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e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
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Food web structure affects the extinction risk of species in
ecological communities
Tomas Jonsson ∗ , Patrik Karlsson, Annie Jonsson
Systems Biology Group, School of Life Sciences, University of Skövde, P.O. Box 408, SE-541 28 Skövde, Sweden
a r t i c l e
i n f o
a b s t r a c t
Article history:
This paper studies the effect of food web structure on the extinction risk of species. We
Received 15 July 2005
examine 793 different six-species food web structures with different number, position and
Received in revised form 24 May
strength of trophic links and expose them to stochasticity in a model with Lotka–Volterra
2006
predator–prey dynamics. The characteristics of species (intrinsic rates of increase as well as
Accepted 16 June 2006
intraspecific density dependence) are held constant, but the interactions with other species
Published on line 4 August 2006
and characteristics of the food web are varied.
Keywords:
extinct in communities with strong interactions as compared to communities with no
Environmental stochasticity
strong interactions where only the secondary consumer went extinct. Extinction of a species
Food webs
directly involved in a strong interaction was more frequent than extinctions of species not
Extinction risk
directly involved in strong interactions (here termed direct and indirect extinctions, respec-
Interaction strength
tively). In model webs where both direct and indirect extinctions occurred, roughly 20% were
Extinctions of producer species occurred but were rare. Species at all trophic levels went
indirect extinctions. The probability of indirect extinctions decreased with number of links.
It is concluded that not just the presence of strong interactions but also their position and
direction can have profound effects on extinction risk of species.
Three principal components, based on 11 different food web metrics, explained 76.6% of
the variation in trophic structure among food webs that differed in the number and position, but not strength, of trophic links. The extinction risk of consumer species was closely
correlated to at least two of the three principal components, indicating that extinction risk
of consumer species were affected by food web structure. The existence of a relationship
between food web structure and extinction risk of a species was confirmed by a regression
tree analysis and a complementary log-linear analysis. These analyses showed that extinction of consumer species were affected by the position of strong interactions and a varying
number of other food web metrics, different for intermediate and top species. Furthermore,
the degree to which the equilibrium abundance of a species is affected by a press perturbation is an indication of the risk of extinction that this species faces when exposed to
environmental stochasticity. It is concluded that extinction risk of a species is determined
in a complicated way by an interaction among species characteristics, food web structure
and the type of disturbance.
© 2006 Elsevier B.V. All rights reserved.
∗
Corresponding author. Tel.: +46 500 448636; fax: +46 500 448499.
E-mail address: [email protected] (T. Jonsson).
0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2006.06.012
94
1.
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
Introduction
Today, human activity leads to increased impacts on ecosystems worldwide and species are thought to go extinct around
the clock. Furthermore, extinction probabilities are not constant across species. In the light of predicted species extinctions in the near future (Hughes et al., 1997; Ceballos and
Ehrlich, 2002; Myers and Worm, 2003; Thomas et al., 2004)
this highlights how urgent it is to learn more about what
affects the extinction risk of different species. Most studies
that relate to extinctions in communities have to this date
focused on trying to identify species attributes that may buffer
population densities against the effects of environmental variability or that correlate with an increased risk of extinction
of a single species. Theoretically, extinction risks of species
has been studied mainly using single-species models that
include intrinsic and/or extrinsic factors such as environmental noise, but without considering other species that the
species may interact with (e.g. Lande, 1993; Heino et al., 2000;
Ripa and Lundberg, 2000; Jonsson and Ebenman, 2001). Studies of extinction risk in multispecies models are few (but see
Borrvall et al., 2000; Lundberg et al., 2000, for studies of the risk
and potential effects of secondary extinctions in multispecies
systems). That species interactions can affect extinction risk
is illustrated by experiments on aquatic food webs that have
shown that presence of a secondary consumer can reduce
variability, and thus extinction risk, of producers (Persson et
al., 2001).
At the species level, apart from effects of life history
characteristics on the risk of extinction, previous studies
have focused on the effect of trophic position, degree of
omnivory and strength of species interactions. Pimm and
Lawton (1978) argued that extinction risk could be related
to the trophic position of a species in the food web. Experiments (Petchey et al., 1999) and simulations (Borrvall et al.,
2000) have indicated that species at higher trophic levels
may be more prone to extinction following a large disturbance than basal species. Number of links and link density (i.e. connectance) of a community have been suggested
to affect the likelihood of a species extinction (MacArthur,
1955; McCann et al., 1998). Furthermore, an uneven distribution of trophic links among species could imply that some
species will have a higher risk of extinction since they are
surrounded by few alternative paths. MacArthur (1955) proposed that omnivory could reduce variation in population
sizes among consumer species and thereby decrease the probability of extinction of species. On the other hand, omnivory
could lead to higher extinction risks of some species. For
example, a consumer species situated in between an omnivore species at the trophic level above and their mutual prey
at the trophic level below will suffer two-fold from omnivory
(Pimm, 1982) since the omnivore will function both as the
consumer’s predator and competitor for resources. Therefore,
omnivory could decrease the extinction risk of omnivorous
species but increase the extinction risk of other species. Theoretical studies have indicated a decreased risk of extinction with small amounts of omnivory (McCann and Hastings,
1997; Borrvall et al., 2000) and experimental results indicate
that increased degrees of omnivory enhance the capacity of a
community to recover from an external perturbation (Fagan,
1997).
Studies indicate that the distribution of interaction
strengths among species can affect the likelihood of extinction of species in food webs. Borrvall et al. (2000), for example,
indicated that stability of food webs, in terms of resistance to
extinction following removal of a species, was lower in food
webs containing a few strong links among many weak links
compared to food webs with a uniform distribution of stronger
link strengths. McCann et al. (1998) argued that time series
of a species not directly involved in a strong interaction vary
less than a species taking part in a strong interaction. The
risk of extinction therefore could be lower for a species not
involved in a strong interaction due to less variable population
fluctuations. It has been suggested that such a distribution of
strong and weak interactions may be an emergent property of
dynamic constraints on the strengths of trophic interactions
in communities (Berlow et al., 2004; Emmerson and Yearsley,
2004).
