Title: Network structure, predator-prey modules, and stability in large food webs

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1 Editorial Manager(tm) for Theoretical Ecology Manuscript Draft Manuscript Number: THEE-0R1 Title: Network structure, predator-prey modules, and stability in large food webs Article Type: Original Research Keywords: Predator-prey; complexity/stability; food webs; sign-stability; weak interactions Corresponding Author: Professor Stefano Allesina, Corresponding Author's Institution: University of Michigan First Author: Stefano Allesina Order of Authors: Stefano Allesina; Mercedes Pascual Abstract: Large, complex networks of ecological interactions with random structure tend invariably to instability. This mathematical relationship between complexity and local stability ignited a debate that has populated ecological literature for more than three decades. Here we show that, when species interact as predators and prey, systems as complex as the ones observed in nature can still be stable. Moreover, stability is highly robust to perturbations of interaction strength, and is largely a property of structure driven by predator-prey loops with the stability of these small modules cascading into that of the whole network. These results apply to empirical food webs and models that mimic the structure of natural systems as well. These findings are also robust to the inclusion of other types of ecological links, such as mutualism and interference competition, as long as consumer-resource interactions predominate. These considerations underscore the influence of food web structure on ecological dynamics and challenge the current view of interaction strength and long cycles as main drivers of stability in natural communities.

2 Manuscript Click here to download Manuscript: Theor_Ecol_Allesina_Pascual.tex Noname manuscript No. (will be inserted by the editor) Network structure, predator-prey modules, and stability in large food webs Stefano Allesina Mercedes Pascual the date of receipt and acceptance should be inserted later Abstract Large, complex networks of ecological interactions with random structure tend invariably to instability. This mathematical relationship between complexity and local stability ignited a debate that has populated ecological literature for more than three decades. Here we show that, when species interact as predators and prey, systems as complex as the ones observed in nature can still be stable. Moreover, stability is highly robust to perturbations of interaction strength, and is largely a property of structure driven by predator-prey loops with the stability of these small modules cascading into that of the whole network. These results apply to empirical food webs and models that mimic the structure of natural systems as well. These findings are also robust to the inclusion of other types of ecological links, such as mutualism and interference competition, as long as Department of Ecology and Evolutionary Biology, University of Michigan, Natural Science Building, 0 North University, Ann Arbor, MI - USA. stealle@umich.edu consumer-resource interactions predominate. These considerations underscore the influence of food web structure on ecological dynamics and challenge the current view of interaction strength and long cycles as main drivers of stability in natural communities. Keywords Predator-prey complexity/stability food webs sign-stability weak interactions 1 Introduction Large number of species interact and coexist in the intricate networks of consumer-resource interactions known as food webs (Pimm 1; Dunne 00; Montoya et al 00). Until May s mathematical argument suggested the opposite (May 1, 1, 00), ecologists held the view that more complex ecosystems, those with a larger number of species and connections, were also more likely to be stable (Hutchinson 1). May s result initiated the complexity-stability

3 debate and stimulated extensive theoretical and empirical research to address the discrepancy between mathematical insight and empirical ecological knowledge. Two general themes were the addition of biological realism to the original random structure of species interactions in the food web (Daniels and MacKay 1; Deangelis 1; Lawlor 1; Yodzis 11; Pimm 1; Haydon 1) and the consideration of other definitions of stability, such as persistence and variability, that are not limited to the proximity of equilibria (Ives 1; McCann et al 1; McCann 000). Complexitystability questions are still pervasive today in efforts to understand the relationship between structure and dynamics in large ecological networks of interacting species (McCann 000; Pascual and Dunne 00). May (May 1, 1) considered a community of S species with connectance C measuring the number of realized links over the number of possible links. Complexity in the form of a more diverse or connected community would promote instability. Here we show that when the interaction between species is constrained to consumer-resource relationships, large and very interconnected communities exhibit a high probability of stability compared to the random case. We demonstrate that this result is based largely on the signs of interactions rather than on their strengths. Moreover, we show how stability of the whole network can be constructed in a bottom-up way by assembling small stable modules, such as stable predatorprey ones. Materials and methods May s original argument is based on the concept of a community matrix: let the dynamics of species densities in an ecosystem be governed by the set of equations dx i dt = f i (X 1,...,X S ) (1) where X i is the abundance of species i and S is the total number of species. The community matrix A[a i j ] is obtained by linearizing the system around a nontrivial equilibrium solution X (with X i > 0 i): a i j = f i X j where X X () specifies that the partial derivative is evaluated at the feasible equilibrium point. If the matrix A is stable, that is to say if its eigenvalues λ i all have negative real parts (Re(λ i )<0 i), then the equilibrium X is Lyapunov asymptotically stable. In May s food webs, the diagonal coefficients of the matrix were set to 1 (self-regulation) imposing a reference time for self-damping, while off-diagonal coefficients were given a value of zero with probability 1 C. Non-zero coefficients were chosen from a normal distribution (with mean 0 and variance α). We repeated May s analysis (May 1) with a minor modification: diagonal elements were assigned negative values and chosen from the same distribution than that of the other coefficients so that stability has no dependence on α.

