Download Interspecific interaction strength influences population density more

Ecological Modelling 332 (2016) 1–7
Contents lists available at ScienceDirect
Ecological Modelling
journal homepage: www.elsevier.com/locate/ecolmodel
Interspecific interaction strength influences population density more
than carrying capacity in more complex ecological networks
Chuan Yan, Zhibin Zhang ∗
State Key Laboratory of Integrated Management of Pest Insects and Rodents in Agriculture, Institute of Zoology, Chinese Academy of Sciences,
Beijing 100101, China
a r t i c l e
i n f o
Article history:
Received 18 December 2015
Received in revised form 26 March 2016
Accepted 29 March 2016
Keywords:
Population density
Species composition
Population dynamics
Species interaction
Ecological network
Communities
a b s t r a c t
Understanding the mechanisms determining population density of species in communities and ecological
networks is an important task in ecological studies. Interactions and carrying capacity largely determine
population density of species and then community structure. However, their impacts on population density have not been fully investigated in ecological networks. In this study, we examined the associations
of interspecific interaction strength and carrying capacity with population density in three kinds of theoretical and empirical ecological networks with different complexity. We firstly demonstrated both the
net direct and indirect interaction strength of a species received from the other species showed positive
associations with population density of the species in all ecological networks (except for in predation
networks), particularly in more complex ecological networks. Direct interaction was more important
than indirect interaction in determining population density. Carrying capacity showed a positive association with population density, particularly in less complex ecological networks. Our results suggest
that interspecific interaction strength is more important than carrying capacity in determining species
dominance in more complex networks.
© 2016 Elsevier B.V. All rights reserved.
1. Introduction
Understanding species composition and structure of communities is a great challenge in ecology. Species dominance,
which is measured by population density of each species, is an
important indicator of community. It has been well known that
population density of a species is determined by biotic interactions
and environments (Krebs, 2008). Various theoretical models like
Lotka–Volterra models and their derivatives were developed
in two-species systems, and served well for ecologists in early
literature for assessing the impacts of species interaction and
carrying capacity on population density of interacting species
(Lotka, 1925; Volterra, 1926; Pimm, 1982). In literature on food
chains or webs, the bottom-up and top-down effects have been
realized and debated over their relative importance for a long time,
and a synthesis of both forces considering heterogeneity within
or across trophic levels and flexibility is believed to an answer
(Power, 1992). Three- or four-species models were developed to
account for both bottom-up and top-down effects (Hastings and
Powell, 1991; Abrams, 2005). In general, the effects on population
∗ Corresponding author. Tel.: +86 10 64807099; fax: +86 10 6480 7099.
E-mail address: [email protected] (Z. Zhang).
http://dx.doi.org/10.1016/j.ecolmodel.2016.03.023
0304-3800/© 2016 Elsevier B.V. All rights reserved.
densities within a community are mostly studied in the systems
of a few species, not in ecological networks.
In natural ecosystems, species interact directly with each
other in a variety of ways, which forms ecological networks. The
network thinking is a prevalent systematic way of studying community and ecosystem (May, 1972; Patten, 1985). Network analysis
contributed to identifications of some primary properties of ecosystems by examining the pathways, flows, storages, and net flows of
static ecosystem models (Fath and Patten, 1999). Recently, there
are increasing studies on food webs and mutualistic networks in
literature (May, 1972; Bascompte, 2010; Thébault and Fontaine,
2010; Bascompte and Jordano, 2013), addressing various questions
including properties of ecological networks, description of realistic
webs, and stability of ecological networks, etc. Ecological network
analysis, incorporating species composition and interaction, has
provided a novel method to analyze community composition and
dynamics (Proulx et al., 2005). However, previous studies mostly
focused on network-level consequence like robustness and stability
(May, 1972; Montoya et al., 2006; Bascompte, 2009; Thébault and
Fontaine, 2010), whereas relations of population density of species
with species interaction strength and carrying capacity within a
network have not been investigated. It is necessary to investigate
these relations so as to understand the role of abiotic and biotic
factors in determining community composition. We hypothesized
2
C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7
that in more complex networks, the role of interspecific interaction
strength would be more important than that of carrying capacity in explaining population densities within a community because
species received more effects from other species.
Besides direct species interactions, impacts of indirect interaction on community dynamics have also been noticed (Menge, 1995;
Křivan and Schmitz, 2004). Indirect effect requires the presence
of intermediary species. Effects of indirect interactions included
competition for common resource species (Levine, 1976), and predation cascade in food chains (Oksanen et al., 1981) on population
density have been explored in theoretical models of a few species.
