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Transcript
Review
Integrating food web diversity,
structure and stability
Neil Rooney1,2 and Kevin S. McCann3
1
School of Environmental Sciences, University of Guelph, Guelph, ON, N1G 2W1, Canada
Saugeen Ojibway Nation, R. R. #5, Wiarton, ON, N0H 2T0, Canada
3
Department of Integrative Biology, University of Guelph, Guelph, ON, N1G 2W1, Canada
2
Given the unprecedented rate of species extinctions
facing the planet, understanding the causes and consequences of species diversity in ecosystems is of paramount importance. Ecologists have investigated both
the influence of environmental variables on species diversity and the influence of species diversity on ecosystem function and stability. These investigations have
largely been carried out without taking into account
the overarching stabilizing structures of food webs that
arise from evolutionary and successional processes and
that are maintained through species interactions. Here,
we argue that the same large-scale structures that have
been purported to convey stability to food webs can also
help to understand both the distribution of species
diversity in nature and the relationship between species
diversity and food web stability. Specifically, the allocation of species diversity to slow energy channels within
food webs results in the skewed distribution of interactions strengths that has been shown to confer stability
to complex food webs. We end by discussing the processes that might generate and maintain the structured,
stable and diverse food webs observed in nature.
The triad of food web ecology
Of the many characteristics that can be used to describe
food webs, species diversity, structure and stability (see
Glossary) are three that have driven much recent ecological
research. These three domains are fascinating unto themselves, but some of the most groundbreaking ecology occurs
at the intersection of these realms (Figure 1). As we outline
below, ecologists have long been fascinated by the link
between species diversity and food web stability (the diversity–stability domain). Results from empirical, experimental and theoretical studies regarding this link have been
inconsistent, leaving the direct role of species diversity in
the stability of food webs elusive. Early efforts to address
this conundrum focused on using empirical data from qualitative food web networks to explore the relationship between food web complexity and stability (the structure–
stability domain). An excellent review of these efforts was
published by Dunne in 2006 [1]. More recently, ecologists
have begun to explore the intersection between quantitative
food web structures that do take into account the interaction
strengths among species and the stability of whole food
webs. Similarly, early research on the relationship between
Corresponding author: Rooney, N. ([email protected])
40
food web structure and diversity (the structure–diversity
domain) focused on the relationship between qualitative
food web structure and diversity. In this article, through
a synthesis of theory and data, we make a case for linking
species diversity to food web stability through the examination of weighted food web structures, where interaction
strengths have a key role in defining that relationship.
Below, we briefly synthesize progress that has been
made within each of the domains identified in Figure 1,
and then present a general integrated framework for
diversity, stability and structure in food webs that reintegrates diversity directly into the stability of food webs
through the lens of weighted food web structures. Finally,
we present a synthesis of research that has focused on how
diversity is generated and maintained in both space and
time, and demonstrate how our framework fits within this
larger ecological context.
The diversity–stability debate
The study of the relationship between diversity and food
web stability has a long and storied history in ecology. Both
intuition and observation led many ecologists to hold
confidence in a causal link between diversity and stability.
In his groundbreaking early work, Elton [2] noted that
‘simple communities are. . .more easily upset. . .than richer
ones, that is, more subject to destructive oscillations in
Glossary
Energy channel: a food web structure that comprises a set of highly interacting,
lower trophic-level species that derive their energy largely from the same basal
resource. These energy channels tend to be coupled by higher order, mobile
predators.
Interaction strength: the interaction strength (ai,j) of species j on species i, is
generally defined as the per capita measure of the instantaneous rate of
population change of species i owing to changes in species j. This can be
thought of in the classical sense as the terms of the elements of the community
matrix (sensu Robert May [7]), or more directly as the rates of biomass flux
between species j and i (e.g. the interaction strength, in terms of biomass flux,
of predator j on prey species i is the per capita functional response).
Qualitative food web: a representation of feeding interactions within an
ecosystem that simply illustrates who eats whom, with no information about
the strength of the feeding interactions.
Quantitative food web: a representation of feeding interactions within an
ecosystem that illustrates both the presence and strength of feeding
interactions within that ecosystem.
