Download The form of direct interspecific competition modifies

Oikos 000: 001–009, 2013
doi: 10.1111/j.1600-0706.2013.00346.x
© 2013 The Authors. Oikos © 2013 Nordic Society Oikos
Subject Editor: Carlos Melian. Accepted 9 April 2013
The form of direct interspecific competition modifies
secondary extinction patterns in multi-trophic food webs
Mike S. Fowler­
M. S. Fowler ([email protected]), Inst. Mediterrani d’Estudis Avançats (IMEDEA, UIB-CSIC), C/ Miquel Marquès 21, ES-07190
Esporles, Spain. Present address: Centre for Sustainable Aquatic Research, Dept of Biosciences, Wallace Building, Swansea Univ., Singleton Park,
SA2 8PP, UK.­
Forcibly removing species from ecosystems has important consequences for the remaining assemblage, leading to changes
in community structure, ecosystem functioning and secondary (cascading) extinctions. One key question that has arisen
from single- and multi-trophic ecosystem models is whether the secondary extinctions that occur within competitive communities (guilds) are also important in multi-trophic ecosystems? The loss of consumer–resource links obviously causes
secondary extinction of specialist consumers (topological extinctions), but the importance of secondary extinctions in
multi-trophic food webs driven by direct competitive exclusion remains unknown. Here I disentangle the effects of extinctions driven by basal competitive exclusion from those caused by trophic interactions in a multi-trophic ecosystem (basal
producers, intermediate and top consumers). I compared food webs where basal species either show diffuse (all species
compete with each other identically: no within guild extinctions following primary extinction) or asymmetric competition
(unequal interspecific competition: within guild extinctions are possible). Basal competitive exclusion drives extra extinction cascades across all trophic levels, with the effect amplified in larger ecosystems, though varying connectance has little
impact on results. Secondary extinction patterns based on the relative abundance of the species lost in the primary extinction differ qualitatively between diffuse and asymmetric competition. Removing asymmetric basal species with low (high)
abundance triggers fewer (more) secondary extinctions throughout the whole food web than removing diffuse basal species.
Rare asymmetric competitors experience less pressure from consumers compared to rare diffuse competitors. Simulations
revealed that diffuse basal species are never involved in extinction cascades, regardless of the trophic level of a primary
extinction, while asymmetric competitors were. This work highlights important qualitative differences in extinction patterns that arise when different assumptions are made about the form of direct competition in multi-trophic food webs.
Extinction cascades have become a classical theme in ecosystems ecology following almost half a century of research.
Secondary (cascading) extinctions have been observed in
both empirical and theoretical studies after the primary
removal of a species from the ecosystem (Paine 1966,
Estes and Palmisano 1974, Pimm 1979, 1980, Fox and
McGrady-Steed 2002, Kondoh 2003, Petchey et al. 2004,
Ebenman and Jonsson 2005, Montoya et al. 2006) and affect
both species diversity and ecosystem functioning (Thébault
et al. 2007, Zavaleta et al. 2009). If the remaining members
of an ecosystem persist following primary deletion, it can be
termed ‘species deletion stable’ (Pimm 1979) or ‘permanent’
(Law and Morton 1993, 1996); otherwise, secondary extinctions will occur as the ecosystem fragments to a state with
fewer species.
Two general approaches have been adopted in ecosystem
modelling studies, based on single trophic-level (competitive)
communities (Pimm 1979, Lundberg et al. 2000, Fowler
and Lindström 2002, Fowler 2005, 2010, Ruokolainen
et al. 2007) or multi-trophic food web models, which may
or may not include interaction strengths and population
dynamics (Pimm 1979, Borrvall et al. 2000, Dunne et al.
2002, Ebenman et al. 2004, Quince et al. 2005, Borrvall and
Ebenman 2006, 2008, Thébault et al. 2007, Petchey et al.
2008, Kaneryd et al. 2012). These approaches are related,
as multi-trophic models often incorporate fully connected
competitive communities (guilds) as a logistically growing
basal (autotrophic/producer) level, with consumer species
requiring feeding links to species on lower trophic levels to
persist. Petchey et al. (2004), however, questioned the applicability of insight from single trophic level models to multitrophic food webs.
