[advances in ecological research] advances in ecological research volume 24 volume 24 || food webs:...

Food Webs: Theory and Reality

S . J . HALL and D . G . RAFFAELLI

I . Introduction . . . . . . . . . . . . . . . . . . . I1 . A Brief Review of Patterns Catalogued to Date . . . . . . . .

A . Intervality . . . . . . . . . . . . . . . . . . . B . Topological Holes . . . . . . . . . . . . . . . .

I11 . New and Better Data . . . . . . . . . . . . . . . . . IV . Do the Patterns Hold up with Better Data? . . . . . . . . .

B . Scale Invariant Patterns . . . . . . . . . . . . . . C . S-C Relationships . . . . . . . . . . . . . . . . D . The Length of Food Chains . . . . . . . . . . . . . E . Taxonomic Resolution . . . . . . . . . . . . . . .

V . Theory and Data . . . . . . . . . . . . . . . . . . A . Lotka-Volterra: an Industry Standard . . . . . . . . . B . Model Predictions and the Effects of Data Re-evaluation . . . C . Community Assembly . . . . . . . . . . . . . . . D . The Cascade Model . . . . . . . . . . . . . . . .

VI . Profitable Pursuits and Blind Alleys . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

C . Compartments . . . . . . . . . . . . . . . . .

A . Omnivory . . . . . . . . . . . . . . . . . . .

187 189 190 192 193 194 195 196 201 203 206 208 213 214 216 226 231 232 235 236

I . INTRODUCTION A principal goal for ecology is to understand the nature of species interactions and to determine the extent to which they can explain the observed patterns and dynamic properties of biological communities . Perhaps the most obvious interaction of all is predation. and it is not surprising that from very early in the history of ecological research documenting what eats what has been a priority . Formal presentation of the feeding links between species has often been in the form of a food web graph and these images have led some to draw an analogy between food webs and road maps (Pimm et al., 1991) . Implicit in such an analogy is a message that food webs can help us find our way through the ecological complexity that confronts us .

ADVANCES IN ECOLOGICAL RESEARCH VOL . 24 ISBN 0-12-013924-3

Copyright 1993 Academic Press Limited A / / righrs of reproduction in any form reserved

187

188 S. J. HALL A N D D. G. RAFFAELLI

Recently, a number of authors have questioned the utility of food web patterns as a route toward a better understanding of natural systems (Paine, 1988; Winemiller, 1989, 1990; Martinez, 1991). A particularly contentious issue is whether, given the variable quality of data on real food webs, we should even attempt to relate observed static patterns or properties of documented webs to inferences about dynamical processes derived from modelling studies. Many food web models either incorporate assumptions about specific properties of webs or make predictions about those properties. Clearly, if our current perception of these properties is not correct, then the foundation on which some models are based is shaky. Also, models which have made predictions thought to be consistent with observed patterns will require re-evaluation.

Two recent reviews have sought to address these problems (Lawton, 1989; Pimm et al., 1991). These will deservedly remain as landmarks in the food web literature for some time to come and those who have read them may wonder why yet another review is warranted. We can think of three good reasons.

First, several large, well-documented webs have been published since Lawton’s review and analysis of their properties suggests non-conformity with the patterns previously described. This is, perhaps, unsurprising because the previous catalogue of webs is viewed as being of “highly variable quality, hardly any of which is really good” (Lawton, 1989). Aspects of two of these new data sets were discussed by Pimm et al. (1991), but these authors did not have the opportunity to explore their implications fully, and were unaware of other large webs subsequently published. We will show that con- sideration of these new data forces a re-evaluation of several web properties.

Second, several papers have recently addressed a major concern of food web theorists, i.e. the variable taxonomic resolution of web entities, both between and within webs. Sensible searching for recurrent patterns in real webs usually rests on the presumption that all the webs used in the analyses are documented to a similar degree, and that all trophic levels are resolved to the same degree. This is untrue for the collection of webs which have been analysed for patterns to date. One approach to overcoming this problem has been to try and standardize this aspect of webs by grouping entities into “kinds” of species (Cohen et al., 1990). Since the reviews of Lawton (1989) and Pimm et al. (1991), there have been two independent studies of the effects of taxonomic resolution on web properties (Martinez, 1991; Hall and Raffaelli, 1991) which indicate that this problem may be more significant than was previously assumed (Pimm et al., 1991).

Thirdly, although this review covers much the same ground as those by Lawton (1989) and Pimm et al. (1991), we have arrived at different conclusions as to the severity of the problems concerning the use of real web data for parametrizing and testing food web models. Pimm et al. (1991)

FOOD WEBS: THEORY AND REALITY 189

concluded that “present evidence suggests that most patterns appear across all webs” and that “recent studies to address limitations in the data have generally confirmed the patterns”. We do not agree. Hence, our third motive for writing this review is to present an alternative perspective on the usefulness of some of the food web data for deriving patterns and properties and to discuss the implications this has for web models.

A review like this will inevitably have some hard things to say about particular analyses or approaches if we are to fulfil our remit. We are conscious that some aspects of the subject have suffered from an overoptimistic inter- pretation of data on the one hand, and from overzealous criticism on the other. It is only too easy to be nihilistic with the data on food web patterns, but this is not the spirit in which our criticisms are offered. We believe that, if we are to progress in this exciting area, recognition of the considerable shortcomings in much of the data (e.g. Lawton, 1989; Pimm et al., 1991) is only the first step. We argue that alternative approaches to the analysis of natural systems are likely to be much more profitable than further re-work- ing of the existing catalogue of food web data.

11. A BRIEF REVIEW OF PATTERNS CATALOGUED TO DATE

Published food webs combine two sets of information about the community of animals and plants they represent: a list of the elements in the system, and the distribution of feeding links (information on who eats whom). For all practical purposes, feeding links are represented as a simple “yes” or “no”, since data on how much of a prey a given predator eats are rarely available. Data are presented in the familiar form of a food web graph with feeding links represented by vertices between predator and prey. Alternatively, a binary interaction matrix with predators as rows and prey as columns is used.

Three broad categories of web can be found in the literature: sink, source and community webs. Sink webs are constructed by tracing trophic links downwards from a single species, usually a top predator. The predator’s prey are identified, then their prey, and so on. Source webs are constructed by tracing trophic links upwards from a species, using a basal resource. Its consumers are identified, then their consumers, and so on. For both sink and source webs, the trophic pathways identified define which species will be included in the web. This is in contrast to a community web where a collection of species is defined, usually within a habitat, and then the trophic links between them identified. Sink and source webs usually represent only part of much larger community webs.

Armed with this binary data, a variety of simple statistics can be calculated to describe food web properties (Table 1). Often web elements which are perceived to have similar prey and predators are combined into “trophic

190 S. J . HALL A N D D. G. RAFFAELLI

Table 1 Statistics or properties that can be calculated from food web data

Statistic/property Definition

Web size

Connectance

Linkage density

Numbers of top intermediate and basal elements

Number of feeding loops

Food chain lengths

Number of omnivores

Total number of elements represented in the

The proportion of realized links in the web. web (S)

“Lower” connectance is the proportion of realized trophic links, “Upper” connectance is lower connectance plus presumed competitive links between predators which share prey

The number of vertices on a food web graph or 1’s in a matrix (L) divided by the number of web elements (S)

Top species: those that have no predators. Intermediate species: those that have both

Basal species: those that have predators,

A feeding loop is where feeding links can be

predators and prey.

but no prey

traced in a circuit back to the starting point, i.e. element A eats B, B eats C and C eats A

Food webs are composed of food chains which run from each top predator to the basal elements. The chain length is the number of links in this path and is one less than the number of elements in the chain. A number of chains of different lengths can run between a top and basal element, depending on the links with intermediate elements

than one trophic level Omnivores are organisms which feed on more

species” o r “kinds of species” prior to any analysis of web statistics. The search for general patterns in these statistics for the growing population of available webs has been a cornerstone of food web research, and summary lists of the general patterns have appeared in a number of publications (e.g. Lawton and Warren, 1988; Lawton, 1989; Pimm et al., 1991). Table 2 paraphrases these earlier summaries, but for clarity we deal in more detail with three of the less obvious concepts below.

A. Intervality

If, instead of drawing a line to link predators to prey as in a conventional food web graph (Fig. l (a)) , one links elements that share common prey, the graph obtained is termed a “predator overlap graph” (Fig. l(b)).

FOOD WEBS: THEORY AND REALITY

Table 2 Patterns found in food web statistics

191

1. The average proportion of top, Patterns 1, 2 and 3 are independent of the number of elements in the web (Cohen and Briand, 1984; Briand and Cohen 1984) and are termed “scale invariant”. Despite showing high variance about the mean, values for these statistics are often referred to as being “roughly constant”

intermediate and basal elements

2. The average proportion of feeding links between top-intermediate, top-basal, intermediate-intermediate and intermediate-basal elements

3. The ratio of predators to prey Independent data sets gathered by a number of authors support the conclusion that the ratio of predators to prey is “roughly constant” (Evans and Murdoch, 1968; Arnold, 1972; Cameron, 1912; Moran and Southwood, 1982; Jeffries and Lawton, 1985)

4. Linkage density

5. Feeding loops

6. Food chain lengths

7. Omnivory

Once thought to be constant, but now believed to increase for large webs (Cohen et al., 1990). (Constant density leads to a hyperbolic decline in connectance with increasing web size)

Rarer than would be expected by chance (Pimm and Lawton, 1978, 1980; Pimm, 1982)

Food chain lengths tend to be short, typically with only three or four links between basal and top elements. Chains involving more than six species are rare (Hutchinson, 1959; Pimm, 1982; Cohen et al., 1986)

Omnivory is less common in some kinds of real webs than in randomly generated webs (Pimm, 1982)

Predator overlap graphs were originally termed “niche overlap graphs” because two vertices (elements) in the graph are only joined if they share a common resource, and the graphs therefore represent the trophic overlap or potential competitive structure of the community. Study of the graphical properties of predator overlap graphs was first begun by Cohen (1978) who showed that these pictures could often be collapsed into one-dimensional representations known as “interval graphs” (Fig. 2(a)). This could be achieved more often than would be expected by chance. Webs with predator overlap graphs that can be collapsed in this way are described as “interval”.

192 S. J. HALL AND D. G. RAFFAELLI

(a) Food web (b) Predator overlap (c) Prey overlap

@ 6

Fig. 1. Food web, predator and prey overlap graphs. The food web graph (a) links predators to prey, the predator overlap graph (b) links predators which share prey and the prey overlap graph (c) links prey which share predators.

Predator overlap graphs from interval webs often show another property known as “rigid circuitry” (Fig. 2(b)). A circuit exists if a path can be traced between some or all of the vertices along the links in a niche overlap graph and the path returns to its starting point. A niche overlap graph is said to be “rigid circuit” if, for every circuit with more than three vertices, a shorter circuit can be traced. Although most rigid circuit graphs are interval, configurations which have radiating arms (termed “asteroidal graphs”, Fig. 2(c)) lead to non-intervality.

