How to Invade an Ecological Network

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Opinion

How to Invade an Ecological Network Cang Hui1,2,* and David M. Richardson3 Invasion science is in a state of paradox, having low predictability despite strong, identifiable covariates of invasion performance. We propose shifting the foundation metaphor of biological invasions from a linear filtering scheme to one that invokes complex adaptive networks. We link invasion performance and invasibility directly to the loss of network stability and indirectly to network topology through constraints from the emergence of the stability criterion in complex systems. We propose the wind vane of an invaded network – the major axis of its adjacency matrix – which reveals how species respond dynamically to invasions. We suggest that invasion ecology should steer away from comparative macroecological studies, to rather explore the ecological network centred on the focal species. The State of Paradox in Invasion Science To differentiate the introduction status of species, to align this with ecological and biogeographical concepts, and to standardise terminology in invasion ecology, Richardson et al. [1] proposed a conceptual scheme based on a series of major barriers that mediate the progression of alien species through different stages. This barrier scheme, which echoes the filter paradigm that has shaped theories of community assembly [2], has become a mantra in invasion science. Building on this scheme, Blackburn et al. [3] proposed a unified framework for invasion biology that further elucidates the factors that mediate progression along the linear introduction–naturalisation–invasion continuum [hereafter, the invasion continuum (see Glossary)]. Management can target different stages with specific mitigation strategies [4]. Adopting the filtering scheme metaphor for biological invasions, substantial effort has gone into applying correlative statistics to identify the functional traits of species able to penetrate specific filters [5]. A macroecological approach is often applied for this purpose to compare, for example, the niches of alien species in invaded versus native ranges or the traits of alien and (ecologically similar) native species or to identify traits and environmental factors associated with alien species with contrasting introduction statuses [6,7]. This avenue of research has to some extent unified management practice and has yielded reasonable insights on invasion dynamics, but it suffers from low predictability. The filter scheme that underpins most research in invasion science and the filter paradigm that does the same in community ecology are, however, facing mounting challenges [2,8]. Managing invasions as part of increasingly complex social–ecological systems in the face of rapid global change demands innovative approaches to elucidate invasion-driven reinforcing feedback loops [9,10], such as those created by introduced ecosystem engineers or formed by intransitive interactions among species [11]. Reinforcing feedback loops can push systems toward a state of paradox: although patterns derived from observations are highly structured (e. g., invasive traits identified from comparative studies), predictions based on such observed structures perform poorly and often suffer from high levels of uncertainty [12,13]. Feedback/ coupling loops are central to the functioning of all living systems [14]. Invasive species not only negotiate realised niches in recipient ecosystems, but also imprint their impact onto the resident

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Highlights Invasion performance can be tentatively explained by the traits of alien species relative to those of natives, recipient site characteristics, and introduction pathways. The rush to identify invasive traits from comparative studies has not yet led to predictability at a satisfactory level. Synergies among invasion science, network ecology, and community ecology warrant increasing attention. Winners and losers in recipient ecosystems are the results of the multiplayer game between natives and aliens, as well as human factors. Statistical tools that can handle multispecies interactions are on the horizon.

1 Centre for Invasion Biology, Department of Mathematical Sciences, Stellenbosch University, Matieland 7602, South Africa 2 Mathematical and Physical Biosciences, African Institute for Mathematical Sciences, Cape Town 7945, South Africa 3 Centre for Invasion Biology, Department of Botany and Zoology, Stellenbosch University, Matieland 7602, South Africa

*Correspondence: [email protected] (C. Hui).

https://doi.org/10.1016/j.tree.2018.11.003 © 2018 Elsevier Ltd. All rights reserved.

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species, interactions, and processes in ever-evolving ecological systems. To tackle the state of paradox facing invasion science, and in line with rapid advances in community ecology in relation to metacommunity theory [15] and metanetwork theory [16–19], we have tentatively suggested the need to shift the metaphor of biological invasions from one based on a linear filtering scheme to one that invokes complex adaptive networks [20]. We follow this thread here by elucidating the relationship between invasion performance and network topology and the loss of network stability. In so doing, we propose the wind vane for invaded networks: an approach that elucidates how each species (native or alien) responds dynamically to biological invasions. We also propose a list of research priorities rooted in the network metaphor that we believe could yield new insights on effective conservation and invasion management.

