followed by predator and prey populations obeying equations 19 and 20 (a=O.OS, b=O.OOOS, c=0.2, d=O.Ol).

Thick lines show predator and prey zero isoclines. The small 'x' shows the origin of the trajectory. 54 dependence in each of them, assuming a variant of the disc equation functional response that allowed for a range of ratio dependence from 0 (prey dependence) to 1

(complete ratio dependence).

One general explanation for ratio dependence is mutual interference (Arditi and

Ak9akaya 1990). Predators shift part of their time from prey procurement to interacting with other predators. This shift increases with predator density. This has been represented as a decrease in the area of discovery of the Nicholson and Bailey model with an increase in predator density (Hassel and May 1973), and is roughly equivalent to multiplying N in the above functional responses by a coefficient that is a decreasing function ofP. The theoretical boundaries of stability for this modified model are presented by Hassel and May and indicate that even a small amount of mutual interference can make an otherwise unstable system stable. It has also been proposed that if a predator's distribution within a unit of space is aggregated, then

"pseudo- interference" could result (May 1978). The effects of spatial variation in predation intensity will be discussed later. Ratio dependence in the predator's functional response has been invoked as an explanation for observed stability in certain consumer/resource interactions (Berryman 1992) where large fluctuations are predicted by prey dependent models ( eg. the biological control paradox). Though intuitive in its predictions (Arditi and Ginzburg 1989), ratio dependence has eluded strict biological explanation (Murdoch 1994). Models that depend solely on the ratio of consumers to 55 prey disregard spatial considerations. As consumers and prey become diluted in space, the per-consumer consumption rate of prey must decrease if consumers can only search a limited area per unit time, regardless of their ratio. Ratio dependence as a result of time delays in the conversion offood to offspring has been proposed (Arditi and

Ginzburg 1989) and convincingly disputed (Oksanen et al. 1992). There are, however, mechanisms that lead to predator isoclines with positive slopes analogous to isoclines that result from including ratio dependence in the functional response (Murdoch 1994).

All conventional deterministic predator-prey models show either a stable limit cycle set by the stabilizing resource limitation term (in Lotka-Volterra models K -+oo) and the destabilizing predator functional and numerical response terms (May 1972,

Gilpen 1972). In models considered unstable, the predicted limit cycle generally carries one of the populations below one, which means extinction in ecological terms. Gil pen

(1972), using isoclines, shows how one model system including logistic prey self limitation and a type II functional response gives rise to a stable equilibrium or a stable limit cycle, depending on whether the mutual equilibrium point lies to the right or left of the humped prey isocline's peak (see figures 7 and 8). An increase inK relative to the predator isocline increased the cycle amplitude, ie. was destabilizing (see also

Rosenzweig 1971). If the prey's K is below the predator isocline's intersection with the prey axis, ie. the prey never attain a density that could support a predator population.

The predator goes extinct. 56

To what degree is this theory descriptive of real systems? May (1972) states that stable limit cycles, due to their theoretical ubiquity, are sure to underlie many of the cycles observed in nature. It is less clear, however, how environmental stochasticity would affect the distinctiveness of such cycles. Gilpin (1972) notes that limit cycles with randomly varying parameters are susceptible to random extinction.

The well known Canadian lynx-hare cycles are likely caused by a time lag in the availability of high quality food. Thus they may be analogous to a fluctuation inK or the parameters of the predator functional response, merely hastened by predators, rather than caused by them (Smith et al. 1988).

One of the greatest discrepancies between theory and observed dynamics has been termed "the biological control paradox", the suppression of certain insect pest populations to extremely low but stable levels by natural predators. As stated above, prey suppressed far below their resource carrying capacity by predators are predicted to show large fluctuations. This incongruity has left some researchers scrambling for explanations ( eg. Murdoch 1994, see below).

Refuges for prey, or prey densities in a particular habitat at which predation becomes nearly zero, can increase the stability of a system (Rosenzweig and

MacArthur, 1963). At most, according to graphical predation theory, refuges within the habitat of interest can limit the amplitude of predator-prey oscillations (see also 57

125 Predator 1 00 ..... -...... -......

----0.. -~ 75 0 Ctl "'0 50 ....Q) 0.. 25 Prey

0 0 100 200 300 400 500 600 Prey (N)

Figure 7. Predator/prey graph showing trajectory (thin line) followed by predator and prey populations. Prey show logistic self-limitation (a=O.OS, K=SOO) and predators have a type II functional response (c=0.2, m=OA, w=300, d=O.Ol). Trajectory converges on a stable-limit cycle. Thick lines represent predator and prey zero- isoclines. 'X' represents origin of trajectory. 58

100

Predator ...... 75 ._...o_ (/) '- 0 ...... ro 50 "0

0 100 200 300 400 500 600 Prey (N)

Figure 8. Predator/prey graph showing trajectory (thin line) followed by predator and

prey populations. Population equations the same as in figure 7 but d=0.03. Trajectory

converges on equilibrium point. Thick lines represent predator and prey zero-isoclines.

'X' represents origin of trajectory. 59

McAllister et a!. 1972). Murdoch (1994) investigated the effects of a refuge external to the study system on that system's stability. Such an external refuge can provide a low, stabilizing influx of immigrants, although this has proved not to be stabilizing in

Murdoch's system. A constant immigration rate has an increasing influence on population size as the population size decreases and is therefore density dependent.

The two locations of refuges, external and internal to the study system, are similar in that their representation by a prey isocline incorporating a standard type II functional response is a more or less sharp upward tum near some low prey level. An external refuge, that influences the prey population only through a trickle of immigrants, makes the negatively sloping isocline asymptotic to the predator axis. A refuge whose prey numbers are included in the study system makes the prey isocline asymptotic to the carrying capacity of the refuge if there is an exchange of individuals between the protected and unprotected areas. The isocline is simply a vertical line at the refuge's carrying capacity if no exchange of individuals occurs. Stable limit cycles are buffered against crossing or merging with the predator axis (N=O) by the vertical segment of the prey isocline (Rosenzweig and MacArthur 1963, McAllister eta!. 1972). Refuges can include physical refuges from predation (Murdoch 1994), feeding thresholds in predators (McAllister et a!. 1972), and, similarly, metabolic reduction in predators at low prey densities (Salt 1967). 60

Density dependence in the predator population can be stabilizing whether it occurs in the predator's recruitment or death rates. Parasitoid wasps feeding on already parasitized hosts has been suggested as a density dependent factor in the wasps' recruitment rate (Murdoch 1994). Recruitment density dependence produces a positively sloping isocline similar to that of ratio dependent models. The above mentioned mutual interference, or "squabbling" (Rosenzweig 1972), amongst predators introduces density dependence to their recruitment rate via the functional response.

Beddington (1975) has proposed a functional response equation derived from assumptions of this behavior.