B. Topological Holes The conjugate, or inside-out version, of the predator overlap is the prey overlap or resource graph (Sugihara, 1982); these graphs link prey elements which share one or more predators (Fig. l(c)) . Prey overlap graphs from webs with small numbers of species have been shown to lack features known as topological holes. To understand the idea of a topological hole, consider four prey which share a predator. In a prey overlap graph, each prey will be connected by a line to every other. Now consider a physical analogy where the lines which connect any three of these prey form a triangular plane, and arranging all four prey forms a solid tetrahedral structure. This structure is the prey overlap graph for this single predator. Other predators may share some of these prey, however, and eat other prey not represented in the tetrahedron. Graphs for these other predators can also be considered as forming solid structures which must be joined to the original. When they are joined, vertices representing the same prey cannot occur in two places so some corners of the structure must co-occur. Combining these solids could sometimes produce a structure with holes in it. The equivalent in the mathematics of graph theory is the topological hole, which can occur in multi- dimensional structures for which there is no physical analogue. The signifi- cance of this property has yet to be fully explored.


@-@ 3-3 2-2 4-4 1-1 5-56-6 @

Predator overlap

(b) Rigid circuit

Predator overlap

Not rigid circuit

ps: Predator overlap

Interval graph

1-1 2-2 3-3

4- 4

Interval graph

1 cannot be placed

2-2 3-3 4-4

Not an interval graph

(c) Asteroidal

s 1 cannot be placed

2- 2 4-4 6-6

5-5 3- 3 7- 7

Fig. 2. (a) A predator overlap graph can often be represented as an interval graph where segments on a line represent graph elements and overlap between line segments indicates shared resources. (b) Overlap graphs from webs which are interval often show rigid circuit properties, but asteroidal configurations can be rigid-circuit yet still non-interval. (Part (c) redrawn from Pimm et al. (1991).)

C. Compartments If feeding links in a food web are arranged in blocks such that there are many links within blocks, but few between them the web is said to be divided into “compartments”, or “modules”. May (1973) suggested that, “as an


interesting corollary” which “should not be taken too seriously”, this could confer stability on systems, a comment which has prompted several discus. sions of whether compartments occur in nature (e.g. Goh, 1979; Paine, 1980; Pimm and Lawton, 1980; Yodzis, 1982; Moore and Hunt, 1988; Tho- mas, 1990; Winemiller, 1990). Paine (1980) provides a dynamical explana. tion for the evolution and maintenance of compartments based on his extensive work in the marine intertidal. In his model, keystone predators keep superior competitors in check allowing groups of other species to co-evolve sophisticated mutualisms, thereby generating web infrastructure. There is compelling evidence for comparmentation in some rocky shore systems (Raffaelli and Hall, 1992), and habitat-based compartments seem intrinsically reasonable.

111. NEW AND BETTER DATA

Since Lawton’s (1989) review and the major synthesis by Cohen et al. (1990), several analyses of new webs have been published which, by virtue of their large size and/or high degree of documentation, offer an opportunity to re- evaluate catalogued patterns. Since much of the following discussion is based on the analyses of these webs it is appropriate to set the scene by briefly describing the natural history of these systems.

Coachella Valley web ( Polis, 1991 ). This analysis of a North American desert system is the most intensive and extensive of any terrestrial community web to date. The web comprises thousands of species described over a large geo- graphical area. Interactions between species are documented to an extra. ordinarily high degree with good taxonomic resolution at the lower trophic levels. Many of the species in the web are arthropods but vertebrates are also included. Not all the interactions were documented by the observer and many are reasonable assumptions based on a detailed knowledge and familiarity with the system. However, most of the important statements about web patterns are based on a simplified web of only 30 “kinds of species”.

Little Rock Lake web (Martinez, 1991). This system is a large North Amer- ican freshwater community web consisting of 93 trophic species based on a 182 taxa web. Taxonomic resolution is good at all trophic levels. The web has been partly constructed on the basis of discussions with other experts as to the likely feeding links given the list of species recorded from the lake.

Ythan Estuary web (Hall and RafSaelli, 1991). The Ythan Estuary in north- east Scotland comprises just under 100 species with over 400 interactions documented. There are no hypothesized links. Taxonomic resolution is good at all but the basal levels.

Tropical South American Freshwater webs ( Winemiller, 1990). This analysis


comprises a large swamp and a small stream in Costa Rica and a similar swamp and stream in Venezuela. The webs presented are complex and well documented and are based on gut contents analysis of fish. Sink webs were fleshed out from information in the literature on the likely diets of the prey in the stomachs. However, these latter interactions were omitted from the author’s calculations of food web statistics.

Temperate freshwater pond webs ( Warren, 1989). These are seasonal versions of the web found in a large pond in Yorkshire, England comprising 12-32 species. The webs are largely complete and the interactions between species have been extensively documented using gut contents analysis, feeding trials and inference from the literature.

Pelagic freshwater webs (Havens, 1992). This is a catalogue of 50 pelagic communities of small lakes and ponds in New York State, ranging in size from 10 to 74 species. Taxonomic resolution is good (genus and species level), Pelagic assemblages were censused and linkages inferred from the literature, rather than direct observation.

Insect webs (Schoenly et al., 1991 ). This is a catalogue of 95 insect dominated webs spanning a range of web sizes, habitat types and of variable data quality. Of the webs 75% are new to previous compilation efforts, and 61 of the webs are considered community webs. Taxonomic resolution is generally good. This catalogue contains all 60 arthropod dominated webs previously analysed by Sugihara et al. (1989), although several pairs of webs listed in the earlier analysis are amalgamated in Schoenly et al. (1991). These two analyses are not therefore independent, and only Schoenly et al.’s findings, which include and extend those of Sugihara et al. (1989) are reported here.

IV. DO THE PATTERNS HOLD UP WITH BETTER DATA?

Table 3 shows values for several food web properties derived from these new data sets. Values from Cohen et al. (1986) and Briand (1983) are included for comparison. There are of course shortcomings in all of these new analyses. For instance, do the hypothesized interactions in the Little Rock web actually occur in the system? How much is lost from the Coachella Valley analysis by reducing the web to a matrix of 30 elements? Are the dung and parasitoid webs analysed by Schoenly et al. true webs or pieces of a much larger web? Does the poor resolution of basal species in the Ythan web render analyses of its web properties unwise? These shortcomings (and others) can be found in all the webs reported in the literature, but these new data sets do suffer less than most in these respects. Nevertheless, some of the new webs are more appropriate for the analyses of particular properties than others.

196 S . J . HALL AND D. G. RAFFAELLI

Here we argue that these new studies have important implications for the status of now well-accepted web patterns and for the pattern-seeking approach to the study of food webs. In developing these arguments we have chosen to focus on four major patterns: omnivory, scale invariance, connectance and the length of food chains. The same points could be illustrated with most of the patterns described in the previous section, but we have selected these four patterns, either because they are particularly open to criticism or because, as we show later, the pattern has particular importance for food web modelling. We then go on to consider the sensitivity of these properties to the degree of taxonomic resolution in the data.

A. Omnivory

A number of definitions of omnivory are possible, but in the broadest terms food web analysts define an omnivore as an animal that feeds on more than one trophic level. (As we shall see, however, this broad definition is sufficiently imprecise to allow a variety of methods for assigning species the status of omnivore.) Omnivory is less common in real webs (except those characterized by insects and parasitoids) than in randomly generated webs (Pimm, 1982). For the food webs in Cohen et al. (1986) and Briand (1983) about 27% of species are omnivores, and in Schoenly et al.’s (1991) insect web catalogue this figure is 22% (Table 3). For the Ythan Estuary web, 21% or 34% of species are omnivores, depending on how the trophic structure is initially arranged, i.e. whether epibenthic invertebrate predators, such as crabs, shrimps and some polychaetes are included with fish to give a four- level structure, or are kept separate to give five trophic levels (Hall and Raf- faelli, 1991). At first sight these values for the Ythan are not at variance with webs in general. However, if an alternative metric (the degree of omnivory) is used for comparing levels of omnivory between webs, a rather different pic- ture emerges. The degree of omnivory is estimated, as the number of closed omnivorous links divided by the number of top predators (Sprules and Bowerman, 1988). Closed omnivorous links (or same chain omnivory) exist when a feeding path can be traced from a predator to a prey more than one trophic level away and then back to the predator through at least one other prey occupying an intermediate trophic level. The degree of omnivory for the four-level structure of the Ythan web is 2.77 and for the five-level structure is 7.65. Of Briand’s original webs, 70% have a degree of omnivory less than that observed for the four-level structure and 90% for the five-level structure, indicating that the Ythan has a much greater prevalence of omnivory than in other webs. Either the Ythan really does have a much higher degree of omnivory, or omnivory is seriously underestimated in most food webs. The latter explanation seems the most likely in view of the high incidences of omnivores reported from freshwater systems (Sprules and Bowerman,

Table 3 Values for several food web properties derived from new data sets (for cases where data for more than one web are available mean

values are shown)

Previous Inset Ythan Coachella Little Rock South Pelagic Yorkshire Web property cataloguea websb Estuaryc Valleyd Lakee Americaf websg Pondh

Web size, S Range

No. of links, L Range

Chain length Max. Min. Mean

Omnivory Y O

Degree Compartments Predator/prey ratio Yo Top predators Yo Intermediate

% Basal species species

LIS sc

17

34 2-48

2- 138

24

43 3-90

92 30 93 75 50- 104

514

38

169 10--74

17-571

22 12-32

380 409 289 1023

5.2 2.2 2.8

7 1 2.9

9

5 ~

12 3 7.3

5-9

3.8

27 0-12

0.88 -

29

22 -

-

0.64 31-65

21-34 2.8-7.7

0.72 No

28

78

No 1.1 1 0

- 15-25

~

No

11 ~

-

1.14 1

-

0.66 14

-

6

86 13 11.0

(36.3) -

52 19 1.9 2-6

25-59 2-3 1 2.2 4.3

68 4 4.1 9.0

(36.9)

44 50 3.5

77 9

10.8 ~

9.6 14-7

Values in parenthesis are SC products based on values for upper connectance (see Section IVC). Tohen et al. (1986), Briand (1983). eMartinez (1991). fWinemiller (1990). gHavens (1992). hWarren (1989).

bSchoenly et al. (1991). ‘Hall and Raffaelli (1991). dPolis (1991).


yJt .... ... .

dragonfly spA and B lumped

Poor taxonomy lumps species and creates sarne-

mayfly spA and B lumped

Fig. 3. The effects of taxonomic lumping on perceptions of same-chain omnivory.

1988; Vadas, 1990) and the Coachella Valley, with 78% of species classed as omnivores (Polis, 199 1). Polis' numerous examples include life-history omnivory, opportunistic feeding and consumers which eat food in which other consumers live (e.g. scavengers, frugivores, granivores and detriti- vores). Unfortunately, it is not possible to calculate the degree of omnivory for the Coachella web because it apparently lacks top predators (Table 3; see also below).


These new data strongly suggest that omnivory may have been previously underestimated. However, some high estimates of omnivory may be due to lumping of species into very coarse taxonomic categories (Fig. 3). Polis’ detailed accounts of the trophic relationships in the Coachella web are sufficiently persuasive to convince the reader that his web really does have a high proportion of omnivores, despite the criticism concerning lumping.

