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Vol.:(0123456789)
1 3
Population Ecology
https://doi.org/10.1007/s10144-018-0628-3
REVIEW
Complexity andstability ofecological networks: areview ofthetheory
PietroLandi1,2 · HenintsoaO.Minoarivelo1,3· ÅkeBrännström2,4· CangHui1,5· UlfDieckmann2
Received: 20 April 2017 / Accepted: 29 June 2018
© The Author(s) 2018
Abstract
Our planet is changing at paces never observed before. Species extinction is happening at faster rates than ever, greatly
exceeding the five mass extinctions in the fossil record. Nevertheless, our lives are strongly based on services provided by
ecosystems, thus the responses to global change of our natural heritage are of immediate concern. Understanding the rela-
tionship between complexity and stability of ecosystems is of key importance for the maintenance of the balance of human
growth and the conservation of all the natural services that ecosystems provide. Mathematical network models can be used to
simplify the vast complexity of the real world, to formally describe and investigate ecological phenomena, and to understand
ecosystems propensity of returning to its functioning regime after a stress or a perturbation. The use of ecological-network
models to study the relationship between complexity and stability of natural ecosystems is the focus of this review. The
concept of ecological networks and their characteristics are first introduced, followed by central and occasionally contrasting
definitions of complexity and stability. The literature on the relationship between complexity and stability in different types
of models and in real ecosystems is then reviewed, highlighting the theoretical debate and the lack of consensual agreement.
The summary of the importance of this line of research for the successful management and conservation of biodiversity and
ecosystem services concludes the review.
Keywords Biodiversity· Community· Complex networks· Ecosystem· Resilience
Introduction
In the geological era of the Anthropocene, our planet is
changing at paces never observed before (Millennium
Ecosystem Assessment 2005). Pollution, natural resources
exploitation, habitat fragmentation, and climate change are
only some of the threats our biosphere is facing. Species
extinction is happening at faster rates than ever, greatly
exceeding the five mass extinctions in the fossil record. Even
if sometimes we do not realize it, our lives are strongly based
on services provided by ecosystems, thus the responses to
global change of our natural heritage are of immediate
concern for policy makers. As ecosystems are composed
by thousands of interlinked species that interact directly or
through their shared environment, such as nutrients, light,
or space, a holistic perspective on the system as a whole is
normally required to predict ecosystem responses to global
changes (Wolanski and McLusky 2011). A systems-analysis
approach is thus often crucial for acquiring an understand-
ing of all the dynamical feedbacks at the ecosystem level
and for accurately managing the biodiversity that we rely on
in terms of ecosystem services. In particular, mathematical
network models can be used to simplify the vast complex-
ity of the real world, to formally describe and investigate
ecological phenomena, and to understand how ecosystems
react to stress and perturbations (Dunne 2006).
Complex-networks models are composed of a set of com-
partments, describing either species or coarser functional
groups, and a set of links that represent interactions or
energy or biomass flows among compartments. Thus, such
models can describe both biotic and abiotic interactions
* Pietro Landi
landi@sun.ac.za
1 Department ofMathematical Sciences, Stellenbosch
University, Stellenbosch, SouthAfrica
2 Evolution andEcology Program, International Institute
forApplied Systems Analysis, Laxenburg, Austria
3 Centre ofExcellence inMathematical andStatistical
Sciences, Wits University, Johannesburg, SouthAfrica
4 Department ofMathematics andMathematical Statistics,
Umeå University, Umeå, Sweden
5 Mathematical andPhysical Biosciences, African Institute
forMathematical Sciences, Muizenberg, SouthAfrica
Population Ecology
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among species, i.e., both interactions among the species
themselves and interactions with their external environ-
ment, and consequently they can often successfully be used
to assess ecosystems stability to perturbations. Stability of
an ecosystem can be understood as its propensity of return-
ing to its functioning regime after a stress or a perturbation
in its biotic components (e.g., decline in species abundances,
introduction of alien species, and species extinction) or abi-
otic components (e.g., exploitation, habitat fragmentation,
and climate change). A challenging and central question
that has interested ecologists and systems analysts alike for
decades is how the stability of an ecosystems depend on its
complexity, as roughly measured by the ecosystems’ diver-
sity in species and their interactions (Johnson etal. 1996;
Worm and Duffy 2003; Dunne etal. 2005; Hooper etal.
2005; Kondoh 2005; Loreau and De Mazancourt 2013).
To appreciate the importance of this question, we first rec-
ollect and differentiate between the major different functions
that ecosystems continuously provide. Natural ecosystems
sustain life and provide services that can be divided into four
areas (Millennium Ecosystem Assessment 2005): provision-
ing, such as the production of food and water; regulating,
such as the control of climate and disease; supporting, such
as nutrient cycles and crop pollination; and cultural, such as
spiritual and recreational benefits. For the management and
conservation of ecosystems services it is important to know
how the complexity of an ecosystem is related to its stability,
thus how the diversity of species in the ecosystem and the
network of their interactions can contribute to maintaining a
stable supply of services. This is especially important in an
era in which the pressure exerted on natural ecosystems is
becoming stronger and stronger, influencing their structure
and functioning, while the services they provide are vital for
a continuously increasing number of people. In particular,
human activities, directly or indirectly, tend to simplify the
composition and the structure of natural ecosystems. There-
fore, understanding the relationship between complexity and
stability of ecosystems is of key importance for the mainte-
nance of the balance of human growth and the conservation
of all the natural services that ecosystems provide. Using
ecological-network models to study the relationship between
complexity and stability of natural ecosystems is the focus
of this review. We first introduce the concept of ecologi-
cal networks and their characteristics, followed by central
and occasionally contrasting definitions of complexity and
stability. After that, we review the literature on the relation-
ship between complexity and stability in different types of
models and in real ecosystems, highlighting the theoretical
debate and the lack of consensual agreement. We continue
with describing the importance of considering the dynamic
adaptation of species behaviour and the resulting changes in
ecosystems structure, after which we conclude by summariz-
ing the importance of this line of research for the successful
management and conservation of biodiversity and ecosystem
services in the current era of the Anthropocene.
Ecological networks dened
An ecological network describes interactions among species
in a community (Pascual and Dunne 2006). There are dif-
ferent types of interactions, e.g., trophic interactions (feed-
ing), mutualistic interactions (pollination, seed dispersal,
etc.), and competitive interactions (interference for common
resources). Ecological networks can be represented as a set
of S nodes, characterizing the species, connected by a set
of L links, characterizing possible interactions among each
ordered pair of species (Newman 2010; Estrada 2012). Links
can be described by either a binary variable (0 or 1, absence
or presence of interaction) or by a real number characterizing
the weight (or strength) of the interaction. In the first case
the network is called unweighted, while in the second case
it is called weighted. Moreover, interactions can be undi-
rected (or symmetric), meaning that species i affects species
j to a certain amount and equally vice versa, or directed
(or asymmetric), meaning that species i can affect species
j differently from how species j affects species i (Fig.1).
Moreover, interactions can be described by their sign (+ or
−). For example, trophic networks (food webs) are charac-
terized by the fact that one species is feeding on the other,
thus the coefficients aij (describing the effect of species j on
species i) and aji (describing the effect of species i on species
j) will obviously have opposite signs (thus their product will
be negative, aijaji < 0), i.e., one species is benefiting while
the other is suffering from the interaction. In mutualistic
networks both species are benefiting from the interaction,
thus both coefficients aij and aji will be positive (and so their
product, aijaji > 0), while in competitive networks both spe-
cies are suffering from the interaction, thus both coefficients
aij and aji will be negative (thus their product will be again
positive, aijaji > 0) (Fig.2). Notice therefore that trophic
networks cannot be undirected (symmetric), since the two
coefficients describing the interaction always have opposite
sign (and typically also different absolute values).
The structure of the ecological network can be described
by the S
×
S matrix A = [aij], where each element aij describes
the link between species i and species j, i.e., the effect that
species j has on species i. In the most particular case of
unweighted and undirected network, matrix A is symmetric
(i.e., aij = aji) and its elements are either 0 or ± 1 (Newman
2010; Estrada 2012). In the most general case of weighted
and directed network, matrix A can have any composition of
real values. For bipartite networks (i.e., those formed by two
disjoint groups of respectively m and n species, S = m + n,
with interactions only between two species of different
groups), such as mutualistic networks of plants and their
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pollinators or antagonistic networks of host-parasite interac-
tions, the matrix A = [aij] is a m
×
n matrix(Fig.2).
Unfortunately, there is no unique quantification of the
elements aij. Depending on the scope (theoretical vs. empiri-
cal), several measures and indexes have been used to quan-
tify the matrix A. For example, theoretical studies mostly
refer to aij as the effect of a perturbation from equilibrium
of the abundance of species j on the population growth rate
of species i (elements of the Jacobian matrix describing the
linearized dynamics of the model ecosystem around equi-
librium, see also ‘Network stability’). In such cases, the
matrix A has been called community matrix (Novak etal.
2016). Another option is to define the elements of matrix
A as the effect of a single individual of species j on the per-
capita growth rate of species i: in such case, matrix A has
been called interaction matrix, and its elements are called
interaction strengths (Kokkoris etal. 2002). Unfortunately,
such coefficients are well defined in theory, but very hard
to measure in the field or in lab experiments. On the other
hand, empirical observations mainly quantify magnitude of
energy and biomass flows between compartments in model
ecosystems, or consumption rate for resource-consumer and
prey-predator interactions, or visiting probabilities in pol-
lination networks. Such quantities are relatively easy to be
estimated empirically, but they are not directly related to
elements of the theoretical Jacobian (community) matrix
as they are independent of species equilibrium abundances.
For example, in empirical studies of mutualistic interac-
tions, the degree of species dependence on another species
(see ‘Network complexity’ for a definition) has been used
to quantify the link among involved species (Jordano 1987).
See also Berlow etal. 2004, Wootton and Emmerson 2005,
and Novak etal. 2016 for reviews on the different definitions
of strength of interaction and of the matrix A.
Network complexity
Species richness S, or the total number of interacting spe-
cies in the network, also known as the network size, has
been used as the simplest descriptor of network complexity
(MacArthur 1955; May 1972, 1973; Pimm 1980a; Table1).
In the particular case of bipartite networks, species richness
S is expressed as S = m + n. In food-web studies, the use of
trophic species (a functional group of species sharing the
same set of predators and preys) as a replacement of taxo-
nomic species (i.e., when species are distinguished based
on morphological and phylogenetic criteria) is a widely
accepted convention (Schoener 1989; Pimm etal. 1991;
Goldwasser and Roughgarden 1993; Williams and Mar-
tinez 2000; Dunne etal. 2002a). The use of ‘trophic species’
has indeed been shown to reduce methodological biases in
food web datasets because it reduces scatter in the data and
avoids redundancy of interactions (Pimm etal. 1991; Mar-
tinez 1994). Sometimes, the use of morphospecies (species
distinguished from others by only their morphologies) as a
replacement of taxonomic species is also considered because
of a lack of taxonomic distinction between species (Olito and
ab
cd
Fig. 1 Categorisation of ecological networks, according to link
directionality and weight. Black (white) entries in matrices a, c rep-
resent presence (absence) of interaction. a Unweighted undirected;
b weighted undirected; c unweighted directed; d weighted directed.