Invasion Performance and Network Topology We are witnessing a rapid rise of ‘network thinking’ that is moving us away from linear hierarchical thinking in community ecology [21,22]. Network ecology has long emphasised the critical role of the multitude of biotic interactions in forming complex feedback loops and thus affecting species persistence and coexistence [23,24]. With increasing emphasis on higher-order and intransitive interactions [11,25,26], network ecology is philosophically breaking away from both the hierarchical filter paradigm and individualism [27]. A network of biotic interactions is typically represented as a graph of L links connecting specific pairs of S nodes (Figure 1A). Quantitatively, the topology of this network can be captured by its adjacency matrix with its element representing the link/interaction strength of connected nodes (Figure 1B). With the community matrix, a number of network topology metrics can be defined, some of which are widely discussed in ecology, such as local network indices of node degree and centrality and global network indices of connectance, nestedness, and modularity [21,28]. Network ecology has increasingly moved beyond the graph theory that manipulates static network topology to the dynamical systems of complex dynamic networks that can capture the dynamic behaviour of open and adaptive networks (Box 1). Humans act not only as an influencer of this network by modifying interaction strengths but also as an active node in the system. Understanding how human decisions, perceptions, and management efforts interact in ecological networks is crucial for forecasting network response to invasions [29]. Consequently, nodes of an invaded network should be expanded from only representing native and alien species to also include human-mediated factors. Moreover, the performance of a species can feature multiple nodes in a network, representing different facets of performance such as the traditional demographic/population dynamics and the potential of often-rapid trait evolution under density-dependent selection from ecoevolutionary feedbacks in invaded networks [30,31]. Following this trend, interaction strength, traditionally recorded only as a binary variable (presence/absence of a link between two nodes), is increasingly measured quantitatively, reflecting both adaptive changes and the density/interaction dependence of interaction strength [32–34]. Evidence that shows how network topology can be used to profile invaders and estimate invasibility, most of it theoretical, is accumulating, but remains divided. First, the level of generalisation [measured by the number of species (i.e., node degree) with which a focal species interacts] and trait distinctiveness relative to resident species (related to the centrality that defines the importance of a node in a network) [35–38], as well as trophic positions (e.g., intraguild predators and omnivores; related to both metrics) [39,40], are correlated with invasiveness. Unfortunately, emerging insights in this regard are highly context dependent and thus lack congruent support [41–43]. Second, high connectance in recipient ecosystems suggests that a large portion of all possible biotic interactions has been established; it has been 2

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Glossary Adjacency matrix: a square matrix with its element depicting the linkage weight between two nodes in a graph or network. Conceptually, it is equivalent to both the interaction matrix of interaction strength when depicting a dynamic network and the linearised Jacobian matrix of a dynamical system. Dynamical systems: a modelling framework that depicts a system with multiple interacting parts, often formulated using a set of differential equations describing each part’s time dependence. Eigenvector: a special vector that will not be distorted by a matrix operator but only amplified or dwarfed along the original direction in the vector space by a factor corresponding to its eigenvalue. Interaction strength: the sensitivity (partial derivative) of species A’s population growth rate to changes in species B’s population size; it can be measured as the covariance between A’s growth rate and B’s density. It is measured in a variety of ways in empirical ecology, from trait matching to the frequency of interaction events, from level of impacts (e.g., scars) to cooccurrence. Invasion continuum: a conceptualisation of the progression of stages and phases in the status of species introduced to a new environment. It posits that alien species must negotiate a series of barriers and pass through a set of filters to become established (reproducing regularly) and invasive (spreading). Lead eigenvalue: the Jacobian matrix of a dynamical system possesses multiple eigenvalues. Eigenvalues can be complex numbers, with each comprising a real part and an imaginary part. The eigenvalue that has the largest real part indicates how a system responds to perturbations near an equilibrium. If the real part of the lead eigenvalue is negative, the perturbation will be exponentially dampened and the system will converge back to its original dynamic regime; if it is positive, the perturbation will be amplified, leading to an eventual regime shift.