Although at first sight Winemiller’s estimates of omnivory fall within the range previously documented (Table 3), they may in fact be conservative. He assigned fish species to trophic levels using the formula provided by Adams et al. (1983):

j= 1

where Ti is the trophic level of fish species i, is the trophic level of prey species j , and p i j is the fraction of the diet of fish species i consisting of prey species j . Plants are coded as T = 0.0, herbivore prey as T = 1.0 and carnivore prey as T = 2.0. Thus, a fish with a diet comprising 50% plant tissue and 50% herbivore prey would be assigned to a trophic level of T = 1.5. The Tvalues for herbivorous fish range from 1 to 1.5, those for primary carnivores from 2 to 2.5, and those for secondary and tertiary carnivores are > 2.5. Omnivores have T values in the range 1.5-2.0.

In this scheme, a fish with a diet of 40% herbivore prey and 60% plant material would be classified as a herbivore ( T = 1.4), whilst a diet of 60% herbivore prey and 40% plant would classify a fish as an omnivore (T = 1.6). One might argue that both are omnivores, since both diets comprise significant amounts of animal and plant tissue, and that Winemiller’s estimates of omnivory would be greater if defined by T values in the range 1.1-2.0.

A second difficulty which arises in discussions of omnivory is whether the analyses were carried out on a community web or restricted to a trophic subset of species. Winemiller (1990) and Vadas (1990) focused on fish assemblages and their prey, yet these represent only one or two tiers of the community web: fish have avian and mammalian predators and their invertebrate prey may be several links away from the base of the web. One might ask, therefore, if the high prevalence of omnivory estimated for fish assemblages (or any other trophic grouping) reflects that for the overall web? We address this question by analysing different trophic groupings within the Ythan Estuary food web.

Omnivory on the Ythan was assessed for four groupings: piscivorous birds, fish, shorebirds and invertebrates (including epibenthic predators such as shrimps, crabs and predatory polychaetes). Figure 4 shows the number of species within each of these groupings which are omnivores. For the invertebrate grouping only three (7%) of the 43 species are omnivores and


r

No. of closed omnivorous links Fig. 4. The number of species in different taxonomic groups in the Ythan web which are same-chain omnivores.

each of these is involved in only a small number of such interactions. Six (66%) of the nine species of piscivorous birds are omnivores. Most are involved in one or two closed omnivorous links, although the heron is involved in five such links. For shorebirds, a grouping which embraces waders, wildfowl and gulls, 13 (72%) of the 18 species are omnivores and are involved in 1-10 closed links. Eleven (65%) of the 17 fish species are omnivores and the number of closed links in which they are involved ranges from 3 to 16.

Our analysis clearly shows that the estimated prevalence of omnivory is similar (65-72%) for three of the tiers within the Ythan web, but that for species feeding towards the base of the web the prevalence is low (7%). This is not surprising because there is less opportunity for these invertebrates, most of which are primary consumers, to form closed links. Perhaps

FOOD WEBS: THEORY AND REALITY 20 1

more surprising is the similar proportion of omnivorous species amongst fish and piscivorous birds, since the latter are at a higher trophic level than fish and, therefore, have more opportunity to feed at several trophic levels. An extension of this argument might be that “taller” food webs would have more omnivores than “shorter” webs. Our own data for the Ythan are consistent with this idea (Hall and Raffaelli, 1991). The proportion of omnivores and the degree of omnivory were both greater in our five-level compared with the four-level version of the web (34% and 7.65 as opposed to 21 % and 2.77). Note, however, that we can get a “taller” web and more omnivory with a simple rearrangement of the same data. Another feature of the data in Fig. 4 is the larger number of closed omnivorous links for fish species compared to piscivorous birds. This could be interpreted as fish having a more general diet than birds, or it may simply reflect the exhaustive diet analysis that can be done for fish compared with birds. (People are less willing to shoot birds than catch fish.)

In conclusion, it seems likely that high estimates of omnivory will be obtained by focusing on a particular group of predators in a system and that the prevalence of omnivory will be dramatically reduced when all species are considered because there are likely to be far more species towards the base of a community web where ominivory is less common. This problem may be exacerbated by the generally poor taxonomic resolution of these species, leading to an overestimate of omnivory involving these species (Fig. 3). Clearly, an accurate estimate of the prevalence of omnivory will require much greater rigour than characterizes many of the studies to date. Nevertheless, several of the new data sets in Table 3 indicate that omnivory is probably more common than appears from analysis of the existing web catalogue.

B. Scale Invariant Patterns As described earlier, the proportions of different trophic types in food webs are claimed to be constant across a range of web sizes. These trophic types are: “predators”, which feed on other species; “prey”, which are fed upon; “top” species, which have no predators; “basal” species, which have no prey; and “intermediate” species which have both predators and prey. Plots of the numbers of predators and prey in real webs do seem to conform to a monotonic relationship, in that webs with more predators have more prey (Fig. 5). In this analysis a substantial proportion of web species are both prey and predators, but a more restrictive analysis which excludes such species also shows this pattern (Jeffries and Lawton, 1985).

Predator/prey ratios (although commonly referred to as predator/prey ratios, these are in fact prey/predator ratios) are less than 1.0 for documented webs (Table 3) , but intuitively one might expect real communities

202 S . J. HALL AND D. G. RAFFAELLI

A

A

A

b- 0

g 20

10

A

0 ‘ I I I I I

0 20 40 60 80 100

No. of Predators Fig. 5. The relationship between the numbers of prey and predators in food webs. 0, Briand and Cohen (1984); A Schoenly et al. (1991).

to have more prey, including plants, than predators. Although there is no way of demonstrating this for the majority of published webs, we concur with Polis (1991) that the ratios reported for most webs are likely to be too low because of the coarser taxonomy operating at lower trophic levels, especially amongst basal “species”. For instance, in our own work on the Ythan web we calculated a ratio of 0.72 (Table 3), but recognized that our three basal “species”, including detritus, were fairly gross categories. Detritus alone might include 10, 100 or 1000 species of bacteria, and this would change our estimates of the predator/prey ratios to 0.85, 1.9 and 12, respectively.

The problem of poor taxonomic resolution of basal species also has relevance for a second web property: scale invariance of the proportions of top, intermediate and basal species. These proportions are typically 29%, 52% and 19%, respectively (Table 3), but if lumping of basal taxa is severe, as argued above, then these proportions have little real meaning. Not only are basal taxa poorly documented in many literature webs, but the designa- tion of web elements as basal simply because there are no prey documented for them in the original web, can verge on the ridiculous. For instance, in Briand and Cohen’s (1 984) analysis, basal species include ostracods, nematodes, a grasshopper, a marine snail, polychaete worms, crustaceans, zooplankton, marine animals, a passerine bird, and two species of wildfowl.


This nonsense clearly illustrates the limitations of the existing web data sets and is a disappointing reflection on overzealous food web analysis. Similarly, defining top predators may not be as easy as it seems. Polis’ extensive documentation of the Coachella desert web suggests that if we look hard enough, even those species we traditionally assign as top are in fact preyed upon, a point also made by Winemiller (1990), Martinez (1991) and Havens (1992) (Table 3). If top predators really do exist in webs, then by their very nature they are likely to be transient and poorly documented in most webs (Paine, 1988).

Finally, we feel there is a very real problem with scale-invariant patterns because of the lack of rigour applied to their description. Plots of the proportions of top, intermediate and basal species against web size display enor- mous scatter (Fig. 6) and there are clearly no relationships in the data, a point acknowledged by the original authors (Briand and Cohen, 1984). Unfortunately, this non-relationship has been hailed as evidence of indepen- dence of the proportions of these trophic types with web size. While such a phrasing may be technically acceptable, it is not particularly rigorous or useful in that it defines a non-property rather than a property. More alarming, however, is the leap from “no relationship” to “scale invariance”. Inspection of the scatter in Fig. 6 shows that the proportions of top, basal and intermediate species are anything but invariant. In the many source papers and reviews of web patterns, authors have discovered several interesting ways of expressing this non-relationship. Perhaps the most misleading is that the proportions are “constant”, although this is often toned down to somewhat meaningless phrases like “more-or-less constant” or “roughly constant, but with high variance”.

In summary, the poor taxonomic resolution of elements towards the base of literature webs and our lack of technical competence at identifying basal and top predators renders analysis of so-called “scale-invariant properties” questionable. Given the argument that at least one of the relationships is in fact a non-property we wonder why and how it has attracted so much interest.

C. S-C Relationships The notion of a decreasing hyperbolic relationship between connectance C and the number of web elements S and an SC product (links per species) of between 2 and 6 fits well with theoretical studies on food webs described in later sections. Whilst Schoenly et d ’ s (1991) analyses of these properties agree well with previous work (Table 3, Fig. 7(b)), data from other webs do not (Fig. (7(c, d)) . Thus the SC product for the Ythan web is 9 and that for the Coachella Valley is 14.7 (Table 3). Little Rock Lake has a SC product of 36.3 which, since it is based on calculating C as upper connectance which includes trophic and presumed competitive links, is not dissimilar to the

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equivalent Ythan value of 36.9 (Table 3), and therefore higher than would be expected. If a mean and standard error are calculated for the SC products of the webs originally used by Briand, then the Ythan value lies many standard errors above the mean value.

Nevertheless, an argument could be made that addition to existing S-C plots of the single data points for the Ythan, Coachella Valley and Little Rock Lake webs would not significantly affect the overall appearance of the relationship. This impression may be more due to the large number of older data points than to conformation of the new data. A more rigorous examination of the S-C relationship comes from two independent studies


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featuring a range of web sizes (Fig. 7). These analyses suggest that C is independent of S (Warren, 1989) or may even increase with S (Winemiller, 1989; Martinez, 1992b). The authors, among other points, agree with Paine's (1988) contention that much of the decline in C may be due to sampling effects. If the same effort is put into deriving C for a range of webs of different size, then the relationship is not hyperbolic.

More recently, most workers have abandoned plots of C versus S in favour of plots showing the number of trophic links L versus S. This is because C is not independent of S. Links per species ( L / S ) average about 2 for the webs catalogued by Briand and Cohen et al., and Schoenly et al.'s webs have a mean linkage density of 2.2 (Table 3). However, the Ythan, Coachella Valley and Little Rock Lake webs have values considerably larger than 2 (Table 3). Again, the reason for this may be the greater effort put into documenting the trophic interactions between species in these new webs and/or because of a non-linearity in the relationship. The original hyperbolic relationship observed between S and C implied that linkage density remained constant regardless of web size, although even more recent analysis of the


expanded food web data collection (Cohen et al., 1986; Havens, 1992) shows that L increases with Saccording to the relationship L = 0.6713 x S 1’36. This point is discussed further below.

D. The Length of Food Chains Maximum and average food chain lengths are around 5-7 and 3, respectively, for the catalogues analysed by Cohen et al. (1986), Briand (1983) and Schoenly et al. (1991) (Table 3). These values are considerably lower than those derived from several of the new data sets. In the smaller webs documented by Warren (1989) an average chain length of 3.8 is slightly higher than that documented from the web catalogues, although some versions of Warren’s webs have an average length of 5 and a maximum length of 8 links. Average chain lengths for the larger webs (Ythan Estuary, Coachella Valley and Little Rock Lake) range from 5 to 7, raising the possibility that chain length is a function of web size. There is indeed a relationship between web size and food chain length in Warren’s data (Fig. 8) (see also Martinez (1992b)). Although this trend is not apparent in Briand’s (1983) and Cohen et al.’s (1986) data it is thought to be obscured by different authors using different linkage criteria and uneven aggregation in the original webs (Martinez, 1991). It seems likely, therefore, that the lower values for food chain statistics recorded in previous analyses are at least in part due to the inclusion of small webs in the analyses and that the greater values of these statistics for the new webs noted here may be attributed to their larger size. However, it also seems likely that these higher values are due to the better degree of documentation for the new webs, both in terms of the resolution of the elements and the effort put into documenting linkages. This is explored further in the following section.