Abrams et al. (1996) reviewed theoretical and empirical studies on
indirect effects in food webs, suggesting that magnitude of indirect effects might be smaller than direct effects in most of cases
because of density dependence and stochastic disturbance, but
large enough in magnitude that they cannot be ignored. However, by applying economic input–output analysis in ecosystems
(Network Environ Analysis) (Patten, 1978), studies have demonstrated that the effects of indirect interactions, usually quantified
by flows of currency (energy or matter), often exceed the effects of
direct connections on the overall throughflow in large ecosystems
(Patten and Higashi, 1984; Patten, 1985; Fath and Patten, 1999;
Borrett et al., 2010). The dominant indirect effect in ecosystem
models raised the question whether it applies to population densities, making it crucial to test the performance of indirect effects
in explaining population densities within the complex ecological
networks, and more importantly, how the relative effects of indirect and direct interactions change with complexity of ecological
networks.
In this study, we examined the impacts of both direct and
indirect interaction strength and carrying capacity on population density of species in three kinds of theoretical and empirical
ecological networks through numerically solved multiple-species
Lotka–Volterra equations that are widely used in the studies of population and community ecology. There are two kinds of indirect
interactions in natural communities. The first one is that one species
affects population density of another species through affecting population density of an intermediate species, i.e., density-mediated
indirect interaction such as food chains; the second one is that one
species affects another species’ abundance through affecting the
interaction between the second and third species (Abrams et al.,
1996). In this study, we mainly focused on the density-mediated
indirect interaction. We have the following three predictions. First,
in more complex networks, the role of interaction strength in determining population density will be more important than that of
carrying capacity. Second, population density of a species is positively related to sum of all direct interaction strengths (net direct
interaction strength, NDIS) which come from its directly connected
partners, and also positively related to the net indirect interaction
strength it received (net indirect interaction strength, NIIS). Third,
NDIS would have a better performance in explaining population
densities in networks than NIIS. It is notable that the “interaction
strength” may be referred to different metrics in previous studies
(Berlow et al., 2004) and here we defined it to the per capita effect of
one species on another species in classic Lotka–Volterra equations,
and it had “+” or “−” signs corresponding to any positive or negative
effect caused by competition, amensalism, antagonism, neutralism,
commensalism and mutualism.
2. Material and methods
2.1. Ecological networks
We used two sets of systems: three types of simulated
networks (random network, cascade prey–predator network and
mutualistic network), and empirical mutualistic networks by
Vázquez and Simberloff (2002) and food webs by Thompson
and Townsend (2003), downloaded from an online Interaction
Web DataBase (www.nceas.ucsb.edu/interactionweb/). The random networks, mutualistic networks and cascade prey–predator
networks were constructed by following May (1972) and Allesina
and Tang (2012), with randomly distributed coefficients carrying capacity (K) sampled from uniform distributions (100–200
limited) and interaction strength (IS) sampled from normal distributions (mean = 0, standard deviation (SD) = 0.01, 0.03 or 0.05).
The normal distribution of interaction coefficients (with mean = 0)
corresponds to right-skewed distributions of interaction magnitudes that are commonly discovered in empirical works (Wootton,
1997; McCann et al., 1998). The interaction coefficients in networks
have signs corresponding to its positive or negative effects. For
simulated networks, we set different species number (S = 30,
80 or 130), connectance (C = 0.05, 0.1, 0.3 or 0.5) and interaction strength level (for random and prey–predator networks,
SD = 0.01, 0.03 or 0.05; for mutualistic networks SD = 0.01, 0.02
or 0.03). 50 networks were simulated for each combination of
S, C, and SD. For more details, see Yan and Zhang (2014). For
realistic networks, we only used their structures, and the interaction coefficients were similarly drawn from random distributions
(see above). We used average links per species (ALPS = (S − 1) × C)
and SD to indicate complexity of simulated networks. The larger
the SD value, the larger the interaction coefficients (absolute
value).
2.2. Metrics of interaction
The definition of interaction effects varies according to the purposes of different studies. Our current study focus on population
densities in ecological networks, so we used classic Lotka–Volterra
population dynamic models to quantify the interaction effects.
Based on analytical results (Lawlor, 1979, and also see Appendix
S1 and S2), the population density of a species at equilibrium was
determined by the direct and indirect interact strength it receives
and carrying capacity. We calculated two species-level metrics
of interaction strength in networks. The net direct interaction
strength was calculated
by summing all the
interaction strengths of
species i received
NDISi =
J
a
/ i ij
j=1,j =
, where J is the number
of species interacting with species i. The net indirect interaction
strength (NIIS) was calculated by summing all the indirect interaction strengths species i received. The indirect strength of one
species on another species involves all potential indirect paths
through the rest of community, and includes various interactions
like mutualism, predation, competition, etc. The sign of indirect
effect might be the same or opposite to that of direct effect. We
derived the indirect strength by following Lawlor (1979), details
in Appendix S1. Besides, we also calculated a first-order indirect
interaction strength (see Appendix S2), which is the sum of interaction strengths multiplied by itsdirectly connected species’
net
interaction strength respectively
FIISi =
J
a NDISj
/ i ij
j=1,j =
.