Here we argue that extinction risk of a particular species
not only depends on characteristics of that species (such as
body size, generation time or dispersal ability), but also on
interactions with and characteristics of other species (Ives and
Cardinale, 2004). The reason for this is that the sensitivity of a
species to disturbance depends not only on direct effects of a
perturbation on that species but also on indirect effects caused
by changes in abundances of other species. Furthermore, Xu
and Li (2002) showed that the responses of populations in a
food web to environmental variability is affected by food web
structure, internal dynamics and the environmental noise.
Thus, here we focus on how food web structure affects the
risk of extinction of species in a multispecies setting.
We examine 68 different six-species food web structures
with dynamics described by simple Lotka–Volterra equations
and expose them to environmental stochasticity in order
to highlight extinction risks of species in relation to food
web structure. We keep the number of species constant but
vary the number and position of trophic links and the location of one strong interaction. Thus, the characteristics of a
species (intrinsic rate of increase, as well as intraspecific density dependence) are constant, but the interaction with other
species and characteristics of the food web is varied. We do not
consider the effects of demographic stochasticity on extinction risk, instead our focus is on extinctions triggered by environmental stochasticity (which may reduce population sizes
to levels where demographic and genetic stochasticity eventually causes extinction). To our knowledge, how the food web
and species, that a particular species interact with, combine
to affect the extinction risk of a species has not been studied
directly before.
2.
Methods
2.1.
Construction of model communities and food web
metrics
From a basic triangular-shaped community module with six
species at three trophic levels we constructed seven network
types by varying the number of trophic links from 5 to 11 (Fig. 1).
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e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
Fig. 1 – Schematic view of how a basic community module with six species at three trophic levels was used to generate
network types, food webs and subwebs. Network types were created by varying the number of links from 5 to 11. A food
web is a specific constellation of trophic links within a network type (food webs that represented a mirror image of one
already existing food web were excluded). Within a food web the particular location of a strong interaction was varied to
generate different subwebs (bold lines indicate strong interactions). Extinction of species were studied by simulating 100
replicates of the dynamics of each subweb, exposed to stochastic perturbations, using Eq. (1).
In every network type the number of species and number of
trophic links were constant. By varying the position of the
trophic links in each network type, in total 68 different food
webs were produced. Within each food web one strong interaction between two species, 10 times greater in magnitude
than the basic parameter setting (see below), was added and
its position was varied. In total this procedure generated 757
different subwebs, which all had the same number of species
but varied in the number of trophic links and position of one
strong interaction. Thus, network types differ in the number
of trophic links, food webs within a particular network type
differ in the position of the trophic links, and subwebs within
a food web differ in the position of one strong interaction. As
a reference to study the effect of having one strong interaction, for each food web we also created one subweb without
any strong interaction. Altogether this resulted in a total of
825 different subwebs.
In order to characterize the structure of a food web and
enable an analysis of the relationship between extinction
of species and food web structure a number of food web
metrics (Table 1) were calculated for every model web. Here,
connectance, proportion of weak interactions and average
generality of consumer species were perfectly positively correlated to the number of trophic links (this is due to the method
used here to construct model communities, i.e. varying the
number of links and not the number of species, but this
should also be the case among natural food webs with the
same number of species). Thus, the effect of these web
metrics on extinction risk of a species cannot be separated
here and we have chosen to present results as the effect of
number of links on the risk of extinction.
2.2.
The model
Direct (interference) competition, mutualism or migration was
not included in the model. Environmental stochasticity can be
included in a mathematical model of the dynamics of species
either by making one or more model parameters stochastic
variables (with specified means and variances) or as a new
multiplicative or additive term in the deterministic model.
Here we have chosen the latter alternative. Thus, the dynamics
of the species were modelled by coupled differential equations
of Lotka–Volterra type with stochasticity:
⎛
⎞
dNi
= N i ⎝b i +
aij Nj ⎠ − ˇi ()
dt
n
j=1
(1)
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e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
Table 1 – Food web metrics used to characterize the structure of 825 model food webs
Food web metric
Irregularitya (I)
I=
Definition
(xL −x̄L )
i
(n−1)
2
Kurtosis (K)
K=
Skewnessc (M)
M=
b
n(n+1)
(n−1)(n−2)(n−3)
n
(n−1)(n−2)
Explanation
1
x̄L
xLi −x̄L 4
s
xLi is the number of links connected to
species i, x̄L the mean number of links
connected to species in a web, n the
number of species in a web, and s is the
standard deviation in number of links per
species
−
3(n−1)2
(n−2)(n−3)
xLi is the number of links connected to
species i, x̄L the mean number of links
connected to species in a web, n the
number of species in a web, and s is the
standard deviation in number of links per
species
xLi x̄L 3
xLi is the number of links connected to
species i, x̄L the mean number of links
connected to species in a web, n the
number of species in a web, and s is the
standard deviation in number of links per
species
s
n
x
i=1 i
n
Trophic height (Th ) of secondary
consumer
Th =
Number of trophic links (L)
Average vulnerabilitye (V)
Total number of predator-prey links in a web
V̄ = L/Sprey
Species-specific generalityf (Gi )
Gi = Lpreyi
Number of omnivorous links
Number of trophic links from the secondary
consumer to primary producers
d
xi is the trophic level of secondary
consumer in food chain i, n the number of
food chains from any producer species to
the secondary consumer in a web
see definition
L is the number of trophic links in a web,
Sprey the number of prey species in a web
Lpreyi is the number of prey links of a
particular predator species i
see definition
Table 3 shows the range for each metric among the food webs studied here. The extinction risk of species under environmental stochasticity
was further studied in 793 of these webs that were feasible and locally stable.
a
b
c
d
e
f
Irregularity (Sokal and Rohlf, 1995) describes the coefficient of variance in the number of links per species.
Kurtosis (Sokal and Rohlf, 1995) describes the peakedness in the distribution of number of links per species (relative to the normal distribution).
Skewness (Sokal and Rohlf, 1995) describes the assymetry in the distribution of number of links per species (relative to the normal distribution).