A major problem in making between-study comparisons of food chain statistics is the variable way in which these statistics are estimated. Some authors count the number of species per chain, others the number of feeding links. The latter will of course be one unit shorter than the former. However, it is not always clear which method has been used. Another variant is the inclusion of loop-forming species in the calculations of chain length. These are species which eat each other, usually through life-history cannibalism, and they are generally absent from existing catalogues. However, life-history cannibalism is prevalent in the Little Rock Lake, Yorkshire Pond and in the Coachella Valley webs (Martinez, 199 1). Excluding loop-forming species from the Little Rock Lake analyses produces shorter food chains, reducing the average to 7 links (Table 3).

I t is clear that analysis of at least some of the new data sets does not con- firm several of the properties previously regarded as general attributes of food webs. Most of the new webs are larger and more completely docu-

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mented than previously available webs. An exception is the catalogue of insect dominated webs described by Schoenly et al. (1991). This description contains many smaller webs, for instance dung and parasitoid systems, and it is a debatable point whether these represent true webs or merely the pieces of a much larger jigsaw of the kind described by Polis (1991). Defining the


appropriate spatial and temporal scales for food web studies is a lively area for discussion and we offer some opinions in the discussion (Section VI). In the following section we consider the more tractable problem highlighted in our foregoing discussions of poor and variable taxonomic resolution.

E. Taxonomic Resolution Four studies (Sugihara et al., 1989; Hall and Raffaelli, 1991; Martinez, 1991, 1992a,b) have specifically examined the effects of variable taxonomic resolution on food web properties. All the studies progressively lump web elements into coarser and coarser taxa and analyse the resulting webs for a range of properties. In the case of Martinez, and Hall and Raffaelli, the starting point for collapsing the web is a large, reasonably well-documented web: Little Rock Lake and the Ythan Estuary, respectively. The starting webs in Sugihara et al.’s analysis are 60 invertebrate dominated systems ranging in size from 2 to 87 elements, although they restricted this particular analysis to webs with more than 10 species. The procedure used to decrease taxonomic resolution is similar in Sugihara et al. (1989) and Martinez (1991). The condition for aggregating web elements is whether they share predators and prey. This is an objective and computationally attractive procedure, but it is biologically unrealistic in that it takes no account of the identity or taxonomic affinities of the elements which are aggregated. For instance, applying this method to the Ythan web would aggregate flounders, crabs and eiders into a single taxon simply because they all feed on the amphipod Corophium and are all prey of the herring gull. Whilst this procedure might have some intrinsic merit for exploring how grouping species into guilds or functional groups affects web properties, it does not reproduce the variable taxonomic resolution which characterizes real food web data sets.

In view of this, we employed different aggregation criteria when progressively collapsing the Ythan web (Hall and Raffaelli, 1991). In our analyses species were lumped subjectively to produce coarser and coarser taxa that have some biological integrity. For instance, crabs, prawns and mysids were grouped as epibenthic crustaceans without reference to their trophic affinities. We also made an effort to retain a refined taxonomy at higher levels (e.g. shorebirds) and a coarser taxonomy at lower levels (e.g. ‘‘difficult” groups of invertebrates). In adopting these aggregation criteria we believe we have more closely reproduced the variable taxonomic resolution found in real webs. We contend that field biologists are more likely to pool animals which look the same, rather than group them on the basis of trophic similarities. This tendency is likely to be especially apparent in pictorial representations of webs where artistic constraints apply (Paine, 1988).

The results of three of these exercises are summarized in Table 4. Of the properties that were examined by all these studies only predator/prey ratios


Table 4 The effects of decreasing taxonomic resolution on food web properties

209

Sugiharaa et al. Martinez Hall and Raffaelli Web property (1989) (1991) (1 99 1, unpublished)

Food chain length Robust Declines Declines Connectance Slight decline Variable Increases Predator/prey ratio Robust Robust Robust % Top species Robust Increases Robust % Intermediate species Robust Decreases Robust % Basal species Slight increase Increases Slight increase LIS Slight decline Declines Declines

were found to be consistently robust. Food chain length declined with increasing aggregation in both Martinez, and Hall and Raffaelli (Fig. 9), but was judged relatively robust by Sugihara et al. (1989). The latter authors were surprised by this result and Martinez (1991) has suggested that their failure to detect any trend in this property may be the result of an incomplete analysis. A full discussion is provided in Martinez (1991, 1992) but the essence of his argument is that Sugihara et al. were inconsistent in their aggregation of species. This obviously has implications for their analysis of other web properties.

Connectance (measured as upper or lower; see above) declined with aggregation in Sugihara et al.’s analysis, was variable and inconsistent in trend in Martinez’s and increased in Hall and Raffaelli’s (Table 4). The latter trend was sufficient to generate the well-documented relationship between S and C described above (Fig. 10; see also Fig. 7). Links per species declined in all the studies. In our analysis values ranged from 2.5 for a 13 species version of the Ythan through to 4.34 for the full version. This supports the contention (Cohen et d., 1986) that L/S is not a constant but a function of S and that the relationship is more properly described by the equation L = 0.6713 x S 1.36. Havens’ (1992) analysis and the collapsed versions of the Ythan web reproduce this relationship well, generating a slope of 1.34 (Hall and Raffaelli, 1991), although another analysis (Warren, 1990), which was not specifically addressing the question of taxonomic resolution, suggests that the slope may be much higher, around 2. In view of these findings it is, perhaps, surprising that Sugihara et al. (1989) found L / S to be only slightly sensitive to aggregation, but Martinez (1991) argues that this can also be explained by inconsistencies in their approach. Martinez re-analysed Sugihara et al.’s data and found that “the same trends observed in differently resolved Little Rock Lake webs were observed among their variably sized 60 webs” (Martinez, 1991; 382).

In Sugihara et al.’s, Hall and Raffaelli’s, and Martinez’s analyses the proportion of basal species increased as the web was collapsed. This is not sur-

210

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prising because basal species are usually already documented to a fairly coarse taxonomic level and it would be difficult to aggregate them further. Thus the number of basal species would remain fairly constant across all web sizes. In contrast, the numbers of the better resolved intermediate and top species will decrease through aggregation. Hence, the proportion of basal species will increase. This increase in the proportion of basal taxa is not apparently mirrored by a decrease in the proportions of intermediate and top species, possibly because the status of intermediate and top species may change inconsistently, and possibly because the authors did not judge any trends significant. Thus in our own analysis it is debatable whether an increase, albeit consistent, in the proportion of basal species from about 0.03 to 0.14 is sufficiently large to affect the validity of previous statements concerning its invariance.

In summary, with the exception of predator/prey ratios, there is evidence that at least some web properties are sensitive to taxonomic resolution. Also, for some properties, the effect of collapsing webs is to reproduce some of the scale-dependent relationships derived from analysis of web catalogues, e.g. the S/C and L/S relationships for the Ythan. Similar patterns were reproduced by Martinez (1991), leading him to state that “this correspondence suggests the intriguing possibility that previously reported webs are severely aggregated versions of more elaborate webs very similar to the trophic species web of Little Rock Lake”. Aggregation of species is an obvious feature of many of the webs in the literature and the problem is only aggra-


Table 5 The sensitivity of web properties to linkage criteria“

Statistic Sensitivity to linkage criteria

Lower connectance Upper connectance Directed connectance (includes feeding loops) LIS SC Average chain length Maximum chain length YO Top species O h Intermediate species O h Basal species Proportion of links

To top species Intermediate to intermediate Intermediate to basal

High High Medium Low Low Low Medium High Medium Medium

High High Medium

“From Martinez (1991 :Table 2)

vated by grouping trophically similar elements into “kinds of species” prior to analysis. In view of the sensitivity of some properties to resolution this does not seem a sensible way forward. Conversely, for those properties thought to be insensitive to taxonomic resolution we would argue that this insensitivity does not necessarily make them robust and meaningful features of biological systems. Rather, their very insensitivity brings into question their values as biological descriptors.

Winemiller (1990) and Martinez (1991) have also examined the related problem of variable documentation of trophic links in webs by varying the thresholds at which prey are included in the diet of predators. In Martinez’s analysis of taxonomic resolution linkages between two aggregated clusters of species (new or collapsed web elements) were assigned on the basis of how many links there were between the two sets of species in the original web matrix. For instance, a minimum linkage web requires every species in one cluster to be linked to every other member of another cluster in order for the two clusters to be linked. In contrast, a maximum linkage web requires only one member of each cluster to be linked for the two clusters to be linked. Average linkage webs require half the species in each cluster to be linked. The sensitivities of several web statistics to these linkage criteria are summarized in Table 5 . Maximum chain length, connectance, the proportion of top species and the proportion of links to top species and between intermediate species were all judged to be highly sensitive by Martinez (1991), whereas L / S , SC and average food chain length were relatively insensitive.

Winemiller (1990) varied linkage according to 10 prescribed abundance thresholds. Prey had to exceed these thresholds in order to be included in a


predator’s diet. Four properties were examined in webs with different linkage thresholds, compartmentation, connectance, mean number of prey per node (trophic position), and mean number of predators per node. The values for all four properties declined when linkage was reduced.

Both Winemiller’s and Martinez’s analyses suggest that the degree to which diets are documented has implications for web properties, those webs which include all linkages having different properties to those including only the major links. It is likely that many of the webs in the literature are incompletely documented with respect to trophic linkages (see for instance the above discussion of the S-C relationship) and this, together with their variable taxonomic resolution, reduces the effectiveness of analyses of web properties. Nevertheless these properties form an important subset of the data used by modellers. These data are used in at least two ways: to construct and parametrize models so that they are plausible representations of the real world which allows the effects of varying a particular property to be explored, and for testing predictions against data. The relative importance of these two aspects differs from study to study. Although static patterns are by no means the only kinds of data used in the development of food web models they are an important component and we now consider the possible sensitivity of models to our altering perception of the robustness and validity of web properties.

V. THEORY AND DATA Since the early 1970s, considerable effort has been expended in exploring the dynamical properties of multi-species predator-prey models using a variety of techniques, and these efforts have focused much of our thinking about the properties of real systems. The majority of food web models fall at the “strategic” end of the continuum between “tactical” and “strategic” models identified by Holling (cited in May, 1973). These models have as their aim the identification of possible ecological principles and the “provision of a con- ceptual framework for discussing broad classes of phenomena” (May, 1973). For strategic modelling to step beyond an abstract academic pur- suit, the insights models provide need to be critically evaluated in the light of available data. Only by doing so can we hope to refine insights to the point where they inform our decisions about the management, protection or restoration of ecosystems. In the case of food webs this is a formidable challenge, not least because most of the available food web data documents (incompletely) static structural properties and offers little or no information on dynamics. There are, however, notable exceptions such as the efforts to follow the dynamics of phytotelmata webs (pitcher plants, water-filled tree holes, etc.) (see review by Kitching and Beaver (1990)).