A predator death rate that increases with predator density can occur when predators are limited by some resource other than prey, such as shelter. Using graphical predation theory, Rosenzweig and MacArthur (1963) have suggested such limitation can be represented by a horizontal predator isocline segment capping the vertical segment. When the prey isocline intersects the horizontal portion of the predator isocline, stability is greatly enhanced, since predators cannot increase long before their food resources become limiting, oscillations will hardly occur. When density dependent mortality is described by a term in the predator equation, its affect on the predator's isocline becomes more complex, and is partly dependent on the functional response. A Latka-Volterra functional response (bN) produces an isocline that continues to rise with an increase in prey density. This is a result of the lack of a 61 maximum attack rate and a potentially infinite instantaneous rate of increase. The inclusion of a type II functional response results in a horizontal or asymptotic predator isocline segment of the type described by Rosenzweig and MacArthur (1963), depending on the term describing the density dependent mortality. It is noteworthy that, if mortality rises continuously from a background level (d) to a linearly increasing term at the predator's carrying capacity (described by Hassel (1974) for single populations), then the above mentioned horizontal asymptote can be well below the predator's actual carrying capacity (see figure 9). Specifically, for a predator equation including a Holling-Michaelis-Menten functional response and a mortality term that is density dependent on top of a background mortality rate:

dP/dt=c[mN/(w+N)]P-[ln(1+P/Kp)+d]P

equation (18) the asymptote (level ofP where cm=ln(l+P/Kp)+d) is at

P=Kpecm-d -Kp.

equation (19)

Kp is the habitat carrying capacity of predators, set by some resource other than prey, c, d, m, and ware as above (equations 17 and 18). The above point is ecologically significant for two reasons. It is likely that all predator populations have size-limiting factors other than prey, be it simply space that individuals can occupy. Theory says that two limitations can act in concert to suppress predator numbers and strongly 62

50

. 40 ...... _...0... ,_(/) 30 0 -(\j "0 Q) 20 ,_ Predator 0... 10

0 0 100 200 300 400 500 600 Prey (N)

Figure 9. Predator/prey graph showing trajectory (thin line) followed by predator and prey populations. Prey equation same as in figure 7. Predators obey equation 18

(c=0.2, m=0.4, w=300, Kp=SOO, d=O.Ol). Trajectory converges quickly on equilibrium. Thick lines represent predator and prey zero-isoclines. 'X' represents origin of trajectory. 63 stabilize predator/prey interactions. Figure 6 also shows that the equilibrium predicted by the equations with the chosen qualities, self limitation in the prey population and both prey limitation and self-limitation in the predator population, is always near the prey carrying capacity. The equilibrium would fluctuate, therefore, as the preys' carrying capacity varied with the environment and the predators' population tracked it.

The effects of environment and predator/prey interactions on their mutual dynamics would be nearly indistinguishable.

Predator aggregation has received much attention in the literature. By analyzing several predator-prey models that incorporate the spatial distribution of predators and with different predator responses leading to aggregation, Hassel and May

(1974) and May (1978) found that increased clumping of predators tended to stabilize otherwise unstable interactions. Pacala et a!. (1990) further established that, specifically in parasitoid-host theory, if the probability of being parasitized for an individual host varied in a patchy environment to the degree that its CV2

2 2 (covariance =variance/mean ) > 1, irrespective of the host's distribution, then stability over the sum of patches resulted. Such aggregation produces 'refuges' from predation or parasitism in areas oflow risk. These 'refuges', unlike actual refuges (above), regulate according to predator density, not host density (Reeve and Murdoch 1986).

Kareiva (1987) demonstrated the importance of the aggregation behavior ofladybird beetles to their successful stable suppression of aphid densities while showing 64 simultaneously how the realization of the generalization, 'aggregation of predators can be stabilizing', depends on the interaction of the predator's specific behavior with its environment. Reeve and Murdoch (1985) failed to find evidence of aggregation in parasitoids controlling red scale.

Temporal density dependence in predation is another concept similar to predator aggregation. It has been proposed that mortality in prey populations due to predation might increase with prey density. Reeve and Murdoch (1986) distinguish between the effects of density dependent mortality and predator aggregation by stating that the latter gives rise to "' pseudointerference', in which the efficiency of individual parasitoids declines with parasitoid density." It is my understanding, however, that a density dependent decline in predator efficiency is true mutual interference (Hassel and

Varley 1969), that is merely "mimicked" by pseudointerference (May 1978), in which predator efficiency remains constant. It seems that a prey population under the influence of negative density dependent predation experiences greater risk of mortality per prey individual at high densities than at low densities, making density dependent mortality simply a result of predator behavior leading to aggregation, given nonsynchronous population fluctuations over space. Temporal density dependence assumes that the entire prey population either fluctuates synchronously, or that no significant exchange of prey or predators occurs between patches to stabilize their fluctuations. To stabilize a prey population, such density dependent mortality must 65 have a relatively short time lag (May 1974), meaning that the predator generation time must be shorter than the prey's. This is the case for the parasitoid Aphytis and its red scale host. However, Reeve and Murdoch (1986) found no evidence for temporal density dependence in this system.

Metapopulation dynamics is another mechanism that relies on spatial heterogeneity that has been proposed to provide stability in otherwise unstable interactions. Vandermeer (1973) has analyzed a metapopulation model in which the dynamics of patches containing predators and/or prey, having characteristic "birth"

(recolonization) and "death" (extinction) rates. Solving for the "patch equilibrium" he then showed that the modeled system tended to converge on the equilibrium if the prey rate of recolonization was high relative to the predator rate of recolonization and/or if the predator extinction rate was high relative to the prey extinction rate. By combining local elements of density dependence and global effects of a heterogenous environment with migration between patches, a predator-prey system can be stable on a large spatial scale (Reeve 1988). Murdoch (1994) has investigated metapopulation dynamics as an explanation for the biological control paradox in his parasitoid-host system by isolating subpopulations but has failed to induce instability. 66

Mzttualism

Mutualism could be discussed in the context of multispecies interactions because almost all mutualisms occur in the context of at least three interacting species

(Freedman et al. 1987). Notable exceptions are the cases of pollination. In a mutualistic interaction, interacting species benefit each other reciprocally. This benefit can take four forms (Boucher eta!. 1982); (1) nutritional, through increasing the supply of, consumption rate of, supplementing, or enhancing the efficient use of a food resource (Freedman et a!. 1987), (2) supply of energy, generally through photosynthesis, (3) protection from enemies or environment, and ( 4) transport of individuals or propagules. Benefit in mutualistic terms can be viewed on either the level of individual fitness or the level of population dynamics (Boucher et all982).

Although the former can lead to shifts in population parameters giving stability over evolutionary time, this review is concerned with the latter view of benefit. Mutualists can be facultative, when one population can exist in the absence of the other, or obligate, when one population cannot survive without the other.

Mutualistic interactions have generally been considered to be destabilizing (May

1973) due to their mutual positive feedback characteristics. When modeled as modified

Lotka-Volterra competition equations where the presence of a mutualist population of size N increases its "partner's" or "partners"' K to K +a:N, both populations grow infinitely (May 1981 ). These models can be stabilized by strong negative density 67 dependence in all involved populations (Goh 1979) and/or curvilinearities in the mutualistic interaction caused by saturation of at least one of the benefits exchanged or threshold effects in the case of obligate mutualists (May 1981). A phenomenological model has been proposed whereby mutualists cause a linearly decreasing increase in one another's carrying capacity which asymptotes at the next resource that becomes limiting (Dean 1983). Depending on where the zero isocline of each mutualist crosses the one or the other axis, a locally stable equilibrium, an unstable equilibrium, both or neither can result. Facultative mutualists are predicted to have a stable equilibrium, obligate mutualists have a stable equilibrium and a mutual origin at zero, and obligate mutualists with thresholds, below which both go extinct, have both a stable and unstable or just an unstable equilibrium. "Ridges" occur in phase space intersecting unstable equilibria, upon which trajectories can tend to either a stable equilibrium or extinction if perturbed, and environmental stochasticity is predicted to cause the location of such "ridges" to shift randomly. It has been suggested that the observed prevalence of obligate mutualism in the comparatively stable tropics might be attributable to this phenomenon (Dean 1983).