Trophic height describes the degree of omnivory of the secondary consumer (or the average number of links from the secondary consumer
to a producer) in a web.
Average vulnerability describes the average number of predatory links per prey species (Schoener, 1989).
Species-specific generality describes the number of prey of a particular predator species i.
Here Ni is the abundance of species i, bi the densityindependent per capita growth or mortality rate of species i, aij
the per capita effect of species j on the per capita growth rate
of species i (being negative if j is a predator on i and positive
if j is a prey to i), aii values are always negative, representing
negative density-dependence, and
ˇi () = ␦()˛˘i
(2)
is the stochastic mortality of species i (with being the integer
value of t). ␦() is a time-specific number drawn at random
(each integer time step) from a uniform distribution between
zero and unity, ˛ the constant and ˘ i is the “equilibrium gross
production of species i”, representing the biomass inflow to a
species at equilibrium. The equilibrium gross production of a
species was for producers defined as
˘i = bi × Ni ∗
(3)
where Ni * is the abundance of species i at equilibrium. For
consumers equilibrium gross production was defined as:
⎛
˘i = Ni∗ ⎝
i−1
⎞
aij Nj∗ ⎠
(4)
j=1
where the summation is across all prey species of consumer
species i. Without stochastic mortality the biomass inflow to
a species is, at equilibrium, exactly matched by an outflow
of biomass due to mortality (predation, density-dependent
and density-independent intraspecific mortality). Here, ˛ was
set to 0.2. Consequently, the growth rate of each species was
reduced by at most 20% of the gross production at equilibrium at a single time step. Thus, the long-term growth rate
of a population was depressed and the stochasticity could be
thought of as representing environmental change that affects
all species negatively.
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
Table 2 – Basic parameter setting used to simulate Eq. (1)
Parameter
Trophic level (species being
affected)
Parameter
value
bi
Basal species i = 1, 2, 3
Intermediate species i = 4, 5
Top species i = 6
+200
−0.01
0.0035
aii
All species i = 1:6
−0.025
aij
Basal species i = 1, 2, 3, j = 4, 5
i = 1, 2, 3, j = 6
Intermediate species i = 4, 5, j = 6
−0.08
−0.07
−0.016
aji
Intermediate species j = 4, 5, i = 1, 2, 3
Top species j = 6, i = 4, 5
j = 6, i = 1, 2, 3
+0.00003
+0.0002
+0.000003
Model parameters include intrinsic growth and mortality rates (bi values), intraspecific density dependence (aii ), effects of predator
on prey (aij -values) and effects of prey on predator (aji -values) of
species in six-species food webs. Parameters that represent links
that were not present in a particular food web were set to zero.
2.3.
Parameter values and simulations
Table 2 shows the basic parameter setting used to simulate
Eq. (1). All producer species have the same growth rate bi
and in absence of interspecific competition they only differ
in which predator species they are consumed by. To reflect the
longer generation times often observed at higher trophic levels (Peters, 1983) the absolute magnitude of bi decreases with
increasing trophic position of consumers. Furthermore, predator effects (aij ) and prey effects (aji ) are not equal in size and
interaction strength ratios (|aij |/aji ) decrease with trophic position of the consumer. This was chosen to reflect the unique
pattern in the distribution of interaction strengths claimed to
be found in real food webs (deRuiter et al., 1995), which could
be caused by differences in body size between consumers
and their resources (Jonsson and Ebenman, 1998; Emmerson
and Raffaelli, 2004) and/or realistic distributions of biomass
(Neutel et al., 2002).
Subwebs were checked for feasibility and local stability in
the absence of stochasticity. Only subwebs that were locally
asymptotically stable and feasible (all population sizes positive) without stochasticity were further analysed. The parameter choice (Table 2) resulted in 726 of 757 subwebs with one
strong interaction, and 67 of 68 subwebs with no strong interactions, being locally stable and feasible. Thus 793 subwebs
were further examined by simulations. In all simulations population densities started at equilibrium. Each subweb was
numerically integrated for 2000 time steps and replicated 100
times. The stiff ODE solver ode15s in Matlab 6.5 was used
to numerically integrate each subweb. A species was considered extinct at time t if Ni (t) < × Ni∗ (where Ni * is the equilibrium abundance of species i). The exact value of (here
0.05) was chosen as a compromise between obtaining enough
extinctions to allow statistical analysis and not causing extinctions to take place too early during a simulation. Results do
not differ qualitatively for other values of . Only the first
extinct species is recorded, i.e. secondary extinctions were
not considered. We have not explicitly considered the effects
of demographic and genetic stochasticity on extinction risk,
97
instead our focus is on extinctions triggered by stochasticity (e.g. environmental change) and how these are affected by
food web structure. However, the extinction threshold used
can be thought of as a critical level of population abundance
where demographic and genetic stochasticity becomes important and deterministically causes extinction of a species.
The parameter values used to simulate the dynamics of the
model webs here were arbitrarily chosen to represent a system
of three producers and three consumers with the prerequisite
that the basic parameter setting should produce a fair number of feasible and locally stable subwebs within each food
web. Other parameter values where considered but resulted
in a smaller fraction of food webs being feasible and locally
stable. We do not claim that the parameter values used here
necessarily are representative of the values found in real communities. Interaction strength have been measured in very few
real systems (e.g. Paine, 1992; Raffaelli et al., 1996; Wootton,
1997) and with great uncertainty and we simply do not know
what typical parameter values are (but see deRuiter et al., 1994;
Jonsson and Ebenman, 1998; Emmerson and Raffaelli, 2004
for recent developments trying to obtain interaction strengths
using estimates of abundance and an equilibrium assumption,
or relating interaction strengths to body size). Furthermore, we
chose to define the extinction threshold relative to the equilibrium abundance of each species rather than a fixed number
(see above). This means that it is the variability in abundance
of a species relative to its equilibrium abundance that will
affect extinction probability. We did not intend to quantify
extinction risks of species in a specific community. Rather,
our intention was to study, from a theoretical perspective,
how species characteristics and food web structure interact to
affect the vulnerability of single species, using one out of an
infinite number of possible parameter settings. Other studies
will have to study the generality and robustness of the results
presented here, using different systems, sets of parameters
and extinction thresholds.