In this section we outline the basics of the more common types of model


employed, the conclusions drawn from these theoretical studies, and consider the robustness of these conclusions, especially in the light of new data. There are many texts which describe in detail the mathematical techniques on which the models are based and one or two excellent texts which explain the approaches with particular reference to modelling food webs (e.g. May, 1973; Pimm, 1982; Yodzis, 1989). In view of these earlier comprehensive treatments we avoid mathematical formalism as much as possible and concentrate on the underlying principles of the modelling approaches. The reader interested in the methodological details should consult May (1973), Pimm (1982), Yodzis (1989) and references therein.

A. Lotka-Volterra: an Industry Standard

Perhaps the most common mathematical representation of food webs are those employing differential or difference equations which describe the growth rates of species populations based on formulations which incorporate predator-prey interactions. The familiar Lotka-Volterra equations form the basis for many of these models where, in the differential form, the rate of change in the density of species i is given by:

- = xj bj + c a& dXi dt ( )

here X i is the density of species i, the bj terms are positive (growth rates) for basal elements and negative (starvation rates) for non-basals, and the ajj terms are the interaction coefficients between species i and j (the per capita effect of species j on an individual of species i). The sign of each ajj term defines the structure of the food web since, if species i eats species j the prey ( J ) will have a positive effect on the predator i (a j j > 0) whereas aji will be negative (< 0) because the predator has a negative effect on the prey. For species which are not trophically linked a j j = 0 in systems which model trophic interactions only, although it is perfectly possible to model other competitive or mutualistic interactions where the signs for aij and a j j would be (--) and (++), respectively.

One class of dynamic models works from the basic premise that model webs should in some sense be stable. At the very least we expect to find the same number of species in the system at the beginning and at the end. Models which fail to satisfy our various notions of stability run counter to observations in the real world. In essence, those characteristics which confer stability on model webs should be the ones we observe in nature. Since acceptable food web models must be in some sense stable, we need to know how they describe changes in populations through time. A useful distinction can be made between global stability where populations return to


positive equilibrium values after any size of perturbation and local stability where returns to equilibrium are limited to small perturbations from equilibrium levels. For an equilibrium point that is only locally stable, too great a push will tip the population out of the stable domain and the perturbation will amplify itself. Pictures of boulders on valley bottoms and resting pre- cariously in shallow craters on top of volcanoes are often used analogies for global and local stability. In cases where the equations describing population dynamics are linear, global and local stability are the same.

Ideally, it would be useful to characterize the global behaviour of a model system, i.e. for all sizes of perturbation and for all initial conditions. How- ever, linearity rarely holds for biological systems and this makes the mathematics of global stability analyses very difficult and usually impossible. For this reason theoreticians have often searched for properties which confer local stability on food web models.

I . Local Stability

The essence of local stability analyses is that models composed of the non- linear equations which make global analysis so difficult can be assumed to be linear for a local region around an equilibrium point. Linearizing the system in this restricted area makes the mathematics much more tractable and requires more restricted (and more easily guessed) information about the biology of the system. The community matrix summarizes the interactions between species populations in this localized region about an equilibrium; if the community contains three species, it will have i = 3 rows and j = 3 columns and each element c i j represents the effect of the population of each species on the growth rate of every other when all species are at their equilibrium densities. The elements in the matrix can take a variety of forms, but in the case of the Lotka-Volterra model cij = aijX;, where X; is the non-trivial (i.e. Xi # 0) equilibrium density of species i, and aij is the interaction coeffi- cient described above. As we have seen, the signs of the aij terms denote the trophic (+ -), competitive (- -), or mutualistic (+ +) relationship between pairs of species and, since X ; > 0, the sign of each term in the community matrix will reflect these relationships. But, because food web models focus only on trophic interactions, mutualistic and competitive interactions are not included in the community matrix. The only exception to this is for the diagonal ( i = J ) elements of the matrix which represent intraspecific interactions; these are always negative. Without such self-limiting interactions the models fall apart.

Analyses of local stability about an equilibrium point are normally described using the community matrix, because matrix algebra offers a convenient way to abstract the properties of the model web from indices which summarize the dynamic behaviour. If we represent a community


matrix containing n species (i.e. an n x n matrix) by C it is, under most practical circumstances, possible to find an (n x 1) column vector v and a real number I which satisfy the condition:

I v = c v

In this case v is known as an “eigenvector” and the associated real (or complex) number 1 its “eigenvalue”. There are as many eigenvalue/vector pairs as there are species in the community matrix. For a community matrix with small numbers of species (i.e. < 4) these can be calculated explicitly-in a two-species system the pair of eigenvalues are roots of a quadratic equation, and for the three-species case they are the roots of a cubic equation. For larger matrices the solutions must be obtained using numerical methods.

The important point is that the sign and magnitude of the eigenvalues tells us about the dynamical behaviour of the model food web near an equilibrium point. In order for the system to be stable, all the eigenvalues must be negative. (For cases where eigenvalues are complex numbers (i.e. when I = r t iq, where i = n), the real parts of all the eigenvalues must be negative.) Con- versely, if any of the eigenvalues have a positive real part the system will be unstable and a small perturbation from the equilibrium point will amplify without limit. For stable systems, the size of the largest eigenvalue provides a measure of how quickly the system will settle back to the equilibrium point- the more negative the largest number the quicker the return. Using this feature, Pimm and Lawton (1977) proposed the following function:

- I Return time = ; forX,, < 0

Re(Xrnax) where Re(Xrnax) is the real part of the largest eigenvalue. The return time gives the time taken for a perturbation to decay to approximately 37% of its initial value.

B. Model Predictions and the Effects of Data Re-evaluation

1. Stability and Complexity Up till the 1970s, our understanding of the relationship between complexity and stability in ecological systems was essentially based on arguments which used a mixture of observations of nature and reasonable logic. Both suggested that complex systems were more stable than simple ones. For example, Elton (1927) observed larger fluctuations in population sizes and more frequent pest outbreaks in high (species poor) compared with low (species rich) latitude systems, whilst MacArthur (1955) reasoned that the greater connectance and species richness there is in a system, the more alternative trophic pathways there will be, and the less likely it is that the loss of any particular species will result in a system collapse.


May (1972), was one of the first to explore this question using the local stability analyses described above. This work confirmed earlier suggestions (Gardner and Ashby, 1970) that there is a sharp transition between stability and instability as the connectance in systems with large numbers of species increases. May then went on to determine how this sharp transition scales with the size of the web ( S ) , the connectance (C) and the average strength of the interactions between species in the web (a) . This was achieved by selecting values for S and C, assigning a values to randomly chosen off- diagonal elements in the community matrix and finding the largest eigenvalue to reveal whether the system is stable. Repeating the process with many sets of randomly chosen elements gives the proportion of webs that will be stable for a given S , C and average a combination.

May’s analysis suggested that the system would usually be stable if the following condition is satisfied:

a ( S c ) 1 / 2 < 1

In other words, for a given average interaction strength (a ) , the more complex a community is (as determined by the SC product) the less likely it is to be stable. This ran counter to the received wisdom of the time.

One implication of May’s findings is that, assuming constant a, the stability condition a(SC ) ‘I2 < 1 anticipates a hyperbolic relationship between connectance C and species richness S. AS we have seen, the analyses of real food webs reported in the literature are consistent with this expectation (Fig. 7), and it has been argued that species-rich webs retain their stability by virtue of their lower connectance and/or lower average interaction strength (Rejmanek and Stary, 1979; Pimm, 1982; Briand, 1983).

Although the S-C relationship for many of the available food webs corre- sponds with the expectations of May’s analysis, several authors have argued independently (Paine, 1988; Warren, 1989; Winemiller, 1989; Martinez, 1991, 1992a), and we argued above, that the hyperbolic relationship may be spurious. The nub of the argument is that a decline in C with increasing S is an artefact resulting from the practical difficulties involved in documenting all the trophic links that exist in larger webs. If the same effort is put into deriving C for a range of webs of different size, then C becomes independent of S (Paine, 1988; Warren, 1990) or may even increase (Winemiller, 1989) (Fig. 7). Where does this leave the complexity-stability question? If C remains constant or increases with S, then this might imply that complex systems would find themselves near the edge or outside of the stability bounds set by May’s original expression. A preferred alternative may be to continue to accept that food webs should exhibit local stability and either reject May’s conclusion that webs with high C or Swill be unstable, or alter one or more of the underlying assumptions. In reality it is fair to say that discussing the implications of May’s findings in the light of available data on S and C is


(and probably always has been) rather pointless in view of the difficulties and uncertainties in estimating the third variable, average interaction strength.

Critics of May’s approach pointed out that randomly generated webs allow biologically unrealistic structures such as an absence of autotrophs. Lawlor (1978) showed, for example, that it was extremely unlikely that any of the random webs generated by May would have realistic biological structures and argued that it was more important to determine the structural features of real webs that distinguished them from random ones. Other studies using more realistic web structures also showed that the patterns of decreasing stability with increasing complexity proposed by May (1972) need not necessarily hold. For example, DeAngelis (1975) demonstrated that stability increased with complexity if food webs were characterized by low assimilation efficiencies, strong self-regulation in species occupying higher trophic levels, or donor control dynamics where predator populations have very little impact on their prey.

Following criticism of studies based on randomly assembled webs, effort shifted towards stability analyses in which the community matrices (or more correctly the structural properties of food webs that the matrix describes) were constrained to be biologically “reasonable”. In this context the criterion of reasonableness constrains webs to those topological properties that emerge from the published catalogue (specified by the signs of the elements) and biologically defensible parameter ranges for the interaction terms. It is here that the importance of pattern documentation for model for- mulation can be seen most clearly. Yodzis (1981) adopted the term“plausib1e community matrix” to describe such a matrix.

Taking plausible community matrices as the starting point, Yodzis (1981) explored how reasonable a representation of the real world the community matrix actually is. To address this question Yodzis generated many plausible community matrices, each with the same “real” topology, but with values for the interaction strength selected at random from within a range spanning an order of magnitude. This was done for the 40 real webs catalogued by Briand (1983) and the proportion of locally stable matrices for each web was calculated. Each matrix was then “disrupted” by randomly permut- ing all positive off-diagonal elements among themselves and all negative off- diagonal elements among themselves. This process altered the relative strengths of interactions between particular species in the community, while preserving both the topology of the web and the average interaction strength. Yodzis (1981) showed that for the vast majority of real webs the proportion of matrices that was stable fell when the web was disrupted, and concluded that local stability analyses of community matrices is a sensible way to represent real systems. Moreover, this finding implies that the structures observed in real webs confer greater stability than alternative web configurations.


Work by Yodzis (1981) and other work along similar lines (e.g. Pimm, 1982) has confirmed the suggestion by Lawlor (1978) that reasonable structures have a greater chance of being stable. This finding inevitably reinforces the idea that the patterns which emerged from the early food web data are valid. In consequence, considerable effort has gone into exploring the importance of various observed features for the stability of systems. We have no way of knowing how robust these conclusions would be if the structures implied from the more recent food web data were substituted for those from the original catalogue (Briand, 1983).

2. The Length of Food Chains

The earliest explanation for the apparent shortness of food chains was that their length was determined by energetic constraints. The argument goes that, because only about 10% of food intake at one trophic level will be available as biomass at the next, only a few links are possible before the energy to support another trophic level becomes insufficient. In 1977, Pimm and Lawton proposed an alternative explanation based on dynamic stability arguments. They reasoned that energy constraints must ultimately affect the length of food chains, but that chains are truncated before this con- straint can apply. The observation that chains do not appear to be longer in more productive environments is cited as evidence that this may be so. Using local stability analyses, Pimm and Lawton examined five possible web configurations for a four-species system and constructed 2000 community matrices for each. The results of this study showed that systems with longer food chains had longer return times, and Pimm and Lawton argue that this would mean that the system would be less likely to persist in a fluctuating environment. Thus, environmental unpredictability might limit the length of food chains before they became sufficiently long for energetic constraints to be important.