4 We then repeated the same analysis by constraining the random matrices to include only predator-prey (or parasitehost) interactions, thus, in our model the strengths of the interactions are random, but their sign is constrained. Specifically, if the coefficient a i j, indicating the presence of a link between species i and j, is positive, then a ji is negative (when i j). For the results presented below, all the coefficients (including self-regulation) were drawn from a normal distribution (µ = 0,σ = 1). We sampled parameter space by varying the off-diagonal connectance C from 0.0 to 0. with a step of 0.0, and the size of the system S, from to 0 with a step of. For each combination of C and S, we generated 00 matrices. We then studied stability of the generated matrices as a function of the size of the system S and the off-diagonal connectance C by considering the eigenvalues of the community matrix. For each one of the matrices, we also measured the percentage of eigenvalues with negative real part. If a matrix was stable we saved it and built 0 matrices that had the same sign pattern, but chose the coefficients magnitude at random. Measuring the percentage of stable systems after this further randomization allowed us to determine whether stability was due to a particular combination of coefficients or was instead a property of the structure of the network itself. Finally, we repeated the computation using, instead of random structures, the community matrices derived from empirical food webs. Food webs are usually published as adjacency matrices (containing therefore just values of 0 and 1) where a positive coefficient m i j means that i is a prey of j. In order to convert these matrices to community matrices of the type described in this work, we imposed that if m i j > 0 then m ji < 0 for each i j, creating therefore an antisymmetric signed matrix. We also imposed self-limitation (m ii < 0 i). In this process, some links were lost, but their number was small. In particular: a) cannibalism was removed and b) mutual predation was lost (a b, in which case we randomly choose a predator). This procedure resulted in a signed matrix for each published food web. Coefficient strengths were then assigned by extracting values from a standard normal distribution, taking the absolute value and multiplying them for the corresponding sign (i.e. positive if from prey to predator and negative otherwise). We describe in the Electronic Supplementary Material (S1) the details of the procedure. The results presented here are therefore based on random matrices, in order to allow comparison with May s original results. More recently, a series of models for food web structure have been proposed that capture many of the properties of empirical networks. We therefore analyzed, for each combination of C and S, 0 food webs created with the niche (Williams and Martinez 000) model, and its predecessor, the cascade (Cohen et al 10) model. The results of these experiments are reported in the ESM (S), along

5 with the ones obtained by relaxing other assumptions, such as the presence of self-regulation, the normality of interaction strengths, and the lack of other kinds of ecological interactions such as competition and mutualism. Results Figure 1 summarizes the results of the simulations described above to compare the stability of random and predator-prey community matrices: the top left graph (a) shows the percentage of stable systems obtained using the original formulation, while the top right one (b) has been obtained using only predator-prey interactions. The area where stability is highly probable grows dramatically when we constrain the relation between species to be predator-prey (sign +/ ). This result is also confirmed by Figure 1c and 1d where we show the percentage of eigenvalues with negative real part. When just predator-prey interactions are allowed, we are far from 0%, which is what we would expect for a totally random matrix (i.e. with random coefficients on the diagonal as well). In the case of predator-prey matrices, each stable matrix is extremely robust to changes in the coefficients, as shown by a minimum percentage of stable matrices above 0% when we randomize interaction strengths but preserve the sign structure of the original matrix (Figure 1e, 1f). We call this property Quasi Sign-Stability (QSS) recalling that a system is sign-stable if its stability follows uniquely from the sign of the matrix coefficients and not from their actual magnitude (May 1; Jeffries 1; Logofet and Ulianov 1, S). What are the implications of these findings for empirical food webs? In Figure 1, we superimpose the values of S and C for published food webs with less than 0 species (blue dots). Although most webs fall in regions of parameter space where stability is still improbable, the predatorprey constraint makes stability much more likely than in the random case. To confirm these results, we simulated different combinations of coefficient strengths as described above. Figure 1 (compare c to d) shows that for many empirical structures the average fraction of networks whose eigenvalues have negative real parts is much higher than for their random counterparts. For 000 randomizations of these community matrices, Figure shows the number of eigenvalues with negative real part. There is still an effect of SC but in out of food webs complete stability is achievable with random (signed-constrained) coefficients: the Chesapeake Bay system exhibits the higher likelihood of stability (0% of cases); the others six systems have lower probabilities with 0.% for Benguela and.% for Scotch Broom. However, for the three food webs for which we found no stable combinations, the maximum percentage of eigenvalues with negative real part was greater than 0%. These results indicate that predator-prey relationships increase the probability of a system being stable, and that a stable predator-prey system is likely to remain stable under ran-