Note that links point to the affected species. For example, species A
in d is positively affected by species B and D while negatively affect-
ing species B and D
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Fox 2014). Hence, network size often refers to the number of
functional or morphological diversity in the system.
Another commonly-used indicator of complexity is the
connectance C (May 1972, 1973; Newman 2010; Estrada
2012), measuring the proportion of realised interactions
among all the possible ones in a network (i.e., the total num-
ber of interactions L divided by the square of the number of
species S2 or L divided by the product m
×
n in the case of a
bipartite network). It accounts intuitively for the probability
that any pair of species interact in the network. It is probably
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Fig. 2 Examples of real-world ecological networks. First row: food
web from the estuary river of St. Marks, Florida, USA (Baird etal.
1998). Second row: mutualistic network of pollination from the Flo-
res Island, one of the Azores oceanic islands (Olesen et al. 2002).
Left column: network representation. Second column: matrix repre-
sentation. The food web is unweighted directed: in b the black entries
in the matrix represent presence of interaction. The mutualistic net-
work is weighted undirected: the link width in c and the shade of grey
in d are proportional to the weight of the interaction which represents
the number of pollinator visits
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Table 1 Measures of network complexity
Network complexity Definition References
Species richness (S) Total number of species in the network May (1972, 1973)
Food webs: MacArthur (1955), Pimm (1979, 1980a),
Cohen and Briand (1984), Cohen and Newmann
(1985), Havens (1992), Martinez (1992), Haydon
(1994), Borrvall etal. (2000), Dunne etal. (2002a,
b), Dunne and Williams (2009), Banašek-Richter
etal. (2009), Gross etal. (2009), Thébault and
Fontaine (2010) and Allesina and Tang (2012)
Mutualism: Okuyama and Holland (2008), Thébault
and Fontaine (2010), Allesina and Tang (2012) and
Suweis etal. (2015)
Competition: Lawlor (1980), Lehman and Tilman
(2000), Christianou and Kokkoris (2008), Fowler
(2009) and Allesina and Tang (2012)
Connectance (C) Proportion of realized interactions among all pos-
sible ones, L/S2May (1972, 1973)
Food webs: De Angelis (1975), Pimm (1979, 1980a,
1984), Martinez (1992), Haydon (1994, 2000),
Chen and Cohen (2001), Olesen and Jordano
(2002), Dunne etal. (2002a, b), Dunne etal.
(2004), Banašek-Richter etal. (2009), Dunne and
Williams (2009), Gross etal. (2009), Thébault and
Fontaine (2010), Tylianakis etal. (2010), Allesina
and Tang (2012), Heleno etal. (2012) and Poisot
and Gravel (2014)
Mutualism: Jordano (1987), Rezende etal. (2007),
Okuyama and Holland (2008), Thébault and
Fontaine (2010), Allesina and Tang (2012), Suweis
etal. (2015) and Vieira and Almeida-Neto (2015)
Competition: Fowler (2009) and Allesina and Tang
(2012)
Connectivity (L) Total number of interactions Mutualism: Okuyama and Holland (2008)
Competition: Fowler (2009)
Linkage density Average number of links per species, L/SFood webs: Pimm etal. (1991) and Havens (1992)
Mutualism: Jordano (1987)
Interaction strength Weight of an interaction in the interaction matrix Food webs: Paine (1992), McCann etal. (1998),
Berlow (1999), Borrvall etal. (2000), Berlow etal.
(2004), Wootton and Emmerson (2005), Rooney
etal. (2006) and Otto etal. (2007)
Mutualism: Okuyama and Holland (2008), Allesina
and Tang (2012), Rohr etal. (2014) and Suweis
etal. (2015)
Competition: Lawlor (1980), Hughes and Rough-
garden (1998), Kokkoris etal. (1999, 2002),
Christianou and Kokkoris (2008) and Allesina and
Tang (2012)
Jacobian element Weight of an interaction in the community (Jaco-
bian) matrix Food webs: De Angelis (1975), Yodzis (1981),
Haydon (1994), de Ruiter etal. (1995), Haydon
(2000), Neutel etal. (2002, 2007), Emmerson and
Raffaelli (2004), Emmerson and Yearsley (2004),
Allesina and Pascual (2008), Gross etal. (2009),
Allesina and Tang (2012), Jacquet etal. (2016) and
van Altena etal. (2016)
Mutualism: Allesina and Tang (2012)
Competition: Lawlor (1980), Hughes and Rough-
garden (1998), Kokkoris etal. (1999, 2002),
Christianou and Kokkoris (2008) and Allesina and
Tang (2012)
Weighted linkage density Average number of links per species weighted by
interaction strength Food webs: Bersier etal. (2002), Tylianakis etal.
(2007) and Dormann etal. (2009)
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one of the earliest and the most popular descriptors of eco-
logical networks structure. Sometimes, a simpler measure-
ment of interactions, known as connectivity, has been used
instead of connectance. The connectivity of a network is
simply its total number of interactions L (Newman 2010;
Estrada 2012).
To understand the average level of specialization of the
network, i.e., whether the network is dominated by special-
ists (species holding few interactions) or generalists (species
holding many interactions), food web ecologists have intro-
duced linkage density. It is calculated as the average number
of links per species, or the connectivity divided by species
richness, L/S (Montoya etal. 2006).
To increase the information value of these network met-
rics, some theoretical studies have incorporated the strength
of interactions. Thus, quantitative counterparts of linkage
density and connectance, called respectively weighted con-
nectance and weighted linkage density, have been developed
(Bersier etal. 2002; Tylianakis etal. 2007; Dormann etal.
2009). Weighted linkage density considers the proportion of
biomass flow to weight the contribution of each link to and
from all equivalent species. Equivalent species are defined
using the Shannon metric (Shannon 1948) of entropy (or
uncertainty). Weighted connectance is then computed as
the weighted linkage density divided by species richness.
There are several reasons for believing that networks metrics
incorporating the strength of interaction are better suited to
reflect salient ecosystem properties, among which the ability
to give increased weight to strong interactions and the fact
that weighted metrics change continuously with the change
of link strength and even with the eventual removal of the
link. The latter can be particularly important in empirical
food-web studies in which the sampling effort typically
dictates the number of links discovered, with greater effort
often leading to many more additional weak links.
As connectance and linkage density are only community-
average descriptors of network structure, they do not inform
on the relative importance of each species to the overall con-
nectivity. Node degree distribution, i.e., the distribution of the
number of interactions per species, is another widely used
descriptor of network complexity (Newman 2010). The degree
of a node (or a species) refers to the number of links to other
interacting partners in the network. The distribution of node
degree in ecological networks have been shown to differ from
a Poisson distribution that characterises large random networks
(Camacho etal. 2002; Dunne etal. 2002b; Montoya and Solé
2002; Jordano etal. 2003).
A generalization of the node-degree distribution is the inter-
action-strength distribution, taking into account the weights
associated with each link (Newman 2010). The strength (or
weighted degree) of each species is computed as the sum of
all the weighted interaction strengths of that species (Feng and
Takemoto 2014; Suweis etal. 2015). However, particularly
for pollination and frugivory networks, interaction strengths
are often approximated by the number of visits of an animal
species to a plant species (Jordano 1987). A normalized index
for this kind of networks is species dependence on another
species. The dependence of a species i on a species j is defined
as the fraction of interactions (e.g., visits or diet item) between
i and j relative to the total number of interactions of species
i (Bascompte etal. 2006; Vieira and Almeida-Neto 2015). In
this context, species strength refers to the sum of dependences
of the mutualistic partners relying on the species.
Network architecture
Beyond ecological patterns in interaction and strength
distribution, interactions in ecological networks exhibit
even more complex topological features, related to the
Table 1 (continued)
Network complexity Definition References
Weighted connectance Weighted linkage density divided by species rich-
ness Food webs: Haydon (2000), Bersier etal. (2002),
Tylianakis etal. (2007), Dormann etal. (2009) and
van Altena etal. (2016)
Mutualism: Minoarivelo and Hui (2016)
Species degree Number of interactions (links) with other species Food webs: Waser etal. (1996), Memmott (1999),
Solé and Montoya (2001), Camacho etal.
(2002), Dunne etal. (2002b), Montoya and Solé
(2002);Vázquez and Aizen (2003) and Dunne and
Williams (2009)
Mutualism: Jordano etal. (2003) and Rohr etal.
(2014)
Species strength Sum of weighted interactions shared by the species
with others Mutualism: Bascompte etal. (2006), Feng and Take-
moto (2014) and Suweis etal. (2015)
Dependence of species i on species jNumber of visits between i and j divided by the
total number of visits between species i and all
other partners
Mutualism: Jordano (1987), Bascompte etal.
(2006), Feng and Takemoto (2014) and Vieira and
Almeida-Neto (2015)
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architecture of the network (Table2). Among the most
important of these features is the level of modularity or
compartmentalization. Modularity depicts the extent to
which a network is compartmentalized into delimited
modules where species are strongly interacting with spe-
cies within the same module but not with those from
other modules (Olesen etal. 2007). Although a number
of metrics have been developed to quantify the level of
compartmentalization in a network, modularity (devel-
oped by Newman and Girvan 2004) has been the most
widely accepted. This measure assumes that nodes in
the same module have more links between them than
one would expect for a random network and Modules are
thus obtained by partitioning all nodes in the network
in order to maximize modularity. However, see, e.g.,
Rosvall and Bergstrom (2007) and Landi and Piccardi
(2014) for limitations of modularity and other metrics of
compartmentalization.
Another important descriptor of ecological network
architecture, especially for mutualistic networks, is
nestedness. It is a pattern of interactions in which spe-
cialists can only interact with a subset of species with
which more generalists interact. It means that in a nested
network, both generalists and specialists tend to interact
with generalists whereas specialist-to-specialist interac-
tions are rare (Bascompte etal. 2003). To quantify the
nestedness of a network, several metrics have been devel-
oped. Among the most commonly used are for example
the ‘temperature’ metric by Atmar and Patterson (1993)
and the NODF (Nestedness metric based on Overlap and
Decreasing fill) metric by Almeida-Neto etal. (2008).
Despite the existence of several metrics and algorithms,
they are all mainly based on measuring the extent to
which specialists interact only with a subset of the spe-
cies generalists interact with.