Stability: here, we generally define stability as any continuous metric that
attempts to measure the likelihood of the persistence of some set of interacting
species. Thus, different metrics (e.g. CV, maximum real eigenvalue or
minimum population density) can be used to measure how stable the system
is (e.g. a system with a high coefficient of variation is less likely to persist and
so is less stable; a system with a lower minimum population density is less
likely to persist and so is less stable, etc.).
0169-5347/$ – see front matter . Crown Copyright ß 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.tree.2011.09.001 Trends in Ecology and Evolution, January 2012, Vol. 27, No. 1
Review
populations and more vulnerable to invasions’. This inference is not surprising, given that homogenized systems
(e.g. agricultural systems) are more prone to wild fluctuations in population densities (as characterized by pest
outbreaks) compared with their more diverse counterparts
[3,4]. Following Elton’s lead, MacArthur [5] noted that,
among food webs, ‘Stability increases as the number of
links increases. If the number of prey species for each
species remains constant, an increase in the number of
species in the community will increase the stability’. Although MacArthur put forth some plausible explanations
for the relationship between diversity and stability in
nature, he never rigorously tested them.
This entire diversity–stability supposition, however,
was challenged by both Gardner and Ashby [6] and May
[7], who used theoretical approaches to test the relationship between diversity and stability in complex systems.
May’s use of random community matrices importantly
stripped the communities (i.e. food webs) of any structure,
allowing for the direct test of the influence of diversity on
the stability of the communities. In doing so, he demonstrated that diversity in fact tends to destabilize community dynamics in the absence of food web structure. More
specifically, May posited a stability criterion, wherein food
webs that are near equilibrium will tend towards stability
when i(SC)1/2 <1 where i is the average interaction
strength in the web, S is the number of species and C is
the connectance of the web. Thus, a direct causal link
between diversity and stability (in the absence of structure) was falsified, and May challenged ecologists to ‘...elucidate the devious strategies which make for stability in
enduring natural systems.’
There have been many studies, both empirical and theoretical, that have attempted to elucidate the ‘devious strategies’ through which stability is attained in diverse
ecosystems. Following on from May’s stability criterion,
ecologists set out to investigate the relationship between
connectance and species diversity, hypothesizing that stability could be gained in diverse communities as long as
connectance decreased with increasing diversity [1,8,9].
Early explorations of qualitative food web data suggested
that more diverse communities tend to have lower connectance, presenting a plausible mechanism through which
more diverse communities could remain stable, given May’s
stability criterion. In an important contribution, DeAngelis
[10] showed that increased connectance could result in
increased stability if: (i) consumers had low assimilation
efficiencies (i.e. low interaction strengths) (ii) higher trophiclevel species had high levels of self regulation; or (iii) donor
control had a large effect on the ecosystem in question.
Tilman and colleagues [11–15] have effectively argued
for several mechanisms by which the diversity of plant
communities could beget temporal stability. However,
these mechanisms are largely dependent on diversity occurring within the same trophic level (i.e. producers) and
do little to explain the influence of multi trophic-level
diversity. Most recent work on multi-trophic diversity
has taken a theoretical approach and results often vary.
Thebault and Loreau [16], for example, showed that, within multi-trophic food webs, increased diversity can lead to
several outcomes, largely depending on the strengths of
Trends in Ecology and Evolution January 2012, Vol. 27, No. 1
interactions among the species. Similarly, Duffy et al. [17]
have shown that multi trophic-level food webs can result in
a variety of diversity–function relationships. Results such
as these demonstrate the importance of food web topology
and the distribution of interaction strengths in nature in
determining the relationship, or the lack thereof, between
diversity and stability in food webs.
Food web structure and stability
Insight into the importance of structure in nature was
highlighted in the diversity–stability debate by Yodzis
[18]. Yodzis followed May’s mathematical scheme but this
time created community matrices that instead contained
real food web topologies and plausible estimates of interaction strength. Suddenly, these diverse natural food webs
were more stable than the randomly assembled communities in May’s mathematical universe. Yodzis [18] confirmed
that diversity in nature was not random; however, it still
remained unclear what the underlying structure was that
produced stability in these real food webs.
One approach to tackling the problem of how food webs
are structured to confer stability has been to identify stabilizing modules (or motifs) in food webs, whose stabilizing
properties are hypothesized to cascade throughout the larger network of interactions [19,20]. Laboratory analysis of
simplified microcosm food webs has demonstrated the stabilizing properties of weak interactions [21,22]. In soil food
webs, Neutel et al. [23] have shown that, over the course of
ecosystem succession, food webs generate omnivorous loop
structures that are hypothesized to stabilize what would
otherwise be a destabilizing increase in species diversity.