2.4.
Data analyses
To analyze effects of food web structure on the risk of extinction of a species we analyzed simulation results by means
of (Ia) principal component analysis (PCA) to identify major
components (based on the food web metrics in Table 1) responsible for differences in structure among model webs and (Ib)
subsequent correlation analyses of the effect of three major
principal components on the extinction risk of a species, (IIa)
log-linear analysis of the null hypothesis of no relationship
between the position of the strong interaction and extinction risk of a species, and (IIb) log-linear analysis of the null
hypothesis of no interaction between the position of the strong
interaction and various food web metrics on extinction risk of
species and (III) regression tree analysis (De’ath and Fabricius,
2000) to analyze the interaction between the position of the
strong interaction and various food web metrics on extinction
risk of species. Thus, the principal goal of the PCA-correlation
analysis (Ib) was to study how the different food web metrics
affected the extinction probability of species. The focus of the
first log-linear analysis (IIa) was the effect of the position of
the strong interaction on extinction risk of species while the
combined effect of food web metrics and position of the strong
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e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
Table 3 – Range and categories for food web variables in 825 model food webs used in the multivariate log-linear analysis
to test the null hypothesis of no interaction between the position of one strong interaction and various food web metrics
on extinction risk of species (see analysis IIb, Section 2)
Variable
Extinction frequency
Position of strong interaction
Number of trophic links (L)
Number of omnivorous links
Irregularity (I)
Kurtosis (K)
Skewness (M)
Trophic height (Th ) of secondary consumer
Average vulnerability (V)
Generality of species 4 (G4 )
Generality of species 5 (G5 )
Generality of species 6 (G6 )
Range
0–100
5–11
0–3
0–0.50
−1.90–6.00
−2.40–2.40
2.25–3.00
1.00–2.50
1–3
1–3
1–5
Categories
1: 0; 2: >0
1: a41 , a42 , a43 , a51 , a52 , a53 , a61 , a62 , a63 ; 2: a14 , a24 , a34 , a15 , a25 , a35 , a64 ,
a65 ; 3: a46 , a56 , a16 , a26 , a36
1: ≤6; 2: 7, 8, 9; 3: ≥10
1: 0; 2: ≥1
1: <0.3; 2: ≥0.3
1: <−0.8 and >1.0; 2: >−0.8 and <1.0
1: <−0.1 and >0.1; 2: >−0.1 and <0.1
1: ≤2.5; 2: >2.5 and <3; 3: ≥3
1: ≤1; 2: >1
1: ≤1; 2: >1 and <3; 3: ≥3
1: ≤1; 2: >1 and <3; 3: ≥3
1: ≤1; 2: >1 and <4; 3: ≥4
For definition of food web metrics see Table 1.
interaction on extinction risk of species was the focus of the
second log-linear analysis (IIb) and the regression tree analysis.
In order to perform a log-linear analysis of frequency tables
continuous variables in the raw data (here extinction probabilities and food web metrics) needs to be classified into discrete categories. Too many categories in each variable make
it impossible to perform the analysis due to too many cells
with zero frequency while few categories make the analysis
coarse and unable to detect interesting relationships. Here,
two categories of extinction risk (<20 and ≥20%) and all 22
categories of the position of the strong interaction was used in
analysis IIa, while each variable (extinction probability, position of the strong interaction and food web metrics) needed
to be represented by only two or three categories in analysis IIb (see Table 3). In analysis IIb the best log-linear model
was selected by first finding the best two-way combination
of extinction frequency and one other variable (position of
the strong interaction and food web metrics). A better model
was then sought by including other variables, one at a time,
and keeping those variables that significantly improved the
model. The process stopped when it was no longer possible
to significantly improve the explanatory power of the model
by including new variables. In the regression tree analysis the
final best tree was found using cross-validation and choosing
the tree that represented the best compromise between small
prediction error and low complexity (the final best tree was the
smallest tree that had a prediction error within one standard
error of the prediction error of the tree with minimum prediction error). Predictor variables used in the regression tree
analysis were (1) number of trophic links, (2) trophic height
of top species, (3) generality of species 4, (4) generality of
species 5, (5) generality of species 6, (6) average vulnerability, (7) irregularity, (8) skewness, (9) kurtosis, (10) number of
omnivorous links, (11) number of specialist consumers, and
finally, presence of a strong interaction from (12) basal species
to intermediate species, (13) basal species to top species, (14)
intermediate species to top species, (15) intermediate species
to basal species, (16) top species to basal species and (17)
top species to intermediate species. More detailed classifi-
cations of the position of the strong interaction were also
tried but in those cases the regression trees explained less
of the variance in extinction risk of intermediate and top
species.
3.
Results
In subwebs without any strong interaction 15% of model webs
experienced an extinction in at least one replicate but only
the secondary consumer went extinct. In subwebs with one
strong interaction 43% of model webs experienced extinction
in at least one replicate. In 86% of these webs the same species
went extinct in the different replicates but in 14% of the webs
the extinct species and its trophic position differed among
replicates. The distribution of number of extinctions among
subwebs without any, as well as with one strong interaction
was very bimodal, that is either none of the replicates of a subweb showed any extinction or most of the replicates of a subweb showed extinctions. Extinctions of intermediate species
were the most frequent and extinctions of basal species were
rare. Extinction events of the top species were comparatively
rare among subwebs but when extinctions of the top species
occurred the majority of the replicates were included. Extinction risk of top species decreased with the number of links
while extinction risk of intermediate species was more or less
unaffected by the number of links (Fig. 2).
3.1.
Effects of strong interactions
Focusing on the effects of strong interactions revealed that the
majority of extinctions in model webs with one strong interaction involved a species that took part directly in a strong
interaction (73% out of all replicates with a recorded extinction, in only 0.01% of these cases a producer went extinct).