6 domization of interaction strengths (QSS). Similar results hold for predator-prey networks whose structures are not random but closer to those of real systems, as the ones generated by the niche and cascade model (S). Moreover, the results are robust to the inclusion of other types of ecological interactions, such as mutualistic and interference competition (corresponding respectively to +/+ and / signs in the community matrix), provided predator-prey links predominate, and to the inclusion of positive and null coefficients in the diagonal; finally, the choice of the distribution for interaction strengths does not qualitatively alter these patterns (S). To explain this stabilizing effect of predator-prey interactions, we studied the sign patterns of the matrices eigenvalues. These quantities are obtained as the roots of a (characteristic) polynomial of order equal to the number of species in the system. Descartes rule of signs states that a polynomial, in order to have no positive roots, must have all coefficients of the same sign. If the characteristic polynomial of the community matrix has no positive roots the system is stable. How do the consumer-resource cycles influence the signs of the characteristic polynomial? Levins (Levins 1; Puccia and Levins 1) pointed out that each coefficient α i of the characteristic polynomial can be written as the sum of the determinants of all possible i i sub-matrices of the systems (S). This establishes a powerful relationship between the terms concurring to form each determinant and the struc- tures in the graph associated with the community matrix, as illustrated in Figure for a simple case. Each term in the determinant expansion is either a cycle in the graph or a combination of cycles. Therefore, we expect predator-prey loops to influence all the coefficients of the characteristic polynomial of degree S. In a sign-stable system (May 1; Jeffries 1; Logofet and Ulianov 1), all the terms of the expansion of determinants involving i variables would have the same sign (positive for even i and negative for odd i) so that Descartes condition is fulfilled regardless of the actual value of the coefficients in the community matrix. It is therefore important to ask how many terms in the determinant expansion will have the right sign (that is, negative sign for odd i and positive sign for even i) given the structure of the food web. For the simple example of Figure the determinant of the community matrix (that is the last coefficient of the characteristic polynomial) will possess terms with the right sign (1 combination of three self-loops, combinations of a predator-prey loop and a self-loop and 1 cycle involving three species) out of (the only term with the wrong sign is the cycle of size ). Therefore, we expect the last coefficient of the characteristic polynomial, that is simply the sum of these six terms, to be of the right sign with high probability (S). The other terms of the characteristic polynomial are constructed in a similar way, but we immediately see that each subgraph containing just one species would have a determinant of the right sign (negative, because of self regula-

7 tion): this ensures that the second coefficient in the polynomial has the right sign. Also each possible two-species system would yield the right sign (because each determinant is comprised of predator-prey loops and combinations of self regulation loops, yielding both the right sign). Thus, for the system in Figure the stability is completely determined by the determinant of the community matrix and we expect the system to be stable with high probability. Counting the number of terms constituting the coefficients of the polynomial for a larger network quickly becomes prohibitive, even numerically, but a combination of linear algebra and considerations on the roots of polynomials makes this computation possible, if not simple, for middle sized ecological networks (S). The k th coefficient of the characteristic polynomial will be formed by a maximum of ( S) k k! = S! terms, where S is (S k)! the number of species. For example, the maximum number of terms composing the 1 th coefficient of the characteristic polynomial of a network containing 1 species is approximately Clearly, listing all the terms composing each coefficient is not an option. A valid alternative, given that one needs only the signs of the terms forming each coefficient, is to count the total number of terms using the permanent of the minors of the matrix of signs (Dambacher et al 00, S) and the difference between positive and negative terms using determinants. This is the approach we took to obtain the results reported below. Constraining the interactions to be of the predator-prey kind has three main consequences on the right sign pattern: a) the first three coefficients of the characteristic polynomial will be composed only of terms with the right sign; b) for any coefficient, the number of terms with the right sign will always be greater than those with the wrong sign; c) for the same connectance, the number of terms in the expansion of the various determinants will be higher for predatorprey than for random systems. These hypotheses are suggested by theoretical considerations (S) and are confirmed by numerical simulations. Figure (a-b) shows the percentage of terms with the right sign in the sums constituting each of the coefficient α i of the characteristic polynomial of ten 1 1 systems with different connectance levels. Panels (c-d) show the difference between the number of terms with the right and the wrong sign for these same matrices. The left panels correspond to random matrices and the right ones to predator-prey ones. We see that, for random matrices, the percentage of terms with the right sign is in the minority, at least for one coefficient in all cases but two (C = 0.1 and 0.) (a), so that Descartes rule of signs is not likely to be satisfied. By contrast, for the predator-prey matrices (b) the first three coefficients are formed exclusively by right sign terms and the percentage of terms with the right sign is always greater than 0%, meaning that, on average, each coefficient in the characteristic polynomial will have the right sign. This is confirmed by the difference between the num-