Bascompte and colleagues [24] have similarly shown that
the arrangement of interactions in marine food webs makes
for relatively stable systems. Specifically, the occurrence of
two consecutive strong interactions in food webs is less
frequent than would be expected by chance, pointing to
the destabilizing properties of strong interactions. Further,
when two consecutive strong interactions do occur, they tend
to be accompanied by omnivory. Interestingly, it has been
shown that a strongly interacting and unstable food chain is
often greatly stabilized by weak to moderate strengths of
omnivory [16].
The common thread in these results is that they can be
viewed within the light of the arrangement of interaction
strengths in food webs, particularly weak interactions.
McCann et al. [25] showed that weak interactions, which
are overrepresented in real food webs [26], have the potential to be powerfully stabilizing. Recent work [27–29] has
scaled this weak interaction effect up to the food web scale,
documenting the stabilizing properties of fast and slow (i.e.
weak and strong) energy channels in food webs. Within this
structure, fast energy channels tend to have smaller, faster
growing populations that have higher biomass turnover
rates compared with the slow energy channels. The ‘food
web structure’ domain in Figure 1 shows this overarching
architecture, where the left side of the graph represents a
slow energy channel wherein biomass turnover rates of
taxonomic groups and the interaction strengths between
consumers and resources are relatively low when compared with the fast energy channel illustrated on the right
side of the graph. Importantly, these energy channels are
41
(Figure_1)TD$IG][ Review
Trends in Ecology and Evolution January 2012, Vol. 27, No. 1
Food web structure
Trophic position
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80 100
Phytoplankton-derived carbon (%)
Diversity
Non-equilibrium stability
Stability
3
2
1
0
0.0
0.5 1.0 1.5 2.0 2.5
Structural asymmetry
3.0
TRENDS in Ecology & Evolution
Figure 1. The relationships among diversity, food web structure and stability. Food webs are structured such that top predators couple fast and slow energy channels. This
is represented here by the Cantabrian Sea Shelf system [17], wherein top predators, such as tuna, hake and anglerfish, derive energy from both phytoplankton (fast) and
detrital (slow) sources. This structural asymmetry in energy flow was shown to have stabilizing properties through the generation of asynchronous dynamics between
consumers within each energy channel. The figure embedded in the ‘stability’ domain shows how structural asymmetry increased the non-equilibrium stability of a model
system [14]. The relationships between diversity and stability, as well as between diversity and structure, are still unclear. However, we argue that a closer look at food web
structure reveals that they are structured such that increasing diversity within slow channels has the potential to stabilize them.
coupled by the foraging behavior of mobile top predators
that derive energy from each energy channel. There are
several reasons why this food web structure is stabilizing.
When these systems are perturbed, populations in the fast
channel respond quickly, allowing for the rapid recovery of
predator populations. The lagged response of the populations in the slow energy channel sets up a scenario in which
prey populations in the fast and slow energy channels are
behaving in an asynchronous fashion. Such asynchronous
resource dynamics produce a less variable resource base
for predators, allowing for a rapid but damped return to
equilibrium. The weak chain also competes with the strong
chain, muting some of the energy flow that would go up the
strong channel in the absence of the weak channel. This
muting also enhances stability.
The relationship between diversity and structure in
food webs
Here, we argue that we have an opportunity to re-examine
the relationship between diversity and stability through
42
the lens of overarching food web architecture. Within
offshore marine food webs, the flow of energy through
the food web originates either from phytoplankton production (the fast channel) or detritus (the slow channel).
Figure 2a shows data from the Cantabrian Sea Shelf
ecosystem [30], and demonstrates not only that the diversity of taxa within the slow detrital energy channel (defined as any taxonomic group that ultimately derives more
than 50% of its energy from detritus) is higher overall
compared with the phytoplankton channel, but this pattern also holds within each trophic level examined. This
pattern stands in contrast to the patterns for energy flow
within each channel (Figure 2b). Here, even with lower
diversity, the fast phytoplankton channel is more productive than the detrital channel.