2.3. Population dynamics models
The population density change of one species can be represented
by a combination of contributions from logistic growth and positive and negative contributions from other species (Eqs. (1)–(4)).
In these equations, xi is the population density of species i, ri is
the intrinsic increase rate, Ki is the carrying capacity, and aij is the
per capita interspecific (i =
/ j) interaction strength. P is the number
of species showing positive effects on xi , and N is the number of
C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7
3
species showing negative effects on xi .
Population change of xi = Logistic Growth
+ Positive contributions
− Negative contributions
Logistic growth = ri xi 1 −
xi
Ki
(2)
P
Positive contributions = ri xi
a x
/ i ij j
j=1,j =
(3)
Ki
N
Negative contributions = ri xi
(1)
a x
/ i ij j
j=1,j =
Ki
(4)
If aij has either positive or negative sign, the Eqs. (2)–(4) can be
combined as the Eq. (5).
dxi
= ri xi
dt
x
1− i +
Ki
J
a x
/ i ij j
j=1,j =
Ki
Fig. 1. The performance (R2 ) of explaining population densities by different metrics
in the simulated random networks with different average links per species (ALPS)
and standard deviation (SD) of interaction strength. Each dot represents the R2
from a linear mixed model with the one or two metrics as exploratory variables
for networks with a particular parameter set. NDIS, net direct interaction strength;
NIIS, net indirect interaction strength; NDIS + NIIS, both metrics of NDIS and NIIS; K,
the carrying capacity.
(5)
J is the number of species interacting with species i. Analytical
solution is often difficult to achieve for large networks. Thus, we
numerically solved Eq. (5) by using the fourth-order Runge–Kutta
method for 20,000 time units with a step of 0.01, implemented by
deSolve package in R software (R Development Core Team, 2010).
Initial population values for species in networks were randomly
sampled from a uniform distribution between 5 and 10. The equilibrium abundance of each species was then used to infer the effects
of metrics of interaction on population density within a network.
Cycling and vanishing populations may occur in our results, but our
additional analyses indicated that they did not affect our observations (for details, see Appendix S3).
2.4. Statistical analysis
Linear mixed models (LMM) were applied to test the effects of
K, NDIS, NIIS and FIIS on the population density of species at equilibriums within each simulated network. The response variable is the
population densities within each network, the predictor variables
are K, NDIS, NIIS, or both of NDIS and NIIS within each network,
and the random variable is the networks (50 networks for each
parameter set). For example, for simulated networks with a particular parameter set, the statistical model to calculate R2 for K is as
population density = K + random (networks). In realistic webs, linear models (LM) were applied (26 food webs and 59 pollination
webs). Nine pollination webs were not used when testing the effect
of NDIS, because they were too large to calculate the inverse matrix
for NDIS. The R2 of LMM or LM was used to make comparisons of
variance explained between different metrics in different networks
with different ALPS and SD, so as to infer the impact of direct and
indirect interaction strength, and K on population density in ecological networks with different ALPS. Following the method provided
by Nakagawa and Schielzeth (2013), the calculation of R2 for LMM
was implemented by R package MuMIn, which is suggested to outperform other metrics to access model performance.
3. Results
In random networks, the net direct interaction strength (NDIS)
of a species mostly showed significant and positive impacts on population density of the species, while the net indirect interaction
strength (NIIS) only showed significant and positive impacts at the
higher SD level of 0.05 (Table S1, Fig. S1). The R2 of NDIS explained
is higher than that of NIIS at all SD levels (Fig. 1). When taking both
NDIS and NIIS into models (“NDIS + NIIS”), there was higher R2 than
either NDIS or NIIS (Fig. 1). The R2 of NDIS, NIIS and “NDIS + NIIS”
increased with the average links per species (ALPS) and SD of interaction strength (Fig. 1). The results of FIIS are similar to that of NIIS
in random networks (Fig. S4). The carrying capacity showed significant and positive impacts on population density of the species,
and its R2 decreased with increase of the average links per species
(ALPS) and SD of interaction strength (Fig. 1). At the lower SD level
of 0.01, K always showed higher R2 than interaction metrics, but
at higher SD levels of 0.03 and 0.05, K showed only higher R2 than
interaction metrics at low ALPS levels.
In simulated cascade prey–predator networks, NDIS mostly
showed significant positive impacts on population density s, while
NIIS showed small negative impacts at lower SD levels but large
negative impacts at higher SD levels (Table S2, Fig. S2). NDIS had a
higher R2 than NIIS at all cases. Taking both NDIS and NIIS into
models, they showed a similar R2 with NDIS at the SD level of
0.01, but showed a higher R2 than using NDIS or NIIS separately
at higher ALPS and SD levels (Fig. 2a). R2 of NDIS and “NDIS + NIIS”
in simulated cascade prey–predator networks increased with ALPS
at the SD level of 0.01, but showed a dome-shaped relationship with
ALPS at higher SD of interaction strength (Fig. 2a). In realistic food
webs, NDIS mostly showed significant positive impacts on population density, while NIIS showed little impacts at the SD level of
0.01, but the negative impacts increased with increase of SD levels
(Fig. 2b, Table S3). The R2 of NDIS is higher than that of NIIS at all SD
levels, when taking both NDIS and NIIS into models, “NDIS + NIIS”
showed a higher R2 than either NDIS or NIIS separately (Fig. 2b).