8 ber of terms with the right and those with the wrong sign (c and d). This difference is responsible for the mean values of the coefficients of the characteristic polynomial (S). These results on the right sign pattern of these systems are further confirmed by the observation that all predatorprey systems examined in Figure were stable in more than 0% of the simulations, while random matrix systems were never stable for connectance greater than 0.1. Discussion We have shown that predator-prey networks are more likely to be stable than random ones, and that this pattern is conserved regardless of size and density of connections. Thus, the topology (structure) of food webs can play a much more fundamental role than previously considered. This is confirmed by the test we have performed changing the magnitude, but not the signs, of stable community matrices: the resulting matrices are likely to be stable as well with high probability (QSS). Results with more realistic non-random graphs, for empirical food webs and those generated with models of food web structure, the niche and cascade models, confirm these findings. One may question, as it has been done before, the relevance of considering local stability and a relatively simple biological description of food webs, given the rich literature that has incorporated increasing biological realism and other definitions of stability in the complexity-stability debate. Local stability has for example been replaced with more general definitions such as persistence, resilience and reactivity (McCann et al 1; Brose et al 00; Rooney et al 00). Also, the random graphs at the base of May s stability criterion have been substituted by non random networks, such as the ones sampled in nature (Yodzis 11; Emmerson and Raffaelli 00) or by random networks generated by stochastic processes that can reproduce aspects of the structure of empirical food webs (Martinez et al 00; Brose et al 00). We note, however, that results for the community matrix provide the point of reference to consider not only other types of dynamics but also other biological assumptions. Do we need to invoke further biological mechanisms to explain (local) stability? How relevant are equilibria to the dynamics of systems as their complexity increases (Fussmann and Heber 00)? In the same way that May s results provided the null model to gauge additions to the theory, the results presented here should be considered when developing arguments about food webs composed predominantly by consumer-resource interactions. Note that consumer-resources interactions encompass also the behavior of parasites and their hosts. In light of recent findings on the prevalence of parasitism in food webs (Lafferty et al 00), the contribution of parasites could be critical to the stability of food webs. At the same time, addressing the complex life cycles and different ways in which parasites interact in food webs with numerical simulations could be extremely challenging,

9 with models difficult to parametrize. Our approach offers a feasible and simpler way to begin to examine the role of parasitism. More generally, studies addressing nonlinear dynamics are typically limited by the computational feasibility of exploring a huge parameter space, with the usual trade off between the size of the network and the sampling of parameter space (Martinez et al 00; Brose et al 00). The analysis presented here, on the other hand, is based on signs and can be easily performed on systems with realistic number of species. It provides an alternative and first order approximation for extremely large systems, as complex as the ones we observe in nature. Many studies have shown how realistic configurations have higher probability of stability than their random counterpart (Deangelis 1; Lawlor 1; Yodzis 11; Neutel et al 00; Emmerson and Raffaelli 00). Here we have shown that one ubiquitous type of ecological interaction (i.e. consumer resource) alone can stabilize large networks whose structure is completely random. It is then natural that this effect is simply conserved when realistic structures, such as the ones provided by the cascade and niche models, are considered or when we use empirical food webs as a reference. Our results contrast with the recent emphasis on the role of weak interactions as a major stabilizing mechanism in food webs (McCann et al 1; Willmer et al 00; Emmerson and Raffaelli 00; Berlow et al 00; Bascompte et al 00). Weak interactions have been proposed as one way ecosystems achieve species persistence. The original theoretical results concern the variability of population abundances in the context of the full nonlinear dynamics of food webs, and for this reason, were obtained with low-dimensional systems of three or four species (McCann et al 1). When larger systems were considered, stability was evaluated, as we have done here, close to equilibria (Willmer et al 00; Jansen and Kokkoris 00). In particular, we have shown how a stable network tends to remain stable after randomization of its coefficients. It is therefore not necessary to invoke weak interactions to obtain large, (locally) stable food webs. This argument is confirmed by simulations we performed varying the percentage of weak and strong interactions (S). The simulations show no effect of weak interactions on the probability of stability, as expected. Note that this does not mean that weak interaction play no role in food web stability, but rather that for every possible distribution we obtain the same pattern (predator-prey networks are more stable). The actual percentage of stable systems is instead distribution dependent. Weak links in long cycles have also been proposed as a stabilizing mechanism (Neutel et al 00). Here, we have shown that shorter cycles (predator-prey and self-regulation) are more likely to contribute to overall system stability than longer ones. Because short cycles impact all coefficients of the characteristic polynomial, creating hundreds of billions