Yodzis (1989) gives a very lucid account of how one might distinguish between energetic and dynamic stability explanations for the length of food chains, and concludes that empirical evidence is inconclusive at present. We do not propose to paraphrase that account here, but point out that food chains are probably longer than previously thought.

3. Omnivory

One of the first suggestions that omnivory may be an important factor for web stability came from the paper by Pimm and Lawton (1977) described in the previous section. This work showed that 78% of webs with three or four trophic levels were unstable if omnivore links were present. These initial

220 S. J . HALL AND D. G . RAFFAELLI

findings were followed up with a more detailed analysis (Pimm and Lawton, 1978) of a system with four trophic levels and four species. In this system there are eight possible combinations of omnivore links that can be assigned to give configurations with 0-3 omnivore links. Pimm and Law- ton found that the more omnivore links there were in the system the greater the proportion of webs that were unstable. Paradoxically, however, for those webs which were stable the return time decreased with increasing omnivory. These effects work in opposition to one another so it is difficult to determine how omnivory will affect real systems. Neverthe- less, for systems with the interaction coefficients parametrized with best guesses to represent systems containing vertebrates or both vertebrates and herbivorous insects, only a very small proportion of the matrices were stable if they had more than one omnivore link. Moreover, the proportion of stable webs is reduced further if omnivore links miss out the adjacent trophic level and connect at one level removed. In contrast, the stability or return time of models parametrized to represent host-parasitoid interactions was relatively unaffected by the degree of omnivory. These results led the authors to conclude that a high incidence of omnivory should be rare in the real world, except in host-parasitoid systems. In addition, one should rarely find systems where species feed simultaneously both high and low in the web.

To test these expectations, Pimm and Lawton extracted information on the incidence of omnivory in food chains with three or more trophic levels from 19 of the webs in Cohen (1978). The authors found that 11 of the 58 food chains had a high incidence of omnivory, but argue that this is probably due to the grouping of species into gross categories (such as invertebrates, insects, and zooplankton) which will naturally affect the level of omnivory (see Fig. 3). Pimm and Lawton also noted that dynamically unimportant links may have been included in webs. Weak links where the predator does not significantly depress prey populations were not represented in the model, but probably are included in the original web data, thereby increasing the prevalence of omnivory. In response to Pimm and Lawton’s analysis, Yodzis (1984) showed that the rarity of omnivore links in real webs could be accounted for by the lack of animals that feed on both plant and animal tissues. Yodzis argues that basic biological constraints, which make it more difficult for animals to feed on species at widely spaced trophic levels (especially the difficulties for animals which feed on plants and animals), are sufficient to explain the apparent rarity of omnivory and that one need not necessarily invoke dynamic stability arguments to explain the observations. It is probably fair to say that the relative importance of these two explanations remains unresolved. However, our own analysis of the more recent web data suggests that omnivores may be less rare than was previously believed.


4. The Concept of Permanence: An Alternative to Local Stability Analysis

Much of the preceding theory was developed using local stability analysis and was based on the idea that locally stable configurations of species were more likely to be observed in the real world. This type of analysis has greatly stimulated both theory and observation and it is an important tool in the theoretician’s armoury. Nevertheless, it is not altogether clear whether this approach is entirely appropriate and the generality of the conclusions that can be drawn from analysis of system behaviour in a small area around a local equilibrium is unclear. Law and Blackford (1992) point out that there is no reason to suppose that systems start close to equilibrium or that they approach equilibrium from outside the “local” domain. Moreover, analysis of equilibrium systems precludes the inclusion of cyclic and chaotic trajectories which may still be consistent with the persistence of all species (see references cited in Law and Blackford (1992)). I view of these potential shortcomings it is important to explore alternatives to local stability analysis.

One approach concerns the notion of permanence which steps back from the question of whether systems are in equilibrium or not and asks the more fundamental question of whether a particular configuration of species can co-exist. Law and Blackford (1992) have taken the mathematical develop- ments concerning the concept of permanence and have applied them in an ecological context to examine (among other things) the concordance between the permanence of simple Lotka-Volterra systems and their local stability. To understand the idea of permanence, consider a system with a boundary which encloses the region of phase space in which all species densities are positive. The system is said to be “permanent” if species trajectories which approach the boundary (i.e. come close to extinction from the system) are repelled. No statements are made about the trajectories within the boundary, which may exhibit local equilibria, cycles of chaos. It is no surprise to find that Law and Blackford found that permanent systems may be locally unstable, but they also show that, although omnivory increases the likeli- hood of local instability, it often fails to result in the loss of species, tending instead to produce permanent systems with stable limit cycles or chaotic trajectories. This finding illustrates how alternative mathematical methods may alter our view of the importance of food web attributes for the dynamics of the system, and is consistent with our own perception of the prevalence of omnivory in nature. We return to questions of permanence again in Section V.C.

The problems that Pimm and Lawton faced when trying to confront model expectations with real data are fundamental for a range of questions relating to food webs. The cynic would argue that when model predictions and data do not match it is convenient to cite the reasons why the data may be

222 S. J . HALL AND D. G. RAFFAELLI

inadequate, rather than to reject the model. On the other hand, if they do match, one argues that the test is legitimate, despite the inadequacies of the data. We do not adopt such a cynical stance, but it is undeniable that these tendencies are evident in some of the literature on food webs. In fair- ness, however, there is also considerable agonizing and ready acknowledge- ment of the difficulties of work in this field. In the case of omnivory, the inadequacies in the data do indeed seem overwhelming (see earlier sections) and the legitimacy of Pimm and Lawton’s conclusions remain untested. We believe that interpreting the results of theoretical studies on omnivory and other aspects of food web structure is especially problematic in the absence of data on interaction strengths, the one parameter in models that is almost always guessed.

5. Interaction Strength in Models and Nature

Measuring interaction strengths requires much more effort than documenting who eats whom, and theoreticians have had to resort to guessing the relative magnitude of interaction strengths based on intuition about the biology of the species the matrix elements represent. How have such guesses been made? In their study of the effects of omnivory (see above), Pimm and Lawton (1978) suggested a range of values for the interaction between different types of species. For vertebrate predators which are larger than their prey Pimm and Lawton argue that the per capita effect of the predator on the prey (the c i j term) will be much greater than the per capita effect of the prey on the predator; as a result they chose cii values in the range - 10,O for the effect of the predator on the prey and 0, +0.1 for the converse. In contrast, for insects which feed on trees the predator will have much less effect on the prey than the prey will on the predator and values in the range -0.1,O and 0, +10 were chosen to reflect these interactions. For host-parasitoid systems it was argued that the per capita effects of parasitoid on host, and the host on the parasitoid will be more equitable; values in the range - 1 , O (parasitoid on host) and 0 $1 (host on parasitoid) were chosen in this case.

One end-point of the continuum between strong and weak interaction between species is donor control, where the growth rate of the prey population is not limited by the predator. In systems under donor control the predator removes prey which are dead or destined to die through other causes, so that the signs of the predator-prey interaction are ( fO) rather than (+ -). As noted above, DeAngelis (1975) and others (e.g. May, 1973; Pimm, 1982) have shown that donor control tends to stabilize model systems, leading to an increase in stability with increasing complexity. An important question then is whether donor control dynamics are a reasonable representation for real systems. Under donor control, equations describing the growth of the prey population explicitly exclude any effect of predators on prey equilibrium


densities. Thus if donor-control dynamics truly reflect processes in nature, then removal of predators from systems might be expected to have no effect on prey populations. Conversely, if there are significant effects of predator removal on prey populations, then predictions of models based on Lotka- Volterra equations would seem more relevant.

On reading the literature surrounding this issue, Lotka-Volterra and donor-control dynamics seem to have been cast as alternative descriptions for a given web, with the (probably unintentional) implication that all interactions will be of either one form or the other. There is clearly no reason why this should be so and interactions within the web may be variously placed along the continuum between the two extremes. To avoid focusing on these extremes our question can, perhaps, be more helpfully rephrased by asking: What is the distribution of interaction strengths in webs?

Using available field data, food web theorists have usually concluded that donor-control models are inappropriate and that systems are normally characterized by strong interactions. Pimm (1980, 1982) argued this position strongly, basing his conclusion on a review of published data from field experiments in which the effects of predator removal have been examined. Pimm reviewed some of the earlier literature on predator removals, many of which come from marine rocky shore studies and almost all of which are aquatic. This is no coincidence, since hard marine substrata provide the experimenter with a system that is relatively easily manipulated and which responds rapidly to perturbation. Whilst some of Pimm’s examples are not predator removal experiments sensu stricto, it is clear that predation can be important in structuring some kinds of prey communities. Many have recognized, however, that an experimenter’s attention is likely to focus on species which are expected to have an effect and that experiments showing large predator effects are easier to publish than those which do not. Never- theless, there is a growing realization amongst aquatic ecologists that predator removals do not always lead to dramatic effects on prey communities (e.g. Menge, 1992, Hunter and Price, 1992). The situation is better now than it was, but reviewers will probably always be harder on papers arguing non- significant results than on those showing large effects. As Sih et al. (1985) report, non-significant results from predator manipulations are usually only documented in papers where significant effects for other species have also been shown.

The factors described above combine to make the distribution of interaction strengths in a web difficult to judge. Rectifying this situation requires systematic-removal experiments for many predators in the system-a task which few have undertaken. For those studies which are available, however (Paine, 1992; Raffaelli and Hall, 1992), the evidence suggests that weak interactions may be much more prevalent than food web modellers have supposed.


Perhaps the most comprehensive analysis of the distribution of interaction strengths is that by Paine (1992), who studied a herbivore guild in a rocky intertidal community. What is particularly heartening about this study is Paine’s effort to produce operational definitions of strong and weak interactions and to offer a means for calculating per capita interaction strength. Pain argues that the most important interactions in a community are those which prevent a species from dominating an assemblage. Studies on rocky shores support this viewpoint with many examples where a competitive dominant develops as a monoculture when released from predation or disturbance (Paine, 1980). A strongly interacting species can then be defined as one which is capable of preventing the development of a monoculture or of destroying one. With this definition the strong interactor will have measurable effects on the competitively dominant prey and will also have an indirect effect on other species in the assemblage. The importance of each member of a guild of consumers can thus be determined by comparing their potentials for controlling the competitively dominant prey.

At Paine’s study site the laminarian brown alga Alaria sp. dominated the assemblage in the absence of grazers and the effect on this species was, therefore, chosen as the reference point ‘for determining interaction strength. A series of exclusion experiments was conducted on a subset of the grazer species and the effect of each species on the numbers of Aluria sporelings was then calculated as:

Spore density in exclusion - Spore density in control Spore density in control

Dividing this statistic by the density of grazers gives the per capita effect. The results of this study showed that the majority of species (five out of

seven) had either no significant negative effect or had positive effects by con- suming other algae which would otherwise prevent sporeling settlement. More- over, for the two species which did interact strongly (an urchin and a chiton) their effects were not additive because the urchin often excluded chitons by eating them or driving them away. On the basis of these results, Paine concludes that, in his system, the distribution of interaction strengths is strongly skewed with many weakly interacting species and only one or two which have the potential to effect major changes in the structure of the community.