R2 of NDIS and “NDIS + NIIS” in realistic food webs showed a consistent increase with increase of ALPS and SD levels of interaction
strength (Fig. 2b). The results of FIIS are similar to that of NIIS in both
networks (Fig. S5), but it showed positive effects on the population
density. The results of carrying capacity on population density were
similar to those in random networks (Fig. 2).
In both simulated and realistic mutualistic networks, NDIS and
NIIS both showed significant positive impacts on population density of both trophic levels (Fig. 3, Fig. S3, Tables S4 and S5). R2 of
NDIS, NIIS, and “NDIS + NIIS” were similar in simulated mutualistic
networks, with a little higher R2 in “NDIS + NIIS” at high ALPS and
SD levels (Fig. 3). The R2 of NDIS, NIIS, and “NDIS + NIIS” increased
with ALPS and SD levels of interaction strength at both trophic
levels in simulated and realistic mutualistic networks (Fig. 3). The
results of FIIS are similar to that of NIIS in both networks (Fig. S6).
The results of carrying capacity were similar to those in random
networks (Fig. 3).
4
C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7
Fig. 2. The performance (R2 ) of explaining population densities by different metrics
in the simulated cascade prey–predator networks (a) and real food webs (b) with
different average links per species (ALPS) and standard deviation (SD) of interaction
strength. Each dot represents the R2 from a linear (mixed) model with the one or two
metrics as exploratory variables for networks with a particular parameter set (a) or a
real network (b). NDIS, net direct interaction strength; NIIS, net indirect interaction
strength; NDIS + NIIS, both metrics of NDIS and NIIS; K, the carrying capacity.
4. Discussion
In this study, we firstly demonstrated both net direct and
indirect interaction strength of a species received could largely
determine its population density, particularly in more complex
ecological networks, while carrying capacity could largely determine its population density in less complex ecological networks.
We found both the net direct and indirect interaction strength of
a species received were positively associated with population density of the species in all three kinds of ecological networks except
for in predator-prey networks, largely supporting the second prediction. We found net direct interaction strength explained more
variance of population density than indirect interaction strength
including both NIIS and FIIS in randomly and prey–predator
networks, not in mutualistic networks (in which there was little
difference), generally supporting our third prediction. Our results
provide a new insight about the impacts of biotic and abiotic factors on structure and dynamics of ecological networks. The net
direct and indirect interaction strength and the carrying capacity are important criteria in determining population density of a
species, but their roles differ in networks with different complexity;
the former is more important in more complex ecological networks,
while the latter is more important in less complex ecological
networks.
4.1. Effect of direct interaction
Direct interaction is mostly concerned regarding its theoretical conception and empirical measure in literature. In theoretical
models derived from classic Lotka–Volterra models, it is apparent that the interaction strength and K affect population density,
because the equilibrium density is calculated as a sum of parameters including interaction strength and K values as the numerator
part (see Appendix S1 and S2). The role of direct interaction strength
in determining population density has long been recognized in
Fig. 3. The performance (R2 ) of explaining population densities by different metrics
in the simulated mutualistic networks (a) and real pollination webs (b) with different
average links per species (ALPS) and standard deviation (SD) of interaction strength.
Each dot represents the R2 from a linear (mixed) model with the one or two metrics
as exploratory variables for networks with a particular parameter set (a) or a real
network (b). Higher level indicates higher trophic level species, and lower level
indicates lower trophic level species. NDIS, net direct interaction strength; NIIS,
net indirect interaction strength; NDIS + NIIS, both metrics of NDIS and NIIS; K, the
carrying capacity.
models with a few species. Mostly studied at 2–4 species systems,
classic Lotka–Volterra model and its derivatives provided basic
explanation on how species interaction affects population density in mathematic ways (Lotka, 1925; Volterra, 1926). Applying
Lotka–Volterra models in multiple-species systems is also common (Levine, 1976; Lawlor, 1979), but studies on the contribution
of species interaction to population density in ecological networks
have not been fully studied. Our results showed the simple metric
of net direct interaction strength effect turned out to be a good predictor at most cases except when SD levels of interaction strength
were too low, which implied that species that get more positive and
less negative effects would have higher population density. It can
be easily shown that many species are involved in multiple interactions including prey–predation and mutualism. From a network
view, different biotic resources a species occupies are reflected in
the positive interactions that other species interact with, such as
prey, mutualism, and commensalism, while the negative interactions indicate predation, competition and amensalism.