There are several potential proximate explanations for
this observed pattern of increased diversity within the
slower detrital energy channel (Figure 2a). One first prospective reason is that the benthos is more structurally
complex compared with the pelagic zone [31]. Such
(Figure_2)TD$IG][ Review
(a)
Trends in Ecology and Evolution January 2012, Vol. 27, No. 1
(b) 600
30
–1
y )
–2
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Production (g C m
Taxonomic diversity
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400
200
0
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2+
3+
4+
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Trophic position
(c)
Trophic position
(d) 25
2000
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Percent frequency
1500
Number of chains
4+
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Phytoplankton
Detritus
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10 11 12 13 14 15
Number of links
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Figure 2. Diversity, production and network properties of energy channels within food webs. Comparing properties between energy channels in the Cantabrian Sea Shelf
ecosystem reveals that: (a) the majority of diversity within the food web occurs within the slow channel, a property that is repeated at all trophic levels; and (b) productivity
is far higher within the fast channel. Given the increased diversity, (c) the slow channel exhibits a higher degree of reticulation, as exhibited by the increased number of
realized energy pathways from basal resource (detritus) to top predator compared with the fast energy channel; and (d) the detritus-based slow channel also shows an
increase in the average number of links between basal resource and top predator (mean length = 9.2) compared with the fast channel (mean length = 8.6). Key: filled black
bars, detritus; open bars, phytoplankton.
complexity is expected to harbor more diverse communities
than those found in structurally simple habitats, as both
competitive and predatory refuges are found more often in
complex habitats [32,33]. A second but not exclusive, plausible explanation for increased diversity in the benthos is
that the basal resource (detritus) is less labile compared
with phytoplankton, and this increases diversity as there is
an ontogenetic decomposition process that requires more
functional groups to break down the basal resource into
usable energy [34]. Both of these factors point towards an
increased number of niches within the benthos. By contrast, the pelagic zone is more spatially homogeneous, and
the basal resource is far more labile compared with the
recalcitrant detritus. Therefore, this energy channel is hypothesized to be a far more competitive environment wherein a singular strategy of growing fast and reproducing
quickly is the most successful. Under these conditions,
competitive exclusion could have a role in decreasing the
diversity of the energy channel.
There are empirical arguments to suggest that the
above results might be widespread. We note that the
architecture of the coupled energy channels in the Cantabrian Sea Shelf system that results in the observed
differences in both diversity and interaction strengths
has analogies in other ecosystems. The general structure
proposed is that of macrohabitats (energy channels) that
differ in their structure and are coupled by the feeding
behavior of consumers. Similarly, lakes have benthic and
pelagic habitats that are coupled by predatory action, and
there is evidence that both zooplankton and fish communities are more diverse in the benthic compared with the
pelagic habitat [19,35]. There are other food webs where
43
Review
Reintegrating diversity back into food web stability
Above and beyond the aforementioned putative causes of
the patterns in diversity are the possible consequences of
this pattern. Specifically, the resulting pattern holds interesting possibilities for a more direct relationship between diversity and stability in nature. For example, the
network characteristics within the slow channel differ to
those of the fast channel, with the possible number of
pathways from basal resource to top predator being far
higher in the detritus-based slow energy channel compared with the phytoplankton-based fast channel
(Figure 2c). This pattern reflects back to MacArthur’s
hypothesis that the amount of choice there is in following
the pathways up through the food web is a measure of the
stability of the community. Furthermore, the mean number of links between the basal resource and top predator
within the detritus-based slow channel is higher compared
with the fast channel (Figure 2d), representing a less
efficient pathway from basal resource to top predator. It
should be noted that this measure is not reflective of the
mean trophic level of the top predator, but is instead a
network attribute based on qualitative links, many of
which are very weak.
These observations are particularly fascinating when
we revisit MacArthur [5], who stated that, ‘Efficiency
enables individual animals to out compete others, but
stability allows individual communities to out-survive less
stable ones. From this, it seems reasonable that natural
selection operates for maximum efficiency subject to certain necessary stability.’ Here, we propose that the observed coupled energy channel architecture of food webs
provides efficiency (through the fast channel) and stability
(through the slow channel) to larger food webs.
One final but crucial pattern emerges from this analysis.