Paine’s conclusion runs contrary to what most food web modellers have imagined (Lawton, 1992) (Fig. 11) and suggests that more work on modelling webs with a few strong links embedded in a system with many weak ones is required. Interestingly, it is Paine’s earlier demonstrations of strong predator links and the keystone predator concept which arose from it which has probably been most influential in shaping the modellers’ view of food webs to date. It is fitting, therefore, that he should be the one to challenge this view with clear and unequivocal data.


80 - - 70 - 60 - 50 - 40 - 30 - 20 - 10 - 0

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Interaction Strength Fig. 11. (a) Hypothesized distribution of interaction strengths in nature compared with those commonly used in food web models (b-d).

Despite the usefulness of Paines' approach it should be recognized that a competitive dominant which can develop to a monoculture and provide a reference point is often difficult to find. Indeed, such dominants are usually discovered after a keystone predator has been manipulated and these have not been found in many systems (e.g. Raffaelli and Hall, 1992; Menge, 1992; Hunter and Price, 1992). Paine's definition of a strong predator link is restrictive in the sense that, even if a predator has a large per capita effect


on its prey species, it is only a strong interaction if the prey had the capacity to dominate the system. This restriction confuses the notion of strength with that of importance, because it suggests that a strong link will be important for the structure of the community as a whole. If we adopt Paine’s definition of a strong link, one can legitimately argue that some systems may have no strongly interacting species at all!

The reader may well be wondering how the above argument affects the discussion over whether natural systems are best described by donor-control or Lotka-Volterra dynamics. If important functional linkages are absent from many webs, then it might be argued that donor-control dynamics underlie these systems, as argued by ourselves for the Ythan food web (Raffaelli and Hall, 1992). We now realize that this argument is fallacious, because most predator removal experiments focus on the role of predation in structuring communities (importance) rather than assessing the magnitudes of pair-wise species interactions (strength). It seems to us that strength is directly expressed by the aLi term in a Lotka-Volterra system (i.e. the per capita effect of one species on another), but importance embodies the indirect consequences of changes in the density of the prey species for the remainder of the community. Models require interactions expressed in terms of strength, whereas experimentalists usually think about interactions in terms of importance. Clearly, lack of evidence for keystone species in a web does not mean that pair-wise interactions are insignificant. For instance, in our own experiments, exclusion of the eider duck (Somateria mollissima) had no impact on the dynamics of the mussel-bed assemblage and we concluded that predation by the eider plays no role in maintaining the structure and composition of this assemblage. However, eider do have a highly significant impact on the abundance of their main prey, the mussel (Mytilus edulis), virtually eating out the entire production every year (Raffaelli et al., 1990). The interaction between eiders and mussels is therefore strong, but this does not generate the cascading changes through the rest of the assemblage of the kind seen on rocky shores following removal of a keystone predator (Paine, 1980). Given the magnitude of the eider- mussel interaction, and possibly other predator-prey interactions in the Ythan web, we were quite wrong in asserting that this system would be best described by donor-control dynamics.

C. Community Assembly

A second class of food web theory recognizes that communities are not created instantaneously as complete entities but develop, or assemble, from much humbler beginnings. This may be important because the principles which govern the progressive invasion of species into a system may explain the structural properties we observe in webs. In essence, community


assembly models specify a model web and a pool of potential colonists which challenge the system. Successful colonization by a new species occurs according to rules which depend on the emphasis and detail of the particular model in question. Some models explore the consequences of a progressive sequence of challenges in which communities are built up from simple starting points, perhaps a few primary producers (e.g. Yodzis, 1981; Post and Pimm, 1983). Others start somewhat further down the line to investigate the consequences of invasions into more developed web structures (e.g. Mithen and Lawton, 1986).

Yodzis (1981, 1984) modelled community assembly to explore the constraints that the flow of energy through the system might impose on development and hence on the structure of webs. Webs were assembled according to two fundamental principles; first, a species could not invade the web unless there was sufficient energy to meet its needs; and second, the more energy a population consumed the more energy it made available to its consumers. The detailed rules governing the assembly process need not concern us here, but three parameters were varied: N , the number of basal species (these formed the starting point for the assembly process and the value of N is taken as the same as for a given real web); P , the production of the basal species; and E, the ecological efficiency (Slobodkin, 1960) of the non-basal species. For each combination of these three parameters, 100 random webs were allowed to develop according to the specified assembly rules and the range of values for each of 10 common food web statistics was calculated. These ranges were then compared with those documented by Briand (1983) for his collection of 40 real webs. If the value for any of the test statistics from the real web fell outside the calculated range obtained for the 100 modelled webs, the hypothesis that the real web could have been chosen from the set of assembled webs was rejected.

Yodzis obtained satisfactory fits between his model webs and the data for webs classified by Briand as coming from fluctuating environments (i.e. those indicated in the original report as experiencing substantial temporal fluctuations in any major physical parameter). However, the fit was less good for webs from constant environments. Yodzis argues that this result supports the distinction Briand (1983) draws between webs from these two types of environment. Moreover, he hypothesizes that web structure in fluctuating environments can be explained as the product of simple rules for assembly with energetic constraints playing the major role. In constant environments, however, additional or other processes would appear to be involved. It should be noted that this work is not without its critics, particularly concerning the way the assembly process terminates and the choice of parameter values for primary productivity and ecological efficiency in model webs compared to known values for their real counterparts (Lawton, 1989). We would also point out that such attempts to model food webs are also


likely to be devalued by the use of poor data sets to parametrize the model. For example, in matrix 36 from Briand’s 1983 collection, eight species were recorded as basal (parameter N in Yodzis’ model) because no prey were documented for them in the original paper. However, the identity of four of these species (a grasshopper, a marine snail, nematodes and ostracods) shows this value of N to be nonsense. We cannot judge the sensitivity of model outcomes to variations in parameters specified by documented web structure, but the inclusion of such inconsistencies does not instil confidence and needs to be explicitly addressed. (This type of criticism is by no means confined to the model described here, it is simply a convenient vehicle for making the point.)

Sugihara (1982, 1984) considered the process of assembly as a possible explanation for the prevalence of intervality and rigid circuitry in food webs. Although these two properties are a little difficult to grasp, and are often only mentioned in passing, they are unexpected in the sense that such properties are statistically unlikely for random webs (Cohen, 1978). Sugihara showed that rigid-circuit webs will occur if new colonists are “mild specialists” that prey on species which themselves have at least one prey in common. This offers a biologically defensible explanation for the rigid-circuit property since, if we assume that species which share prey are probably biologically similar, it demands only that a new colonist eats similar types of prey. Sugihara also suggests that the property of intervality is an accidental consequence of the high frequency of rigid-circuit webs. If this explanation for the structure of webs is valid, a question arises about the legitimacy of edited webs in which prey are lumped as “similar kinds of species”. On the one hand, it is argued that patterns which are robust with respect to such editing are likely to be important, whereas on the other hand the editing itself aggregates the structural elements “similar types of species” which we assume need to be separate for an explanation of the rigid-circuit property by the assembly of “mild specialists”.

A somewhat different emphasis on the assembly issue has been taken by Post and Pimm (1983) and Drake (1985) who developed models based on Lotka-Volterra formulations which allow the local equilibrium and stability properties of the developing web to be determined relatively easily. In these models an initial web of a few primary producers is defined along with a potential colonist. The growth rate of the invader and the links with species in the existing community are defined by drawing parameter values at random from specified intervals. Successful colonization occurs when an invader population increases from initially low densities, and with each successful invasion the equilibrium densities of species in the system change, with some perhaps going extinct. If the new community is feasible (i.e. has an equilibrium with at least one positive density) and is locally stable, a


new potential colonist is chosen and the sequence repeated. Post and Pimm (1983) found that after a fairly rapid initial rise in the number of species in the system a relatively constant number was maintained because, on average, each successful invasion by one species led to the loss of another. However, as the system aged the period between successful invasions increased. In other words it became more difficult for another species to enter the system. This suggests the intriguing possibility that systems with the same species richness may differ in their susceptibility to invasion, depending on their “ecological age”. Post and Pimm (1983) also showed that the more connected (complex) a system is, the harder it is to invade; a finding which adds a new slant to the stability-complexity debate. That the assembly process itself might be important in this context is illustrated by Pimm (1991). Pimm contrasts Post and Pimm’s (1983) findings with those of Robinson and Valentine (1979) who examined the invasibility of webs which were drawn at random, rather than sequentially assembled. Both studies found that more complex systems were harder to invade, but assembled systems were more difficult to invade for a given connected- ness.

Drake (1988, 1990) obtained similar results to those of Post and Pimm (1983), but also showed that alternative stable species configurations could develop. Drake also used his assembly model to compare the structure of the invasion resistant end-point communities with real data and to examine the landscape patterns which emerge by assembling a number of patches simultaneously. Six food web statistics calculated for end-point model communities were statistically indistinguishable from those found for five different sets of food web data from varying habitats and with varying degrees of completeness. The only difference detected was that model communities had higher than average predator/prey ratios, a finding consistent with the arguments developed earlier. Drake is understandably cautious in interpreting these comparisons, but given that the model was not parametrized to represent any real web the structural similarities are quite striking. Analysis of landscape assembly showed that a high degree of heterogeneity in species distributions could be achieved despite the absence of environmental heterogeneity, gradients or disturbance. This heterogeneity arose through the development of alternative stable states and could be modified by the spatial distribution of these states in the landscape. Various patterns of dispersal were modelled and in some cases patches which were otherwise vulnerable to invasion could not be colonized because of the distribution of invasion resistant communities in the landscape.

The above-described approach to studying community assembly still assumes that local stability properties are important. However, the concept of permanence (described in an earlier section), which does not assume stable equilibria (it simply requires all species in the system to persist), has


also been applied to the assembly of communities. Law and Blackford (1992) argue that in the real world random events will cause the loss of species from communities from time to time. Thus, the communities we observe should, at least in part, reflect the ease with which communities re-assemble after the loss of a species. To explore re-assembly, Law and Blackford took apart model systems which were known to be permanent and then examined the ways in which they might be put back together, arguing that successful re-assembly of the complete system required a sequence of species introductions in which the addition of each new species resulted in a system which was itself permanent. Such a sequence is termed a “permanent path” and ensures that, after the successful establish- ment of one species the system does not collapse into another state before the next one can arrive. Although permanent paths are not the only possible route to the complete community (because species could be lost when a system collapses, but be re-introduced successfully later), Law and Blackford argue that they are probably the most important assembly routes. The number of permanent paths to the complete system gives a measure of the ease with which the system can return to its original state after the loss of species and the sequences of arrivals for each path gives the likely order in which species might assemble. Using permanent paths to study assembly processes requires rather less restrictive assumptions than for the methods described earlier because it does not demand each stage to be locally stable.