Thus, only 27% of all extinctions were characterized as indirect
extinctions (i.e. extinctions of a species not directly involved
in a strong interaction). Among model webs, extinctions were
usually either direct or indirect, i.e. in only 34 out of 315 subwebs with at least one recorded extinction, there were both
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
99
ate species (species 4 and 5) were associated with subwebs
with either a strong effect of species 6 on the intermediate
species or a strong effect of a basal species on the intermediate species (Fig. 3). Indirect extinctions of intermediate species
occurred in subwebs with a strong effect of the basal species
on a top species. Comparing direct and indirect extinctions of
the intermediate species (species 4 and 5), analysis showed
a significantly higher average community extinction probability of intermediate species in subwebs where they were
directly involved in a strong interaction compared to subwebs
where the strong interaction was located elsewhere in the
web (Mann–Whitney adj. for ties, U = 40886.0 and U = 94637.0,
p < 0.001 for species 4 and 5, respectively). Direct extinctions of
the top species (species 6) were mainly associated with webs
with a strong effect of species 6 on an intermediate species
while indirect extinctions of the top species mainly occurred
in webs with a strong effect of an intermediate species on a
basal species. Comparing direct and indirect extinctions of
the top species, analysis showed that, contrary to the intermediate species, species 6 had a significantly higher average
community extinction probability if there was a strong interaction in the web but it did not involve the top species, com-
Fig. 2 – Probability of extinction of (a) intermediate species
(species 4 and 5) and (b) top species (species 6) among
subwebs with one strong interaction in a six-species
Lotka–Volterra food web model exposed to stochastic
perturbations. Subwebs differed in number of links and
position of one strong interaction. The extinction threshold
was set to 5% of the equilibrium abundance of each
species. See Section 2 for simulation details, Table 2 for
description of model parameters and Fig. 1 for food web
configurations used. Bars: Total network type extinction
incidence indicates the proportion of subwebs among all
network types with at least one extinction. Line: Average
network type extinction probability indicates the average
proportion of replicates with extinctions.
direct and indirect extinctions (i.e. some replicates produced
direct and some indirect extinctions). The probability of indirect extinctions decreased with number of links. Furthermore,
the position of a strong interaction also affected the extinction pattern (Fig. 3 and log-linear analysis of no interaction
between extinction frequency and position of strong interaction: p < 0.001 for all consumer species). Strong interactions
between basal and intermediate species produced the highest
average community extinction probability (i.e. proportion of
replicates with extinctions among all subwebs). Strong omnivorous interactions (i.e. interactions between basal and top
species) were associated with the lowest average community
extinction probability.
Extinctions of producers were very few and only occurred
in subwebs with a strong effect of an intermediate on a
basal species. However, the producer species going extinct
did not necessarily take part in the strong interaction (48%
of the 23 recorded extinctions of a basal species were so
called indirect extinctions). Direct extinctions of intermedi-
Fig. 3 – Probability of direct and indirect extinctions in a
six-species Lotka–Volterra food web model exposed to
stochastic perturbations as affected by the type and
position of one strong interaction. Average community
extinction probability is the average proportion of replicates
with extinctions among all subwebs. Subwebs differed in
number of links and position of one strong interaction. The
extinction threshold was set to 5% of the equilibrium
abundance of each species. See Section 2 for simulation
details, Table 2 for description of model parameters and
Fig. 1 for food web configurations used. Interaction
coefficients aX,Y represents a strong effect of a species on
trophic level Y on a species on trophic level X. Significance
levels are presented according to Mann–Whitney U-test
(adj. for ties) with Bonferroni correction.
100
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
Table 4 – Results of principal component analysis on food web metrics using 68 model food webs with the same number
of species that differed in the number and position of trophic links
Food web metricsa
CEPb
PC1
+0.44 × [no. of links]
+0.44 × [vulnerability]
+0.30 × [generality of species 5]
−0.40 × [irregularity]
−0.38 × [no. of specialist consumers]
38.7
PC2
−0.50 × [mean trophic height]
+0.47 × [generality of species 6]
+0.50 × [no. of omnivorous links]
65.0
PC3
−0.38 × [generality of species 4]
−0.61 × [skewness]
−0.51 × [kurtosis]
76.6
Principal component analysis (PCA)
For definition of metrics see Table 1.
a
b
Text in brackets describe the most important food web metric for each of the three principal components (PC1, PC2 and PC3, factor threshold
value: ±0.30) that explained most of the variance in structure among model food webs. Numbers in front of each food web metric indicate
relative power and direction of a food web metric to the variation among food webs.
Cumulative explicatory power (% of variance).
pared to when species 6 was involved in the strong interaction
(Mann–Whitney, adj. for ties, U = 56890.0, p < 0.001).
Not only the frequency of extinctions differed between subwebs with and without a strong interaction, but also which
species went extinct. In subwebs without strong interactions
the top species was the only species that ever went extinct,
whereas in subwebs with one strong interaction extinction
occurred at all trophic levels and intermediate species were
more susceptible to extinctions than the top species. Using
two categories of extinction risk (<20 and ≥20%) and all 22
categories of the position of the strong interaction in a loglinear analysis, an interaction was detected between extinction frequency and position of the strong interaction. That
is, the probability of extinction of any consumer species was
significantly affected (p < 0.001) by the position of the strong
interaction.
3.2.