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argued that this reduces invasibility [44–46], although there have also been notable counterarguments [38,39]. Network size (species richness of recipient ecosystems) has been hotly debated under the banner of the invasion paradox [47,48]. Key questions relate to the relationship between species richness and the number of opportunity niches for invasions [49] and other factors that mediate such opportunity niches [50]. Finally, network models with mixed and adaptive interactions have further complicated the picture, such as models that emphasise the role of resource acquisition [31] and trait tolerance [51] in mutualistic networks facing invasion. With such overwhelming dependence on context, methodology, and taxonomy, it is debatable whether these results, mostly from theoretical models with complex and specific model structures, could be generalised to real-world cases to yield robust predictability of invasion dynamics. The gap between data-hungry complex models in theoretical ecology and the reality of mostly correlative statistics with low-quality data in invasion science seems insurmountable. Progress demands that we move beyond the graph theory that manipulates static network topology and the dynamical systems of complex dynamic networks, to explore statistical tools that can capture the dynamic behaviour of open and adaptive networks.

Invasibility and the Loss of Network Stability Before suggesting the statistical tool, we offer two propositions as a prologue and as a nexus to bind it with existing network theories. First, a relationship between network structure and function often emerges in open and adaptive networks, even as spandrels from network assembly. Second, invasions can cause functional changes in an invaded network, and the altered network function can be manifested by features of network topology in many, often alternative, ways due to the constraint imposed by its emerged structure–function relationship.

Network topology: a set of metrics describing local (for a particular node or link) and global (overall) network structure and architecture. Observed network topologies in ecology are mostly nonrandom and cannot be explained by simple null models. Principal component: Principal Component Analysis (PCA) is a statistical tool to reduce multicollinearity in multiple vectors by converting them into space with orthogonal uncorrelated axes. The PC1 is the major axis that retains the maximum variability. Mathematically, the major axis is the eigenvector of the lead eigenvalue, which it is argued here is the wind vane of invaded ecological networks. Stability criterion: a mathematical inequality, rooted in the circular laws that govern the distribution of eigenvalues, that sets the condition for a random network (also networks with special topology) to be stable with a high certainty.

First, and central to the enduring debate on Elton’s [52] diversity–invasibility hypothesis, lies May’s [53] stability criterion that outlines the boundary on the relationship between system structure (complexity) and function (stability) (Box 2). May’s original stability criterion has been widely challenged because it ignores the feasibility (positivity) of network equilibriums [54] and oversimplifies complex communities into linear systems [34,55]. Later formulations of the criterion have consequently reformed/updated its original form in recent years by taking on different interaction types [56] and nonlinear density-dependent interactions [57], including dispersal [18,19]. Although the expanded stability criterion accommodates diverse forms and the concept of stability boasts multiple layers [58], it is nonetheless instrumental to our understanding of how complex systems organise and function [59]. Like Gause’s law, which posits that interspecific competition needs to be weaker than intraspecific regulation to allow two competing species to coexist, May’s stability criterion and its variants provide for the stable coexistence of resident species in diverse ecological networks (Box 2). Like Gause’s law, it argues that for stable coexistence average interspecific interactions experienced by species in the network need to be weaker than the intraspecific regulation. Resident species in a network thus engage in a grand multiplayer game. To survive, species must have multiple contingency plans to release ecological or evolutionary selection pressures; for example, by weakening interactions through rapid trait evolution and adaptive rewiring [60,61] or by pushing the network toward the edge of marginal stability (i.e., putting inequality equal in the stability criterion) [62], thereby constraining the topology for persistent networks from specific assembly histories [63,64]. Consequently, we should anticipate an emerging relationship between network structure and function from community assembly and invasions. Second, invasion success and local extinctions are the strongest manifestations of network instability. Invasions can cause extinctions and change network topology. This often leads to increased connectance [35] because invasions manifest much more quickly than extinctions Trends in Ecology & Evolution, Month Year, Vol. xx, No. yy