An important issue arising from these approaches concerns the alternative possible routes to secondary extinctions
(Christianou and Ebenman 2005, Ebenman and Jonsson
2005, Thébault et al. 2007, Fowler 2010). If a specialist consumer loses its only resource in a primary extinction event,
this topological breakdown will lead directly to secondary
extinction of the specialist consumer. Resource (basal or
intermediate) species may also be lost through indirect exclusion via apparent or exploitative competition (consumer or
resource mediated coexistence). Here I aim to investigate the
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importance of one extinction route that is apparent in single
trophic-level systems, but has not yet been explicitly studied
in multi-trophic food webs: secondary extinctions via direct
competitive exclusion (Fowler 2010). Thébault et al. (2007)
recently studied direct within guild competition across all
trophic levels, showing quantitative effects of direct, asymmetric competition among consumers, but did not compare
different forms of competition, considering only the presence/
absence of within guild competition. Here, I consider the
impact of (a)symmetric competition among producer/basal
species on secondary food web extinction cascades.
I study this issue using two otherwise identical food web
models, with either asymmetric or symmetric (diffuse) basal
competition, where only asymmetric competition leads to
direct competitive exclusion among basal species. In food
web models of different size or connectivity, basal competitive exclusion via asymmetric competition is shown to alter
extinction cascade patterns across the whole food web, following primary deletion of species from any trophic level.
Different forms of basal competition are associated with
qualitatively different secondary extinction patterns when
primary extinctions are based on species’ relative abundances.
This work illustrates that important differences arise in secondary extinction patterns in food web models, depending
on species traits and the particular assumptions made about
direct interspecific competition.
Methods
Food web assembly
Multi-trophic consumer–resource dynamics were modelled
using a multi-species logistic growth model with linear
(Lotka–Volterra) type interactions between species, as
mT


dxi
 xi  bi  ∑ αij x j  dt


j =1
(1)
where xi is the density of species i in an mT species ecosystem, bi gives the intrinsic rate of increase (or mortality) and
aij gives the per capita effect of species j on the per capita
growth rate of species i. The food webs were assumed to be
triangular, with mP top consumers (carnivores), mH intermediate consumers (herbivores, mH  mP  1) and mN basal
(producer, mN  mH  1) species.
Basal species had positive intrinsic growth rates (bi  0),
set here as bi  1 (i  [mP  mH  1]...mT), while consumer
species were assumed to have negative intrinsic growth rates
(bi  0, i  1...[mPmH]). Consumer species’ mortality rates
(bi) were drawn randomly from a continuous uniform distribution with limits bi ∼U[–0.01, 0]. These values were then
sorted such that top consumers had the lowest intrinsic rates
of decline (bi values closest to zero), while intermediate consumers had relatively faster rates of decline (more negative bi
values). This approach has previously been used (Borrvall and
Ebenman 2006, Thébault et al. 2007, Petchey et al. 2008,
Kaneryd et al. 2012) to model food webs where species at
higher trophic levels are larger (Cohen et al. 2003), while
assuming larger species have lower mortality rates (Brown
et al. 2004).
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Food webs were composed of mT  6, 9, 12, ...24 species.
Connectance among consumer species and their resources
was C  L / mT2 where L is the realised number of consumerresource links. When the number of species in the food web
was varied, connectance was set as C  0.2. Connectance was
varied between 0.1 and 0.3 (the range reported for natural
food webs; Dunne et al. 2002), illustrated here with an 18
species food web. Consumer species experienced relatively
weak intraspecific feedback (aii  0.1), basal species had
stronger intraspecific feedback (aii  1).