Most of these models are unspecific as to the mechanisms of the exchanged benefits. When the actual mechanisms are known, the conditions for stability might become clearer. A well known mutualism, the association of nitrogen fixing bacteria with legumes, has been shown to benefit one species oflegume on the population level 68 by conditionally altering the outcome of a competitive interaction (de Wit et al. 1966, reviewed in Began et al. 1990). The presence of nitrogen fixing bacteria increases the value ofKlogumo, but only under limiting resource conditions. The stability oflegume populations is further complicated by the fact that they alter the resource nitrogen availability in their environment with a time lag based on decay rates. The eventual result can be their competitive replacement. Another common mutualism, the interaction of "farming" ants and honeydew-producing insects, may or may not be dependent on the presence of other species. Ants have been shown to protect membracids from predators (Cushman and Whitham 1989) and the strength of this relationship varies from year to year and with membracid age class. The degree of benefit can increase with aggregation size (positive density dependence- Cushman and

Whitham 1989) but decreases with the number of neighboring aggregations (negative density dependence-Cushman and Whitham 1991). Mutualism between ants and aphids has been shown in one case to not rely on a multispecies context but to probably arise from ants increasing the feeding rates of aphids and to be negatively density dependent

(Breton and Addicott 1992). In both systems the result of the mutualism on the ant's

"partner" was to increase the net population growth rate, not K. The benefit to the ants may have been an increased K. These interactions have not been shown to stabilize either population and membracid populations have been observed to substantially fluctuate interannually (Cushman and Whitham 1991). The degree to which a 69 mutualistic interaction affects the stability of one or both involved populations is probably an increasing function of its obligatory nature and a decreasing function of the number of species present that could fulfill the role of either mutualist. Certain flowers, for instance, are dependent on insects for pollination, but are visited by several species.

Each of those insect species might be expected to visit many species of flowers. IV. Stability ofMultispecies Interactions

Almost all species exist and interact with more than one other species. The goal of the "community matrix" (Levins 1968) approach to ecology has been to establish the nature of all of the paired interactions in an assemblage of interacting species, present them in a grid of every species versus every other species and, using matrix algebra, predict the dynamics of the ensemble. The local stability of the system, if a joint equilibrium point emerged, could be ascertained by assuming all pairwise interactions were linear with the slopes of the instantaneous interactions at the joint equilibrium point. This approximation holds only at the equilibrium point unless all pairwise interactions are truly linear, in which case the local stability describes the global stability. The stability criteria for a theoretical matrix of pairwise interactions are summarized by May (1973).

Indirect effects

The addition of one or more species to a two species system can alter the outcome and even nature (Wilbur and Fauth 1990) of the two species interaction via indirect effects. An indirect effect involves at least three species. The effect is indirect when one species causes it in a second by directly affecting a third species which has a direct effect on the second. Following the terminology of Abrams (1987), species one,

70 71 two, and three will be called the donor, receiver, and transmitter respectively. The effect on the second species can further affect a fourth species, in which case species two would be both a receiver and a transmitter. The nature of the effect is generally considered to be the equilibrium population size (Holt 1977), but can also be the instantaneous growth rate or population stability of the affected population (Abrams

1987). The consideration of an additional species, then, can modify the stability of a two species interaction in a variety of ways.

Indirect effects can be divided into two basic groups based on how they come about, interaction chains and interaction modifications (Wootton 1993). Interaction chains occur by the transmitter species' abundance being a function of the donor species'. This has also been termed a trophic linkage indirect effect (Miller and Kerfoot

1987) and simply an indirect effect (Worthen and Moore 1991). Although modeled as a direct interaction by the Latka-Volterra competition equations, pure exploitation competition is actually an example of an indirect effect, when the contested resource is a food organism (Abrams 1987). The keystone predator effect (Paine 1966) is another example of an important interaction chain. Interaction chains come about in the normal course of pair- wise population dynamics (Abrams 1991) and have been discussed in the context of the community matrix (Levins 1975). Interaction modifications have been largely discussed under the term 'higher order effects' and occur when the donor affects the receiver by influencing the nature of the direct interaction between the 72 receiver and transmitter. Interaction modifications have been further divided into behavioral indirect effects and chemical response indirect effects, based on the mechanistic explanation for the modification (Miller and Kerfoot 1987). Miller and

Kerfoot (1987) use as an example of the former predatory bass causing bluegill sunfish to forage in areas of a pond that provide cover, thereby partially freeing the sunfish's prey, zooplankton, from predation. An example of a chemical response indirect effect, according to Miller and Kerfoot (1987), is a type of beetle feeding on willows which provide the beetle with toxins useable in defensive secretions. In this case the predators on the beetle are the receivers of the effect donated by the willow and transmitted by the beetle. Such interaction modifications cannot be predicted from the knowledge of pair-wise interactions.

Whereas interaction chains have been demonstrated in several studies (Paine

1974, Lubchenco 1978, Brown eta!. 1986), interaction modifications are more difficult to detect in the field and laboratory studies (Wilbur 1972, Neill1974, Wilbur and Fauth

1990) and positive results have been contested (Pomerantz 1981). Detection of interaction modifications is difficult because they are predicted to be caused by their interspecific interaction coefficients ( eg. ex of competition equations) being functions of other species (Abrams 1983) but can resemble nonlinearities in the receiving species' intraspecific growth function (Pomerantz 1981). A significant interaction term in an analysis of variance (anova) can be used to detect an interaction modification, provided 73 the experiment is properly designed. An experiment designed to distinguish interaction chains from interaction modifications among fiuitflies (Worthen and Moore 1991) found evidence of only the former. Experiments with two predators, salamanders and dragonfly larvae, and two prey species, frog tadpoles, found evidence of interaction modifications over the short period in their life histories when these species would co­ occur (Wilbur and Fauth 1990). Wootten (1993) found evidence of an interaction modification in the field whereby barnacles decreased the rate of predation on limpets by birds by giving barnacles a background in which barnacles could camouflage themselves.

One example of how a species can, via amensalism, affect the outcome of an interaction between two other species has been suggested above. Using a variety of consumer resource equations that describe various kinds of interactions, Abrams

(1987) shows how the stability of two noncompeting prey populations and a predator population can be variously affected through indirect interactions. When a second prey population is introduced to a predator/prey system, where the prey shows an Allee effect (tends to extinction below a threshold density) and the predator shows a linear

(Latka-Volterra) functional response, the result is to decrease the range of population parameters that can give a stable population (ie. can be destabilizing or, if those parameters fluctuate with environmental stochasticity, is destabilizing). In the familiar case where prey are logistically self-limited and the predator shows a type Il functional 74 response, adding a second prey species is destabilizing. This can be explained by the spreading of the risk of predation between the prey species, as has been demonstrated to occur (Wilbur and Fauth 1990). Because predators have a choice between two prey items, a lower density of either prey item is required to allow the predator to balance its per capita rate of increase (rate of consumption) against its per capita rate of decline.

The result is analogous to Rosenzweig's (1971) "paradox of enrichment".

It is unclear from the literature I have reviewed what the stability effects would be of increasing the carrying capacity for two prey populations that share one predator but have different carrying capacities or per capita growth rates. The amplitude and period of the oscillations predicted by Rosenzweig (1971) in an overly-enriched habitat are functions of the predator's and prey's population growth parameters. A predator with the option of two prey items with different population growth parameters would be partially uncoupled from either prey's population dynamics. Switching on the part of the predator can be further stabilizing (Oaten and Murdoch 1975b). Switching occurs when predators "specialize" on the most abundant prey item, leading to a type ill functional response. Each prey population is provided with a refuge when it reaches low densities and when in the presence of the other prey population.