Effect of food web metrics
Our results clearly show that extinction risks of different
species are affected by the structure of the food web. Number of trophic links, irregularity, kurtosis, skewness, trophic
height, average vulnerability and generality and distribution
of interaction strengths are all different aspects of the trophic
structure of a community. Table 4 shows the three major components responsible for differences in structure among model
webs as identified by a principal component analysis (PCA)
based on the food web metrics in Table 1. The three principal
components, PC1, PC2 and PC3, were interpreted to mainly
reflect number of trophic links, number of omnivorous links
and irregularity, respectively. A factor analysis (results not
shown here) corroborated this interpretation by identifying
the same metrics as the three most important factors. The
close coupling between extinction probability and food web
structure (as characterized by the three principal components)
was verified by a correlation study (Table 5). For the model
webs studied here the scores of PC1 were significantly negatively correlated to number of extinctions of the top species
in subwebs with one strong interaction, as well as in subwebs without any strong interaction (the positive correlation
was close to, but not significant, p = 0.083, for the intermediate species). The scores of PC2 were significantly positively
correlated to number of extinctions of intermediate, and significantly negatively correlated to number of extinctions of top
species, in subwebs with one strong interaction (and significantly negatively correlated for the top species in subwebs
Table 5 – Spearman rank order correlations (rs ) between frequency of extinctions of species in (i) food webs with one
strong interaction or (ii) food webs without any strong interaction and scores of principal components (PC)
Extinctions
(i) Food webs with one strong interaction (n = 68)
Intermediate species
PC1
PC2
PC3
0.212 (p = 0.083)
0.698 (p < 0.001)
−0.024 (p = 0.1845)
Top species
−0.454 (p < 0.001)
−0.610 (p < 0.001)
−0.229 (p = 0.061)
(ii) Food webs without any strong interaction (n = 67)
Top species
−0.418 (p < 0.001)
−0.596 (p < 0.001)
−0.223 (p = 0.070)
Values in parentheses are the significance levels. Analyses were performed at the food web-level (n = 68) since subwebs created within a certain
food web only differed in position of their strong interaction, not in values of their food web metrics. As a consequence, the average number of
replicates with an extinction was used for subwebs with one strong interaction within a specific food web configuration.
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
101
Fig. 4 – The final best regression tree for explaining variation in extinction risk of (a) intermediate species (species 4 and 5)
and (b) top species (species 6) in a six-species Lotka–Volterra food web model exposed to stochastic perturbations.
Regression trees were generated using V-fold cross validation where the final best tree was chosen as the smallest tree that
had a prediction error within one standard error of the prediction error of the tree with minimum prediction error. For
intermediate and top species the best tree explains 90.93 and 95.15%, respectively of the variance in extinction risk among
different subwebs. In each node of the tree the number and short name of the predictor variable used for a binary split of
the data is listed (see Section 2 for a detailed explanation) and the split values are given on the outflow of each node. For
example, “(13) basal → top” is predictor variable 13 that details the absence (=0) or presence (=1) of a strong interaction from
a basal to a top species. In each terminal node, the expected (average) extinction probability (E) and number of observations
(n) is given. See Section 2 for all predictor variables used and simulation details, Table 1 for definition of predictor variables,
Table 2 for description of model parameters and Fig. 1 for food web configurations used.
102
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
Fig. 4 – (Continued ).
without any strong interaction). In fact, the top species never
experienced extinction as an omnivore. The scores of PC3 were
not significantly correlated to frequency of extinctions among
replicates for any of the species. Note however that PC3 is
close to significantly correlated (p = 0.061) to extinction risk of
the top species (in webs with one strong interaction). Thus,
the results show that extinction risk of consumer species is
affected by food web structure.
3.3.
Food web metrics and strong interactions
When combining food web metrics and position of strong
interactions in the same analysis, using the categories in
Table 3 a log-linear analysis show that there are significant
effects of food web structure on the risk of extinction of
primary and secondary consumers. This analysis identifies
generality of species 4, position of the strong interaction,
irregularity in number of trophic links per species and the
total number of trophic links as significantly affecting the
extinction risk of intermediate and top species. These results
are confirmed by the regression tree analysis (Fig. 4). The final
best tree explained 91% of the variation in extinction risk for
intermediate species and 95% for top species. Apart from the
variables identified as important by the log-linear analysis
the regression tree analysis also used number of omnivorous
links, trophic height of the top species, generality of the top
and intermediate species, skewness and kurtosis to construct
the best regression tree. Note that the final best regression
tree for top species is smaller (contains fewer predictor
variables) than that for intermediate species and despite
the smaller size, slightly more of the variance in extinction
risk was explained for the top species than intermediate
species.
4.
Discussion
Here we have found that response of a particular species to
stochastic perturbations depends on the trophic structure of
the food web and the strength and distribution of the intraspecific interactions among the species. The mechanisms leading
to the extinction of a species include direct as well as indirect
effects. More specifically, the extinction of consumer species
was mainly affected by the position of the strong interaction,
the number of prey of the consumer species (i.e. generality),
irregularity in the number of trophic links per species, the total
number of trophic links and the interaction among these (different combinations for different species).
From Eqs. (2) and (4), consumer mortality is expected to
be correlated with the number of prey species consumed
and the strength of the interaction with these prey species.
Thus, it could be hypothesized that the extinction risk of
consumer species is positively correlated with these two
variables. Analyzing this hypothesis explicitly reveals that,
contrary to prediction, for top species the correlation (Spearman rank correlation of extinction probability versus number
of prey species) actually is significantly negative (p < 0.001).
For intermediate species the correlation is non-significant
(p > 0.2). The coefficient of variation is low or very low (r2 = 0.23,
0.0008, respectively), leaving a lot, or most of the variation
in extinction probability unexplained. Also, contrary to prediction, for intermediate species extinction probability is
significantly negatively correlated (p < 0.001) to the average
strength of the trophic interactions with their prey species
and for top species the correlation is non-significant (p > 0.4).
Thus, it can be concluded that it is not the magnitude
of the potential consumer mortality, as defined in Eq. (4),
103
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
that explains variation in extinction risk among consumer
species.
Most extinctions today and in the near future are considered to be the result of human activity leading to environmental change (e.g. habitat degradation, climate change and soil
or water acidification). Ives (1995b) developed techniques to
predict the long-term responses of species to such directed
environmental change. He found that the response of a particular species depends on how a stressor affects the different
species and the strength and distribution of the intraspecific
interactions in the community. Intraspecific interactions are
important since they transmit positive and negative feedbacks
that produce indirect effects and lead to varying degrees of
compensation among the species (increase in the abundance
of a species when its competitors and/or predators decrease)
as a result of a perturbation. Despite the potential for very
complex responses Ives results suggested that species that
interact strongly with other species would be more strongly
buffered against changes in abundance than weakly interacting and functionally redundant species. Mechanisms behind
long-term responses of species to environmental stochasticity, which is the focus in this study, are likely to be as complex
as those governing the effects of environmental change but
are the results as simple?