As noted above, patterns of interaction strengths in nature
have long been hypothesized to influence the stability of
food webs. Specifically, it has been noted that the distribution of interaction strengths in nature is such that there
are many weak and few strong interactions [26,35,39], and
theoretical studies have shown this skewed distribution to
be stabilizing [40]. We suggest here that this pattern can be
explained by combining food web architecture and the
distribution of diversity within food webs. One of the
characteristics of feeding interactions within the slow
channel is that they tend to be weaker than those in the
fast channel [27]. Thus, we would predict that the majority
of the weak interactions observed in nature would occur
within these slow channels. This is supported in the analysis of the Cantabrian Sea Shelf ecosystem, where we
determined the per-capita standardized measure of the
strength of the interaction of predators on their prey (IS) as
44
calculated by Bascompte and others [24] as (Equation 1):
IS ¼
ðQ=BÞ j DCi j
Bi
(1)
where (Q/B)J is the number of times a population of
predator j consumes its own weight per day, DCij is the
proportion of prey i in the diet of predator j, and Bi is
biomass of prey i. The well-documented pattern of interaction strengths in food webs being skewed towards many
weak interactions holds true (Figure 3). This pattern,
however, is primarily driven by the many weak interactions observed within the slow channel. Thus, we have a
pattern in nature (the skewed distribution of interaction
strengths) that has been purported to result in increased
stability, and appears to be driven by the distribution of
diversity within the food web.
One factor that has been explored in determining food
web structure and interaction strengths in nature has been
body size. Body size ratios between predators and prey
have been associated with trophic structure [41,42] and
stability [43,44]. Our earlier work on macroecological relationships within and between energy channels showed
clear differentiation in body size relationships between
nested coupled energy channels [28], suggesting that the
observed patterns of interactions strengths is repeated at
different scales within food webs. These body size relationships were observed between coupled terrestrial and
aquatic ecosystems, within aquatic ecosystems between
benthic and pelagic energy channels, and finally within
the pelagic energy channels between bacterioplankton
and phytoplankton energy channels. It would follow that
[(Figure_3)TD$IG]there would be differences in mean interaction strengths
50
45
40
35
30
Frequency
coupling has been observed, but the distribution of diversity has not been documented. Such systems include canopy–understory predatory coupling in forest ecosystems
[36], fungal and bacterial energy channel predatory coupling in soil ecosystems [37] and resource coupling by
frugivorous birds in shrubland communities [38]. It will
be fascinating to follow up the distribution of diversity
within these systems to determine how general our observed pattern is in nature.
Trends in Ecology and Evolution January 2012, Vol. 27, No. 1
25
20
15
10
5
0
–3.0
–2.5 –2.0 –1.5 –1.0
–0.5
0.0
0.5
1.0
Log interaction strength
Key:
Detritus Phytoplankton
TRENDS in Ecology & Evolution
Figure 3. Distribution of integration strengths by energy channel. It has been
established that the distribution of interaction strengths (defined here as the percapita, standardized measure of the strength of the interaction of predators on
their prey [24]) is highly skewed towards many weak interactions and few strong
interactions. Here, we show that this pattern is generated in the Cantabrian Sea
Shelf ecosystem through the abundance of weak interactions found in the slow
detrital channel. Key: filled black bars, detritus; open bars, phytoplankton.
Review
between these energy channels at all levels, reflecting the
larger observations made in this paper. This pattern suggests that the stabilizing distribution of species and interaction strengths in food webs can be iterative and nested,
conveying stability to food webs at many different spatial
scales.
Taken together, these patterns point towards a plausible resolution of the diversity–stability debate. Specifically,
increased diversity can indeed lead to increased stability, if
that diversity is in the right place within the food web
architecture. Although May’s conclusion that diversity in
the absence of structure begets instability was no doubt
correct, our analysis suggests that food webs are indeed
structured such that higher diversity in slow energy channels increases the relative proportion of weak interactions
in food webs, leading to increased stability.
We argue that, if this pattern holds, there are real
insights that can be gained that will shed light on the
causes and consequences of diversity in nature. Take, for
example, tropical seagrass food webs. These ecosystems
are both diverse and productive when not impacted by
anthropogenic activities. However, increased human activities have been associated with decreased species diversity
in seagrass beds. Tewfik et al. [38,45] examined the structure and diversity of seagrass food webs across a gradient
of enrichment. As human density (and associated nutrient
concentrations) increases, these ecosystems lose top predators, detritus, specialist consumers and edible seagrass.