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Figure 1. Dynamic Responses of an Ecological Network to Biological Invasions (A) The graph diagram of an ecological network of biotic interactions (50 species with interaction strengths randomly assigned from a normal distribution with zero mean and standard deviation s = 0.1, with diagonal elements
m =
1. The network is illustrated by high-dimensional embedding using Wolfram Mathematica, with unfilled black dots representing nodes and directed green lines pairwise interactions. (B) The adjacency matrix of the invaded network, with the orange lines dividing the matrix into four blocks (Box 1) representing respectively the original native–native adjacency matrix M (top left; dimension: 50 by 50), invasion impacts on resident species (matrix P, top right; dimension: 5 by 50), biotic resistance to invasions (matrix R, bottom left, with s = 0.01 for illustration; dimension: 50 by 5), and alien–alien interactions (matrix L, bottom right, with diagonal zeros for illustration; dimension: 5 by 5). The colours of matrix elements correspond to interaction strength; darker shading denotes stronger interactions. (C) Visualisation of all eigenvalues in the original network (50 green discs) and invaded network (55 red dots). Each eigenvalue comprises a real part Reðli Þ and an imaginary part Imðli Þ; the two parts of an eigenvalue allows us to locate it in the complex plane. Note, the real part of the lead eigenvalue for the original network (the rightmost green disc) is negative and becomes positive for the invaded network (the rightmost red dot), where the alien species defines the lead eigenvalue due to its low abundance. (D) The major axis (i.e., the eigenvector of the lead eigenvalue) of the joint matrix, revealing how each species dynamically responds to the invasion (species 1–50 are the original resident species; species 51–55 are the newly introduced species). Note, introduced species 51 became highly invasive; species 52 and 53 experienced invasion failures; species 54 and 55 are established yet not invasive. Some resident species responded positively, others negatively, and many resident species are insensitive to the invasion.

[44,65], and most invasions cause changes only in abundances [66]. When there are no new (alien) species that can establish and invade an ecological network (note, here we are not discussing invasions driven by massive propagule pressure), we can consider this scenario to be equivalent to the assumption that the network can repel any invasions with small initial propagule pressure. Invasions can succeed only if some alien species can grow in abundance from the initially small propagule pressure. In other words, invasions can succeed only if both the resident species and the newly introduced alien ones form an unstable mixed network 4

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Box 1. Ecological Networks as Dynamical Systems The dynamic behaviour of an ecological network can be depicted mathematically using dynamical systems of differential equations, _ ¼ F ðNÞ; N

[I]

_ the time where N = 〈N1, N2, . . . 〉 is the abundance vector of resident species and 〈〉 its transpose; N, derivative of N; F ðNÞ ¼ hF 1 ðNÞ; F 2 ðNÞ; . . . iT , is a set of smooth, often nonlinear population growth functions. The system can be approximated linearly at a given point N* as n_ ¼ Jn, where n  N
N* is the standardised vector of abundances and J its Jacobian matrix, J ¼ h@F i =@Nj jN¼N i. Note, N* does not have to be a particular equilibrium [F ðN Þ ¼ 0] for this linearisation. The element on the i-th row and j-th column, @F i =@Nj jN¼N , measures the sensitivity of species i’s population growth rate to the abundance change of species j. In practice, we could use the Jacobian matrix as the linearisation of the adjacency matrix of the network, J = M; that is, we define interaction strength aij ¼ @F i =@Nj jN¼N . However, for highly nonlinear networks, we also need the Hessian matrix, H = 〈@2Fi/@Ni@Nj〉 for adjusting the linearisation. In the following, we discuss only the linear stability of a network based on its Jacobian matrix (adjacency matrix). T

T

When an ecological network is invaded by a number of alien species, we can formulate this invaded network using a joint dynamical system: Z_ ¼ F 0 ðZÞ, where Z = 〈N, A〉T is the joint abundance vector of native species N and alien species A. After linearisation, we have Z: ¼ M0 Z, where the joint adjacency matrix M0 includes four submatrices (Figure 1B in main text):


 _ A _ ¼ M P N ; N [II] R L A where M represents the original native–native adjacency matrix with its element on the i-th row and j-th column representing the interaction strength of native species j on native species i; P, the alien–native matrix with its element representing the interaction strength of alien species j on native species i; R, the native–alien matrix with its element representing the interaction strength of native species j on alien species i; and L, the alien–alien matrix with its element representing the interaction strength of alien species j on alien species i. To assess which alien species can invade or be repelled, and which resident species can be favoured or suppressed as a result of such biological invasions, we need to know the dynamics of the invaded network at the point of invasion, Z* = 〈N*, A*〉T, where N* is the abundance of native species before invasions and A* is a near zero vector representing the small initial propagule sizes of introduced species. We could further expand the dynamical systems to considering potential trait evolution from biological invasions, Z_ ¼ F 0 ðZÞ, where Z = 〈N, A, x, y〉T is a joint vector with x and y trait vectors of natives and aliens. The trait _ is often formulated according to frequency- or density-dependent evolutionary game dynamics, for instance x, theories as being proportional to the selection gradient, defined as the partial derivative of mutant fitness with respect to changes of the trait in the framework of adaptive dynamics and covariance between fitness and traits in the Price equation [89]. This expansion, albeit inevitably complicating the system dynamics, could allow us to test many trait-dependent invasion hypotheses [51,74,85] and potential ecoevolutionary feedbacks in invaded networks [30,31], thus allowing us to explore synergies between invasion science and evolutionary ecology. The dynamical system can also be expanded to including human factors, Z = 〈N, A, h〉T, with h a vector of human decisions, perceptions, and management efforts [29], which allows us to explore potential synergies between invasion ecology and social sciences. We here leave threads of pursuit to future research and focus our attention here only on demographic dynamics.