Links were assigned randomly between consumerresource pairs, based on C, with certain constraints, i.e. all
consumers required a link to at least one resource species in
the adjacent (lower) trophic level. Top predators could behave
as omnivores, feeding strongly on at least one intermediate
consumer species and weakly on at least one basal resource,
with the effect of consumer species i on resource species j
given as aji  0.4. Generalist consumers had weak links
to one or more intermediate or basal species (aji  0.1 /
[n – 1], where n is the total number of resource species for
generalist predator i). Alternatively, specialist predators fed
only on a single resource species in the intermediate trophiclevel (aji  0.5). The effect of the resource species (j) on
a consumer (i) includes a conversion efficiency, e, given
as aij  eaji. Here, e  0.2 for resources on the adjacent
trophic level and e  0.02 for omnivorous links.
Interspecific competition between basal species was either
assumed to be equal across all species pairs (aij  aji  a,
i ≠ j), which has been referred to as ‘diffuse’ competition
(Hughes and Roughgarden 2000), or asymmetric (aij ≠ aji,
i ≠ j). These two scenarios allow exploration of the effects of
secondary extinctions driven by competitive exclusion in the
basal trophic level, in the context of a trophic food web. Primary deletion of species in a diffuse competitive community
(here a  0.5 between basal species) does not directly lead
to secondary extinction cascades via direct basal competitive
exclusion, while primary deletion of species in an asymmetric competitive community (aij ∼U[–1, 0], i ≠ j) can directly
induce secondary extinction cascades via basal competitive
exclusion.
Food webs were initially selected for further analysis
by ensuring that they were feasible (all xi* 0) and locally
(asymptotically) stable. The mT by 1 column vector of species’ equilibrium densities is found as x*  A1-b, where A
is the mT by mT matrix of aij values, and b is the mT by
1 column vector of intrinsic growth (mortality) rates, bi.
Local stability occurs when all real parts of the eigenvalues
from the Jacobian matrix (J) evaluated around the equilibrium are less than zero (ℜ{li}  0), where the elements of the
Jacobian are Jij  xi*aij.
Food web topology for both diffuse and asymmetric
competition was based on diffuse web architecture. For
each persistent diffuse web that was assembled, the same
topological consumer–resource structure was then used
to assemble an asymmetric web. Here, all aij values corresponding to basal species were sampled simultaneously
and repeatedly from the corresponding random distribution until a feasible, locally stable food web was found. This
approach ensured results were not driven by differences in
the underlying topology of diffuse and asymmetric food
webs. Persistent food webs were replicated 10 000 times
for each level of species richness (mT), connectance (C) and
form of competition.
Secondary extinctions
For each persistent food web that was assembled, equilibrium densities were used to rank each species according to
relative abundance, within a trophic level. Each species was
removed independently (primary deletion) and the feasibility and local stability of the remaining (mT – 1) species food
web was assessed. If the remaining food web was feasible and
locally stable, it was assumed to be ‘species deletion stable’
(Pimm 1979), i.e. persistent with no further species loss. If
not, secondary extinctions were assumed to occur. The probability of secondary extinctions in these models was therefore determined initially by post-primary extinction food
webs being classified as either unfeasible, or locally unstable
if feasible. These numerically based results were tested and
corresponded to simulated food webs, with at least one species showing exponential decline in density in unfeasible or
locally unstable communities. Therefore ‘species deletion
stable’ food webs here were also the only permanent assemblages (Law and Morton 1993, 1996). Consumer–resource
interactions with type II functional responses were tested
using simulations, following the assembly approach outlined
in Borrvall and Ebenman (2006), but did not generate qualitatively different results (results not shown).
The probability of secondary extinction cascades following a primary extinction in food webs with asymmetric
[P(ExtA)] or diffuse basal competition [P(ExtD)] was calculated from numerically derived results based on (Eq. 1). Each
replicate food web that was species deletion stable following
the primary extinction was scored 0 (no secondary extinctions occurred), while each food web that was not species
deletion stable was scored 1 (at least one secondary extinction occurred). P(ExtA,D) was calculated as the mean of this
score across all replicate food webs for a given parameter set
and form of competition. The impact of basal competitive
exclusion on secondary extinction cascades is simply defined
as P(ExtA) – P(ExtD). In other words, the difference in the
probability of extinction cascades that occurs as a result
of the presence of competitive asymmetries amongst basal
species is assumed to be driven, directly or indirectly, by
competitive exclusion within that guild, compared to the
baseline of secondary extinctions in food webs with diffuse
basal competition, where direct basal competitive exclusion
does not occur.