A common explanation of population density of species or
species dominance in plants is the resources species can occupy
C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7
(McKane et al., 2002). Thus previous studies on plants mostly
focused on inter-specific competition, especially exploitative
competition, while less concerned interactions are the biotic interactions such as the top-down effect and mutualistic interaction (e.g.
pollination). The resources-partition hypothesis is also linked to the
negative relationship between body size and population density in
animals (White et al., 2007). It is apparently shown in our results
that if species receive more positive effect and less negative effect
from other species, species will be stabilized at a higher population
density.
The interaction strength used in this study is the per capita
interaction strength in Lotka–Volterra models, which is not easy to
measure in reality. In nature, variation of both time and space can
affect estimation of interaction strength (Wootton and Emmerson,
2005). Experimental and observational approaches have been
proposed in previous studies. Experimental approaches usually
compare the absolute difference in the densities of target species
to a manipulated species after some period of time (Paine, 1980).
Observational approaches mainly include static regressions of population density s at different sites (Hallett and Pimm, 1979),
measurements of energy flow by diets and biomass (Bascompte
et al., 2005), direct observation (Wootton, 1997), modeling time
series data (Ives et al., 2003), etc. In bipartite mutualistic networks,
visiting frequency is a commonly used surrogate for interaction strength (Vázquez et al., 2005; Fang and Huang, 2016). The
interaction strength is proportional to the frequency of interaction between the two species relative to all other interactions
(Bascompte et al., 2006; Vázquez et al., 2007). It is apparent
that abundance would determine this sort of interspecific interaction frequency or strength because abundant species should have
more frequent encounters with individuals of many other species
(Vázquez et al., 2005). Thus interaction frequency is not suitable to
explain population density in communities or networks because
of apparent autocorrelation problem. We suggest the observed
interaction frequency should be corrected by using the observed
population density data before it is used as proxy of per capita interaction strength for calculating the net direct or indirect interaction
strength. The interaction strength in our study is not affected by
population density or abundance (Berlow et al., 2004). We provided clear evidence that per capita interaction strength could
well explain the population density of species in communities or
networks, particularly in complex networks.
4.2. Effect of indirect interaction
Our results in large ecological networks are consistent to observations in some previous studies which were achieved in systems of
a few species, that indirect interactions showed a relatively minor
effect (Abrams et al., 1996). However, this result is inconsistent
with the dominant indirect effect in ecosystem models (Fath and
Patten, 1999). This difference could mostly come from the different study focuses and ways to quantify inter-species connections.
Network analysis in ecosystem models focused on indirect energy
flow and storage of the whole system, while in classic population
and community models, the focuses are the population abundance
of a targeted species and the indirect influence of another species
on it through other species. There could be a large energy flow in
the system, but this does not necessarily mean it will outweigh the
direct effects on population density. How to integrate these two sets
of models and contexts needs more crossing-field work in future
studies.
Previous studies have proposed several reasons for the minor
effect of indirect interaction on population density. If interaction strength is small, then indirect interaction throughout the
path way from one species to the targeted species would be
small (Abrams et al., 1996). There are general mechanisms which
5
explain the reduction of indirect interaction effect: density dependence and stochastic disturbance (Wootton, 1994a; Abrams et al.,
1996). Schoener (1993) and Menge (1995) both found that indirect effects were smaller than direct effects in many cases, but
large enough not be ignored. However, some studies have documented important indirect interaction effects in systems with a
few species (Dungan, 1986; Wootton, 1994b). Our results showed
that the differences between effects of direct interactions and indirect interactions varied across network types. A big difference in
the explained variances of population density between direct interactions and indirect interactions was found in simulated random
and prey–predator networks and realistic food webs, but there was
little such difference in mutualistic networks. Because the absolute value of interaction strength in our study is much smaller than
1.0, the value of indirect interaction strength multiplying direct
interaction strength with the net strength its “partners” received
would become smaller than direct interactions in magnitude,
which may partially explain the relatively smaller impact of indirect interaction. For mutualistic networks, they had only positive
interaction strengths, which made the direct interact strength and
indirect interaction strengths correlated and then leaded to similar
effects.
The negative effect of NIIS on population density in
prey–predator networks is inconsistent to our prediction, which
implies that structure of network affects the effect of net indirect
interaction strength on abundance. It is not surprising that the
net indirect effects can show different signs with direct effects
(Lawlor, 1979). This might explain why NIIS showed negative
effects on population density in prey–predator networks. However, FIIS showed positive effects on population density. It is more
easily calculated that NIIS, particularly when the network is large.
Therefore, we propose FIIS might be a better metric as an indirect
interaction in predicting population density or species dominance
in communities or networks.