Indirect mutualism has been predicted to occur when two consumers feed on separate resource populations which are in competition with each other (Vandermeer

1980). By reducing the intensity of competition with the other consumer's resource 75 population, the first consumer benefits the second consumer indirectly. So long as the competition between resource populations is not so asymmetrical that one quickly displaces the other, then the benefit should be reciprocated. It certain cases, eg. consumers with type II or type ill functional responses, the result would be an increase in the resource K and might be destabilizing in the sense ofRosenzweig (1971) for all involved populations. A case of extremely asymmetrical competition between macroalgae and diatoms has been documented that leads to indirect commensalism; a one-way positive effect of a chiton on limpets by removing the competitively dominant macroalgae (Dethier and Duggins 1984). This intertidal system has exhibited short term constancy in that, whereas no long-term perturbation studies of the system were conducted to demonstrate stability, the observed effect was consistent for the duration of the experiment and there is little reason to expect the interacting populations to oscillate. As demonstrated by Stimson and Black (1975), limpet populations tend to converge on a single per-unit-area carrying capacity when disturbed. V. Stability of Assemblages

A review of stability, built around the backbone of levels of organization, might be discontinued after the consideration of multi species interactions. The reductionist approach to ecology assumes that the behavior of the assemblage of all organisms in a unit area can be predicted from the detailed knowledge of the population characteristics and pairwise interactions of involved species. The admission of the existence of higher -order interactions, where it occurs, is a step towards holism. They clearly remain within the domain of population ecology, however, because, once demonstrated, they can be directly incorporated into models of the three or more interacting species, without invoking community-level "emergent properties"

(Underwood 1986). There have been several discussions of assemblage stability, however, that have included influences of characteristics ( eg. species diversity) or been based on measures ( eg. total productivity) that only exist on an assemblage-wide basis.

It seems wise to consider the arguments that have occupied much of ecological thought, which debate the importance of assemblage-level influences on assemblage stability.

When an arbitrarily defined assemblage is considered as a whole, what characteristic measure should define its stability? In his analysis of the "community matrix" May (1973) discussed the stability of the joint equilibrium point in the

76 77 multidimensional phase space of all species, using the value of the community matrix element 'A'. The variable of interest, then, was species abundance and the variation of any one species population would imply that the whole was unstable. The result could be multidimensional oscillations in phase space or the extinction of one or more populations, and the end of the assemblage as defined by its component species.

MacArthur (1955), by contrast, also defined stability in terms of species abundance, but considered a large variation in one component species oflesser importance if other component species were little affected. Some sort of mean divergence from equilibrium seemed to be the key variable. Overall stability has been qualitatively judged by collectively measuring individual species' divergences from control abundances (Farrell 1988). A measure comparable to abundance is canopy cover of the visually dominant assemblage species, where the remaining assemblage is inferred

(Turner 1985, Johnson and Mann 1988). Other investigators have defined assemblage stability using measures such as total productivity (McNaughton 1977) or energy flow rate (Margalef 1963). If the assemblage is a useful unit of study with regards to overall stability, then there should be one measured property, the stability of which is considered and that is a consistent function of some assemblage-level characteristic for that kind of assemblage, eg. the stability of mean abundance could consistently increase with diversity. 78

Diversity

It is an "old intuition" (Strong 1992) that diversity begets stability in an assemblage. Disturbances in the form of extreme abundances of individual species have been claimed to be "spread out" amongst various species in the affected trophic level, decreasing the total impact (MacArthur 1955). If a herbivore species becomes abundant, then a diversity of primary carnivores will be less altered in abundances than one major primary carnivore and low herbivore abundance will impact any one carnivore less if there is a diversity of alternate herbivore species to choose from.

While this seems logical, it assumes a great deal with regards to population dynamics, including total overlap of resources and thus total competition at any given trophic level. Theory predicts instability of such system due to competitive exclusion. In giving the example of an introduced weed being suppressed by an introduced herbivorous beetle, which, in turn, ignored all other weed species that took the consumed plant's place, Strong (1992) points out that herbivory, at least, "is rarely if ever unified in terrestrial ecosystems".

When diversity leads to a "trophic tangle" of omnivory and generalism the result is predicted to be increased stability in terms of resistance of total biomass at a given trophic level to change given an alteration in primary productivity or number of trophic levels (Strong 1992). Detailed descriptions of complex food webs seem to even draw the utility of "trophic levels" as a concept into question (Polis 1991 ). Even highly 79 complex and interconnected food webs seem to show basic trophic level effects, however, if certain species exert a strong influence on other important species (Paine

1980, reviewed in Power 1992). Many species exert only minor effects on the structure of the surrounding assemblage, as can be demonstrated by their removal without inducing significant changes in other populations (Paine 1980). By randomly assembling interaction webs ofvarious numbers of species, May (1973) has demonstrated theoretically that stability, in fact, decreases with complexity.

McNaughton (1977) argues for the stabilizing of community properties by diversity based on the results of experiments where nutrient input was increased

(discrete disturbance), presence and absence of grazing was tested, and the response of total productivity to variable rainfall was measured in diverse and simple plant assemblages. More diverse systems were less impacted in terms of biomass. The relative abundances of constituent species changed, however, and it is unclear why total productivity should be considered a "community function". MeNaught on points out that such stability in a community property, mediated by "compensatory" shifts in species abundance can have a destabilizing effect on higher trophic levels, presumably also in terms of biomass. A species-oriented explanation of McNaughton's results might be similar to the following. Less diverse assemblages were early successional and probably had a high proportion of species selected for rapid growth. As succession proceeded species were added and evenness increased (Mellinger and McNaughton 80

1975), leading to higher diversity and a lower proportion of early successional species contributing to the total biomass. It is likely that both diversity and the assemblages'

"functional stability" were functions of their successional state. McNaughton (1977) goes on to give an example of a rapidly growing ephemeral species by chance being a major constituent species in a high diversity assemblage. This assemblage showed low stability in terms of total biomass compared to assemblages of equal diversity where the study species was only a minor component, and was given as an example of a decoupling of the usual diversity-stability relationship. It seems possible that high diversity assemblages composed mostly of early successional species could occur in nature and that these would respond in a manner similar to that described for

McNaughton's low diversity assemblages.

High diversity can also be caused by heterogeneity in microhabitats. Certain

"functional properties" in such a mixed assemblage might also be found to be stable, but due to the representation of a wide variety of plant species adapted to different environmental extremes rather than the assemblages diversity, per se. Orians (1974) points to the variety of ways stability (as related to diversity) in the ecological literature is defined, discusses the difficulties with experimentally supporting or refuting theory, and suggests that attempts to find a general relationship between diversity and stability will be fruitless. He states that emphasis should be put on understanding the interaction mechanisms that produce observed instances of stability in nature. 81

Altemative stable states

The occurrence of alternate stable states in assemblages has been long debated in the literature (Holling 1973, Sutherland 1974, 1981, Connell and Sousa 1983,

Peterson 1984, Paine 1985). Alternate stable states are theorized to be the result of multiple points of attraction, or stable equilibria, occurring in multispecies phase space

(Sutherland 1974). Theoretically, multiple equilibria can result when intraspecific or interspecific interactions are nonlinear (Holling 1973). Nonlinearities can include thresholds in vital rates or saturation effects on predation producing "breakpoints", where productivity exceeds consumers' capacity to keep it in check (May 1981).