10 of terms in large networks, while long cycles contribute just to the final, low degree coefficients of the characteristic polynomial, we expect short cycles to have a major impact on eigenvalues compared to long cycles. Moreover, because long cycles are rare compared to the combinations of shorter cycles, the influence of short cycles should prevail even in the last few terms of the characteristic polynomial. This is clearly demonstrated in Figure (and in the more analytical argument - S). To further show that the results are not determined by weak long cycles, we computed the loop weight (Neutel et al 00) of random and predator-prey matrices. According to the weak cycles hypothesis, loop weight would be inversely correlated with stability. This is not what happens: predator-prey systems have heavier cycles but are nevertheless much more stable than random systems containing lighter cycles (Figure, for details on the simulation see S). Community matrices dominated by consumer-resource interactions would possess a stability backbone that ensures the presence of many eigenvalues with negative real parts and therefore, the return to equilibrium after small perturbations. Other mechanisms could then fine tune the strength and distribution of coefficients to achieve stability. Our results emphasize the importance of sign patterns in food webs, a topic that inspired earlier techniques such as Loop Analysis, including recent applications and extensions in ecology (Levins 1; Puccia and Levins 1; Bodini 000; Dambacher et al 00, 00). These linear ap- proaches focused on equilibria should be seen as complementary to the study of the nonlinear dynamics of these large systems. For example, Martinez et al. (Martinez et al 00) have recently mapped a nonlinear set of equations for the dynamics of consumers and their resources onto the static structure of links produced by the niche model. Numerical simulations of these large systems and comparisons to their random counterparts, have shown that a higher number of species are able to persist for these more realistic structures. However, the exploration of the huge parameter space of these systems is still computationally prohibitive. Here, we have shown that their equilibria, by virtue of the predominance of predator-prey links, would also be more stable. Ultimately, it is important to address whether and how these results, based on static networks, matter to the current, more dynamic view, of ecosystems in which robustness at large scales is effectively maintained by instability at lower scales (Tilman 1; May 1; Sole et al 00). Stochastic assembly models (e.g Sole et al 00) provide a natural framework to address these issues. Finally, our results highlight the role of small modules in building the stability of large networks, making the search for such structures in empirical data a relevant pursuit despite the computational difficulties (Milo et al 00). It has recently been observed that some small motifs have significantly higher frequency than expected at random in many biological networks (Milo et al 00; Prill et al 00). Motifs

11 have been also studied in food webs theory (Arim and Marquet 00; Bascompte and Melian 00). A proposed explanation is that motifs that have higher probability of stability (if taken in isolation) are also more heavily represented than those with less stability (Milo et al 00; Prill et al 00). We have shown here that assembling stable building blocks, specifically those composed by two nodes (predator-prey) so that each sub-network containing one or two species is stable, results in systems with a high probability of stability. As we show in Figure, the presence of stable -nodes blocks makes the first three coefficients of the characteristic polynomial of the right sign. This consideration naturally extends to higher-order building blocks: if we build a network using blocks of (or ) nodes so that any subnetwork comprising up to (or ) nodes is stable, this would make the first (or ) coefficients of the characteristic polynomial of the right sign. Because the stable building blocks are present in combinations in all the other coefficients of the characteristic polynomial, we expect that networks built using larger stable modules would possess higher likelihood of stability than those built from smaller such modules. In other words, the stability of the modules contributes in a bottomup (from small to large) direction to that of the whole system. If stable networks are also more persistent and therefore more likely to be observed in nature, then small stable modules should be more frequent than expected at random in biological networks, an observation that is confirmed by the study of motifs (Prill et al 00). Acknowledgements We thank D. Alonso, A. Bodini, J.M. Dambacher and A.P. Dobson for stimulating discussions. This work was supported by a Centennial Fellowship by the J.S. McDonnell Foundation to M.P. References Arim M, Marquet PA (00) Intraguild predation: a widespread interaction related to species biology. Ecol Lett : Bascompte J, Melian CJ (00) Simple trophic modules for complex food webs. Ecology : Bascompte J, Melian CJ, Sala E (00) Interaction strength combinations and the overfishing of a marine food web. Proc Natl Acad Sci USA (1): Berlow EL, Neutel AM, Cohen JE, de Ruiter PC, Ebenman B, Emmerson M, Fox JW, Jansen VAA, Jones JI, Kokkoris GD, Logofet DO, McKane AJ, Montoya JM, Petchey O (00) Interaction strengths in food webs: issues and opportunities. J Anim Ecol (): Bodini A (000) Reconstructing trophic interactions as a tool for understanding and managing ecosystems: application to a shallow eutrophic lake. Can J Fish Aquat Sci :1 00 Brose U, Williams R, Martinez N (00) Allometric scaling enhances stability in complex food webs. Ecol Lett :1 1 Cohen J, Briand F, Newman C (10) Community food webs: data and theory. Springer-Verlag, Berlin, Germany Dambacher JM, Li HW, Rossignol PA (00) Relevance of community structure in assessing indeterminacy of ecological predictions. Ecology ():1 1 Dambacher JM, Luh HK, Li HW, Rossignol PA (00) Qualitative stability and ambiguity in model ecosystems. Am Nat ():

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14 Fig. 1 Comparison between random matrices (left) and random predator-prey matrices (right). (a-b) Percentage of stable systems for each pair (C, S). (c-d) Average percentage of eigenvalues with negative real part. (e-f) Percentage of stable matrices produced by randomizing 0 times the coefficients of each stable matrix in (a-b). The number of matrices concurring to form this average is not fixed, as in the previous graphs, depending on the number of stable matrices (a-b). The dots represent the values of empirical food webs.