Estimating indirect interaction effect in nature is much similar
to estimation of direct interaction. Empirical studies to test indirect interaction effects by manipulating populations were proved
to be applicable in a system of a few species (Wootton, 1992; Miller,
1994). It is not realistic to manipulate every species in a large
community, and many studies included only short chains (Abrams
et al., 1996). Another problem is the difficulty of distinguishing
the direct and indirect effects when a change of targeted species
is observed. Path analysis was also applied in estimating indirect
effect (Johnson et al., 1991), but with shortcomings of a need for
explicit causal assumption, large sample size in complex systems,
and data variability (Wootton, 1994a). In our study, the FIIS would
be an alternative method which only uses data of direct interaction
strength.
4.3. Effect of carrying capacity (K)
Carrying capacity (K) in Lotka–Volterra models is a mathematical abstraction of the organisms environment can maximally
support, which is determined by many factors, such as climate, geographic gradients, and the traits of species themselves (Wangersky,
1978). In the Lotka–Volterra population models, K is usually
included in the analytic solutions of population density (Appendix
S1 and S2). In our study, the values of K were positively correlated to population densities at equilibrium in a network, which
is consistent with the observations in models with a few species.
Our novel finding is that under condition of weak interaction
strength and smaller ALPS, population density is mainly determined by the carrying capacity, but as the complexity of ecological
networks increases, the impact of K becomes less and less. In simple ecosystems with a few species or low and weak connectance,
species seldom interact with each other, thus it is reasonable the
6
C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7
carrying capacity plays a more important role than species interaction in determining population density of the species; but in
more complex ecosystems, resources are heavily overlapped, and
species would face predation or parasitism or mutualism from
other species, thus the contribution of the carrying capacity would
become smaller. These results imply that carrying capacity should
play more important roles in shaping structure and dynamics of
more simple ecosystems (e.g. at early succession stages, desert
ecosystems).
4.4. Effect of complexity of ecological networks (ALPS)
Ecological networks have different species richness, connectance, interaction types and strength magnitude in different
ecosystems across geographic regions (Baiser et al., 2012) and temporal scales (Schoenly and Cohen, 1991). Traditionally, studies on
ecologic networks were often focused on different network types
such as food webs and mutualistic networks depending on the
data availability, but recently studies combining different networks
start to increase (Mougi and Kondoh, 2012; Pocock et al., 2012).
Our results showed that the relative importance of direct and
indirect interactions was different in different types of networks
(see above). Regardless of network types, interaction strength of
a species received from other species includes three parts: the
number of interaction links, signs and the strength magnitude.
When species richness and connectance of networks increase, each
species has more interactions with other species on average. The
indirect interactions also increase with web size (Abrams et al.,
1996). However, at present it is still not clear how interaction
strength per species affects population density of a species when
species richness and connectance of networks increase. Our results
firstly demonstrated that increase of average links per species
(ALPS) increased the importance of direct and indirect interactions,
but reduced the contribution of carrying capacity in determining
population density. Nonetheless, in the cascade prey–predation
networks, the dome-shaped relationship between ALPS and R2
of interaction strength indicates that impact of direct interaction
strength decreased, while impact of indirect interaction strength
increased, particularly at high SD levels of interaction strength.
However, in realistic food webs, the relationship between ALPS
and R2 of interaction strength is consistently positive. These results
imply that species interaction should play more important roles
in shaping structure and dynamics of more complex communities
(e.g. at older succession stages, tropical ecosystems).
4.5. Effect of interaction strength magnitude (SD)
Previous studies show that interaction strength has to be weak
to stabilize communities (McCann, 2000; Montoya et al., 2006). It
is known that distributions of observed interaction strengths in
nature skewed toward many weak interactions and a few strong
interactions (Wootton, 1997; McCann et al., 1998). Current theoretical and empirical studies tend to support theory of weak
interaction strength (McCann, 2000). In theoretical studies, weak
interaction strength is necessary to attain high persistence of complex networks. At present, it is not clear how interaction strength
affects population density in large and well-connected ecological
networks. In our study, interaction strengths were randomly drawn
from normal distributions, and they were set to be weak to achieve
stable equilibrium of ecological networks. We have clearly shown
that increase of interaction strengths (larger SD) can increase the
performance of both direct and indirect interactions in explaining population density, but decrease that of carrying capacity.
Increased SD of interaction strengths would increase the effects
of direct and indirect interaction on population density of the
species.
5. Conclusions
Our study firstly demonstrated that both direct and indirect
interaction strengths are more important in determining population density or species dominance in more complex ecological
networks than the carrying capacity. Thus, the interaction strength
and carrying capacity are good indicators in reflecting the properties of ecological networks with different complexity. We predicted
that in less complex ecosystems (e.g. in early stage of succession or
in degenerated systems), species dominance would have a weak
association with interaction strength but have a strong association with the carrying capacity; while in more complex ecosystems
(e.g. in mature or diversified ecosystems), such relations would be
reversed. With accelerated impact of human disturbance which
has caused rapid loss of biodiversity, the impacts of interspecific
interactions on species dominance or community composition will
become weaker, while abiotic factors (e.g. climate warming, habitat
change, etc.) would play more important roles in changing structure and dynamics of ecosystems. In future studies of ecological
networks, methods of measuring per capita interaction strength or
its indices need to be developed, so as to understand the mechanism
of species dominance in ecological networks.