Intuition and theory (e.g. Ludwig, 1996) suggests that
extinctions should occur after periods of negative trend which
takes a population to such small densities where extinction
finally becomes inevitable. This is supported by real data that
indicate that extinction probability is negatively related to
population abundance (Pimm et al., 1988; Pimm, 1991). Ripa
and Lundberg (2000) however show that in some single population models, extinctions occurring rapidly from high densities
(near or above the carrying capacity) is common. Ives and
Cardinale (2004) argue that when exposed to press perturbations (Bender et al., 1984) such as the directed environmental
changes mentioned above, species will go extinct in order of
their sensitivity to the stress in question. In a study where
species in simple model communities were removed one by
one, either at random or in order of their sensitivity to a stress,
Ives and Cardinale (2004) found that consequences of random
and ordered extinctions on the remaining community differed
(stress sensitivity was defined as the change in equilibrium
abundance relative to a small change in the magnitude of the
stressor). Here we have studied the relationship between food
web structure and species extinctions caused by stochastic
perturbations (and not directed environmental change). However, in many places environmental variability may increase
as a result of global warming and environmental stochasticity
may be critical for species that have had their abundance initially reduced by the effects of human activity. Thus, to fully
understand the effect of environmental trends it is important
to also understand how environmental fluctuations affect the
extinction risk of species in ecosystems. To see the link (if any)
between the tolerance of species (sensu Ives and Cardinale,
2004) to directed environmental change and environmental
stochasticity, we explored the possibility that the species most
resistant to extinction in the face of stochastic perturbations
are also those species that would show the greatest tolerance towards directed environmental change. Tolerance of a
species ( i ) was here defined as the change in equilibrium
abundance relative to the magnitude of a sustained disturbance on the species, i.e.
i =
Ni∗
−si
=
∗ − N∗
Ni,o
i,s
−0.5 × max{ˇi ()}
=
∗ − N∗
Ni,o
i,s
−0.1 × ˘i
(5)
∗ is the equilibrium abundance of species i in the
where Ni,o
∗ is the equilibrium abundance
absence of any disturbance, Ni,s
of species i when exposed to a sustained disturbance of magnitude si . The disturbance was here defined as 50% of the
maximum stochastic mortality that the species could suffer
in the simulations (here equal to 0.1 × ˘ i where ˘ i is the equilibrium gross production of species i defined in Eqs. (3) and
(4)). Spearman rank order correlations show that in those subwebs where a consumer species goes extinct, the tolerance
of the species (as defined above) is negatively correlated with
the extinction risk of the species (species 4: rs = −0.178, n = 172,
p = 0.0195; species 5: rs = −0.214, n = 102, p = 0.003; species 6:
rs = −0.678, n = 67, p < 0.001). That is, the degree to which the
equilibrium abundance of a consumer species is affected by
a press perturbation is an indication of the risk of extinction that this species faces when exposed to environmental
stochasticity. The greater the relative change in equilibrium
abundance the greater the extinction risk. The relationship
was found to be complex though and a large relative change
in the equilibrium abundance of a species (i.e. a highly negative tolerance value) was not necessarily associated with a
high extinction probability and high extinction probabilities
were also recorded for small relative changes in the equilibrium abundance of a species (i.e. tolerance values close to
zero). This is probably because stochastic perturbations and
not sustained perturbations were used here and despite a
highly negative tolerance value of a particular species another
species may go extinct instead. Thus, the species most vulnerable to stochastic extinctions need not be the ones most
vulnerable to a press perturbation. Instead, the response of
species to environmental variability may depend in a complicated way on the interaction among species characteristics,
food web structure and the nature of the stochasticity (frequency, amplitude, autocorrelation, etc.).
4.1.
Strong interactions and omnivory
Our results confirm the importance of strong interactions for
species dynamics (e.g. May, 1972; McCann et al., 1998) and
extinctions by showing that the extinction risk of consumer
species in the model webs studied here is higher in webs with
one strong interaction than in corresponding webs without
a strong interaction. This highlights a potentially important
scenario for conservation biology. Natural undisturbed communities are thought to be diverse, have a high degree of
redundancy with some strong interactions embedded among
a large number of weak interactions (e.g. Paine, 1992; Raffaelli
et al., 1996). Perturbations on communities such as environmental change are thought to affect redundant, weakly interacting species most (Ives, 1995b). A decrease in biodiversity,
through the loss of redundant, weakly interacting species
would increase the average interaction strength of a community, potentially making the system more resistant to directed
change (Ives and Cardinale, 2004) but more variable (McCann
104
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
et al., 1998) and, as indicated here, more sensitive to extinctions through environmental stochasticity.
Not only are the presence of strong interactions important but also their position and direction. Data from some real
soil food webs showed that strong links at the lower trophic
levels enhance stability (deRuiter et al., 1995). If community
stability is linked to extinction probability, this runs contrary
to the findings here where the highest extinction probability
was associated with a strong interaction between a basal and
an intermediate species. The extinction probability was highly
dependent on the direction of the strong interaction though.
Strong top–down interactions between intermediate and basal
species (i.e. effect of intermediate on basal species) resulted in
much lower extinction probabilities of intermediate species
than strong bottom–up interactions.
We observed higher average community extinction probabilities of intermediate species when they were involved in
a strong interaction than when the strong interaction was
located elsewhere in the web. Surprisingly, the pattern seemed
to be reversed for the top species because extinction probability of the top species was higher when involved in weak
interactions only. However, detailed analysis showed that this
was an effect of omnivory. After excluding subwebs with a
strong omnivorous interaction (i.e. a link between a basal and
the top species) the extinction probability of top species with a
strong interaction with one of its resources, was not different
from the extinction probability of intermediate species with
a strong resource link. This result underlines the potential
importance of omnivorous links for dampening population
fluctuations and in doing so reducing the risk of extinction.
4.2.
Species abundance
Apart from the intuitive expectation that species with low
intrinsic rates of increase and small population sizes generally ought to be more likely to disappear than taxa with dense,
large populations and higher growth rates, studies have looked
for other characteristics of species that would make them
more resistant to environmental variability and thus, vulnerable to extinction (Roughgarden, 1975; Lande, 1993; Ives, 1995a).