(Figure 1C). The outcome of invasions is thus directly related to the loss of stability in invaded networks. Being unstable makes a network susceptible to invasion. With these two propositions we can tentatively make sense of the relationship between network topology and invasibility. In particular, May’s stability criterion highlights four factors that could disrupt network stability and initiate assemblage reshuffling: stronger interaction strength; higher species richness; more generalists; and weaker intraspecific density regulation. In particular, large propagule sizes of the invader can increase the standard deviation of interaction strength and network connectance, thereby increasing invasibility [67]. More generalist invaders and distinct invaders will increase interaction strength, thereby making the system more susceptible to invasion [51,68]. High levels of species saturation in a community (meaning that all or most niches are occupied and most resources are used) can increase intraspecific Trends in Ecology & Evolution, Month Year, Vol. xx, No. yy

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Box 2. Stability Criterion of Ecological Networks and Its Variants Ecological intuition tells us that complex networks are more stable than simple ones [52] but in mathematics this notion was challenged by the seminal work of May [53], leading to the enduring debate on ecosystem complexity and stability _ ¼ MN, where M is the [54,59]. May depicted the network dynamics using a linearised dynamical system (Box 1), N adjacency matrix aij. The interspecific interaction strength aij (i 6¼ j) was randomly assigned from a normal distribution with zero mean and standard deviation s and the diagonal element (intraspecific interaction) aii was set to
m representing self-regulation. In addition, interaction strengths were assigned zeros with probability 1
C, where C is the connectance. The complexity of this random dynamic network is thus reflected by three factors: species richness (S), the connectance (C), and the average interaction strength (s). The network follows a sharp transition from stable to unstable as the complexity increases, with the ecological network near its equilibrium almost certain of its stability if pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [I] s CðS
1Þ < m: This inequality is known as the stability criterion that ensures all eigenvalues of the adjacency matrix have negative real parts. The proof of this stability criterion thus relies on theorems governing the distribution of eigenvalues of a random matrix in the complex plane. May’s stability criterion has been fine-tuned to incorporate more realistic network topology such as diverse types and distributions of biotic interactions [56] as well as a meta-network setting [19]. To list a few selected works, Allesina and Tang [56] randomly drew interaction strengths from a distribution with mean E ðaÞ and standard deviation s (let E ðjajÞ=s ¼ r; note, r = 0 in May’s stability criterion) and discovered remarkable differences for different types of   interactions: the stability criterion of a predator–prey network becomes s ðSCÞ1=2 < m= 1
r2 , and becomes s ðS
1ÞC < m=r for a network comprising all mutualistic interactions. Gravel et al. [19] extended the stability criterion to metanetworks by adding a factor of dispersal between n local networks. When the dispersal rate is low (small d) in a large metanetwork (large S and n), the stability criterion becomes s ðCðS
1ÞÞ1=2 < m þ d, suggesting the stabilising force of weak dispersal in metanetworks [19]. Rohr et al. [54] challenged May’s stability criterion by differentiating feasibility from stability, while Stone [57] revisited this issue by exploring both the feasibility and the stability of nonlinear density-dependent ecological networks, with the criterion calculated from the ingenuous numerical technique conforming to May’s simple stability criterion even in such complex nonlinear networks. Ongoing work on adaptive interaction switching [60–62] further suggests that allowing species and individuals to adaptively switch their partners in an ecological network can further strengthen network stability by effectively reducing the average interaction strength from s to a lower level. Models of invaded networks that have incorporated adaptive behaviours, to reflect ecoevolutionary feedbacks, could lead to certain network structures that are more susceptible to invasions of alien species with particular traits [31,74]. However, this should not be confused with system-level evolution/selection as each species and individual behaves selfishly on its own terms, while some seemingly adaptive network topology could simply emerge as spandrels or byproducts of network assembly and function [63]. Different assembly histories can lead to alternative network invasibility [64,85], decoupling the network topology–stability relationship by pushing systems toward marginal stability [62]. Nevertheless, this is an active field of research in network ecology, with more realistic models that include different plausible network interaction structures being regularly examined [25,26], continuously unfolding the conceptual layers of network stability and complexity.