The trophic level of species that were lost in extinction
cascades was determined by simulating food web dynamics.
Post-primary extinction food webs that were unfeasible or
locally unstable were simulated over 5000 time units with
a 2nd and 3rd order Runge–Kutta solver using MatLab
R2008b. Any species in a simulated food web that declined
in density over all of the final 50 integrated time-steps (corresponding to at least 100 time-units) was considered to be
on an unavoidable extinction trajectory, therefore scored as
being extinct. These declining species were removed from the
food web and feasibility and local stability were reassessed. If
reduced food webs were unfeasible or locally unstable, simulations were repeated. Further simulation was never required
in practice: i.e. all food webs were persistent following the
removal of species on unavoidable extinction trajectories.
The frequency of extinction for species in each trophic level
(predators, herbivores and basal/producers) was recorded,
and is presented as the mean frequency of extinction within
each trophic level following a primary extinction, when secondary cascades occurred.
The number of consumer links to each basal resource species was also recorded, and used to calculate the ratio of: the
number of consumer links to 1) the least abundant, or 2)
most highly connected basal species: to the average number
of consumer links across the basal trophic level. This assessed
how important trophic and within guild interactions were
for determining basal species’ relative abundances.
Results
Probability of secondary extinctions across the
food web
Asymmetric competition among basal (producer) species
following a primary deletion increased secondary extinction
risk across the whole food web (Fig. 1). Increasing food web
size (mT) increased the probability of secondary extinctions
across the food web under both diffuse and asymmetric competition, with the difference between them (reflecting the
impact of basal competitive exclusion) increasing with the
number of species in the food web (Fig. 1A).
Increasing connectance stabilised food webs slightly following primary species loss, with a decreasing probability
of secondary extinctions under both diffuse and asymme­
tric competition (Fig. 1B, inset). Again, there was always a
higher probability of extinction cascades under asymmetric
competition, but this changed very little with connectance
(Fig. 1B).
These general patterns hold when considering the trophic
level of the species involved in the primary deletion (Fig. 2).
The probability of secondary extinctions increased with increasing food web size, following removal of species from any trophic
levels (Fig. 2A). The effects of basal competitive exclusion also
increased with food web size, following deletion from any
trophic level; the largest increases followed primary removal of
basal species (Fig. 2C).
Secondary extinctions across the food web declined with
increasing connectivity following primary deletion of intermediate consumers, but varying connectivity had little effect
following primary deletion of top consumers or basal species (Fig. 2B). Increasing connectivity was associated with
a slight reduction in the effect of competitive exclusion on
secondary extinctions following removal of top consumers or
basal species, balanced by the slight increase in effect following the removal of intermediate consumers (Fig. 2D). These
results reflect the interaction of top–down and bottom–up
regulation on intermediate consumers.
Removing species with different abundances
Primary deletion of species according to their relative abundance (within a trophic level) revealed contrasting patterns
of secondary extinctions across the food web, under diffuse and asymmetric basal competition. This difference was
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0.15
Food web topology and species’ relative abundances
(A)
P(Ext)
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Impact of basal competitive exclusion, P(ExtA) – P(ExtD)
0.1
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Which trophic levels are and aren’t involved in
secondary extinctions?