15 Fig. Percentage of eigenvalues with negative real part out of 000 simulations for published food web, modified as explained in Methods. The shades of gray stand for different values of the product SC (from light to dark, for small to large values).

16 Fig. Expansion of the determinant of a matrix in terms of directed cycles and combinations of directed cycles of the associated graph (positive coefficients are represented by arrows and negative coefficients by arrows terminating in a circle). A total of terms out of will yield the right sign regardless of the values of the coefficients. The term with the wrong sign is the one inside the box. This means that the last coefficient of the characteristic polynomial will be negative (right sign) with high probability.

17 Fig. Number of right-signed terms in the characteristic polynomial of 1 1 matrices with different levels of C (from 0.1 to 0. with a step of different shades of gray). (a-b) Percentage of terms with the right sign for each coefficient of the characteristic polynomial. (c-d) Sign difference: number of terms with the right sign - number of terms with the wrong sign.

18 Fig. Effects of long cycles weight on local stability. For each network size (left), we built 00 predator-prey (red) or random (blue) networks with connectance 0.1. We then measured the average Loop Weight (Neutel et al 00) for the networks. This measure is related to the weight of long cycles in the network, and is considered a measure of network instability. Predator-prey systems have heavier cycles, a consideration that is totally sensible given that each closed structure in the predator prey case gives rise to two cycle vs. one or none in the random case. When stability is considered, however, we see that a) for a given size, predator prey systems are more likely to be stable and b) that for a given probability of stability, predator prey systems are larger and with heavier loop weights: the stabilizing effect of predator-prey cycles counterbalance these two well known destabilizing effects.

19 ESM Click here to download Manuscript: Theor_Ecol_App_Allesina_Pascual.tex Noname manuscript No. (will be inserted by the editor) Network structure, predator-prey modules, and stability in large food webs: Electronic Supplementary Material (ESM) Stefano Allesina Mercedes Pascual the date of receipt and acceptance should be inserted later S1: Data Manipulation From food webs to community matrices.in Table S1 we report the known name of the food web (Name), the number of nodes/species (S), the original number of links (Total) the number of cannibalistic links removed (Self), the number of double links removed (Double) and the final off-diagonal connectance after the transformation in community matrix (C). We also report the minimum, average and maximum fraction of eigenvalues with negative real part (mine, avge, maxe respectively). For each food web, we averaged the number of eigenvalues with negative real part out of 00 randomizations of the magnitude of the coefficients of the community matrix (Figure ). S: Mathematical Background Terms and coefficients of the characteristic polynomial.we write the characteristic polynomial of the community matrix A as a sum of determinants: charpoly(a) = λ S α 0 + λ S 1 α λ 0 α S (1) where α k = ( 1) k+1 σk det(a σk ), σ k spans all the possible combinations of S terms taken k at a time, and α 0 = 1. In order that the characteristic polynomial has no positive roots, α k < 0, k. We can count the fraction of terms that would yield α k < 0, that is to say, the fraction of negative terms when k is odd and the fraction of positive terms when k is even, by computing the permanent Dambacher et al (00a,b) and determinant of two matrices associated with the community matrix: the adjacency matrix A, where a i j = 1 if a i j 0 and 0 elsewhere, and the sign matrix A S Department of Ecology and Evolutionary Biology, University of Michigan, Natural Science Building, 0 North University, Ann Arbor, MI - USA. stealle@umich.edu where a S i j = 1 if a i j > 0, a S i j = 1 if a i j < 0 and 0 elsewhere. The permanent of A gives us the number of nonzero terms in the expansion of the determinant of A, while the determinant of A S allows us to compute the difference between the number of positive terms P and the number of negative terms N. If we define P k = σk perm(a σ k ), P 0 = 1, and D S k = ( 1)k+1 σk det(a S σ k ), D S 0 = 1, the fraction of terms with the right sign for each coefficient of the characteristic polynomial, F k, will be F k = P k D S k P k. () For example, looking at Figure, the determinant of the matrix corresponds to the last coefficient of the characteristic polynomial (α = 1 det(a )). If we associate 1 to positive coefficients and -1 to negative ones (building therefore A S ), we will find that the determinant is -. When we compute the permanent of the adjacency matrix A (setting every non zero coefficient equal to 1), we find that the value is. Therefore, the fraction of right sign coefficients is ( ( ))/ = /1 = 0., as we expect from the determinant expansion. This procedure was used to compute the fractions and the differences in the number of terms depicted in Figure. An example of the full computation for a simple community is reported below. Sign-StabilityWe can associate to any community matrix A a sign matrix A S [a s i j ] where, as i j = sign(a i j). Given the matrix sign pattern A S, we can state the necessary and sufficient conditions for A S to be sign-stable. If A S is sign stable, then any A whose sign pattern is A S is stable. The conditions for sign-stability were introduced to ecology by MayMay (1), and modified by JeffriesJeffries (1) and LogofetLogofet and Ulianov (1). They can be summarized as follows: 1) a s ii 0 i, as j j < 0 for some j; ) a s i j as ji 0 i j; ) for any sequences of or more