Network stability
In theoretical studies, each entry aij of matrix A usually
quantifies the change in population growth rate of species i
caused by a small perturbation in the abundance of species
j around equilibrium abundances (i.e., stationary regime,
species abundances are constant in time). Thus matrix A is
equivalent to the Jacobian matrix of the dynamical system
that describes species abundance dynamics over continuous
time, evaluated at equilibrium, and it is also called a com-
munity matrix. Such a matrix is very useful for studying
the (local) asymptotic stability of the equilibrium. In fact,
stability is defined by the real part of the leading eigenvalue
of the Jacobian matrix (i.e., the eigenvalue with the largest
real part). If the real part of the leading eigenvalue is posi-
tive, the equilibrium is unstable, i.e., any small perturbation
from the equilibrium will be amplified until convergence to
another ecological regime, at which some of the species in
the community might be extinct. Otherwise, if the real part
of the leading eigenvalue is negative, then small perturba-
tions around the equilibrium will be dampened, and the sys-
tem will converge back to its stationary regime. Therefore,
the sign of the real part of the leading eigenvalue can be a
binary indicator of stability. Moreover, if stable, the inverse
of the absolute value of the real part of the leading eigen-
value gives an indication of the time needed by the system to
return to its equilibrium. Systems that quickly return to equi-
librium after perturbations are called resilient. Resilience is
therefore often measured by the absolute value of the leading
eigenvalue (if negative) of the community matrix. Notice
that resilience is only defined for stable equilibria and it
only gives information about the asymptotic behaviour of the
system (see Neubert and Caswell 1997 for transient indica-
tors). Global (vs. local) stability implies that any (vs. small)
perturbation from the equilibrium will be dampened. Global
Table 2 Measures of network architecture
Network architecture Definition References
Modularity Extent to which a network is compartmentalized into delimited modules Food webs: Moore and Hunt (1988), Ives etal.
(2000), Krause etal. (2003), Thébault and
Fontaine (2010) and Stouffer and Bascompte
(2011)
Mutualism: Olesen etal. (2007), Mello etal.
(2011) and Dupont and Olesen (2012)
Nestedness When specialists can only interact with subset of the species generalists
interact with Food webs: Atmar and Patterson (1993), Neutel
etal. (2002), Cattin etal. (2004), Thébault
and Fontaine (2010) and Allesina and Tang
(2012)
Mutualism: Bascompte etal. (2003), Memmott
etal. (2004), Almeida-Neto etal. (2008),
Bastolla etal. (2009), Zhang etal. (2011),
Campbell etal. (2012), James etal. (2012)
and Rohr etal. (2014)
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stability usually refers to the case of a single equilibrium
(typical of linear systems).
The notion of structural stability of a system is used when
the system’s dynamical behaviour (such as the existence of
equilibrium points, limit cycles or deterministic chaos) is
not affected by small perturbations such as small changes
in the values of its parameters (Solé and Valls 1992). How-
ever, Rohr etal. (2014) extended this definition to the notion
of structural stability of an equilibrium which refers to the
domain (or probability) of coexistence of all the species in
the ecosystem. An equilibrium at which all the species S in
the system coexist with positive abundances is called fea-
sible. Structural stability usually refers to perturbations in
the system itself (i.e., slightly changing one of its param-
eters) rather than perturbations in the state of the system
(i.e., abundances, see previous paragraph). Assuming that a
system is at a feasible equilibrium, a small perturbation in a
parameter (e.g., species carrying capacity, intrinsic growth
rate, predator conversion efficiency, handling time, …) will
generically move the system to a slightly different (in terms
of species abundances) feasible equilibrium, unless the
system is close to a bifurcation point for that parameter. A
bifurcation is indeed a qualitative change in the asymptotic
behaviour of a system driven by a perturbation in one of its
parameters. Such qualitative change could, for example, be
a switch to a non-feasible equilibrium (where one or more
species go extinct), to a non-stationary (e.g., periodic) orbit,
etcetera. The region in parameter space for which the system
has a feasible equilibrium is its domain of stable coexistence,
and gives an indication (or probability) of its structural sta-
bility. The bigger the domain, the more structurally stable
the system (Rohr etal. 2014).
In addition to this, the number of coexisting species at
an equilibrium could trivially be an indicator of stability.
This number will be S at a feasible equilibrium, and will
be smaller than S at an equilibrium at which some species
have gone extinct. If this number is standardized to the total
number of species S we obtain the proportion of persistent
species once equilibrium is reached, that is, persistence
(Thébault and Fontaine 2010).
The notions of asymptotic stability and structural stability
can of course be generalized in the case of non-stationary
asymptotic regimes (such as cycles, tori, and chaotic attrac-
tors), using, e.g., Lyapunov exponents. In such cases, or in
the study of empirical time series, other stability indicators
can however be more useful. For example, temporal stabil-
ity (the reciprocal of variability) quantifies the stability of
fluctuating variables. It is usually defined as the ratio of the
mean over its standard deviation (the inverse of the Coef-
ficient of Variation). A high mean contributes to temporal
stability, as it contributes to values far from 0 (extinction), as
well as a low standard deviation that describes fluctuations
around the mean.
Another approach to stability considers the effect of
removing target species from a system. The extinction
cascade measures the loss of additional species after the
removal of one target species. Robustness (Dunne etal.
2002a)—or deletion stability or resistance (Borrvall etal.
2000)—is indeed the ability of a system to resist extinc-
tion cascades. Species removal can be random or targeted
(e.g., the most connected species or species with low or high
trophic level).
Instead of removing target species, invasibility describes
the propensity of a system (or a resident community) to be
invaded by new species (Hui and Richardson 2017). Non-
invadable systems are thought to be more stable than sys-
tems that are easily invaded by introduced alien species.
Thus, resistance to invasion can be a measure of system sta-
bility. Invasion can simply bring the system in a new stable
and feasible configuration, or in the worst case it could lead
one or more species to extinction (see Hui etal. 2016 for a
recently proposed measure of invasibility).
A summary of the different introduced measures for net-
work stability is given in Table3. See Pimm (1984), Logofet
(2005), Ives and Carpenter (2007),Donohue etal. (2013),
and Borrelli etal. (2015) for additional reviews on different
stability concepts.
Complexity–stability debate
Before the 1970s, ecologists believed that more diverse com-
munities enhanced ecosystem stability (Odum 1953; Mac-
Arthur 1955; Elton 1958). In particular, they believed that
natural communities develop into stable systems through
successional dynamics. Aspects of this belief developed
into the notion that complex communities are more sta-
ble than simple ones. A strong proponent of this view was
Elton (1958), who argued that “simple communities were
more easily upset than richer ones; that is, more subject to
destructive oscillations in populations, and more vulner-
able to invasions”. In fact, both Odum (1953) and Elton
(1958) arrived at similar conclusions based on repeated
observations of simplified terrestrial communities that are
characterized by more violent fluctuations in population
density than diverse terrestrial communities. For example,
invasions most frequently occur on cultivated land where
human influence had produced greatly simplified ecologi-
cal communities; outbreaks of phytophagous insects occur
readily in boreal forests but are unheard of in diverse tropi-
cal forests; and the frequency of invasions is higher in sim-
ple island communities compared to more complex main-
land communities. These observations led Elton (1958) to
believe that complex communities, constructed from many
predators and parasites (consumers), prevented populations
from undergoing explosive growth (e.g., pest outbreaks)
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Table 3 Measures of network stability
Network stability Definition References
Asymptotic stability Perturbations from the ecological regime are dampened. The system return to its
ecological regime after a perturbation in the state of the system May (1972, 1973)
Food webs: De Angelis (1975), Yodzis (1981), De Ruiter etal. (1995), Haydon
(2000), Neutel etal. (2002, 2007), Emmerson and Raffaelli (2004), Emmerson
and Yearsley (2004), Rooney etal. (2006), Otto etal. (2007), Allesina and Pascual
(2008), Gross etal. (2009), Allesina and Tang (2012), Visser etal. (2012) and van
Altena etal. (2016)
Mutualism: Feng and Takemoto (2014)
Competition: Lawlor (1980), Christianou and Kokkoris (2008) and Fowler (2009)
Resilience Return time to ecological regime after a small perturbation Food webs: Thébault and Fontaine (2010)
Mutualism: Okuyama and Holland (2008)
Competition: Lawlor (1980) and Christianou and Kokkoris (2008)
Persistence Proportion of coexisting species (over the total number of species) at ecological
regime. In case of a feasible regime the persistence is equal to 1 (i.e., coexistence
of all species in the community)
Food webs: Haydon (1994), McCann etal. (1998), Krause etal. (2003), Kondoh
(2003, 2006, 2007), Thébault and Fontaine (2010), Stouffer and Bascompte (2011)
and Heckmann etal. (2012)
Mutualism: Ferrière etal. (2002), West etal. (2002), Bascompte etal. (2006),
Bastolla etal. (2009), Olivier etal. (2009), James etal. (2012), Valdovinos etal.
(2013) and Song and Fledman (2014)
Competition: Kokkoris etal. (2002) and Christianou and Kokkoris (2008)
Structural stability Domain or probability of feasible existence of an ecological regime w.r.t. system
perturbations May (1972, 1973)
Food webs: De Angelis (1975), Haydon (1994), Kondoh (2003, 2006, 2007) and
Allesina and Tang (2012)
Mutualism: Rohr etal. (2014)
Competition: Christianou and Kokkoris (2008)
Temporal stability The reciprocal of temporal variability. It quantifies the stability of fluctuations in
time. It is the ratio of the mean value of the variable in time over its standard
deviation (the inverse of the Coefficient of Variation)
Elton (1958)
Food webs: McCann etal. (1998), Ives etal. (2000) and Kondoh (2003, 2006, 2007)
Competition: Hughes and Roughgarden (1998), Lehman and Tilman (2000) and
Fowler (2009)
Deletion stability
(extinction cascade) Loss of additional species after the removal of one target species Food webs: Pimm (1979, 1980b), Borrvall etal. (2000), Dunne etal. (2002a) and
Dunne and Williams (2009)
Mutualism: Memmott etal. (2004), Campbell etal. (2012) and Vieira and Almeida-
Neto (2015)
Robustness Resistance of a system against additional extinction after species removal Food webs: Dunne etal. (2002a) and Dunne and Williams (2009)
Mutualism: Ramos-Jiliberto etal. (2012)
Resistance to invasion Resistance of a system to be invaded by new species Elton (1958)
Food webs: Hui etal. (2016)
Competition: Kokkoris etal. (1999)
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and would have fewer invasions (see Hui and Richardson
2017 for background in invasion science). His ideas were
closely related to MacArthur’s (1955), who hypothesized
that “a large number of paths through each species is neces-
sary to reduce the effects of overpopulation of one species.”
MacArthur (1955) concluded that “stability increases as
the number of links increases” and that stability is easier to
achieve in more diverse assemblages of species, thus linking
community stability with both increased trophic links (e.g.,
connectance C) and increased numbers of species (S). In
other words, multiplicity in the number of prey and predator
species associated with a population freed that population
from dramatic changes in abundance when one of the prey
or predator species declined in density. Additionally, Paine
(1966) also showed that species diversity in foodwebs is
related to the number of top predators, and that increased
stability of annual production may lead to an increased
capacity for systems to support such high-level consumer
species, thus resulting in increased species diversity.