0.3
0
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0.15
0.2
0.25
Food web connectance, C
0.3
Figure 1. Competitive exclusion amongst basal species drives secondary extinction cascades throughout the food web. Increasing
(A) food web size (connectance  0.2) or (B) connectance (18 species food web) amplifies the impact of basal competitive exclusion
(the difference between extinctions under asymmetric or symmetric
competition) on secondary extinctions. Panel inlays show the probability of extinction cascades across the whole food web for diffuse
(black squares) and asymmetric basal competition (grey circles).
coupled with an interaction between abundance and food
web size under asymmetric basal competition (Fig. 3). For
example, removing the least abundant basal species from a 6,
15 or 24 species food web (species rank 4, 10 or 16, respectively) always generated the highest probability of extinction
cascades under diffuse basal competition, with extinction
risk declining if more abundant species were removed from
that trophic level (Fig. 3A–C). Conversely, primary deletion of the least abundant basal species in 15 or 24 species
food webs with asymmetric basal competition was associated
with the lowest risk of secondary extinctions, but the highest
risk in six-species food webs. Species therefore assume different importance according to relative abundance in terms
of maintaining overall food web persistence (or driving secondary extinctions) under diffuse or asymmetric competition. Removing species with low (high) abundance tended to
provoke fewer (more) extinction cascades with asymmetric
compared to diffuse basal competition (Fig. 3D–F).
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Differences in food web extinction probabilities reflect the
different mechanisms that drive relative abundance ranks
under diffuse and basal competition. For a given food web
size, variation in diffuse basal species relative abundances
was determined only by differences in trophic (consumerresource) patterns, while variation in asymmetric basal
abundances was driven by both trophic and within-guild
interactions (Fig. 4). While the least abundant basal species in food webs with either form of competition had more
consumers than average (i.e. a ratio of C–R links  1), the
least abundant asymmetric competitors had fewer consumers than the least abundant diffuse competitors (Fig. 4).
This occurred even though food webs were constrained
to have the same trophic structure for pairs of diffuse and
asymmetric basal competition: the least abundant asymmetric basal species did not necessarily share the same consumers
as the least abundant diffuse basal species.
The least abundant basal species also had fewer links to
consumers than the basal species experiencing most consumer pressure (the species with most consumer–resource
links; upper lines/circles in Fig. 4). This indicates that the
least abundant basal species have low densities due to specialist (strong) consumer interactions. The most highly connected species tended to have higher densities due to the
presence of multiple weak links from generalist consumers.
Further differences between diffuse and asymmetric basal
competition arose in the frequency that species from
different trophic levels were lost in secondary extinctions
(Fig. 5). There were differences in the frequency of secondary
extinctions within a trophic level according to: the trophic
level and relative abundance of the species lost in the primary extinction, and the food web size. The most striking
differences occurred for basal/producer species: diffuse basal
species were never involved in extinction cascades, while
asymmetric basal species were (Fig. 5 G–I). This result confirms the idea that extra extinctions in asymmetric competition food webs were driven by competitive exclusion, which
subsequently drives other extinction routes across the food
web. For example, herbivores were rarely involved in extinction cascades when other herbivores were lost in the primary
extinction from diffuse basal competition webs: mean frequency of herbivore extinctions following herbivore removal
was always  0.022 (Fig. 5 D–F, black circles). Thus, the
form of direct basal competition had important impacts on
the species lost in secondary extinctions that in turn cascaded
through the food web.
Discussion
Secondary extinction cascades driven by competitive exclusion amongst basal (producer) species with asymmetric
competition have important consequences across all trophic
levels in multi-trophic ecosystem models. The effect of
competitive exclusion, demonstrated here by the complete
1
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P(Ext)
(A)
(B)
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24
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Food web connectance, C
Figure 2. Probability of extinction cascades following primary deletion (A, B) and the impact of basal competitive exclusion on secondary
extinction cascades (C, D) after removal of basal species (triangles), intermediate (circles) and top consumers (squares). Black symbols in
(A, B) indicate results from diffuse basal competition, grey symbols indicate asymmetric competition. Connectance  0.2 in panels (A, C),
mT  18 in (B, D).
absence (presence) of basal species loss with diffuse (asymmetric) competition (Fig. 5), increases with food web size,
but is relatively unaffected by connectance. These results
hold regardless of the trophic level of the species involved
in the primary extinction, but patterns of secondary extinctions driven by competitive exclusion were highly sensitive
to the relative abundance of the deleted species within a
trophic level (Fig. 3, 5). Results from food webs with asymmetric basal competition corroborate previous insight based
on single trophic-level, competitive community models:
primary removal of the most abundant basal species is associated with the highest risk of extinction cascades (Fowler
2005, 2010). However, comparison with food webs based
on diffuse basal competition reveals a previously unappreciated, relatively stabilising effect of the least abundant basal
species with asymmetric competition (Fig. 3), which can be
understood by considering the different mechanisms that
drive relative abundances under diffuse and asymmetric
basal competition (Fig. 4).
