LETTER doi:10.1038/nature.10832 Stability criteria for complex ecosystems Stefano Allesina2&Si Tang Forty years ago,May proved'2that sufficiently large or complex Var(X)=2.The diagonal elements of the community matrix,repre- ecological networks have a probability of persisting that is close to senting self-regulation,are set to-d.For large systems,the eigenvalues zero,contrary to previous expectations May analysed large are contained in a circle'#in the complex plane(Fig.1 and Supplemen- networks in which species interact at random'6.However,in tary Information).The circle is centred at(-d,0)and the radius is natural systems pairs of species have well-defined interactions oVSC.In stable systems,the whole circle is contained in the left half- (for example predator-prey,mutualistic or competitive).Here plane (that is,all eigenvalues have negative real parts).Thus,the we extend May's results to these relationships and find remarkable system is stable when the radius is smaller than d:vSC0,then M0 for any distribution of X, diagonal coefficients are drawn from a distribution with mean 0 and the stability criteria form a strict hierarchy in which the mixture matrices variance o with probability Cand are 0 otherwise.For these matrices, are the least likely to be stable,the random matrices are intermediate, the probability of stability is close to 0 whenever the 'complexity and the predator-prey matrices are the most likely to be stable(Fig.2 SC>1.Local stability measures the tendency of the system to and Table 1).Considerations based on qualitative stability2 and return to equilibrium after perturbations.In unstable systems,even numerical simulations'are consistent with this hierarchy. infinitesimal perturbations cause the system to move away from In the three cases above,the mean interaction strength is zero,and equilibrium,potentially leading to the loss of species.Thus,it should the coefficients come from the same distribution.In fact we can shuffle be extremely improbable to observe rich(large S)or highly connected the interaction strengths,thereby transforming a network of one type (large C)persistent ecosystems'.Mathematically,an equilibrium into another:the difference in stability is driven exclusively by the point is stable if all the eigenvalues of the community matrix have arrangement of the coefficients in pairs with random,opposite and negative real parts same signs,respectively.This feature allows us to further derive the Local stability can only describe the behaviour of the system around stability criteria for all intermediate cases by using linear combinations an equilibrium point,whereas natural systems are believed to operate of the three cases above(Supplementary Information). far froma steady state.However,methods based onlocal stability are Two ecologically important cases,however,cannot produce a mean well suited to the study of large systems7,whose empirical para- interaction strength of zero.In mutualistic networks all interactions meterization would be unfeasible.Moreover,the methods are general, are positive,whereas in competitive networks they are negative.In so that they can be applied to any system of differential equations. these cases,for large systems,all the eigenvalues except one (equal to May's matrices have random structure:each pair of species interacts the row sum)are contained in an ellipse(Fig.3 and Supplementary with the same probability.However,this randomness translates,for Figs I and 2).In mutualistic networks in which all interaction pairs are large S,into fixed interaction frequencies,so that these matrices follow positive and drawn from the distribution ofX independently with a precise mixture of interaction types.For example,in May's matrices probability C,the stability criterion becomes (S-1)CE(X)/< predator-prey interactions are twice as frequent as mutualistic ones (Supplementary Table 1).Here we extend May's work to different (that is,row sum<0;Supplementary Information).For competitive types of interaction,starting from the random case. matrices,in which interaction pairs are drawn from the distribution of -X with probability C,the criterion is Suppose that two species jand iinteract with probability C,and that the interaction strength is drawn from a distribution:M takes the 1-CE2(0X)/o2+CE(X)/o<0 value of a random variable X with mean E(X)=0 and variance VSC(1+(1-2C)E2(0X)/a) Department of Ecology and Evolution.University of Chicago,1101 East 57th Street,Chicago,Illinois 60637.USAComputation Institute,University of Chicago,5735 South Ellis Avenue.Chicago,Ilinois 60637,USA 8 MARCH 2012 I VOL 483 NATURE I 205 2012 Macmillan Publishers Limited.All rights reserved