Holling (1973) reviews some of the evidence (mostly from laboratory studies on insects) that nonlinearities exist capable of producing alternate stable states. Movement between equilibria is theorized to occur when environmental variation causes fluctuations in species numbers to exceed the "basin of attraction" for one equilibrium and move into that of the other equilibrium. Primary production, for instance, could increase after a favorable period and swamp slowly reproducing herbivores. If herbivores were limited in their own capacity to increase, then an equilibrium might be approached near the resource's carrying capacity, where it becomes self-limited.

Increases in one population can indirectly decrease predation on another population by providing an alternative food source for predators. The result can again be the

"escape" of a population from predatory control (Abrams 1987). Other equilibria are 82 possible if some of the species are missing from an assemblage (Sutherland 1974) as can happen through extinction in predator-prey interactions or unstable equilibria in linear competitive interactions where initial conditions determine the outcome (Horn and MacArthur 1972). These, by definition, become a different assemblage (Connell and Sousa 1983) as species are removed and will not be considered further here.

Connell and Sousa (1983) review much of the evidence for alternative stable states in the literature. All fall short of their criteria for demonstrated multiple equilibria because assemblages compared often vary in their physical environment, experiments are conducted where artificial controls are maintained for one of the observed states, or evidence is inadequate. Sutherland (1974) cites the argument that, if history plays a role in the present state of populations, then alternative stable states must exist. This assumes that observed states of assemblages are stable, which is often not the case. If alternative stable states do exist, however, they are certainly attributable to historical events. Dense ghost shrimp burrows have been shown to inhibit invasion by certain bivalves (Peterson 1984). When ghost shrimp are removed, bivalves invade and continue to show successful recruitment to plots occupied by conspecifics. Although plots were not followed for an adequate period of time to demonstrate resistance to reinvasion by ghost shrimp, Peterson gave mechanisms observed in other species which could make plots inhabited by bivalves resistant to reinvasion. Peterson argues that, in cases where species can alter their physical 83 environment to be favorable to their persistence, then alternate stable states can occur.

Such a mechanism has been demonstrated in the rocky intertidal zone of Chile (Paine

1985). Seastars are able to maintain surfaces devoid of mussels. When seastars are excluded long enough for mussels to establish themselves and attain a size where they are invulnerable to predation, the result is persistence as observed in fouling communities by Sutherland (1974). The difference, however, is that adult mussels are able to provide new recruits with refuges from predation within the matrix of the mussel bed and mussels were shown to occupy plots for many generations. Paine discussed evidence that starfish were able to continually reclaim space at a rate equal to mussel mortality along the fiinges of the mussel bed. The question of stability of the mussel population thus becomes dependent on time scale. Sutherland (1981) points out that, given long enough time scales, all populations are necessarily unstable. VI. Conclusion

The body of theory reviewed above predicts various outcomes for the dynamics of populations and collectives of populations, depending on environmental conditions and the population parameters of the species involved. These outcomes include instability. The debate in the literature is not over whether or not stability occurs in nature, but over the relative roles of environment and population characteristics in determining the observed dynamics. Recurrent throughout the various theoretical explorations is the importance of the uniqueness of the habitat within the defined spatial unit under study and the larger spatial scale in which that unit area is embedded. Two spatial units of equal size and with identical species and environment can nonetheless differ in their dynamics as predicted by theory due to differing amounts of refuge from the environment or from interactions with predators or competitors. If disturbance is occasional enough to be considered not a part of the environment, then differing disturbance regimes can alter the outcomes of interactions. Differing histories of disturbance in the patches, given identical disturbance regimes but relatively small spatial scales, as well as residual variance in demographic and environmental parameters can all lead to differing population and interaction dynamics and result in differing "stabilities". The influence of regional processes such as immigration on local dynamics can differ between sites and profoundly affect interaction outcomes. Of equal importance is the biology of the species concerned.

84 85

Theory can have little hope of accurately predicting the dynamics of populations and assemblages if it assumes that all life history stages of organisms are identical in their requirements or interactions. Knowledge of an organism's natural history can aid in the prediction of the outcome of its interactions with other species ( eg. metabolic change in zooplankton at low prey densities, "cannibalism" in parasitoids).

The continued presence offamiliar species is evidence of persistence, not stability. It has been argued that parameters leading to stability in commonly occurring intra- and interspecific interactions will be selected for in natural populations, because the alternative is fluctuations leading inevitably to extinction (Holling 1973, Hassel et al. 1976, Berryman and Millstein, 1989). Den Boer (1991), however, has shown that even random walk populations can persist for up to hundreds of years. Populations can persist while fluctuating nearly randomly within wide bounds when they are provided with "ceilings" ( eg. resource limitation) and "floors" ( eg. background immigration)(Strong 1986). Populations may even go extinct on a local scale but recolonize from some regional refuge when conditions become more favorable

(Andrewartha and Birch 1954). The literature provides examples of populations with nearly every degree of fluctuation (Connell and Sousa 1983). Connell and Sousa

(1983) suggest that emphasis be put on identifYing mechanisms that lead to persistence of populations or assemblages within certain bounds rather than attempting to identifY an equilibrium. Other researchers (eg. Diamond 1975, Gilpin et al. 1986) have 86 concerned themselves with the identification of "assembly rules" that "forbid" the coexistence of certain species and "allow" it to others. The result is a set of possible alternative assemblages that could arise in a given area immersed in a certain species pool. Contrary to Gilpin et a!. (1986), such alternative assemblages differ from alternative stable states or multiple equilibria in that the actual assemblages differ within a g1ven area.

The debate over ecological stability is a philosophically instructive example of the disagreement between scientists regarding the roles of mathematical theory and field studies. Field ecologists have no occasion to attempt to describe organisms' distribution and abundance without some theoretical basis. Otherwise the explanation would be 'why should it be otherwise?'. The incorporation of mathematics is a step beyond intuition or conceptual models toward a more powerful understanding of ecological systems. All that is discovered or implied by field studies can be represented mathematically as replacement of or elaboration on basic assumptions of simpler models. The result is predictions that, upon testing, either secure the power of prediction, or, more often, reveal the incompleteness of scientists' understanding. The generality of any model, mathematical or conceptual, relies on the applicability of its assumptions to the systems where it might be employed. When these assumptions regard the state of an environment (e.g. presence ofrefuges, carrying capacities), their validity is likely to vary. Assumptions regarding the interactions between individuals of 87 species (e.g. predator functional responses, competition coefficients) would be expected to be generally applicable. Often, however, mathematical representations of interactions are oversimplifications that, although descriptive in one situation, become unraveled in another context. The upper extent of an intertidal alga might change relative to another's if it is set not solely by interactions with grazers or the

environment but rather an interplay between the two. The roles of predator and prey might reverse with age or significant shifts in their mutual ratio. Most models are lacking without an understanding of the uniqueness of the local environment and theory is powerless in terms of generality without field studies to define the nature of the interactions between individuals in all of its complexity.