Previous work has identified various routes to secondary extinctions in model ecosystems (Christianou and
Ebenman 2005, Ebenman and Jonsson 2005, Thébault
et al. 2007, Fowler 2010). Direct competitive exclusion
in the context of a multi-trophic food web has received
little attention previously (but see Thébault et al. 2007,
discussed below). The results here show that secondary
extinction cascades can occur through at least three other
important routes: 1) removal of a basal species can lead
directly to the loss of other basal species through competitive exclusion amongst the remaining basal species;
2) removal of an intermediate or top consumer releases a
basal species from predation pressure, which then allows
it to outcompete and exclude (an)other basal species; 3)
removal of a top consumer reduces pressure on an intermediate consumer, which in turn increases pressure on
basal species, which can in turn trigger direct competitive
exclusion. Routes 2) and )3) both involve trophic cascades
and more complex, indirect routes; however, the secondary extinctions highlighted here arise through previously
unconsidered processes.
Relative abundance has previously been proposed as a useful metric for characterising species lost in primary deletions,
in terms of driving secondary extinction cascades in simple,
competitive communities (Fowler 2005, 2010). Christianou
and Ebenman (2005) used the number of strong consumer–
resource links to (or from) the species lost in the primary
deletion to predict food web responses to primary deletion,
in a nine species, rectangular, but otherwise similar food web
model. They showed a trend for increasing secondary extinction risk with an increasing number of strong links from a
resource species, where an increasing number of links was
also positively (negatively) correlated to consumer (resource)
abundance. Here I showed that the response of the food web
to removal of rare species in any trophic level is strongly
dependent on the form of basal competition. However,
while the least abundant basal species have more consumer
EV-5
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(C)
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–0.6
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Relative abundance of primary deletion
Figure 3. The probability of secondary extinctions is strongly affected by the relative abundance of the species involved in the primary
extinction, in food webs with diffuse (black squares) or asymmetric competition (grey circles). Connectance  0.2, food web size is 6 (A,
D; mP  1, mH  2, mN  3), 15 (B, E; mP  4, mH  5, mN  6) or 24 species (C, F; mP  7, mH  8, mN  9), continuous lines connect
species within the same trophic level. The impact of basal competitive exclusion on secondary extinctions (D–F) reveals a relatively stabilising effect of the least abundant basal species with asymmetric basal competition: P(ExtA) – P(ExtD)  0.
links than average, they are not the most highly connected
of basal species.
While relative abundance was used here to characterise
species lost in primary extinctions, another possibility for
predicting secondary extinction risk is by characterising
Ratio of C-R links
2.2
1.8
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Food web size, mT
24
Figure 4. The ratio of consumer–resource links to the least abundant (squares) or most highly connected basal species (circles) compared to the average number of consumer–resource links in the
basal trophic level, in food webs with diffuse (black) or asymmetric
basal competition (grey).
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‘trophic uniqueness’ of the species lost in the primary extinction (Petchey et al. 2008), a measure that requires accurate
estimates of species interaction strengths for calculation,
which can be very difficult to gain from the typically short,
noisy data available from field populations (Berlow et al.
2004, but see Berlow et al. 2009, O’Gorman et al. 2010).
Estrada (2007) and Jordán (2009) also review different network measures that can used to identify keystone species
in food webs. A full comparison of the predictive ability of
these different methods would provide an interesting avenue
for future study.