These early intuitive ideas were challenged by the work
of May (1972, 1973). He used mathematics to rigorously
explore the complexity–stability relationship (see a first
review by Goodman 1975). By using linear stability analy-
sis (asymptotic stability of the Jacobian matrix) on models
constructed from a statistical universe (that is, randomly
constructed Jacobians with randomly assigned elements),
May (1972, 1973) found that complexity tends to destabilize
community dynamics. He mathematically demonstrated that
network stability decreases with diversity (measured as the
number of species S), complexity (measured as connectance
C), and the standard deviation of the Jacobian elements σ.
In particular, he found that more diverse systems, com-
pared to less diverse systems, will tend to sharply transition
from stable to unstable behaviour as the number of species
S, the connectance C, or the average Jacobian element σ
increase beyond a critical value, i.e., the system is stable if
𝜎√
SC <
1
, unstable otherwise.
In his seminal study on community stability, May (1972,
1973) measured asymptotic local stability. In this analysis,
it is assumed that the community rests at an equilibrium
point where all populations have constant abundances. The
stability of this equilibrium is tested with small perturba-
tions. If all species return to the equilibrium—monotonically
or by damped oscillations—it is stable. In contrast, if the
population densities evolve away from the equilibrium densi-
ties—monotonically or oscillatory—they are unstable. In a
community of S species, this approach is based on the S
×
S
Jacobian matrix, whose elements describe the perturbation
impact of each species j on the growth of each species i at
equilibrium population densities. The S eigenvalues of the
Jacobian matrix characterize its temporal behavior. Specifi-
cally, positive real parts of the eigenvalues indicate perturba-
tion growth, while negative real parts indicate perturbation
decay. Accordingly, if any of the eigenvalues has a positive
real part the system will be unstable, i.e., at least one of the
species does not return to the equilibrium. The mathematical
proposition, thus, contradicts the ecological intuition.
Food webs
The use of random community matrices in May’s (1972,
1973) work has attracted much criticism (Table4). It was
shown to be extremely unlikely that any of these random
communities could even remotely resemble ecosystems with
a minimum form of ecological realism, such as containing
at least one primary producer, a limited number of trophic
levels and no consumers eating resources that are two or
more trophic levels lower (Lawlor 1978, but see; Allesina
and Tang 2015 for a review on the random matrix approach).
The non-randomness of ecosystem structure has been dem-
onstrated in detail by more recent food-web topology studies
(e.g., Williams and Martinez 2000; Dunne etal. 2002a, b,
2004, 2005; Dunne 2006). Accordingly, subsequent work
added more structural realism to those random community
matrices by including empirical patterns of food web struc-
ture and Jacobian elements distributions (see Allesina and
Tang 2012; Allesina etal. 2015; Jacquet etal. 2016 for the
most recent advances and; Namba 2015 for a review). Sev-
eral simple models have played an important role in char-
acterizing the non-random structure of food webs, includ-
ing the cascade model (Cohen etal. 1990), the niche model
(Williams and Martinez 2000), and the nested-hierarchy
model (Cattin etal. 2004). The niche and nested-hierarchy
models have been able to capture several structural proper-
ties of empirical food webs.
Species richness
In general, food web features vary with species richness.
Although empirical datasets of ecological networks do not
display any consistency regarding their size, it has been
observed that ecological networks have much smaller size
than other published real-world network datasets, such as
co-authorships between scientists or the World Wide Web
(Dunne etal. 2002b).
Haydon (1994) discussed some of May’s hypothesis (such
as the measure of stability, the consideration of unfeasible
models, and the self-regulatory terms on the diagonal of
the community matrix describing intraspecific interactions)
but still found that (asymptotic) stability and feasibility of
(generalized Lotka–Volterra) model ecosystems is reduced
by the number of species. Gross etal. (2009) found that
smaller model ecosystems follow other rules than larger eco-
systems. Indeed, they studied artificial food webs generated
by the niche model and considering nonlinear functional
responses of different kinds. Thus, adding more details to
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Table 4 Complexity–stability relationship in food webs
References Complexity–stability measures Methods and assumptions Additional results
Negative complexity–stability relationship
Haydon (1994)S-Asymptotic, feasibility Lead eigenvalue of random and plausible
model Jacobian at feasible equilibria Stability is reduced by donor control interac-
tions
Pimm (1979, 1980b)S, C-extinction cascades Simulation of plausible food web model (gen-
eralized Lotka–Volterra) If carnivores are removed
Gross etal. (2009)S, C, Jacobian elements-Asymptotic Lead eigenvalue of Jacobian of realistic food
web model (niche) Increasing strength of interaction destabilizes
large networks
Allesina and Pascual (2008) and Allesina
and Tang (2012)S, C-Asymptotic Lead eigenvalue of random, empirical, and
model (cascade and niche) Jacobian of
antagonistic interactions
Krause etal. (2003) and Thébault and Fon-
taine (2010)C, interaction strength-Resilience, persistence Simulations of model and real food webs
(nonlinear functional responses in Thébault
and Fontaine 2010)
Stability is enhanced in compartmentalized and
weakly connected architectures
van Altena etal. (2016) Jacobian elements-Asymptotic Lead eigenvalue of Jacobian of model
obtained from real food web data Skew toward weak interactions enhances stabil-
ity
Neutel etal. (2002, 2007) and Emmerson
and Yearsley (2004)Jacobian elements-Asymptotic, resilience,
feasibility, persistence Lead eigenvalue of Jacobian of model (cas-
cade) and real food webs Weak interactions in long feedback loops of
omnivorous species is stabilizing
McCann etal. (1998) Interaction strength-Persistence, temporal
stability Nonlinear models away from equilibrium Weak links and intermediate interaction
strengths are stabilizing
Positive complexity–stability relationship
Ives etal. (2000)S-Temporal stability Simulation of model community under envi-
ronmental variation Increasing the number of modular subcommu-
nities increases stability
Pimm (1979, 1980b)S, C-Extinction cascades Simulation of plausible food web model (gen-
eralized Lotka–Voterra) If herbivores are removed
Stouffer and Bascompte (2011)S, C-Persistence, extinction cascade Simulation of model (niche) Compartmentalization increases stability
De Angelis (1975)C-Asymptotic Lead eigenvalue of Jacobian of plausible food
web model Stability is increased by donor control interac-
tions
Dunne etal. (2002a) and Dunne and Wil-
liams (2009)C-Robustness Simulation of model obtained from real food
webs Skewness of degree distribution increases
robustness
Haydon (2000) Weighted C-Asymptotic Lead eigenvalue of Jacobian of plausible food
web model
van Altena etal. (2016) Weighted C-Asymptotic Lead eigenvalue of Jacobian obtained from
real food webs No relationship between unweighted C and
stability
Allesina and Pascual (2008) and Allesina
and Tang (2012)S, Jacobian elements-Asymptotic Lead eigenvalue of random, empirical, and
model (cascade and niche) Jacobian of
antagonistic interactions
Weak interactions are destabilizing
Borrvall etal. (2000)S per functional group, interaction strength-
Extinction cascades Simulation of plausible generalized Lotka–
Volterra food web model with three trophic
groups
Higher risk of extinction if autotrophs (rather
than top predators) are removed; Skewness
towards weak interactions is destabilizing;
Omnivory is stabilizing
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May’s (1972, 1973) stability criteria, they showed that the
strength of predator–prey links increase the stability of
small webs, but destabilize larger webs. They also revealed
a new power law describing how food-web stability scales
with the number of species. Pimm (1979, 1980b) showed
that extinction cascades are more likely in model (general-
ized Lotka–Volterra) communities with larger total number
of species, contrasted by Borrvall etal. (2000) that found
the robustness (resistance) of the same model food web to
increase with network redundancy (number of species per
functional group). Considering the topology of realistic
(Dunne etal. 2002a) and generated model (Dunne and Wil-
liams 2009) food webs, other authors found the same result,
i.e., positive relationship between number of species and
robustness, however ignoring strength of interactions and
community dynamics. Therefore, such contrasts may result
from dynamical properties of food webs.
Connectance
Exploring how the number of interactions varies with the
number of species has been one of the most basic questions
for ecologists trying to find universal patterns in the struc-
ture of ecological networks. Contradicting previous works
which found that the number of interactions increases lin-
early with the number of species (Cohen and Briand 1984;
Cohen and Newmann 1985), Martinez (1992) claimed the
constant connectance hypothesis in food webs: trophic links
increase approximately as the square of the number of spe-
cies. However, with the improvement of methodological
analysis and datasets, the constant connectance hypothesis
has been called into question by later studies (Havens 1992;
Dunne etal. 2002b; Banašek-Richter etal. 2009). One of the
most generally accepted rule on food web connectance is
that food webs display an average low connectance of about
0.11 (Havens 1992; Martinez 1992; Dunne etal. 2002b),
which is however still relatively high compared to that of
other real-world networks (Dunne etal. 2002b).
Since connectance has been used by May (1972, 1973) as
a descriptor of network complexity, it has become central to
early works on the complexity–stability debate (De Ange-
lis 1975; Pimm 1980b, 1984) and continues to be widely
used as a descriptor for network structure (Havens 1992;
Dunne etal. 2002a; Olesen and Jordano 2002; Tylianakis
etal. 2010; Heleno etal. 2012; Poisot and Gravel 2014).
Depending on the way stability is defined, the quality of
empirical datasets, or the methods used to generate theoreti-
cal networks, contradiction has been observed in the rela-
tionship between network stability and connectance. While
some studies reinforced May’s hypothesis of a negative
relationship between connectance and stability (Pimm 1979,
1980b; Chen and Cohen 2001; Gross etal. 2009; Allesina
and Tang 2012), others found that connectance enhances
Table 4 (continued)
References Complexity–stability measures Methods and assumptions Additional results
Haydon (1994)C, Jacobian elements-Asymptotic, feasibility Lead eigenvalue of random and plausible
model Jacobian at feasible equilibria Stability is reduced by donor control interac-
tions; Stability is increased by increased
interaction strengths
Yodzis (1981) Jacobian elements-Asymptotic Lead eigenvalue of Jacobian of empirically
inspired food webs Intraspecific competition is stabilizing whereas
interspecific competition tends to be destabi-
lizing
de Ruiter etal. (1995) Jacobian elements-Asymptotic Model (generalized Lotka–Volterra) and time
series of real and experimental food webs Asymmetries in strength of interaction (i.e.,
strong consumer control interactions at lower
trophic levels and strong donor control inter-
actions at higher trophic levels) are stabilizing
Rooney etal. (2006) Interaction strength-Asymptotic Model (nonlinear functional response and
predator adaptive behaviour) and time series
of real and experimental food webs
Asymmetries in interaction strength (i.e., slow
and fast energy fluxes coupled by top-preda-
tors) convey both local and non-local stability
Gross etal. (2009) Jacobian elements-Asymptotic Lead eigenvalue of Jacobian of realistic food
web model (niche) Increasing strength of interaction stabilizes
small networks
Population Ecology
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network stability (De Angelis 1975; Dunne etal. 2002a;
Dunne and Williams 2009). For example, using extinction
cascade as stability measure, Pimm (1979, 1980b) found that
complex model food webs are more likely to lose additional
species following the extinction of one species than simple
food webs: complexity is negatively correlated with stability.