density regulation, leading to strong resistance to opportunistic invasions [69]. Disturbance and fluctuating resources will allow more species to capitalise on limiting resources and become established, thereby significantly increasing the network size [70,71]. These factors will enhance network invasibility, although caution is needed when generalising results from such simple systems [54,55,63,64]. We anticipate more realistic and contextual findings when using recently updated structure–function relationships for specific networks [31,72–74]. Moreover, by extending the linear system initially assumed when deriving May’s stability criterion to more realistic nonlinear systems, interaction strength can be unpacked to include both per capita interaction strength and encounter rate (related to population densities and interaction preference/efficiency), suggesting that both are crucial determinants of network function and stability [31,57]. Although per capita interaction strength has often been argued to reflect the level of trait matching between involved species [37], the variation explained is normally low [75]. Missing the component of encounter rate explains why trait comparison is but one piece of the puzzle of interaction strength and thus invasion performance in networks. 6

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A Wind Vane of Dynamic Response in Invaded Networks Until now we have argued that relationships between network topology and invasion performance could arise from constraints on network function and assembly (e.g., the stability criterion and its many variates) and that the consequence of invasion in an ecological network is related to its instability when the standing resident species are confronting initially rare alien incursions. However, it remains unclear how each species responds dynamically to invasions. We need to go further by asking which introduced species can establish and invade, and which resident species will benefit and which will be suppressed by the invasion. In other words, we need a practical estimator of species-level invasion performance and response. We argue that such a wind vane of how each species behaves in an invaded ecological network exists and can be statistically estimated as the major axis [i.e., the first principal component (PC1)] from a principal component analysis (PCA) of the adjacency matrix (Box 3). As the invaded network is unstable at the point of successful invasion, the major axis PC1 of the adjacency matrix indicates how the dynamical system will instantaneously deviate from the current standing point when facing invasions, thereby revealing the dynamic trajectory of each species (native or alien) involved in the network (Figure 1D). Note that when the invaded network functions as a linear system, the major axis is equivalent to the eigenvector vm of the lead eigenvalue lm of the Jacobian matrix and thus could indicate both the short- and long-term dynamics. However, when the invaded network is nonlinear, the major axis could indicate only short-term dynamics; forecasting medium- to long-term dynamics in nonlinear systems also requires information on at least the Hessian matrix. For short-term invasion prediction and risk assessment, the major

Box 3. Major Axis as the Wind Vane of Network Instability _ ¼ F ðNÞ and its linearisation N _ ¼ JN. For simplicity let us consider the dynamical system of an ecological network, N Mathematically, the matrix J is an operator that transforms the abundance vector N into a new vector space. To understand this, we now bring in the eigenvalues and their eigenvectors of matrix J. By definition, we have JV = VL, where V = 〈v1, v2, . . . 〉 is the eigenvector matrix with vi the vertical eigenvector of eigenvalue li satisfying Jvi = vili and L the eigenvalue matrix with the diagonal elements li and the rest zero. Note, an eigenvalue li can be a complex number comprising a real part Reðli Þ and an imaginary part Imðli Þ. We can now transform the abundance vector space into a new one using the eigenvector matrix: let n = Ve, where n is standardised N (Box 1) and e the new vector space. This transformation is necessary as the dynamics of each species is not independent from the others (the issue of multicollinearity). After this transformation, each transformed node behaves independently from others. We have e_ ¼ Le and thus ei = cielit, where ci ¼ ei ð0Þ is a constant and represents the initial value of a perturbation to the system. Consequently, we have the dynamics of species i: X v c elj t ; [I] ni ¼ j ji j where vji is the i-th element of the eigenvector vj. With the elapse of time t, we can see that ni ! vmicmelmt, where lm is the lead eigenvalue (the one with the largest real part). It is evident that only if the real part of the lead eigenvalue is negative, Reðlm Þ < 0 (see Figure 1C in main text), do we have an asymptotically stable network. As stated in the main text, invasion performance and invasibility are related to the loss of network stability, more precisely network instability. When an abundance vector N* is unstable [Reðlm Þ > 0], the abundance of species i will move away from this vector. That is, at an unstable point (either representing an unstable equilibrium or at a transient state), the response of species i, both its direction and its magnitude, is solely determined by vmi, noting that cmelmt is the same for all species. Therefore, each species’ response to the invasion is revealed by the major axis (the eigenvector of the lead eigenvalue, vm) of the adjacency matrix calculated for the standing abundances of resident species and the initial abundances of alien species vector. Ecologically speaking, eigenvector vm is the PC1 from a PCA of the adjacency matrix J and its i-th element vmi is the coefficient of PC1 for species i, representing the projection of the principal component on species i’s vector. The PC1 can be readily calculated once the adjacency matrix is available; for instance, by running the default R function prcomp for J. If vmi > 0, species i will increase; if vmi < 0, species i will decline; if vmi  0, species i is insensitive to the perturbation (see Figure 1D in main text). Therefore, the PC1 (i.e., the major axis) of the adjacency matrix for the invaded network will simply reveal how each species, native or alien, dynamically responds to the biological invasion.