20 indices i 1 i... i m the product a s i 1 i a s i i l...a s i m 1 i m = 0; ) the standard expansion of the determinant contains at least a nonzero term; ) the system fails a color test. In what follows, we briefly describe the ecological interpretation of these conditions and consider whether they apply to the case of: a) random matrices with negative diagonal elements and b) random predator-prey matrices with negative diagonal elements. Condition 1 states that any species cannot have a positive feedback to itself, and that at least one species must be self-regulating. This condition applies to all matrices presented in this article, as the diagonal elements are all negative. Condition states that the type of interaction between species can either be neutral (a s i j = 0,as ji = 0), or commensalistic (0, +1), amensalistic (0, 1) or predatorprey (consumer-resource, parasite-host) ( 1, +1). Mutualism (+1,+1) and interference ( 1, 1) are forbidden while competition can be modeled explicitly by including the resource as a separate variable. While the random matrix model can violate this third condition, the predator-prey system cannot. Condition forbids cycles of length or more, and therefore precludes omnivory. It can be violated by both random and predator-prey matrices. In particular, in the predatorprey case this condition is always violated when the offdiagonal connectance is greater than (S 1)/S, as the corresponding undirected graph must be a tree to lack cycles of length or more. For example, any predator-prey matrix with connectance greater than 1% will violate this condition. In the random matrix case, on the other hand, the maximum possible connectance for the absence of cycles is S(S 1)/S. For example, the maximum achievable connectance without long cycles is % for a matrix. Condition is somehow more technical, and involves the decomposability of the food web in sets of disjoint cycles that involve all the nodes/species. Note however that, because the first term of the standard expansion of the determinant is given by the product of the diagonal elements of the matrix, the condition is always fulfilled in all the matrices considered here. Condition is explained in detail in JeffriesJeffries (1). As for condition, the fact that all diagonal terms are negative ensures that the associated graph fails the color test. Summarizing, Conditions 1, and are always met by the matrices examined here. Predator-prey matrices will also fulfill condition, while both predatorprey and random can violate condition depending on the connectance and topology of the system. The permanent of a matrix.the permanent is an analog of the determinant, in which all the signs in the expansion by minors of the matrix are taken as positive. The determinant of an S S matrix M can be written, using Liebniz formula, as: det(m) = sign(σ) σ p S S 1 m i,σ(i) () where the sum is computed over all permutations (σ) of {1,...,S} and sign(σ) assumes two possible values, 1 or +1, for an odd or even-permutation, respectively. Using the same notation, we can write the permanent as: perm(m) = S σ p S 1 m i,σ(i) () Thus, the only difference between the permanent and the determinant is that there is no sign change in the former. This lack of symmetry makes the computation of the permanent much harder than that of the determinant. Permanent and determinant of signed binary matrices.one can exploit the possibilities given by the permanent to compute the number of positive (and negative) terms in the expansion of the determinant of a matrixdambacher et al (00a,b). We can associate with each community matrix A two matrices: the adjacency matrix A, where a i j = 1 if a i j 0 and 0 elsewhere, and the signed adjacency matrix A S where a S i j = 1 if a i j > 0, a S i j = 1 if a i j < 0 and 0 elsewhere. The permanent of A gives us the number of nonzero terms in the expansion of the determinant of A, while the determinant of A S allows us to compute the difference between the number of positive terms P and the number of negative terms N. As an example, we report the full computation for the matrix in Figure : a AA a AB a AC A = a BA a BB a BC () a CA a CB a CC where all a i j are considered positive. The determinant of this matrix is: det(a) = a AA a BB a CC a AB a BC a CA + a AC a BA a CB a AA a BC a CB a AB a BA a CC a AC a BB a CA and the permanent would be: perm(a) = a AA a BB a CC a AB a BC a CA + a AC a BA a CB +a AA a BC a CB + a AB a BA a CC + a AC a BB a CA Therefore, the determinant is composed by negative terms and 1 positive term. The matrix A associated with A is: A = () so that its permanent will be S! =, as all the terms are 1. The permanent of matrix A is the number of terms in the expansion of the determinant of A. The matrix A S associated with A is A S = () 1 1 1