By using different measurements of network stability (resil-
ience and persistence), Thébault and Fontaine (2010) also
confirmed the negative relationship between connectance
and stability in food webs (however, the opposite holds for
mutualistic networks, see next Section). Gross etal. (2009)
also revealed a negative power law to describe how food-
web stability scales with connectance.
The opposite view is sustained, among others, by De
Angelis (1975): using plausible food web community
matrix models, he showed that the probability of stability
can increase with increasing connectance if the food web
is characterized by a bias toward strong self-regulation
(intraspecific competition) of higher trophic level species,
low assimilation efficiencies, or a bias toward donor con-
trol. Also Haydon (1994), improving May’s assumptions
but still relying on community matrices, found stability to
increase with connectance. However, in contrast with De
Angelis (1975), stability is found to be reduced by the preva-
lence of donor control interactions. Furthermore, robustness
increases with connectance considering only the topology
of real food webs (Dunne etal. 2002a; Dunne and Williams
2009).
Weighted connectance
Only a recent study by van Altena etal. (2016) started its
use into the complexity–stability context, although they
found that there is no relationship between food web sta-
bility and unweighted connectance and that a high level of
weighted connectance stabilizes food webs. Following a
different perspective, Haydon (2000) focused on communi-
ties constructed to be as stable as they could be, and show
that communities built in this way require high levels of
weighted connectance, in agreement with van Altena etal.
(2016). According to these studies, high stability requires
high connectance, especially between weakly and strongly
self-regulated (intraspecific competition) elements of the
community.
Degree distribution
Degree distribution in food webs differ from a Poisson dis-
tribution (typical of random networks). However, there is
no universal shape that fits food webs degree distribution.
Most of the webs display exponential degree distribution
(Camacho etal. 2002; Dunne etal. 2002b) and those with
high connectance show a uniform distribution. Power-law
and truncated power-law with an exponential drop-off in the
tail also fit few of food webs degree distribution (mostly
those having very low connectance) (Dunne etal. 2002b;
Montoya and Solé 2002).
The skewness of degree distribution, especially exponen-
tial-type degree distribution (Dunne and Williams 2009)
makes food webs more robust to targeted removals (from
the most generalists) (Solé and Montoya 2001; Dunne etal.
2002a). However, the hierarchical feeding feature due to size
scaling laws imposes a cost to food web robustness (Dunne
and Williams 2009). Allesina etal. (2015) showed that broad
degree distributions tend to stabilize (in term of asymptotic
stability of the community matrix) large size-structured food
webs, obtained either empirically or with the cascade and
niche models.
Strength ofinteractions
In contrast with May’s (1972, 1973) findings, Haydon (1994)
found stability to increase with elements of the Jacobian.
Yodzis (1981) also found that the networks were far more
likely to be stable when such elements are chosen in accord
with real food web patterns rather than strictly at random.
Neutel etal. (2007) showed how non-random Jacobian ele-
ments patterns in naturally assembled communities explain
stability. They used below-ground food webs, whose com-
plexity increased along a succession gradient. The weight
of the feedback loops of omnivorous species characterized
stability (omnivory: feeding on more than one trophic level).
Low predator–prey biomass ratios (biomass pyramid, a fea-
ture common to most ecosystems) in these omnivorous loops
were shown to have a crucial role in preserving stability as
complexity increased during succession. However, Allesina
etal. (2015) showed that it is intervality, i.e., the propensity
for each predator to feed upon all the species in a certain size
interval, to be the driver of stability in large size-structured
foodwebs.
Variability in link strengths have also been found to be
related with stability, but only for relatively small webs,
whereas larger webs are instead to be destabilized by link-
strength variability (Gross etal. 2009). Stability is enhanced
when species at a high trophic level feed on multiple prey
species and when species at an intermediate trophic level are
consumed by multiple predator species. Using an energetic
approach, de Ruiter etal. (1995) and Rooney etal. (2006)
found that different types of structural asymmetry in energy
fluxes is key to stability of real food webs. In particular, de
Ruiter etal. (1995) used empirically estimated community
matrices of the generalized Lotka–Volterra model to show
that simultaneous occurrence of strong top-down effects
(consumer control) at lower trophic levels and strong bot-
tom-up effects (donor control) at higher trophic levels in the
patterns of Jacobian elements in real food webs is important
Population Ecology
1 3
to ecosystem stability. The pattern is a direct result of the
energetic organization of the food web. Rooney etal. (2006)
used empirical food web data into a nonlinear model with
functional responses and predator adaptive switching behav-
iour to show that slow and fast energy fluxes coupled by
top-predators in real food webs convey both local and non-
local stability to food webs. In conclusion, even with very
different approaches, complexity does not lead to instability.
A skewed distribution of interaction strength has been
widely observed in food webs, i.e., there are many weak
interactions and few strong ones (Paine 1992; Berlow 1999;
Berlow etal. 2004; Wootton and Emmerson 2005). This
skewness towards weak interactions has been related to sta-
bility. For example, McCann etal. (1998) found that weak
links and intermediate strength of interaction (measured as
the likelihood of one species to be consumed by another),
taking into account nonlinear saturating consumption, non-
equilibrium dynamics, and empirical strengths and patterns
of interaction, reinforce the stability and the persistence of
the community as they dampen the oscillation in preda-
tor–prey dynamics. Neutel etal. (2002) showed that weak
interactions are more likely observed in long loops in real
food webs. Specifically, Jacobian elements are organized in
trophic loops in such a way that long loops contain relatively
many weak links. They showed and explain mathematically
that this patterning enhances stability, because it reduces the
amount of intraspecific interaction needed for matrix stabil-
ity. On the same line, Thébault and Fontaine (2010) showed
that stability of trophic networks is enhanced in weakly
connected architectures. van Altena etal. (2016) confirmed
the role of weak interactions for stability of real food webs.
However, given skewed distributions of Jacobian elements
towards weak interactions, they found that stability was pro-
moted by even distribution of fluxes over links, in contrast
with de Ruiter etal. (1995) and Rooney etal. (2006) who
emphasized the role of strong asymmetry. In a recent paper,
Jacquet etal. (2016) disproved the association between Jaco-
bian elements and (asymptotic) stability in empirical food
webs, but showed that the correlation between the effects
of predators on prey and those of prey on predators, com-
bined with a high frequency of weak interactions, can sta-
bilize food web dynamics. In agreement with Neutel etal.
(2002) and Neutel etal. (2007), Emmerson and Yearsley
(2004) showed that a skew towards weak interactions in
feasible community matrices promotes local and global sta-
bility only when omnivory is present. A feedback is found
between skewness toward weak interactions and omnivory,
i.e., skewed Jacobian elements are an emergent property of
stable omnivorous communities, and in turn this skew cre-
ates a dynamic constraint maintaining omnivory. Borrvall
etal. (2000) however found that omnivory stabilizes food
webs, but the skew towards weak interaction is destabiliz-
ing (they however use interaction strengths, not Jacobian
elements). Omnivory appears to be common in food webs
(Polis 1991; Sprules and Bowerman 1988). A previous
theoretical work (Pimm and Lawton 1978) predicted that it
should be extremely rare to find species that feed simulta-
neously both high and low in real-world food web, and also
webs with a large number of omnivores should be rare in real
world. However, the authors ignored feasibility of the com-
munity matrices they used to estimate resilience (asymptotic
stability), possibly underestimating omnivorous interactions.
By contrast, Allesina and Pascual (2008) found that
stability is highly robust to perturbations of Jacobian ele-
ments, but it is mainly a structural property driven by short
and strong predator–prey loops, with the stability of these
small modules cascading into that of the whole network.
These considerations challenge the current view of weak
interactions and long cycles as main drivers of stability
in natural communities. In addition to that, Allesina etal.
(2015) showed that average Jacobian element in large size-
structured foodwebs have smaller influence on stability
compared with variance and correlation. Also, Allesina and
Tang (2012) showed that preponderance of weak interactions
(measured by Jacobian elements) decreases the probability
of food webs to be stable. In particular, trophic interactions
are shown to be stabilizing (as opposed to mutualistic and
competitive) but, counterintuitively, the probability of sta-
bility for predator–prey networks decreases when a realistic
food web structure is imposed or if there is a large prepon-
derance of weak interactions. However, stable predator–prey
networks can be arbitrarily large and complex (positive com-
plexity–stability relationship), provided that predator–prey
pairs are tightly coupled (i.e., short loops and high Jacobian
elements). Same negative relationship between stability and
skewness of strength of interaction distribution has been
found by Borrvall etal. (2000), although using different
measures of strength (interaction strengths) and stability
(extinction cascade).
Predator–prey body mass ratio, affecting the interac-
tion strength distribution, contributes largely to food-web
stability. Emmerson and Raffaelli (2004) empirically esti-
mated Lotka–Volterra interaction strengths and equilibrium
population densities and used such a community matrix to
evaluate (asymptotic) stability, showing that using empiri-
cal scaling laws the resulting food webs are always stable in
contrast with statistical expectations from random matrices
(May 1972, 1973). Otto etal. (2007) used a bioenergetic
model combining nonlinear functional responses and body
mass ratios, showing that such scaling may promote the sta-
bility of complex food webs.
Network architecture
The effect of network architecture, in particular modular
structures, has been observed in food webs and related to
Population Ecology
1 3
their stability. Moore and Hunt (1988) showed that food
webs may contain tightly coupled subunits whose numbers
may increase with diversity. Communities may be arranged
in resource compartments and within them species strength
of interaction would decline as diversity increased. Same
result has been found by Krause etal. (2003) and Thébault
and Fontaine (2010), who showed that stability of trophic
networks is enhanced in compartmented and weakly con-
nected architectures. Also, Ives etal. (2000) showed that
increasing the number of modular subcommunities in a sto-
chastic discrete time generalized Lotka–Volterra foodweb
model increases stability through different species reactions
to environmental fluctuations (insurance hypothesis; Yachi
and Loreau 1999). Similarly, Stouffer and Bascompte (2011)
demonstrate that compartmentalization increases the persis-
tence of food webs. Compartments buffer the propagation of
extinctions through the community and increase long-term
persistence. The latter contribution increases with the com-
plexity of the food web, emphasizing a positive complex-
ity–stability relationship. However, the recent study of Grilli
etal. (2016) shows that the stabilizing effect of modularity
is not as general as expected.
Nested diets have been observed in food webs: top pred-
ators are very generalists and prey upon all over species,
while the next predator exploiting all but the top predators
(in the niche model by Williams and Martinez 2000, and the
nested-hierarchy model by; Cattin etal. 2004). Generalist
top predators prey upon intermediate specialist predators
also in the results of Neutel etal. (2002).
Mutualistic communities
As the interaction between a plant and its insect pollinator
has often been used as a straightforward illustrative exam-
ple of a reciprocal coevolution (Darwin 1862), early studies
on mutualistic interactions were mainly dedicated to under-
standing coevolutionary processes (e.g., Ehrlich and Raven
1964; Brown etal. 1978; Wheelwright and Orians 1982;
Herrera 1985). However, coevolution is often considered as
a diffuse mechanism involving several species. Thus, ecolo-
gists started to study mutualism as a whole network of inter-
actions for which tools provided by complex network theory
can be used (Table5).