Acknowledgements
This work is supported by the grants of China Natural Science
Foundation (Grant Nos. 31330013, 31500347), the “Strategic Priority Research Program” of the Chinese Academy of Sciences (Grant
No. XDB11050300), and the Key Research Program of the Chinese
Academy of Sciences (Grant No. KJZD-EW-TZ-L01).
Appendix A. Supplementary data
Supplementary material related to this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.ecolmodel.2016.
03.023.
References
Abrams, P.A., 2005. The consequences of predator and prey adaptations for topdown and bottom-up effects. In: Ecology of Predator–Prey Interactions. Oxford
University Press, Oxford, UK, pp. 279–297.
Abrams, P.A., Menge, B.A., Mittelbach, G.G., Spiller, D.A., Yodzis, P., 1996. The role of
indirect effects in food webs. In: Food Webs. Springer, pp. 371–395.
Allesina, S., Tang, S., 2012. Stability criteria for complex ecosystems. Nature 483,
205–208.
Baiser, B., Gotelli, N.J., Buckley, H.L., Miller, T.E., Ellison, A.M., 2012. Geographic variation in network structure of a nearctic aquatic food web. Global Ecol. Biogeogr.
21, 579–591.
Bascompte, J., 2009. Disentangling the web of life. Science 325, 416.
Bascompte, J., 2010. Structure and dynamics of ecological networks. Science (Washington) 329, 765–766.
Bascompte, J., Jordano, P., 2013. Mutualistic Networks. Princeton University Press.
Bascompte, J., Jordano, P., Olesen, J.M., 2006. Asymmetric coevolutionary networks
facilitate biodiversity maintenance. Science 312, 431–433.
Bascompte, J., Melián, C.J., Sala, E., 2005. Interaction strength combinations and the
overfishing of a marine food web. Proc. Natl. Acad. Sci. U. S. A. 102, 5443–5447.
Berlow, E.L., Neutel, A.M., Cohen, J.E., De Ruiter, P.C., Ebenman, B., Emmerson, M.,
et al., 2004. Interaction strengths in food webs: issues and opportunities. J. Anim.
Ecol. 73, 585–598.
Borrett, S.R., Whipple, S.J., Patten, B.C., 2010. Rapid development of indirect effects
in ecological networks. Oikos 119, 1136–1148.
Dungan, M.L., 1986. Three-way interactions: barnacles, limpets, and algae in a
Sonoran Desert rocky intertidal zone. Am. Nat., 292–316.
Fang, Q., Huang, S.Q., 2016. Plant-pollinator interactions in a biodiverse meadow are
rather stable and tight for three consecutive years. Integr. Zool.
Fath, B.D., Patten, B.C., 1999. Review of the foundations of network environ analysis.
Ecosystems 2, 167–179.
Hallett, J.G., Pimm, S.L., 1979. Direct estimation of competition. Am. Nat., 593–600.
Hastings, A., Powell, T., 1991. Chaos in a three-species food chain. Ecology, 896–903.
Ives, A., Dennis, B., Cottingham, K., Carpenter, S., 2003. Estimating community stability and ecological interactions from time-series data. Ecol. Monogr. 73, 301–330.
C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7
Johnson, M.L., Huggins, D.G., DeNoyelles Jr., F., 1991. Ecosystem modeling with
LISREL: a new approach for measuring direct and indirect effects. Ecol. Appl.,
383–398.
Krebs, C., 2008. Ecology: The Experimental Analysis of Distribution and Abundance,
6th ed. Benjamin Cummings, San Francisco.
Křivan, V., Schmitz, O.J., 2004. Trait and density mediated indirect interactions in
simple food webs. Oikos 107, 239–250.
Lawlor, L.R., 1979. Direct and indirect effects of n-species competition. Oecologia 43,
355–364.
Levine, S.H., 1976. Competitive interactions in ecosystems. Am. Nat., 903–910.
Lotka, A., 1925. Elements of Physiological Biology, Dover, New York.
May, R.M., 1972. Will a large complex system be stable? Nature 238, 413–414.
McCann, K., Hastings, A., Huxel, G.R., 1998. Weak trophic interactions and the balance
of nature. Nature 395, 794–798.
McCann, K.S., 2000. The diversity-stability debate. Nature 405, 228–233.
McKane, R.B., Johnson, L.C., Shaver, G.R., Nadelhoffer, K.J., Rastetter, E.B., Fry, B.,
et al., 2002. Resource-based niches provide a basis for plant species diversity
and dominance in arctic tundra. Nature 415, 68–71.