21 and its determinant: det(a S ) = ( 1 1 1)+(1 1 1)+( 1 1 1) ( 1 1 1) (1 1 1) ( 1 1 1)=, giving the number of positive terms P minus the number of negative terms N in the determinant expansion of A. The fraction of negative terms in the expansion of the determinant of A, f N, is therefore: f N = perm(a ) det(a S ) perm(a ) = ( ) = 0. () Terms and coefficients of the characteristic polynomial.if we write the characteristic polynomial as the sum of the determinants of any possible submatrix of A: charpoly(a) = λ S α 0 + λ S 1 α λ 0 α S () where α k = ( 1) k+1 σk det(a σk ), σ k spans all the possible combinations of S variables taken k at a time, and α 0 = 1, we can count the fraction of terms in the expansion of the determinants yielding the right sign. We define P k = σk perm(a σ k ), P 0 = 1, and D S k = ( 1)k+1 σk det(a S σ k ), D S 0 = 1, the fraction of terms with the right sign for each coefficient of the characteristic polynomial, F k, will be F k = P k D S k P k. () In what follows we illustrate the procedure for the system depicted in Figure. First, α 0 = 1 by definition. To compute the fraction of terms with the right sign that compose α 1, we have to analyze all the possible 1 1 matrices composed by just one variable. These will be: [ a AA ] [ a BB ] [ a CC ] () All the three matrices will have determinant 1 (when transformed in signed matrices) and permanent 1 (when transformed in adjacency matrices). Therefore, α 1 will be formed just by terms with the right sign. In order to study the sign of the terms composing α we have to produce all the possible submatrices of A containing two variables. There are three matrices: [ ] [ ] [ ] aaa a AB aaa a AC abb a BC (1) a BA a BB a CA a CC a CB a CC All these matrices will have determinant and permanent. D will therefore be ( 1) (++)= and P =. α will be formed only by terms with the right sign. The computation for α has been presented in the previous paragraph. The right sign pattern, together with the sign difference, is presented in Figure S1. Figure was produced in the same way. Number of right sign coefficients in predator-prey food webs.the determinant expansion of a matrix can be related to set of disjoint cycles in the directed graph associated with the matrixlevins (1); Puccia and Levins (1). For example, the coefficients on the diagonal represent self-loops, while predator-prey relationships represent cycles of length. Importantly for our argument, these short cycles are also likely to appear in different combinations in the expansion of the determinants of sub-matrices of all sizes. Thus, each determinant of a k k matrix can be seen as composed by cycles of length k and combinations of shorter cycles that, combined, involve k distinct nodes. As an example, the determinant of a matrix can be written as the sum of a) cycles of length, b) combinations of two cycles of length and 1, and c) combinations of three self-loops (see Figure ). The sign of these terms is given by the sign of the coefficients involved, multiplied by ( 1) k m where m is the number of cycles involved. The number of different ways of combining cycles is given by the partition function of k (i.e. the number of ways of writing the integer k as a sum of positive integers). In the matrices representing predator-prey systems, every cycle of length 1 (self-loop) is negative, given the negative values on the diagonal. In the same way, every cycle of length is negative, because only predator-prey interactions are allowed. Therefore, α = ( i, j i< j a ii a j j a i j a ji ) > 0, since the product a ii a j j is positive and a i j a ji either positive or zero depending on whether species i and j interact or not. Every cycle of length or more is associated with another cycle, involving the same nodes taken in the opposite direction. If the signs of a cycle are ++ +, for example, the signs of the associated cycle will be ++. The total sign of the cycle is given by the multiplication of the signs of the coefficients involved. We can therefore divide cycles of length or more into: a) odd-sized cycles containing, as their name indicates, an odd number of nodes; every odd-sized cycle will be associated with a cycle of the opposite sign. b) even-sized cycles, always associated with a cycle of the same sign. Therefore, the total number of coefficients of the right sign in the expansion of the determinant of a k k matrix will be given by: a) all possible combinations of disjoint cycles of size 1 and that involve k nodes; b) half of the possible combinations of disjoint cycles involving at least an odd-sized cycle and other cycles that may be of size 1, or odd; c) roughly half of the possible combinations involving even-sized cycles of length or more. Note that the last statement is just an approximation: even-sized cycles are paired so that both cycles yield the same sign. Assuming that there are many cycles, we may expect that half of them will bear the right sign. These considerations suggest three hypotheses on the sign pattern of a predator-prey system: a) α 0,α 1,α < 0 because just predator-prey interactions are allowed and every species is self-regulating; b) for each α k, k, there will