Law and Blackford (1992) found that the number of permanent paths in a system increased with the incidence of omnivory, suggesting that food webs with greater connectance will re-assemble more easily following the tempor- ary loss of species. This is something of a new development for the stability- complexity arguments outlined above and questions the conclusion usually drawn from local stability analyses that complex systems are likely to be less stable. Indeed, the result has echoes of MacArthur’s earlier reasoning (MacArthur, 1955) , except that here the conclusion is based on population dynamic considerations rather than energy flow. Since an implicit justifica- tion for much of the work on stability stems from the need to conserve communities, it can be argued that knowing how readily a community can repair after the loss of species is more important than knowing its local stability properties. Law and Blackford’s analysis suggests that non-specialized feeding habits (denoted in this instance by omnivory) may be an important determinant for successful recovery-a conclusion which suggests, for example, that marine soft-sediment communities (where many taxa have highly plastic feeding habits) should re-assemble easily after the loss of species.

The processes by which communities assemble seem to offer intuitively reasonable explanations for many of the static patterns that have been observed


in food webs and, most importantly, community assembly provides a con- ceptual framework which is likely to generate testable hypotheses. This framework is attractive, perhaps because it is easier to see the importance and relevance for the real world when questions are framed in terms of invasions and extinctions.

D. The Cascade Model An alternative type of model used to explain the patterns observed in webs has been proposed in a series of papers by Cohen and co-workers (Cohen and Newman, 1985, Cohen et al., 1985, 1986, 1990; Newman and Cohen, 1986). This model differs radically from the dynamic models described above because it is based solely on the static pattern of feeding relationships between elements in a food web. The model has been called the “cascade model” because its central assumption is that elements can be ordered a priori into a feeding hierarchy. The hierarchy is organized such that a species can never feed on those above it and feeding links between a given species and those below it are chosen at random with the probability c/n, where c is a positive constant less than n (the number of species in the web). This probability is independent for each pair of species. To generate a food web the cascade model also requires that the density of links per species d be specified, using available data.

Quantitative predictions from the cascade model have been compared with the static patterns observed in real webs and, for many of them, good agree- ment has been found; the average proportion and variance in the proportions of basal, intermediate and top predators, the fraction of linkages between these groups, the modal length of chains from basal to top species and the decline in the frequency of intervality and rigid circuit properties with increasing web size are all predicted by the model (Pimm et al., 1991). Notwithstanding this suc- cess, the model does make some apparently erroneous predictions such as the high frequency of very long and very short food chains (Pimm et al., 1991). Moreover, there is some debate regarding the predictions the cascade model makes about the frequency of omnivory in webs.

Clearly, the assumption of constant linkage density is no longer tenable (Table 3) . The relationship between L and S is better described as an increasing exponential, a problem anticipated by Cohen (1990) who produced a variant of the cascade model incorporating this new feature. This variant produces a poorer fit to some web patterns, compared with the original model, as do all but one of 12 other model variants (Cohen, 1990). How- ever, in view of our rapidly changing perception of web patterns such as food chain length, scale invariance in proportions and omnivory, it may prove difficult to decide which of these model variants, if any, is the better predictor of the characteristics of real webs.

232 S. J . HALL A N D D. G. RAFFAELLI

1. Biological Basis of Triangularity

An obvious factor which might order species into the trophic hierarchy required by the cascade model is body size since, in general, predators are larger than their prey (Warren and Lawton (1987) and references therein). This property is revealed by a food web matrix in which the elements are ordered by increasing size along the rows and down the columns. With a web arranged in this way feeding links for species which only feed on those smaller than themselves will all appear above the diagonal, giving an “upper triangular” web. Warren and Lawton (1987) showed that body size could indeed lead to the type of trophic hierarchy used in the cascade model and suggested that many food web patterns could be a product of body size. Lawton (1989) also points out that food webs where predators are smaller than their prey, such as those describing host-parasitoid and parasite webs, could also still be ordered into a trophic hierarchy. In this case the web would be lower triangular, but it still fulfils the assumptions of the cascade model. Of course webs which include parasites within complete systems would prevent ordering in a trophic hierarchy on the basis of body size. However, a reasonable catalogue of such webs has yet to be constructed and we can have little idea what patterns will emerge from them or how they might be explained.

VI. PROFITABLE PURSUITS AND BLIND ALLEYS The documentation of static patterns in food webs has been a cornerstone of the discipline and one which has stimulated many theoretical insights. How- ever, it has long been recognized that new data may indicate that particular patterns are not robust or that they need to be modified. One purpose of this chapter was to examine the extent to which the properties of recently published well-documented webs challenges conventional wisdom and to consider the implications of these findings. Several patterns from the new data sets differ substantially from those previously described, including connectance, linkage density, chain length, scale-invariant properties and the prevalence of omnivory. There is persuasive evidence that these discrepan- cies are a product of the better taxonomic resolution of web elements and the greater effort put into documenting linkages in the new webs, rather than their larger size per se.

Many have been quick to point out the limitations and difficulties in using food web data to extract patterns and our own analyses, to some extent, rein- force these criticisms. Nevertheless, if we are to construct models to help us understand and manage the environment we need to know which structural properties are important and why. An obvious difficulty with additions to the published catalogue is the question of how much new data is required before


a pattern should be abandoned. A good example of this problem is the question of connectance. If new data are simply added to the cloud of points formed by the existing catalogue of webs, we are unlikely (in any statistical sense) to alter the accepted pattern, despite the fact that better data suggest the pattern may be wrong. Quality data from studies such as Warren’s (1989) should be given far more weight than a large number of data points from webs of dubious quality. In this case, and for many others, we would argue that, rather than simply adding the new data to the existing catalogue, we should be replacing poor data. Instead of making the catalogue bigger, let us make it better. Webs which contain a duck as a basal species (Briand, 1983) and webs with two or three species (Sugihara et al., 1989) are of questionable value and we can see no good reason why webs such as these should continue to be included in analyses. There is an understandable desire to see the number of webs in the available catalogue increase, but careful gardening might be more profitable in the long run.

In view of the effort required to document larger webs fully, it is reasonable to ask whether such data sets are likely to add substantially to our understanding of natural systems. Our own feeling is that they will not. We obviously need to continue in our efforts to identify the structural properties of communities, but we remain to be convinced that efforts to document webs more completely could not be better placed. Pimm et al. (1991) point out that efforts at more complete web descriptions are likely to result in the inclusion of rare and, perhaps, dynamically unimportant links. This tendency argues for the inclusion of data on the relative frequency of feeding interactions. Being realistic, however, we doubt that such data will ever be available for more than a few webs and that the kind of web collection envisaged by those calling for more and better data is unlikely to be achieved. Also, dynamically unimportant links may contribute significantly to web infrastructure (such as compartments), and omitting them from web analyses might merely lessen the probability of finding such structures (see earlier discussion of Winemiller’s (1990) analysis of linkage threshold effects on web properties in Section 1V.E).

This is not to say that efforts to document patterns are a waste of time. Rather, that data need to be specifically collected with a question in mind. Pimm and Kitching (1988), in their discussion of the criticisms of food web studies made by Paine (1988), also appear to adopt this view. They point out that “neither of us has published a primary paper examining existing food web data since 1980”, and they describe the rationale for studying phytotelmata (tree-hole webs) as the choice of a simplified model system to test general theories “that are often unapproachable through study of more abundant but less tractable examples”. We would be tempted to

234 S. J . HALL AND D. G. RAFFAELLI

replace “often” with “almost always”, and suggest that efforts to document the links in webs ever more completely are probably misplaced in view of the alternative avenues that might be pursued. It is surely more sensible to focus on specific questions which have emerged from food web models using systems which are especially amenable to study. Phytotelamata offer exciting potential in this respect and examples, such as the efforts to test competing theories to explain the length of food chains (Pimm and Kitching, 1987), are encouraging.

Focusing on smaller, tractable systems brings with it the perennial problem of deciding on appropriate scales for such investigations. Whilst tree- hole webs might represent entire systems, other small webs clearly do not. These are usually source webs based on dung, carrion, felled logs, or even individual flower heads (e.g. Sugihara et al., 1989). An argument could be made that these are merely pieces of a much larger jigsaw and that the bound- aries of webs should be drawn at a more appropriate, larger spatial scale. Deciding on the scale to use is not easy, except perhaps for isolated freshwater bodies. Most systems are to some extent spatially open and delimit- ing their extent becomes somewhat arbitrary. Top predators will move between several such webs. The problem increases when temporal scales are considered. More species will be accumulated in predator diets the longer the diets are documented (Cohen, 1989), and web linkage then becomes a function of time and effort available. Polis (1991) suggests that even after about 5000 h of observation over 10 years there was no real sign of a decrease in the number of new species observed per field-day in the Coa- chella web. The Ythan web is based on about 20 years of observations. Over such time-scales one would expect to census a greater proportion of top predators compared with other species, since these tend to be transient features of natural systems (Paine, 1988). All the large webs in the new data sets are examples of such cumulative webs. Attractive as these webs are in terms of their size and completeness, there remains the difficulty of relating their large-scale, long-term-generated patterns to dynamical processes, such as predation, which generally operate on much smaller spatio-temporal scales (Paine, 1988).

We see two avenues of research that seem particularly promising, one focusing on community assembly and the other on documenting the strength of trophic interactions between elements in webs. Viewing the structure of communities from the assembly perspective seems to us to be one of the most profitable ways to try and understand the relevance of the patterns we observe in nature and the mechanisms which lead to their development. This belief is in part a consequence of the fact that the approach appears to offer the most practical means for developing close interactive relationships between the development of theory and experimental test. Drake (1988) articulates this view by suggesting that a profitable course would be


to identify the catalogue of assembly trajectories which are capable of pro- ducing structures of interest. Any regularities which emerge from the analysis of these trajectories (for example, the attainment of invasion resistance after reaching a given predator/prey ratio or the existence of forbidden species combinations) are candidates for assembly rules. These possible rules may then be explored and tested by manipulative experiments in the system of interest.

One could argue that, if theoretical studies are to have any practical use in the real world they should make measurable predictions for natural systems based on measurable model parameters. With the types of model outlined above one practical problem is knowing what “local” stability, “large” or “small” perturbations, or “strong” interactions might actually mean in measurable terms. Operational definitions of modelled properties are required before adequate tests can be expected. Yodzis (1989) argues that it is unlikely that we will ever do better than construct plausible community matrices, because to try to measure these quantities for all species would be far too difficult given the many sources of variability in natural communities. Moreover, because species which share similar predators and prey are usually grouped together as trophospecies, the parameters themselves may not be meaningful in any practically measurable way. Yodzis is probably right about the level of detail achievable for most community matrices, but as specifying the distribution of interaction strengths is one of the core requirements for food web models, measuring these values in nature must surely be a central goal for food web ecologists. Such a goal is certainly formidable.

An important first step towards this goal is a better specification of web structure. In this respect, the application of stable isotope techniques for tracing the flow of matter between species is an exciting development. Such methods may even have potential for determining the strength of trophic interactions (Kling et al., 1992), although difficulties remain in translating measures of the flow of matter through a food web into per capita interaction strength (Paine, 1980; Hall et al., 1990; Raffaelli and Hall, 1992). The experimental manipulation of species densities will probably remain the least equivocal tool for measuring interaction strength. Even then con- ducting such experiments with sufficient rigour will probably be impossible for many types of food web, emphasizing the need to capitalize on systems which are experimentally tractable.

ACKNOWLEDGEMENTS We thank Neo Martinez, Phil Warren, Henrik Moller, John Ollason and Roger Kitching for their comments on earlier drafts, and Neo Martinez for making data available to construct Fig. 9.


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