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axis can serve as a wind vane of the invaded network. Research is needed to elucidate the factors behind the lead eigenvalue lm and its eigenvector vm of invaded networks. Several clues point to rare species, especially those that lack self-regulation, determining the lead eigenvalue lm. Suweis et al. [61] showed that lm is proportional to the product of the level of negative density dependence and the abundance of the rarest species in the network (also see [57]). As introduced species are often initially rare (and lack density regulation), biological invasions can often easily flip the invaded system from stable to unstable (Figure 1C). This means that rare species that lack density dependence hold the key to network stability (lm) and thus invasibility. Understanding how a community regulates rare species will explain how introduced species will fare in such a network. Although much research has focussed on the estimation of lm and its crucial role in system resilience [56,76–78], we must stress that the dynamics of a particular species (native or alien) is determined by the major axis (vm), and not by the lead eigenvalue lm. The lead eigenvalue, lm, simply sets the gear of changes in abundance (elmt) while the corresponding element of a species in the major axis (vmi) controls both the direction and the speed of changes in abundance (Box 3), DNi  vmielmt. Therefore, once the adjacency matrix is quantified, we can run a PCA and identify the PC1 as the wind vane for the invaded ecological network. To date, no attention has been paid to the major axis of the network adjacency matrix, due largely to the overwhelming focus on the lead eigenvalue and its role on network stability. This highlights one immediate research priority: what is and what determines the major axis (vm) of the network adjacency matrix? Here, we provide only a few tentative clues. The eigenvector of the lead eigenvalue has two layers of meaning. First, it indicates the direction of the steepest decline in the effective potential of a dynamical system [78], analogous to the path of water flow over uneven terrain. Second, in terms of signal processing, the network adjacency matrix can be considered as the cross-covariance matrix that converts the signal of relative abundances into the signal of population change rates; the eigenvector vm is simply a signal that will not be distorted from the converter but only be amplified by a factor of lm (Box 3). Along this principal component, the covariance between the abundance and the population change rate of a species becomes the greatest. We could potentially estimate the adjacency matrix of interaction strength as the cross-covariance matrix of species abundances and species change rates, with the element on the i-th row and j-th column being the covariance between species i’s   change rate and species j’s abundance, cov r i ; nj . The diagonal element of the matrix, covðr i ; ni Þ, is essentially the growth–density covariance that has been shown to greatly affect species persistence [79,80]. With the proposal of using the major axis as the wind vane for invaded networks, we can highlight several research priorities. First, biological invasions are driven by the demographic dynamics of abundance and population growth [81]. This systematically breaks away from invasion assessment based solely on trait matching [7,37]. We are not suggesting that traits are irrelevant, but call for further work to assess the demographic effects of trait-based metrics on density and population growth and to explore the meaning of interaction strength beyond simple trait matching [8,38,74]. A standard well-supported metric of interaction strength is clearly needed. Second, considering the role of abundance (and its change) in both vm and lm, we need to revisit Rabinowitz’s [82] classification of species rarity in three dimensions: geographical range, local density, and habitat specificity. Hanski [83] classified resident species based on the first two dimensions and outlined the core–satellite hypothesis that core species (those with a large range and high local density) are expected to interact with other species strongly and also likely to persist under constant perturbation (including from invasions), 8