Species richness
Network size or the total number of species in the network
has been considered as an important determinant of mutu-
alistic networks stability. By using a theoretical model with
empirically informed parameters, Okuyama and Holland
(2008) found a positive relationship between community
size and community resilience. They mainly attributed this
positive relationship to the use of a nonlinear functional
response and its saturating positive feedback effect on
population growth. Their finding was later supported by
Thébault and Fontaine (2010) while they used a population
dynamics model with a nonlinear Holling Type II functional
response in which benefits gained from mutualism saturate
with the effective densities of the interaction partners. They
confirmed that a high number of species diversity promotes
not only the resilience of mutualistic communities but also
their persistence.
Connectance andconnectivity
Contributing to their complexity, mutualistic networks have
been observed to display non-random structural patterns.
Motivated by the finding of scale invariance in food webs
(Cohen and Briand 1984; Cohen and Newmann 1985),
Jordano (1987) studied patterns of connectance and spe-
cies dependences observed in a large dataset of pollination
and seed-dispersal networks. He found that connectance
decreases with species richness but the average number
of links per species (or linkage density) stays invariant to
changing network size. Using empirical mutualistic net-
works spanning different biogeographic regions (including
those used in Jordano 1987), Olesen and Jordano (2002)
observed that connectance indeed decreases exponentially
with species richness. After controlling for species richness
(network size), they also observed that connectance differed
significantly between biogeographic regions. On average,
mutualistic networks exhibit higher connectance than food
webs and other real-world networks do. However, mutualis-
tic networks still have low to moderate level of connectance
(average of 0.11 in Olesen and Jordano 2002 and 0.18 in;
Rezende etal. 2007).
The implication of connectance patterns to the stability
of mutualistic networks has gained attention only recently.
When extending the theoretical work of May (1972, 1973)
so as to incorporate realistic network structures and to dif-
ferentiate between different types of interactions (preda-
tor–prey, mutualistic or competitive), Allesina and Tang
(2012) found that connectance negatively affects the local
stability of mutualistic community networks. The analyti-
cal study by Suweis etal. (2015) is in agreement with this
statement when they correlated connectance with the degree
of localization (a system is defined to have a high degree
of localization when perturbations cannot easily propagate
through the network). They found that mutualistic networks
are indeed localized and the degree of localization decreases
with connectance. Moreover, Vieira and Almeida-Neto
(2015) extended a previously existing model that explores
patterns of species co-extinction (Solé and Montoya 2001;
Dunne etal. 2002a; Memmott etal. 2004) to study the rela-
tionship between connectance and extinction cascades in
mutualistic networks, emphasizing the role of the variation
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Table 5 Complexity–stability relationship in mutualistic communities
References Complexity–stability measures Methods and assumptions Additional results
Negative complexity–stability relationship
Vieira and Almeida-Neto (2015)C-Extinction cascade Stochastic coextinction model applied to a set of
empirical networks Extinction cascades occur more likely in highly
connected mutualistic communities
Feng and Takemoto (2014) Heterogeneity of degree, species strength, and
strength of interaction (visiting frequency
adjusted for uneven species abundance) distribu-
tions-Asymptotic
Theoretical study based on the analytical expres-
sion of the dominant eigenvalue Heterogeneity of node degree and strength of inter-
action distribution primarily determine the local
stability of mutualistic ecosystems; Nestedness
additionally affects it
Suweis etal. (2015)C-Localization Evaluation of the components of the eigenvalues
of a set of empirical networks Mutualistic communities are localized; Localization
is negatively correlated with connectance
Allesina and Tang (2012)S, C, σ, nestedness-Asymptotic Analytical analysis of artificial networks with
realistic structure Mutualistic interactions are destabilizing; Stability
is negatively affected by nestedness
Campbell etal. (2012) Nestedness-Extinction cascade Dynamic binary network-based model of plant-
pollinator community formation High nestedness may in extreme circumstances pro-
mote a critical over-reliance on individual species
and enhances extinction cascade
Thébault and Fontaine (2010) Modularity-Resilience, persistence Simulation of model and real pollination networks A highly connected and nested architecture pro-
motes community stability in mutualistic networks
Positive complexity–stability relationship
Okuyama and Holland (2008)S, L, symmetry of strength of interaction
(similarity of pairwise half-saturated constants),
nestedness-Resilience
Theoretical analysis with empirically informed
parameters; Non-linear functional response Community resilience is enhanced by increasing
community size and connectivity, and through
strong, symmetric strength of interaction of highly
nested networks
Thébault and Fontaine (2010)S, C, nestedness-Resilience, persistence Simulation of real pollination networks under a
model using a nonlinear Holling type II func-
tional response
A highly connected and nested architecture pro-
motes community stability
Memmott etal. (2004) Nestedness-Extinction cascade Topological coextinction model; Explored the
effects on plant extinction of the preferential
removal of the most linked pollinators
Plant species diversity declined most rapidly with
removal of the most-linked pollinators; Declines
were no worse than linear, because of the nested
architecture
James etal. (2012) Species degree, C, nestedness-Persistence Population dynamics model that incorporates both
competition and mutualism Species degree is a much better predictor of indi-
vidual species survival and hence, community
persistence; Nestedness is only of secondary
importance to community persistence
Bascompte etal. (2006) Heterogeneity of species strength distribution,
asymmetry of species dependences-Domain of
coexistence
Population dynamics model (generalized Lotka–
Volterra); Species dependences estimated from
empirical quantitative networks
The asymmetry of plant-animal dependences
enhance long-term coexistence and facilitate
biodiversity maintenance
Suweis etal. (2015)S, heterogeneity of species strength distribution-
Localization Evaluation of components of the eigenvalues of
empirical interaction matrices Mutualistic communities are localized; Localization
is positively correlated with network size and the
variance of the weighted degree distribution
Bastolla etal. (2009) Nestedness-Domain of coexistence Theoretical model of population dynamics (inter-
specific competition among species in the same
trophic level is also incorporated); Nestedness
of simulated networks informed from empirical
networks
Nestedness reduces effective interspecific com-
petition and enhances the number of coexisting
species
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of species dependences after each extinction event. They
used a stochastic co-extinction model in which a species
does not necessarily need the extinction of all its interac-
tion partners to go itself extinct. The chance of a species to
survive indeed depends on its level of dependence upon its
interaction partners. Contradicting previous pattern observed
in food webs (Dunne etal. 2002a), they found that extinction
cascades were more likely to happen in highly connected
communities. However, highly connected communities were
also shown to be persistent (James etal. 2012) and resilient
(Okuyama and Holland 2008), which was also confirmed by
Thébault and Fontaine (2010).
Degree distribution
Although network size and connectance partially deter-
mine the complexity of the network, they discard impor-
tant information regarding individual species connectivity
as well as distribution of the overall connectivity among
species (node degree distribution). Early studies on species
connectivity in mutualistic networks mainly concentrated on
how interactions are distributed among species. Attentions
were mainly focused on the prevalence of either generalists
or specialists in mutualistic networks (Waser etal. 1996;
Memmott 1999; Vázquez and Aizen 2003). Stimulated by
these early studies, Jordano etal. (2003) found generalized
patterns in the node degree distribution of a large number
of plant-pollinator and plant-frugivore networks. Most of
the networks showed a distribution of node degree that fits
a truncated power-law regime, suggesting the prevalence of
specialists and the rarity of super generalists. Few of the
networks showed a power-law or an exponential distribution
in their node degree. Moreover, gamma distribution was also
found to best fit the distribution of node degree in mutual-
istic networks (Okuyama 2008). The heterogeneity of node
degree distribution was found years later to be a primary
factor affecting negatively the local stability of mutualistic
networks (Feng and Takemoto 2014). However, when node
degrees are considered individually for each species, they
were shown to be a good predictor of species own survival
and thus of the community persistence (James etal. 2012).
Strength ofinteractions
Instead of only considering qualitative interactions (pres-
ence or absence), quantitative measurement of strength of
interaction also prevails in mutualistic network studies. In
plant-pollinator as well as in plant-frugivory interactions,
strength of interaction often refers to the relative number of
visits of the animal to the plant. Jordano (1987) observed
an extremely skewed distribution of species dependences in
mutualistic communities: weak dependences greatly exceed
in number strong ones. By including more datasets in their
Table 5 (continued)
References Complexity–stability measures Methods and assumptions Additional results
Rohr etal. (2014)Species degree, interaction strength, nestedness-
Structural stability Population dynamics model; Explored the range of
parameters necessary for stable coexistence A maximal level of nestedness, a small trade-off
between the number and intensity of interactions
a species has, and a high level of mutualistic
strength are factors that maximize stability
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study, Bascompte etal. (2006) confirmed Jordano’s (1987)
finding of a skewed distribution of interaction dependences.
Additionally, mutualistic networks were also found to be
highly asymmetric in terms of species roles: while animals
depend strongly on the plants, plants rely poorly on their ani-
mal pollinators or seed dispersers (Bascompte etal. 2006).
The maintenance of biodiversity was suggested to be
facilitated by both the heterogeneity of species strength
distribution and the asymmetry of species dependences
(Bascompte etal. 2006). Localization, or the ability of the
system to reduce the propagation of perturbations through
the network, has also been shown to be enhanced by the
heterogeneity of species strength distribution (Suweis etal.
2015). Notice that Suweis etal. (2015) defined the strength
of a species as its weighted degree or the sum of interac-
tion strengths in which the species is involved. Opposed to
the finding of Bascompte etal. (2006) who used a linear
functional response in their model, Okuyama and Holland
(2008) argued that the asymmetry of species dependences,
implying an asymmetry of strength of interaction between
animals and plants (measured as similarity of the pairwise
half-saturation constants), has a negative effect (although
small) on the resilience of mutualistic communities when a
nonlinear functional response is used. Feng and Takemoto
(2014) also showed that the heterogeneity of the distribution
of strength of interaction (estimated from visiting frequen-
cies adjusted for uneven species abundances) indeed impacts
negatively on the local stability of mutualistic communities.
Moreover, by also using a saturating functional response,
Rohr etal. (2014) demonstrated that regardless of the dis-
tribution of interaction strength, mutualistic communities
that have on average a high level of interaction strength are
more likely to be structurally stable (have a wider domain
of feasible and stable coexistence).
Network architecture
Although modularity or compartmentalization is a feature
commonly observed in food webs, mutualistic networks also
exhibit a certain level of modularity. A test for modularity
in a wide datasets allowed Olesen etal. (2007) to affirm
that pollination networks with a relatively high number of
species are indeed modular. Moreover, the observed level
of modularity increases with network size. The number of
modules and the level of modularity observed in pollination
networks are found to be invariant to sampling efforts at dif-
ferent time (Dupont and Olesen 2012). Mello etal. (2011)
also noticed a high level of modularity in seed-dispersal
networks. Little is known about the implication of modular
structure to mutualistic network stability. Thébault and Fon-
taine (2010) emphasized that structural patterns favouring
stability fundamentally differ in food webs and mutualistic
networks: while the modularity pattern enhances food web
stability, it has a negative effect on the persistence and resil-
ience of mutualistic networks.