In effect, what is happening is that the fast phytoplankton
energy channel begins to dominate the system, and the
diversity associated with the benthic detrital energy channel is lost. As opportunistic consumers, sea urchins capitalize on this ready supply of energy and begin to dominate
the systems. At extreme levels of nutrient enrichment, the
systems are reduced to a homogenized system dominated
by a single, relatively inedible seagrass and a predominance of sea urchins. Similar patterns of decreased diversity of ecosystems associated with homogenization and
decreased stability have been observed in a variety of other
ecosystems. Habitat fragmentation in tidal creeks reduces
the structural complexity of the benthos, shifting food webs
to becoming cyanobacterial based as opposed to seagrass
and macroalgae based [4]. Algal bloom formation in lacustrine and oceanic ecosystems is largely associated with
increased nutrient inputs and the resulting increased
pelagic production [46]. Finally, agricultural ecosystems
have the tendency to lose species that reside within the
fungal energy channels in soil ecosystems [17,47]. Each of
these processes represents a decrease in diversity that is
targeted at the organisms that reside within slow energy
channels, and result in decreased temporal stability of the
remaining taxa.
Temporal aspects of species diversity, structure and
stability
It is interesting to take one step back at this point to view
our findings in a larger ecological context. As ecosystems
mature, there are several characteristics that change in a
rather predictable manner. Seminal work by Margalef
[48] and Odum [49] highlighted several ecosystem characteristics that change over time that fit very well with
Trends in Ecology and Evolution January 2012, Vol. 27, No. 1
our proposed framework for diversity, structure and stability. From an energetic point of view, more mature
ecosystems (i.e. systems in later successional stages)
have lower primary production per unit biomass (P:B
ratio). Associated with this decreased P:B ratio is the
increased importance of detritus as a basal resource for
consumers. Furthermore, food webs tend to increase in
complexity and diversity with time and are more resistant to external perturbations [23,34,40]. Thus, we propose that an increased importance of detritus in
ecosystems results in an increase of species diversity in
that channel. This increased diversity results in an increase in the number of weak interactions within food
webs (Figure 3), conveying stability to the larger web.
Furthermore, as these slow energy channels tend, on
average, to have larger sized organisms compared with
their fast counterparts, allometric relationships differ
between the energy channels [28].
Along with this ecosystem development come changes in
the behavior of the organisms that reside within. Recent
research has brought the concept of connectance and food
web stability back into the spotlight by highlighting the
role of foraging biology in predicting food web complexity
[50,51]. In these food webs, diet breadth based on body size
determines food web connectance. It would be interesting,
but beyond the scope of this paper, to examine how the
energy channel differences in allometric relationships
translate into diet breadth characteristics and the ensuing
ramifications for differences in connectance between fast
and slow energy channels.
Concluding remarks
In this paper, we have argued that the history of food web
ecology has primarily been driven by two of the three
domains identified in Figure 1: the diversity–stability
domain and the structure–stability domain. A clear and
natural area for research exists in studying the relationship between diversity and food web structure, which is the
diversity–structure domain (Figure 1). Beyond pointing
out this axis, we have identified where structure and
diversity appear tightly linked, and propose proximate
and ultimate explanations for this observed pattern. The
developments that we have presented in this paper point to
a novel way of viewing the causes and consequences of
species diversity in food webs. In contrast to approaches
that have looked for environmental variables that predict
species diversity on the landscape scale, we suggest that
understanding the distribution of species diversity can be
greatly aided by viewing ecosystems through the lens of
food web structure. We suggest that the very factors that
result in differential energy flow though food web energy
channels also result in different potential for species richness within these energy channels, where slow energy
channels tend to be more species rich. These slow energy
channels also exhibit, on average, more and weaker interactions among species, a putative causal mechanism for
increased persistence and, hence, enduring diversity. Although further empirical and theoretical work will advance
understanding of the complex relationships among diversity, food web structure and stability, we feel that the
material presented here is an important step in resolving
45
Review
one of the most enduring conundrums in ecology: the
diversity–stability debate.
Acknowledgments
We would like to thank Jacqueline Powers, Gabriel Gellner and Andrew
Beckerman for helpful comments on this manuscript. This research was
sponsored by an NSERC discovery grant to KSM.
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