Menge, B.A., 1995. Indirect effects in marine rocky intertidal interaction webs: patterns and importance. Ecol. Monogr. 65, 21–74.
Miller, T.E., 1994. Direct and indirect species interactions in an early old-field plant
community. Am. Nat., 1007–1025.
Montoya, J.M., Pimm, S.L., Solé, R.V., 2006. Ecological networks and their fragility.
Nature 442, 259–264.
Mougi, A., Kondoh, M., 2012. Diversity of interaction types and ecological community
stability. Science 337, 349–351.
Nakagawa, S., Schielzeth, H., 2013. A general and simple method for obtaining R2
from generalized linear mixed-effects models. Methods Ecol. Evol. 4, 133–142.
Oksanen, L., Fretwell, S.D., Arruda, J., Niemela, P., 1981. Exploitation ecosystems in
gradients of primary productivity. Am. Nat., 240–261.
Paine, R.T., 1980. Food webs: linkage, interaction strength and community infrastructure. J. Anim. Ecol., 667–685.
Patten, B., 1985. Energy cycling, length of food chains, and direct versus indirect
effects in ecosystems. Can. Bull. Fish. Aquat. Sci. 213, 119–138.
Patten, B.C., 1978. Systems approach to the concept of environment. Ohio J. Sci. 78,
206–222.
Patten, B.C., Higashi, M., 1984. Modified cycling index for ecological applications.
Ecol. Model. 25, 69–83.
Pimm, S., 1982. Food webs. In: Food Webs. Springer, Netherlands, pp. 1–11.
Pocock, M.J., Evans, D.M., Memmott, J., 2012. The robustness and restoration of a
network of ecological networks. Science 335, 973–977.
7
Power, M.E., 1992. Top-down and bottom-up forces in food webs: do plants have
primacy. Ecology 73, 733–746.
Proulx, S.R., Promislow, D.E., Phillips, P.C., 2005. Network thinking in ecology and
evolution. Trends Ecol. Evol. 20, 345–353.
R Development Core Team, 2010. R: A Language and Environment for Statistical
Computing. R Foundation for Statistical Computing, Vienna, Austria, ISBN 3900051-07-0, http://www.R-project.org/.
Schoener, T.W., 1993. On the relative importance of direct versus indirect effects in
ecological communities. In: Mutualism and Community Organization. Oxford
University Press, Oxford, pp. 365–411.
Schoenly, K., Cohen, J.E., 1991. Temporal variation in food web structure: 16 empirical cases. Ecol. Monogr., 267–298.
Thébault, E., Fontaine, C., 2010. Stability of ecological communities and the architecture of mutualistic and trophic networks. Science 329, 853–856.
Thompson, R., Townsend, C., 2003. Impacts on stream food webs of native and exotic
forest: an intercontinental comparison. Ecology 84, 145–161.
Vázquez, D.P., Melián, C.J., Williams, N.M., Blüthgen, N., Krasnov, B.R., Poulin, R., 2007.
Species abundance and asymmetric interaction strength in ecological networks.
Oikos 116, 1120–1127.
Vázquez, D.P., Morris, W.F., Jordano, P., 2005. Interaction frequency as a surrogate
for the total effect of animal mutualists on plants. Ecol. Lett. 8, 1088–1094.
Vázquez, D.P., Simberloff, D., 2002. Ecological specialization and susceptibility to
disturbance: conjectures and refutations. Am. Nat. 159, 606–623.
Volterra, V., 1926. Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560.
Wangersky, P.J., 1978. Lotka–Volterra population models. Annu. Rev. Ecol. Syst.,
189–218.
White, E.P., Ernest, S.M., Kerkhoff, A.J., Enquist, B.J., 2007. Relationships between
body size and abundance in ecology. Trends Ecol. Evol. 22, 323–330.
Wootton, J.T., 1992. Indirect effects, prey susceptibility, and habitat selection:
impacts of birds on limpets and algae. Ecology 73, 981–991.
Wootton, J.T., 1994a. The nature and consequences of indirect effects in ecological
communities. Annu. Rev. Ecol. Syst., 443–466.
Wootton, J.T., 1994b. Predicting direct and indirect effects: an integrated approach
using experiments and path analysis. Ecology 75, 151–165.
Wootton, J.T., 1997. Estimates and tests of per capita interaction strength: diet,
abundance, and impact of intertidally foraging birds. Ecol. Monogr. 67, 45–64.
Wootton, J.T., Emmerson, M., 2005. Measurement of interaction strength in nature.
Annu. Rev. Ecol., Evol. Syst., 419–444.
Yan, C., Zhang, Z., 2014. Specific non-monotonous interactions increase persistence
of ecological networks. Proc. R. Soc. Lond., Ser. B: Biol. Sci., 281.