A widely accepted topological feature proper to mutual-
istic networks is nestedness. Bascompte etal. (2003) started
to explore this feature in a meta-analysis of empirical mutu-
alistic communities and found that mutualistic networks
are indeed highly nested. They also found that nestedness
increases with network complexity expressed in terms of
species richness and connectivity. Nestedness has always
been believed to be the most important determinant of
mutualistic network stability. For example, extinction cas-
cades following the removal of the most generalist pollina-
tor in a pollination community have been shown to happen
only linearly because of the stabilizing effect of nestedness
(Memmott etal. 2004). The nested structure of mutualistic
networks also enhances the number of coexisting species by
reducing effective interspecific competition (Bastolla etal.
2009). Nestedness also has a positive effect on the persis-
tence and resilience of mutualistic communities (Okuyama
and Holland 2008; Thébault and Fontaine 2010). Rohr etal.
(2014) showed that the parameter domain leading to both
dynamically stable and feasible equilibrium, i.e., the domain
of stable coexistence of species (an extended measurement
of the system’s structural stability) is maximized when artifi-
cial networks are assumed to have a high level of nestedness.
However, some recent studies started to discard the impor-
tance of nestedness to network stability. James etal. (2012)
indeed found that nestedness is, at best, a secondary covari-
ate rather than a causative factor for species coexistence in
mutualistic communities and has no significant effect on
community persistence. By means of analytical analyses of
artificial networks with realistic structure, Allesina and Tang
(2012) affirmed that local stability is negatively affected by
the nestedness of mutualistic community matrices. Campbell
etal. (2012) also showed that extreme nestedness facilitates
sequential species extinctions (extinction cascades).
Competitive communities
Competitive interactions have sometimes been considered
together with trophic interactions in food webs models, and
their contribution to stability assessed. There is common
agreement that self-regulating interactions due to intraspe-
cific competition (i.e., negative terms on the diagonal of
the interaction or community matrix) increase stability. For
example, De Angelis (1975), using plausible food web mod-
els, showed that the probability of stability increases if the
food web is characterized by a bias toward strong self-regu-
lation (intraspecific competition) of higher trophic level spe-
cies. Haydon (1994) found a similar result, i.e., that consid-
ering intraspecific competition increases food web stability.
Again, Haydon (2000) focused on communities constructed
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to be as stable as they could be, and show that communities
built in this way require high connectance between weakly
and strongly self-regulated (intra-specific competition) ele-
ments of the community. Neutel etal. (2002) showed and
explain mathematically that the patterning of real food web
Jacobian elements enhances stability because it reduces the
amount of intraspecific interaction needed for matrix sta-
bility. Yodzis (1981) also found that the presence of self-
regulatory terms (intraspecific competition) in some con-
sumer species stabilizes the network. However, the role of
interspecific competition is less clear. For example, Yodzis
(1981) showed that interspecific competition tends to be
destabilizing: assuming that interspecific competition only
happens among consumers sharing resources, he found that
the fraction of stable community matrices decreases with the
number of competitive pairs. By contrast, Allesina and Tang
(2012) showed that competitive interactions are destabiliz-
ing, in the sense that the number of species S and the con-
nectance C that result in an unstable equilibrium are lower
for competitive rather than trophic communities.
In the 1970s, it was believed that ecological communities
were structured by competitive interactions (e.g., MacArthur
1972). Theoretically speaking, purely competitive commu-
nities are simpler compared to food webs and mutualistic
networks because they are composed of only one trophic
level. Thus, the simplicity of competitive communities
makes them an ideal theoretical framework for studying the
relationship between community complexity and stability,
and studies relating biodiversity and ecosystem function
tended to focus on the diversity of primary producers (e.g.,
Hooper etal. 2005). For these reasons, extensive experi-
mental (Lawlor 1980; Tilman and Downing 1994; Tilman
1996; Lehman and Tilman 2000) and theoretical (Lawlor
1980; Tilman etal. 1997, 1998; Doak etal. 1998; Tilman
1999; Cottingham etal. 2001) studies have been done on the
relation between species richness of plants and community
stability (Table6).
Species richness
Species richness has been reported to affect the stability
of competitive communities. Tilman and Downing (1994)
empirically showed that primary productivity in more
diverse plant communities is more resistant and recovers
fully after a major drought. Tilman etal. (1997) confirmed
such findings using theoretical competition models. Doak
etal. (1998) however showed that such result could be sta-
tistically inevitable using the temporal variation in aggregate
community properties as indicators of stability (see also Til-
man etal. 1998 for a reply). Lehman and Tilman (2000)
analysed different models of multispecies competition and
empirical data (Tilman 1996), finding that greater diversity
increases the temporal stability of the entire community but
decreases the temporal stability of individual populations.
Specifically, temporal stability of the entire community
increases fairly linearly without saturation with increased
diversity. Species composition of each community was also
predicted to be as important as diversity in affecting com-
munity stability. The work by Tilman (1999) summarizes
the empirical and theoretical positive relationship between
species diversity and community stability, primary pro-
ductivity, and invasibility in grassland competitive com-
munities (see, however, the critical review in Cottingham
etal. 2001). Lawlor (1980) compared observed communi-
ties (defined by symmetric interaction matrices where each
competition coefficient is given by a measure of overlap of
resource utilization) with analogous randomized versions
of them (note that he randomizes the resources utilization
spectra rather than the competition coefficients themselves):
he found that stability of observed communities decreases
with the number of species, however, observed communities
are generally more stable than randomly constructed com-
munities with the same number of species. The higher stabil-
ity of observed (compared to random) communities is due
to lower similarities among consumer species, suggesting
that interspecific competitive processes are very important
in shaping communities. Christianou and Kokkoris (2008)
reported that increasing the number of species in the com-
munity also decreases the probability of feasibility of the
system, however, species richness does not significantly
affect the probability (proportion of random communities)
of local stability and the resilience of feasible competitive
communities. By contrast, Fowler (2009) demonstrated that
increasing the number of species in a discrete-time competi-
tion model (both symmetric and asymmetric, with a skew
towards weak interactions) results in an increased probabil-
ity (species growth rate parameters region) of local stability
in competitive feasible communities: increasing the competi-
tive negative feedbacks adding more species or links in the
network dampens oscillatory dynamics and contributes to
equilibrium stability.
Connectance
Fowler (2009) also showed that an increase in network con-
nectance and in the number of competitive links (connectiv-
ity) reduces per-capita growth rates through an increase in
competitive feedback, thus stabilises oscillating dynamics.
Furthermore, he affirmed that these results stay robust to
changes in species interaction strengths.
Strength ofinteractions
Most studies on competitive communities focused on the
implication of competition coefficients, i.e., interactions
strengths, on community stability. However, different
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Table 6 Complexity–stability relationship in competitive communities
References Complexity–stability measures Methods and assumptions Additional results
Negative complexity–stability relationship
Lawlor (1980)S-Asymptotic stability Lead eigenvalue of random vs. observed symmet-
ric interaction (overlap) matrices Observed communities are generally more stable
than randomly constructed communities with the
same number of species
Lehman and Tilman (2000)S-Asymptotic, temporal stability Lead eigenvalue and simulations of three different
models (mechanistic, phenomenological, statisti-
cal) and empirical time series
Greater diversity decreases the temporal stability of
individual populations
Christianou and Kokkoris (2008)S, interaction strength-Asymptotic, feasibility,
persistence, structural stability Model of competitive community Asymptotic stability is not affected by the number
of species S, but structural stability (domain
of stable coexistence) decreases with species
richness; Weak interaction strengths enhances
structural stability
Kokkoris etal. (1999) Interaction strength-resistance to invasion Community assembly model from a regional spe-
cies pool Weak interaction strengths enhances resistance to
invasion
Kokkoris etal. (2002) Interaction strength-Asymptotic, feasibility,
persistence, structural stability Model of competitive community Weak interaction strengths enhances species coex-
istence
Positive complexity–stability relationship
Lehman and Tilman (2000)S-Asymptotic, temporal stability Lead eigenvalue and simulations of three different
models (mechanistic, phenomenological, statisti-
cal) and empirical time series
Greater diversity increases the temporal stability of
the entire communities
Fowler (2009)S, C, connectivity-Asymptotic, structural stability Lead eigenvalue and simulation of discrete-time
model of competitive community Result robust to change in interaction strengths
Hughes and Roughgarden (1998) Interaction strength-Temporal stability Discrete-time two-species competition model Stability independent on the magnitude but related
to asymmetry of interaction strengths
Allesina and Tang (2012)S, C, Jacobian elements-Asymptotic Lead eigenvalue of random, empirical, and model
Jacobian of competitive interactions Equilibrium becomes unstable for smaller S and C
in competitive compared to trophic communities
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measurements of stability have been used. For example,
Hughes and Roughgarden (1998) studied temporal stability
measured as the aggregate community biomass in a discrete
time two-species competition model. They found that tem-
poral stability is relatively independent of the magnitude of
interaction strengths but the degree of asymmetry of inter-
actions is the key to community stability. Quantifying the
stability of the community by its invulnerability to invasion,
Kokkoris etal. (1999) studied the distribution of interac-
tion strengths (competition coefficients)—not Jacobian
elements—during the assembly process of theoretical com-
petitive communities. They found that the mean interaction
strength drops as assembly progresses and most interactions
that are formed are weak. It suggests that communities that
are invulnerable to further invasion are those where inter-
specific interactions are weaker than the average interac-
tion strength between competing species of a regional pool.
In a later study (Kokkoris etal. 2002), the same authors
explored how the number of coexisting species vary with
the average interaction strength. Confirming their previous
finding on the importance of weak interactions to commu-
nity stability, they found that the preponderance of weak
interactions indeed allow many species to coexist. Moreo-
ver, correlation in the interaction matrix, mainly a result of
trade-offs between species characteristics, can increase the
probability of species coexistence. Christianou and Kokkoris
(2008) even deepened the study on the importance of weak
interactions to stability by considering system feasibility of
a competitive community. Consistent with previous findings,
they showed that the probability of feasibility decreases with
increasing interaction strength.
Recent developments
In this section we briefly introduce the most recent develop-
ments in the theory of the complexity–stability relationship.
Including more details and making models more realistic
seem to give more space for a positive complexity–stability
relationship. These extensions include, but are not limited
to, considering multilayer networks (accounting for differ-
ent interaction types varying in space and interconnected
communities, see Pilosof etal. 2017 for a recent review),
or describing trait mediated-interactions and adaptive net-
works. After a brief description of multilayer networks, spe-
cific focus will be given to the latter two extensions.