If extinction risk is correlated with population abundance
and intrinsic rates of increase this implies that extinctions
should be most common near the top of food webs, followed
by intermediate trophic levels, with only few extinctions at
the basal level (since species abundance is closely correlated
with trophic position of a species, Jonsson et al., 2005). This
prediction was met here, but only for subwebs without strong
interactions. In subwebs with a strong interaction intermediate species were instead more susceptible to extinctions than
the top species. That extinction probability here is not simply
correlated to trophic position of a species (via the abundance
and intrinsic rate of increase of a species) is probably due to
the way the extinction threshold was defined here. The extinction threshold was defined not as an absolute value but as
a fraction of the equilibrium abundance of the species (here
5% of the equilibrium abundance). This means that a species
will not have a higher extinction probability simply because
the equilibrium or average abundance of the species is low. If
an absolute extinction threshold would have been used (as in
many other studies) it would have been difficult to study other
mechanisms important in the extinction process than those
that affects the equilibrium abundance of a species. To conclude, by choosing a relative rather than an absolute extinction
threshold, we chose to ignore the possibility that abundance
per se may be important for the probability of extinction and
instead we highlight other mechanisms that may be important for determining which species are more prone to extinction than others.
4.3.
Conclusions and future directions
This study has highlighted the effect of food web structure on
the probability of extinction of species in food webs. We show
that the same species can have different extinction risks in
different food web positions or structures. What determines
the probability of extinction of any species, under a particular environmental regime, is the combined effect of species
characteristics and food web structure. Even though there are
complicated interactions among all the factors that affect the
extinction risk of a species, we have here tried to partition the
total risk into its components. We found that the most important aspects of the structure of a food web, in terms of affecting
extinction risks of species, were the presence of, position and
direction/type of strong interactions, number of prey of consumer species (generality), irregularity in number of links per
species, total number of trophic links as well as the presence
of omnivory.
Varying the number of links in a six species community
from 5 to 11 links, as have been done here, roughly corresponds to variation in connectance between 0.3 and 0.73.
Although we acknowledge that these values are higher, or
much higher, than in most observed food webs the purpose
of this study was not to study the effect of realistic variation in connectance on extinction risks of species. Instead we
sought to investigate the relationship between food web structure and extinction risk using a simple community module.
To obtain realistic connectance values in a six species food
web the number of links should not exceed 5, which is the
minimum number of links needed to ensure that no species
is disconnected from the rest of the web. Thus, using a sixspecies module it is not possible to have a food web with
no species disconnected at the same time as connectance
should not exceed 0.3. Future studies will have to show if
the results are sensitive to the range of connectance values
used.
Here we have focused on factors that affect the extinction risk of single species in communities when exposed to
environmental variability. But often the process may not end
with one particularly vulnerable species going extinct. Due
to interdependences among species in ecological communities the loss of one species can trigger a cascade of secondary
extinctions with potential dramatic effects on the functioning and stability of communities. Theoretical studies (Borrvall
et al., 2000; Lundberg et al., 2000; Solé and Montoya, 2001;
Dunne et al., 2002; Ebenman et al., 2004) have predicted the
existence of secondary extinctions and these have also been
observed in many real communities (Paine, 1966; Estes and
Palmisano, 1974; Jackson, 2001; Koh et al., 2004). Community
viability analysis (CVA, Ebenman et al., 2004; Ebenman and
Jonsson, 2005) is a newly developed tool to analyze the prob-
e c o l o g i c a l m o d e l l i n g 1 9 9 ( 2 0 0 6 ) 93–106
ability and likely extent of secondary extinctions caused by
the initial loss of one species. Recent CVA studies suggest
that species-rich communities are on average less vulnerable to species loss than species-poor communities (Dunne
et al., 2002; Ebenman et al., 2004). Jordán et al. (2002) studied extinction dynamics in simple food web models without
stochasticity. They found that species positions and interactions with other species affected extinction probability and
concluded that extinction probability of a species was more
frequent if the species had (i) an intermediate number of
trophic links and/or (ii) the number of resources was identical/similar to the number of consumers of the species. They
also looked for a relationship between the number of disconnected species after the removal of one species and extinction
probability, i.e. whether positional/static and dynamical key
stone status of species were connected, but did not find any
simple relationship. A future interesting development would
be to combine an analysis of the interaction among food web
structure, environmental variability and extinction of single
species with a community viability analysis of the secondary
extinctions that these primary extinctions may cause. Such
analyses could try to identify keystone species (Paine, 1966;
Mills et al., 1993; Menge et al., 1994; Power et al., 1996) as
species with a higher than average extinction probability that,
if going extinct, causes a greater than average number of secondary extinctions. This is a definition of keystone species
somewhat different from the usual one but with a clear quantitative meaning that is free from arbitrary interpretations (e.g.
“species whose impact on their communities is disproportionately large relative to their abundance”).
Direct competition among species was not included in this
model. Intraspecific competition among basal species introduces the possibility of compensatory changes in species
abundances as a result of changes in the abundance of one
species. By using three basal species without intraspecific
competition we could say that we model a community with
three distinct types of primary producers that use different
resources (or resource ratios, Tilman, 1985) and do not compete with each other. Each such group may be composed of a
number of taxonomic species but we only focus on the total
abundance of each primary producer category, not the dynamics within each group. How intraspecific competition among
basal species would affect the extinction probabilities reported
here we do not know, other studies will have to analyse
this.
The arrangement of trophic links and weak and strong
interactions among the species in a community may introduce
compartmentalisation (modularity) in food webs. A highly
compartmented web, for example, is organized into strongly
integrated modules with few and weak links between modules. Theoretical work (Teng and McCann, 2004) shows that
increased degree of compartmentalisation in simple model
food webs decreases variability in population densities over
time and increases the minimum densities of top predators.
This can affect extinction probabilities of species as well as
persistence of food webs (Rozdilsky et al., 2004). The model
food webs studied here were too small to allow an investigation of the effect of compartmentalisation on extinction
probability of species but this is an interesting question for
future studies.
105
Acknowledgements
We thank Bo Ebenman, Noél Holmgren, Per Lundberg and one
reviewer for valuable comments and discussion.
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