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whereas satellite species (those with a small range and low local density) are expected to interact weakly and behave dynamically unstable. This fits with our argument that rare species hold the key to network instability (and invasibility). To this end, we highlight the need to explore abundance gradients in ecological networks and their covariates. In particular, we need to clarify the relationships between demography, traits, and abundance in local networks and the role of environmental heterogeneity in these relationships in metanetworks. Moreover, because interaction specificity can greatly interfere with the cascade effect of species removal and invasion in ecological networks [84], we also need to consider the third dimension of rarity – interaction specificity (and the related interaction promiscuity) – when exploring these potential covariates with and effects on the abundance gradient. Finally, although we should aim to build the adjacency matrix for the full species assemblage of the recipient ecosystem, the option of a smaller matrix depicting a focal species-centric network (FSCN) is more feasible given the scarcity of data [85]. In the absence of a quantitative adjacency matrix, we could even opt for a qualitative matrix or even a binary interaction matrix. Once habitat specific food/host plants and natural enemies have been identified [85], for instance, the more feasible option of FSCN has been shown to be capable of providing habitat-specific assessment of the threat posed to resident species of the aphidophagous guild from the invasive harlequin ladybird, Harmonia axyridis [40,86].

Concluding Remarks Species are not isolated but engage with other species in multiplayer games involving numerous mutualistic and antagonistic interactions. The interaction strength of these biotic interactions can arguably be captured by the growth–density covariance of involved species. The performance of each species, native or alien, in such multiplayer games is measured by its demographic features; these are obviously related to, but clearly go beyond, simply functional traits. Through coevolution and ecological fitting [20], ecological networks emerge from such multiplayer games, with feasible network topologies constrained by assembly and invasion histories [63,64]; rare species, especially those experiencing weak self-regulation, hold the key to network invasibility. This underscores the importance of examining abundance gradients and their covariates in assessing network invasibility [87]. Importantly, the major axis of the network adjacency matrix can be used as the wind vane to reveal how each species, native or alien, responds dynamically to invasions. Consequently, the major axis can not only forecast invasion success and failure but also identify resident species facilitated and suppressed by such invasions. Two avenues of research are needed to build on this approach (see Outstanding Questions). First, we need to standardise the concept and metrics of interaction strength for different types of interactions, both within and between guilds, for all categories of species interactions, from mutualism to antagonism. The measurement uncertainty and spatiotemporal variability of interaction strength, its covariates, and the effects of scaling and sampling, need to be explored. Second, we need to develop practical methods for mapping out network adjacency matrices and exploring the minimum size for practical use of a FSCN. To this end, we call on ecologists to collectively compile their representative local network profiles and to add the adjacency matrix as part of the essential biodiversity variables for monitoring biological invasions [88]. With such advances we could move beyond the linear filter scheme to address the state of paradox in invasion ecology by forging coalitions with network and community ecology.

Outstanding Questions What are the potential strong covariates of the major axis of invaded networks so that we can still inform species-specific response and performance even without the adjacency matrix? What is the minimum size for the FSCN to allow reliable estimates of the major axis and invasion performance? How do the two concepts of interaction strength – the demographic sensitivity (measured as covariance) of population growth rate to population density versus the current standard trait-mediated metrics in function ecology – relate to each other? What correlates with the abundance gradient (or the level of rarity) in a network (e.g., demographic rates and traits)? What does it mean to network invasibility when the correlation is strong (or weak)? Can invasion performance and invasibility be explicitly formulated in stability criteria for open and adaptive ecological networks with more realistic topology?

Acknowledgments We are grateful to Caeul Lim, Jonathan Chase, Ulf Dieckmann, Åke Brännström, Jessica Gurevitch, Gordon Fox, Melodie McGeoch, Mirijam Gaertner, Oonsie Biggs, Ony Minoarivelo, Pietro Landi, Wolf Saul, Guillaume Latombe, and two anonymous reviewers for constructive discussions on the ideas expressed in this Opinion article and to the DST-NRF

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Centre of Excellence for Invasion Biology and the National Research Foundation of South Africa for funding (grants 89967 and 109683 to C.H., 85417 to D.M.R.).

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