Considering multiple interaction types, i.e., trophic,
mutualistic, and competitive in the same community, can
alter ecological networks dynamics, complexity, and stabil-
ity (see review by Fontaine etal. 2011). Melian etal. 2009
combined mutualistic and antagonistic (herbivorous) inter-
actions in an empirically derived model of such ecological
network, showing that species persistence is increased by the
correlation between strong species dependences and the ratio
of the total number of mutualistic to antagonistic interac-
tion per species. Mougi and Kondoh (2012), using random,
cascade, and bipartite (Thébault and Fontaine 2010) models,
showed that a moderate mixture of antagonistic and mutu-
alistic interactions can stabilize community dynamics, and
increasing complexity (species richness and connectance)
leads to increased (asymptotic) community-matrix stability.
Mougi and Kondoh (2014) confirmed their previous results
in an extended version of their model also considering com-
petition, adding that the hierarchically structured antagonis-
tic interaction network is important for the stabilizing effect
of mixed interactions to emerge in complex communities.
Mougi (2016a), using interaction strengths and (asymptotic)
stability of community matrices, showed that overlooked
unilateral interactions (where only one species affects the
partner species, e.g., amensalism or commensalism) greatly
enhance community stability. Such unilateral interactions are
however more stabilizing than symmetric interactions (com-
petition and mutualism) but less stabilizing than asymmetric
interactions (antagonistic), confirming previous results in
Mougi and Kondoh (2014).
The effect of spatial dynamics have been shown to be
stabilizing in classical theoretical ecology. However, only
few recent contributions considered space into the com-
plexity–stability debate, describing meta-communities, i.e.,
networks of networks. Considering local food webs con-
nected through dispersal, both Mougi and Kondoh (2016)
and Gravel etal. (2016) showed that indeed intermediate
dispersal and the number of local patches can increase the
asymptotic stability of the meta-community matrix.
Trait‑mediated interactions andadaptive networks:
food webs
The discussion thus far has implicitly assumed that links
among species remain unchanged over time. This is often a
simplifying assumption, as adaptive foraging (see review in
Valdovinos etal. 2010) or other forms of adaptive behaviour
in response, e.g., to environmental changes (Strona and Laf-
ferty 2016) can often cause links to form, change in strength,
or disappear as time progresses. Adaptive networks has been
shown to reproduce realistic food-web structures (Nuwagaba
etal. 2015), to promote stability (Nuwagaba etal. 2017), and
to allow for positive complexity–stability relationships. For
example, Kondoh (2003) and Kondoh (2006) showed that
foraging adaptation enhances stability of trophic communi-
ties. Without adaptation, complexity is destabilizing, while
adaptive foragers help buffering environmental fluctuations
resulting in a positive relationship between complexity and
persistence. Visser etal. (2012) examined the effect of adap-
tive foraging behaviour within a tri-trophic food web and
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demonstrated that adaptive behaviour will always promote
stability of community dynamics.
Predator–prey body mass ratio, affecting the interaction
strength distribution, contributes largely to food-web stabil-
ity (Emmerson and Raffaelli 2004; Brose etal. 2006; Otto
etal. 2007). Heckmann etal. (2012) combined this allomet-
ric body-size structure and adaptive foraging behaviour in
random and niche food web models with nonlinear func-
tional responses, showing that both body-size structure and
adaptation increase the number of persisting species through
stabilising interaction strength distributions Moreover, adap-
tive foraging explains emergence of size-structured food
webs (in which predators tend to focus on prey on lower
trophic levels and with smaller body sizes) from random
ones, linking these two stabilising mechanisms.
Trait adaptation can also be modelled and give rise to
complex trophic interaction networks (Brännström etal.
2011, 2012; Landi etal. 2013, 2015; Hui etal. 2018), and
their complexity–stability relationship assessed (Kondoh
2007; Ingram etal. 2009). In particular, Kondoh (2007)
studied adaptation in predator-specific defence traits,
reporting its unimodal effect on the complexity–stabil-
ity (connectance-persistence and robustness) relationship,
while species richness always has a negative impact on sta-
bility. Ingram etal. (2009) studied body size and niche width
adaptation in different environmental conditions, emphasiz-
ing a positive correlation between omnivory with temporal
variability and species turnover through extinctions and
invasions–speciations.
Trait‑mediated interactions andadaptive networks:
mutualistic communities
Pioneering studies addressing the effect of mutualistic
community structure to community stability often utilized
dynamic models of changing population abundance such as
those based on extensions of the Lotka–Volterra model for
mutualism, with various types of functional response (e.g.,
Okuyama and Holland 2008; Bastolla etal. 2009; Thébault
and Fontaine 2010 all used a nonlinear functional response).
Although these models have expanded our knowledge about
the structure and dynamics of complex mutualistic systems,
they disregarded important biological processes associated
with plant-animal interactions. One important biological
process is adaptation. Recent studies incorporated adapta-
tion into the foraging behaviour of animal pollinators and
seed dispersers. One way to reflect adaptive foraging is
through rewiring of interactions. In a study by Zhang etal.
(2011), the emergence of nestedness pattern in pollination
and frugivory networks has been well reproduced when
species are allowed to switch their mutualistic partner for
another one providing higher benefit, as a consequence of
adaptive foraging strategy. Going beyond the importance
of adaptive rewiring to the emergence of network structure,
other studies even explored its implication to network sta-
bility. Ramos-Jiliberto etal. (2012) used a spatially explicit
model in which species occupy an infinite number of patches
as habitats, and showed that when animal pollinators have
the ability to rewire their connections after depletion of
host plant abundances, the resistance of the network against
additional extinction induced by primary species removal
(i.e., network robustness) is enhanced. Moreover, preferen-
tial attachment to host plants having higher abundance and
few exploiters enhances network robustness more than other
rewiring alternatives. Foraging effort of pollinators can also
be incorporated directly as an evolving trait affecting pol-
linator’s growth rate. Indeed, Valdovinos etal. (2013) devel-
oped a population dynamics model based on pollinator’s
adaptive foraging and projected the temporal dynamics of
three empirical pollination networks. In their model, asym-
metries between plants and animals were considered based
on the fraction of visits that end in pollination events, the
expected number of seeds produced by a pollination event,
and the amount of floral resources that the animal extracts
in each visit to a plant. They found that incorporation of
adaptive foraging into the dynamics of a pollination network
increases network persistence and diversity of its constituent
species. Moreover, Song and Fledman (2014) constructed a
mathematical model that integrates individual adaptive for-
aging behaviour and population dynamics of a community
consisting of two plant species and a pollinator species. They
found that adaptive foraging at the individual level, comple-
menting adaptive foraging at the species level, can enhance
the coexistence of plant species through niche partitioning
between conspecific pollinators.
Adaptation in mutualistic networks has also been mod-
elled through the evolution of functional traits determi-
nant of the interactions. Such traits are often those that can
impose important constraints on the interactions, such as the
proboscis lengths of a pollinator and the flower tube length
of a plant (Eklöf etal. 2013; Zhang etal. 2013; Hui etal.
2018). For instance, Olivier etal. (2009) showed that toler-
ance traits (those responsible for minimising fitness cost but
not reducing encounter rate), as opposed to resistance traits
(those acting to reduce encounter rate between the inter-
acting partners) are an important factor promoting stability
of mutualisms. Moreover, they argued that a tolerance trait
such as the phenotypic plasticity in honeydew production
can prevent escalation into an antagonistic arms race and
led to mutualistic coevolution. Using a theoretical model
based on the interplay between ecological and evolutionary
processes, Minoarivelo and Hui (2016) studied the evolution
of phenotypic traits in mutualistic networks. By assuming
that interactions are mediated by the similarity of phenotypic
traits between mutualistic partners, they generated certain
realistic architectures of mutualistic networks. In particular,
Population Ecology
1 3
they showed that a moderate accessibility to intra-trophic
resources and cross-trophic mutualistic support can result in
a highly nested web, while low tolerance to trait difference
between interacting pairs leads to a high level of modular-
ity. Moreover, the similarities between functional traits can
be approximated by phylogenetic similarities, allowing the
architecture of bipartite mutualistic networks to be shaped by
the phylogenies (coevolutionary history) of resident species
(Rezende etal. 2007; Minoarivelo etal. 2014).
More abstract traits have also been used in modelling
mutualistic coevolution. For instance, Ferriére etal. (2002)
defined a trait measured as the per capita rate of commodi-
ties trading which represents the probability per unit time
that a partner individual receives benefit from a mutualistic
interaction. They found that the existence of ‘cheaters’, or
individuals that reap mutualistic benefits while providing
fewer commodities to the partner species, can lead to the
coexistence of mutualistic partners and thus is a key to the
persistence of mutualism. In contrast to their study, West
etal. (2002) showed that one of the factors that may stabi-
lize mutualistic interactions is when individuals preferen-
tially reward more mutualistic behaviour and punish less
mutualistic (i.e., more parasitic) behaviour. The stability
of the plant-legume mutualism was also explained by this
cost/reward process. Plants that are selected to supply pref-
erentially more resources to nodules that are fixing more
N2 can be crucial to the establishment of effective legume-
rhizobium mutualisms during biological invasions (Le Roux
etal. 2017).
Finally, Mougi (2016b) also considers adaptive behav-
ioural network dynamics in a two-interaction (antagonistic
and mutualistic) community. While adaptive partner switch
is destabilizing single-interaction communities and does not
reverse the negative complexity–stability relationship (con-
trary to Kondoh 2003), it stabilizes hybrid communities with
multiple interaction types and reverts the complexity–stabil-
ity relationship to positive (with complexity measured by
number of species and connectance, while stability meas-
ured by species persistence).
Conclusions
More than 40years after May’s (1972, 1973) pioneering
work, there is still no complete agreement on the complex-
ity–stability relationship in ecosystems. The main issues
(or rather progress) could be related to the use of different
definitions and measures for both complexity and stability,
and the use of model vs. real ecosystem data. Moreover,
the adaptation and evolution of resident species has only
recently started being explored, and their contribution to
the debate is no doubt important, as foreseeable in the rapid
changes that are affecting our planetary ecosystems in which
all ecological networks are embedded. Generic evolution-
ary models and models that implement adaptive processes
thus serve as a promising tool for resolving the debate and,
importantly, furthering our understanding and better man-
agement of biodiversity in the era of the Anthropocene.
Acknowledgements PL and CH acknowledge support from the
National Research Foundation (NRF) of South Africa (Grant 89967).
CH further acknowledges support from the NRF (Grant 109244) and
the Australian Research Council (Discovery Project DP150103017).
HOM acknowledges support from DST-NRF Centre of Excel-
lence in Mathematical and Statistical Sciences (CoE-MaSS; Grant
BA2017/136). The contribution of two anonymous reviewers greatly
improved the quality of the manuscript. This review is based on Landi
etal. (2018) published as a chapter in the book “Systems Analysis
Approach for Complex Global Challenges” (Mensah etal. 2018).
Open Access This article is distributed under the terms of the Crea-
tive Commons Attribution 4.0 International License (http://creat iveco
mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribu-
tion, and reproduction in any medium, provided you give appropriate
credit to the original author(s) and the source, provide a link to the
Creative Commons license, and indicate if changes were made.
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