Christianou and Ebenman (2005) noted very few secon­
dary extinctions amongst basal species following primary
deletion in their food web models. The difference between
their results and those presented here can be understood
through the specific choice of parameter values for interspecific basal competition (aij). By selecting aij ∼ U[–0.5, 0],
Christianou and Ebenman (2005, their Table 1) generate
basal species that behave more like diffuse competitors, with
a lower variance (and mean) amongst competitors than
examined here (s2(U[–0.5,0])  1/48; s2(U[–1,0])  1/12).
Thébault et al. (2007) examined the same range of within
guild competition strengths as examined here (aij ∼ U[–1,
0]), including direct competition between consumer species, but compared this with the absence of direct consumer
competition, rather than diffuse competition. Reducing the
variance of basal competition strength (e.g. aij ~ U[–0.75,
(A)
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Mean extinction frequency
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Figure 5. How often are different trophic levels involved in secondary extinction cascades? Panels show the mean frequency of predators
(A–C), herbivores (D–F) and producer/basal species (G–I) that are lost when secondary extinctions occur following the primary extinction
of predators (squares), herbivores (circles) or producer species (triangles). Columns shows results based on (left) 6, (centre) 15 and (right)
24 species food webs. Black symbols/lines show results from food webs with diffuse basal competition, grey symbols/lines are asymmetric
competition. Connectance  0.2 in all cases. The complete lack of any diffuse basal species extinctions (black symbols, lower panels) is of
particular note.
–0.25]) also tends to generate P(Ext) results more similar to
diffuse competition (results not shown; see Moen 1989 and
Kokkoris et al. 2002 for further discussion). Goldberg and
Barton (1992) analysed the available literature on plant field
experiments, and found no evidence that interspecific competition was weaker than intraspecific competition, which
suggests that even when including relatively stronger interspecific competition, food web models such as those examined here may still underestimate the impact of competitive
exclusion on ecosystem stability.
More complex approaches to modelling food webs have
incoporated predator switching and/or included allometric
relationships that constrain food web structure (Kondoh
2003, Quince et al. 2005, Staniczenko et al. 2010, Blanchard
2011, but see Naisbit et al. 2012). Recent work has also
investigated the impact of stochastic environmental variation on community extinction patterns (Enberg et al. 2006,
Ruokolainen et al. 2007, Borrvall and Ebenman 2008,
Ruokolainen and Fowler 2008, Kaneryd et al. 2012), which
has supported and extended results based on deterministic models. Here I focused on deterministic dynamics to
simplify the analysis, but future work should incorporate
predator switching and stochastic environmental variation
to investigate if and/or how these factors influence extinction cascades or species persistence in the face of direct competitive exclusion amongst basal species, or higher trophic
levels when direct competition exists. For example, it would
be interesting to establish under what conditions the storage effect (Chesson and Warner 1981, Chesson 1994) could
promote coexistence following primary species deletion,
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reducing the probability of secondary extinction cascades.
Likewise, recent work examining communities assembled
with qualitatively different species level dynamics (Fowler
2009, Fowler et al. 2012) demonstrates that including such
dynamical variation can generate novel diversity–stability
relationships at the community level. Including species
with high growth rates (leading to cyclic or more complex
dynamics in the absence of species interactions) represents
another avenue for future research in extinction cascades.
Other future research should also include mutualistic or
parasitic species interactions in the food web structure
(Fontaine et al. 2011).
Assumptions about the form of direct competition
between species on any trophic level should be based on
the best available knowledge from empirical studies. Much
research effort has so far been focussed on characterising the
number and strength of trophic (consumer–resource) links
in food webs (Berlow et al. 2004, Kéfi et al. 2012). The
results presented here show that the form of direct competition amongst species, e.g. diffuse or asymmetric, has important effects on the patterns of secondary extinction cascades
in food webs, and should be also considered in future studies
examining ecosystem responses to perturbation.­­
Acknowledgements – Thanks to Bo Ebenman and GEP members
for feedback and discussion. The CSIC JAE-Doc programme,
the Spanish Ministry of Science (grant ref. CGL2009-08298) and
the Regional Government of Balearic Islands (FEDER) provided
financial support.
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