It has been my impression, drawn from a review of much of the literature on

ecological stability and, more significantly, the processes that could give rise to stability

or stable cycles in a constant-enough environment, that the theory based on local interactions and incorporating equilibria has been extremely productive. Whereas few

systems have been described well using theoretical predictions as a starting point, a myriad of testable hypotheses have been derived from this body of theory. Emphasis

should be placed on testing them without losing site of the uniqueness of the system at hand. Existing theory provides a framework within which exceptions can be found and perhaps explained. It is important that the issues of spatial and temporal scale as well

as habitat differences be kept at the forefront of ecological thought as these provide the 88 arena in which species interact. BIBLIOGRAPHY

Abrams, P (1980) Some comments on measuring niche overlap. Ecology 61: 44-49

Abrams, P (1983a) Arguments in favor of higher-order interactions. Am Nat 121: 887-891

Abrams, P (1983b) The theory of limiting similarity. Ann Rev Ecol Syst 14: 359-376

Abrams, P (1987) Indirect interactions between species that share a predator: varieties of indirect effects. Pages 38-54 in WC Kerfoot and A Sih, eds. Predation: direct and indirect impacts on aquatic communities. Univ Press ofNew England, Hanover, N.H.

Agami, M, and KR Reddy (1990) Competition for space between Eichhomia crassipes (Mart.) Solms and Pistia statiotes L. cultured in nutrient enriched water. Aquat Bot 3 8: 195-208

Andrewartha, HG and LC Birch (1954) The distribution and abundance of animals. Univ of Chicago Press, Chicago London

Andrews, RM (1991) Population stability of a tropical lizard. Ecology 72: 1204-1217

Arditi, R, and HR Ak~akaya (1990) Underestimation of mutual interference of predators. Oecologia 83: 3 5 8-3 61

Arditi, R, and AA Berryman (1991) The biological control paradox. Trends Ecol Evol 6:32

Arditi, R, and LR Ginzburg (1989) Coupling predator-prey dynamics: ratio dependence. J Theor Bioi 139: 311-326

Arditi, R, and H Salah (1992) Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73: 1544-1551

Atkinson, WD, and B Shorrocks (1984) Aggregation oflarval diptera over discrete and ephemeral breeding sites: The implications for coexistence. Am Nat 124: 336-351

Beddington, JR (1974) Mutual interference between parasites or predators and its effect on searching efficiency. J Anim Ecol 44: 331-340

89 90

Begon, M, JL Harper, and CR Townsend (1990) Ecology: individuals, populations, and communities. 2nd ed., Blackwell Scientific Publications, Boston, Oxford, London, Edinburgh, Melbourne

Ben-Shahar, R (1993) Does fencing reduce the carrying capacity for populations of large herbivores? J Trap Ecol 9: 249-253

Berryman, AA (1991) Stabilization or regulation: what it all means! Oecologia 86: 140-143

Berryman, AA (1992) The origins and evolution of predator-prey theory. Ecology 73: 1530-1535

Berryman, AA, and JA Millstein (1989) Are ecological systems chaotic-and if not, why not? Trends Ecol Evol4: 26-28

Blank, TH, TRE Southwood, and DJ Cross (1967) The ecology of the partridge. I. Outline of population processes with particular reference to chick mortality and nest density. J Anim Ecol37: 549-556

Boucher, DH, S James, and KH Keeler (1982) The ecology of mutualism. Ann Rev Ecol Syst 13: 315-347

Bovbjerg, RV (1959) Density and dispersal in laboratory crayfish populations. Ecology 40: 504-506

Branch, GM (1975) Intraspecific competition in Patella cochlear Born. J Anim Ecol 44: 263-281

Breton, LM, and JF Addicott (1992) Density-dependent mutualism in an aphid-ant interaction. Ecology 73: 2175-2180

Brown, JH (1986) Variation in desert rodent guilds: patterns, processes, and scales. Pages 185-205 in JHR Gee and PS Giller, eds. Organization of communities past and present. Blackwell Scientific Publications, Oxford

Bulmer, MG (1975) The statistical analysis of density dependence. Biometrics 31: 901-911

Connell, JH, and WP Sousa (1983) On the evidence needed to judge ecological stability or persistence. Am Nat 121: 789-824 91

Crete, M (1989) Approximation ofK carrying capacity for moose in eastern Quebec. Can J Zool67: 373-380

Cushman, lli and TG Whitham (1989) Conditional mutualism in a membracid-ant association: temporal, age-specific, and density-dependent effects. Ecology 70: 1040-1047

Cushman, lli and TG Whitham (1991) Competition mediating the outcome of a mutualism: protective services of ants as a limiting resource for membracids. Am Nat 138: 851-865

Dean, AM (1983) A simple model of mutualism. Am Nat 121: 409-417

Den Boer, PJ (1989) Comment on the article "On testing temporal niche differentiation in carabid beetles" by MLoreau. Oecologia 81: 97-98

DenBoer, PJ (1991) Seeing the trees for the wood: random walks or bounded fluctuations of population size? Oecologia 86: 484-491

DenBoer, PJ and JReddingius (1989) On the stabilization of animal numbers. Problems of testing. 2. Confrontation with data from the field. Oecologia 79: 143-149

Dennis, B, and ML Taper (1994) Density dependence in time series observations of natural populations: estimation and testing. Ecol Monogr 64: 205-224

Dethier, MN, and DO Duggins (1984) An "indirect commensalism" between marine herbivores and the importance of competitive hierarchies. Am Nat 124: 205-219

Dhondt, AA (1988) Carrying capacity: a confusing concept. Acta CEcologica 9: 337-346

Edgar, GJ (1993) Measurement of the carrying capacity of benthic habitats using a metabolic-rate based index. Oecologia 95: 115-121

Edwards, RY, and CD Fowle (1955) The concept of carrying capacity. Pages 589-602 in JB Trefethen, ed. Transactions of the twentieth north american wildlife conference. Wildlife Management Institute, Washington DC

Egerton, FN (1973) Changing concepts of the balance of nature. Quart Rev Biol48: 322-350 92

Eisenberg, RM (1966) The regulation of density in a natural population of the pond snail, Lymnaea elodes. Ecology 47: 889-906

Farrell, TM (1988) Community stability: effects oflimpet removal and reintroduction in a rocky intertidal community. Oecologia 75: 190-197

Forrester, GE (1990) Factors influencing the juvenile demography of a coral reef fish. Ecology 71: 1666-1681

Freedman, ill, JF Addicott, and B Rai (1987) Obligate mutualism with a predator: stability and persistence of three-species models. TheorPop Bioi 32: 157-175

Giles, RH Jr (1978) Wildlife management. Freeman, San Francisco

Gilpin, ME (1972) Enriched predator-prey systems: theoretical stability. Science 177: 902-904

Gilpin, ME, MP Carpenter, and MJ Pomerantz (1986) The assembly of a laboratory community: multispecies competition in Drosophila. Pages 23-40 in J Diamond and TJ Case, eds. Community ecology. Harper and Row, New York

Ginzburg, LR, and HR Akc;akaya (1992) Consequences of ratio-dependent predation for steady-state properties of ecosystems. Ecology 73: 1536-1543

Goh, BS (1979) Stability in models of mutualism. Am Nat 113: 261-275

Gould, JR, JS Elkinton, and WE Wallner (1990) Density-dependent suppression of experimentally created gypsy moth, Lymantria dispar, populations by natural enemies. J AnimEcol 59: 213-233

Gutierrez, AP (1992) Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm. Ecology 73: 1552-1563

Hassel, MP (1975) Density dependence in single-species populations. J Anim Ecol 44: 283-295

Hassel, MP, JH Lawton, and RM May (1976) Patterns of dynamical behavior in single species populations. J Anim Ecol 45: 471-486

Holling, CS (1959) The components of predation as revealed by a study of small-manunal predation of the European pine sawfly. Can Entomol 91: 293-320 93

Holling, CS (1973) Resilience and stability of ecological systems. Ann Rev Ecol Syst 4: 1-24

Hornfeldt, B (1994) Delayed density dependence as a determinant of vole cycles. Ecology 75: 791-806

Hurd, LE, and RM Eisenberg (1984) Experimental density manipulations of the predator Tenodera sinensis in an old field community. I. Mortality, development and dispersal ofjuvenile mantids. J AnimEcol53: 269-281

Ives, AR (1992) Density-dependent and density independent parasitoid aggregation in model host-parasitoid systems. Am Nat 140: 912-937

Ito, Y (1972) On the methods for determining density-dependence by means of regression. Oecologia 10: 347-372

Johnson, CR, and KH Mann (1988) Diversity, patterns of adaptation, and stability of Nova Scotian kelp beds. Ecol Monogr 58: 129-154

Kareiva, P (1987) Habitat fragmentation and the stability of predator-prey interactions. Nature 326: 388-390

Kat, PW (1982) Effects of population density and substratum type on growth and migration of Elliptio complanata. Malacol Rev 15: 119-127

Keller, BD (1983) Coexistence of sea urchins in seagrass meadows, an experimental analysis of competition and predation. Ecology 64: 1581-1598

Krebs, JR (1970) Regulation of numbers in the great tit. J Zoo!, Land 162: 317-333

Lamont, BB, and SM Berg! (1991) Water relations, shoot and root architecture, and phenology of three co-occurring Banksia species: no evidence for niche differentiation in the pattern of water use. Oikos 60: 291-298

Levitan, DR (1989) Density-dependent size regulation inDiadema antillanmz: effects on fecundity and survivorship. Ecology 70: 1414-1424

Lewontin, RC (1969) The meaning of stability. Pages 13-24 in GM Woodwell and HH Smith, eds. Diversity and stability in ecological systems. Brookhaven Symp Bioi, Vol. 22 94

Lidicker, WZ Jr (1975) The role of dispersal in the demography of small mammals. Pages 73-102 in FB Galley, K Petrusewicz, and L Ryszkowski, eds. Small mammals: their productivity and population dynamics. Cambridge Univ Press, Cambridge

Lareau, M (1989) On testing niche differentiation in carabid beetles. Oecologia 81: 89-96

Lotka, AJ (1956) Elements of mathematical biology. Dover Publications, Inc. New York

MacArthur, R (1955) Fluctuations of animal populations and a measure of community stability. Ecology 36: 533-536

MacArthur, R, and R Levins (1967) The limiting similarity, convergence, and divergence of coexisting species. Am Nat 101: 377-385

Maelzer, DA (1970) The regression of log Nt+I on log N, as a test of density dependence: an exercise with computer-constructed density-independent populations. Ecology 51: 810-822

Mahdi, A, R Law, and AJ Willis (1989) Large niche overlaps among coexisting plant species in a limestone grassland community. J Ecol 77: 3 86-400

Margalef, R (1963) On certain unifying principles in ecology. Am Nat 97: 357-374

Margalef, R (1969) Diversity and stability: a practical proposal and a model of interdependence. Pages 25-37 in Diversity and stability in ecological systems. Brookhaven Symp Bioi no. 22

Margalef, R (1975) Diversity, stability, and maturity in natural ecosystems. in WH van Dobben and RH Lowe-McConnell, eds. Unifying concepts in ecology. Dr W Junk BV Publishers, The Hague

Matthews, WJ, RC Cashner, and FP Gelwick (1988) Stability and persistence offish faunas and assemblages in three midwestern streams. Copeia 1988: 945-955

May, RM (1973) Limit cycles in predator-prey communities. Science 177: 900-902

May, RM (1973) Stability and complexity in model ecosystems. Princeton Univ Press, Princeton N.J. 95

May, RM (1978) Host-parasitoid systems in patchy environments: a phenomenological model. J AnimEcol47: 833-843

May, RM (198la) Models for single populations. Pages 5-29 in RM May, ed. Theoretical ecology: principles and applications. Second edition. Saunders, Philadelphia

May, RM (198lb) Models for two interaction populations. Pages 78-104 in RM May, ed. Theoretical ecology: principles and applications. Second edition. Saunders, Philadelphia

May, RM, GR Conway, MP Hassel, and TRE Southwood (1974) Time delays, density dependence and single species oscillations. J AnimEcol43: 747-770

Maynard Smith, J (1973) Models in Ecology. Cambridge Univ Press, London

McAllister, CD, RJ LeBrasseur, and TR Parsons (1972) Stability of enriched aquatic ecosystems. Science 175: 562-564

Mcintosh, RP (1982) The background and some current problems of theoretical ecology. Pages 1-62 in Esa Saarinen, ed. Conceptual issues in ecology. D Reidel Publishing Co., Dordrecht, Holland

McNaughton, SJ (1977) Diversity and stability of ecological communities: a comment on the role of empiricism in ecology. Am Nat 111: 515-525

Miller, TE, and WC Kerfoot (1987) Redefining indirect effects. Pages 33-37 in WC Kerfoot and A Sih, eds. Predation: direct and indirect impacts on aquatic communities. Univ Press ofNew England, Hanover, N.H.

Morris, RF (1959) Single-factor analysis in population dynamics. Ecology 40: 580-588

Murdoch, WW (1970) Population regulation and population inertia. Ecology 51: 497-502

Murdoch, WW (1994) Population regulation in theory and practice. Ecology 75: 271-287

Murray, BJ Jr (1982) On the meaning of density dependence. Oecologia 53: 370-373

Nicholson, AJ (1933) The balance of animal populations. J Anim Ecol2: 132-178 96

Nicholson, AJ (1954) An outline of the dynamics of animal populations. Austr J Zoo! 2: 9-65

Nicholson, AJ, and VA Bailey (1935) The balance of animal populations. Part I. Proc Zoo! Soc London: 551-598

Oaten, A and WW Murdoch (1975a) Functional response and stability in predator-prey systems. Am Nat 109:289-298

Oaten, A and WW Murdoch (1975b) Switching, functional response, and stability in predator-prey systems. Am Nat 109: 299-318

Odum, EP (1953) Fundamentals ofEcology. Saunders, Philadelphia

Oksanen, L, SD Fretwell, J Arruda, and P Niemala(1981) Exploitation ecosystems in gradients of primary productivity. Am Nat 118:240-261

Oksanen, L, J Moen, PALundberg (1992) The time-scale problem in exploiter-victim models: does the solution lie in ratio-dependent exploitation? Am Nat 140: 938-960

Orians, GH (1975) Diversity, stability, and maturity in natural ecosystems. Pages 139-150 in WH van Dobben and RHLowe-McConnell, eds. UnifYing Concepts in Ecology. Dr W Junk BV Publishers, The Hague

Pacala, SW, MP Hassell, RM May (1990) Host-parasitoid associations in patchy environments. Nature 344: 150-153

Paine, RT (1974) Intertidal community structure: experimental studies on the relationship between a dominant competitor and its principal predator. Oecologia 15: 93-120

Paine, RT (1980) Food webs: linkage, interaction strength and community infrastructure. J Anim Ecol49: 667-685

Paine, RT, JC Castillo, and J Cancino (1985) Perturbation and recovery patterns of starfish-dominated intertidal assemblages in Chile, New Zealand, and Washington State. Am Nat 125: 679-691

Persson, L (1985) Asymmetrical competition: are larger animals competitively superior? Am Nat 126:261-266 97

Polis, GA (1991) Complex trophic interactions in deserts: an empirical critique of food web theory. Am Nat 138: 123-155

Pollard, E, KH Lakhani, and P Rothery (1987) The detection of density dependence from a series of animal censuses. Ecology 68: 2046-2055

Pomerantz, MJ (1981) Do "higher-order interactions" in competition systems really exist? Am Nat 117: 583-591

Power, ME (1992) Top-down and bottom-up forces in food webs: do plants have primacy? Ecology 73: 733-746

Real, L. (1977) The kinetics offimctional response. Am Nat 111: 289-300

Reddingius J (1971) Gambling for existence. A discussion of some theoretical problems in animal population ecology. Acta Biotheor (Leiden) 20:(Suppl) 1-208

Reddingius, J and PJ den Boer (1989) On the stabilization of animal numbers. Problems of testing. 1. Power estimates and estimation errors. Oecologia 78: 1-8

Reeve, JD, and WW Murdoch (1985) Aggregation by parasitoids in the successful control of the California red scale: a test oftheory. J Anim Ecol54: 797-816

Reeve, JD, and WW Murdoch (1986) Biological control by the parasitoid Aphytis me linus, and population stability of the California red scale. J Anim Ecol 55: 1069-1082

Ricklefs, RE (1987) Community diversity: relative roles oflocal and regional processes. Science 235: 167-171

Rosenzweig, ML (1971) Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171: 3 85-3 87

Rosenzweig, ML (1972) Reply to McAllister et al. 1972. Science 175: 564-565

Rosenzweig, ML, and RH MacArthur (1963) Graphical representation and stability conditions of predator-prey interactions. Am Nat 97: 209-223 98

Roughgarden, J (1986) A comparison offood-limited and space-limited animal competition communities. Pages 492-516 in JDiamond and TJ Case, eds. Community ecology. Harper and Row, Publish. New York Royama, T (1977) Population persistence and density dependence. Ecol Monogr 47: 1-35

Royama, T (1981) Fundamental concepts and methodology for the analysis of animal population dynamics, with particular reference to univoltine species. Ecol Monogr 51: 473-493

Royama, T (1984) Population dynamics of the spruce budworm Choristoneura fumiferana. Ecol Monogr 51: 429-462

Salt, GW (1979) A comment on the use of the term emergent properties. Am Nat 113: 145-148

Salvanes, AGV, DL Aksnes, and J Giske (1992) Ecosystem model for evaluating potential cod production in a west Norwegian fjord. Mar Ecol Prog Ser 90: 9-22

Savolainen, R, and K Vepsalainen (1989) Niche differentiation of ant species within territories of the wood antFormicapolyctena. Oikos 56: 3-16

Schoener, TW (1970) Non-synchronous spatial overlap of lizards in patchy habitats. Ecology 51: 408-418

Schoener, TW (1974a) Some methods for calculating competition coefficients from resource utilization spectra. Am Nat 108: 332-340

Schoener, TW (1974b) Resource partitioning in ecological communities. Science 185: 27-39

Shorrocks, B, and J Rosewell (1986) Spatial patchiness and community structure: coexistence and guild size of drosophilids on ephemeral resources. Pages 29-52 in JHR Gee and PS Giller, eds. Organization of communities past and present. Blackwell Scientific Publications, Oxford

Sih, A (1979) Stability and prey behavioural responses to predator density. J Anim Ecol48: 79-89

Slade, NA (1977) Statistical detection of density dependence from a series of sequential censuses. Ecology 58: 1094-1102 99

Slatkin, M (1974) Competition and regional coexistence. Ecology 55: 128-134

Sousa, WP (1984) The role of disturbance in natural communities. Ann Rev Ecol Syst 15: 353-391

Spight, TM, and J Emlen (1976) Clutch sizes of two marine snails with a changing food supply. Ecology 57: 1162-1178

Steele, JH (1974) Spatial heterogeneity and population stability. Nature 248: 83

Stimson, J, and R Black (1975) Field experiments on population regulation in intertidal limpets of the genusAcmaea. Oecologia 18: 111-120

Strong, DR (1986) Density vagueness: abiding the variance in the demography of real populations. Pages 257-268 in J Diamond and TJ Case, eds. Community ecology. Harper and Row, New York, New York, USA

Strong, DR (1992) Are trophic cascades all wet? differentiation and donor-control in speciose ecosystems. Ecology 73: 7 4 7-7 54

StAmant, JLS (1970) The detection of regulation in animal populations. Ecology 51: 823-828

Sutherland, JP (1970) Dynamics of high and low populations of the limpetAcmea scabra (Gould). Ecol Monogr 40: 169-188

Sutherland, JP (1974) Multiple stable points in natural communities. Am Nat 108: 859-873

Taylor, LR, and RAJ Taylor (1977) Aggregation, migration, and population mechanics. Nature 265: 415-420

Taylor, RAJ, and LR Taylor (1979) A behavioral model for the evolution of spatial dynamics. in RM Anderson, BD Turner, and LR Taylor, eds. Population dynamics. Blackwell, Oxford, England

Thompson, DJ (1975) Towards a predator-prey model incorporating age structure: the effects of predator and prey size on the predation of Daphnia magna by Ischnura elegans. J Anim Ecol44: 907-916

Turchin, P (1990) Rarity of density dependence or population regulation with time lags? Nature 344: 660-663 100

Turchin, P and AD Taylor (1992) Complex dynamics in ecological time series. Ecology 73: 289-305 Turner, T (1985) Stability of rocky intertidal surfgrass beds: persistence, preemption, and recovery. Ecology 66: 83-92

Underwood, AJ (1986) What is a community? Pages 351-367 in DMRaup and D Jablonski, eds. Patterns arid processes in the history oflife. Springer-Verlag Berlin, Heidelberg

Vandermeer, m (1973) On the regional stabilization oflocally unstable predator-prey relationships. J theor Biol41: 161-170

Varley, GC and GR Gradwell (1960) Key factors in population studies. J Anim Ecol 29: 399-401

Varley GC, GR Grad well, and MP Hassell (1973) Insect population ecology, an analytical approach. Blackwell, Oxford London

Vickery, WL, and TD Nudds (1984) Detection of density-dependent effects in annual duck censuses. Ecology 65: 96-104

Volterra, V (1926) Variazioni e fluttuaazioni del numero d'individui in specie animali conviventi. Mem Accad Lincei 2(Ser. VI): 31-113 (English translation of shortened version in RN Chapman (1931) Animal Ecology, McGraw-Hill, New York)

Wilbur, HM, and JE Fauth (1990) Experimental aquatic food webs: interactions between two predators and two prey. Am Nat 135: 176-204

Walda, H (1989a) Comment on the article "On testing temporal niche differentiation in carabid beetles" by M Lareau and the "Comment on the article ofM Lareau" by PJ den Boer. Oecologia 81: 99

Walda, H (1989b) The equilibrium concept and density dependence tests. What does it all mean? Oecologia 81: 430-432

Walda, H (1991) The usefulness of the equilibrium concept in population dynamics. A reply to Berryman. Oecologia 86: 144-145

Wootton, IT (1993) Indirect effects and habitat use in an intertidal community: interaction chains and interaction modifications. Am Nat 141: 71-89 101

Worthen, WB, and JL Moore (1991) Higher-order interactions and indirect effects: a resolution using laboratory Drosophila communities. Am Nat 138: 1092-1104

Yodzis, P (1986) Competition, mortality, and community structure. Pages 480-491 in J Diamond and TJ Case, eds. Community ecology. Harper and Row, Publish. New York