LETTER doi:10.1038/nature10832 Stability criteria for complex ecosystems Stefano Allesina1,2 & Si Tang1 Forty years ago, May proved1,2 that sufficiently large or complex ecological networks have a probability of persisting that is close to zero, contrary to previous expectations3–5. May analysed large networks in which species interact at random1,2,6. However, in natural systems pairs of species have well-defined interactions (for example predator–prey, mutualistic or competitive). Here we extend May’s results to these relationships and find remarkable differences between predator–prey interactions, which are stabilizing, and mutualistic and competitive interactions, which are destabilizing. We provide analytic stability criteria for all cases. We use the criteria to prove that, counterintuitively, the probability of stability for predator–prey networks decreases when a realistic food web structure is imposed7,8 or if there is a large preponderance of weak interactions9,10. Similarly, stability is negatively affected by nestedness11–14 in bipartite mutualistic networks. These results are found by separating the contribution of network structure and interaction strengths to stability. Stable predator–prey networks can be arbitrarily large and complex, provided that predator–prey pairs are tightly coupled. The stability criteria are widely applicable, because they hold for any system of differential equations. May’s theorem deals with community matrices1,2,6 M, of size S 3 S, where S is the number of species. Mij describes the effect that species j has on i around a feasible equilibrium point (that is, species have positive densities) of an unspecified dynamical system describing the species’ densities through time. In May’s work1,2, the diagonal coefficients are 21, and the offdiagonal coefficients are drawn from a distribution with mean 0 and variance s2 with probability C and are 0 otherwise. For these matrices, the probability of stability is close to 0 whenever the ‘complexity’ s ffiffiffiffiffi SC p w1. Local stability measures the tendency of the system to return to equilibrium after perturbations. In unstable systems, even infinitesimal perturbations cause the system to move away from equilibrium, potentially leading to the loss of species. Thus, it should be extremely improbable to observe rich (large S) or highly connected (large C) persistent ecosystems1,2. Mathematically, an equilibrium point is stable if all the eigenvalues of the community matrix have negative real parts1,2,6. Local stability can only describe the behaviour of the system around an equilibrium point, whereas natural systems are believed to operate farfrom a steady state5,15. However, methods based on local stability are well suited to the study of large systems1,16,17, whose empirical parameterization would be unfeasible. Moreover, the methods are general, so that they can be applied to any system of differential equations. May’s matrices have random structure: each pair of species interacts with the same probability. However, this randomness translates, for large S, into fixed interaction frequencies, so that these matrices follow a precise mixture of interaction types. For example, in May’s matrices predator–prey interactions are twice as frequent as mutualistic ones (Supplementary Table 1). Here we extend May’s work to different types of interaction, starting from the random case. Suppose that two speciesj and i interact with probability C, and that the interaction strength is drawn from a distribution: Mij takes the value of a random variable X with mean Eð Þ X ~0 and variance Var(X) 5 s2 . The diagonal elements of the community matrix, representing self-regulation, are set to 2d. For large systems, the eigenvalues are contained in a circle18 in the complex plane (Fig. 1 and Supplementary Information). The circle is centred at (2d, 0) and the radius is s ffiffiffiffiffi SC p . In stable systems, the whole circle is contained in the left halfplane (that is, all eigenvalues have negative real parts). Thus, the system is stable when the radius is smaller than d: ffiffiffiffiffi SC p vh~d=s. In predator–prey networks, interactions come in pairs with opposite signs: wheneverMij . 0, thenMji, 0.With probabilityC, we sample one interaction strengthfrom the distribution of jXj and the other from2jXj, whereas with probability (1 2 C) both are zero. The eigenvalues of large predator–prey matrices are contained in a vertically stretched ellipse19, centred at (2d, 0), with horizontal radius s ffiffiffiffiffi SC p 1{E2 ð Þ j j X s2 and thus the stability criterion is ffiffiffiffiffi SC p vh 1{E2 ð Þ j j X =s2 (Fig. 1 and Supplementary Information). When we constrain Mij and Mji to have the same sign, and thus impose a mixture of competition and mutualism with equal probability, the eigenvalues are enclosed in a horizontally stretched ellipse19 and the criterion becomes ffiffiffiffiffiffi SC p vh 1zE2 ð Þ j j X =s2 (Fig. 1 and Supplementary Information). Take C 5 0.1, X , N(0, 1/4) (that is, X follows a normal distribution with mean 0 and variance 1/4), and d 5 1. The criterion becomes ffiffiffiffiffi SC p v2 for random matrices, and is violated whenever S $ 41. For predator–prey we find ffiffiffiffiffi SC p v2p=ð Þ p{2 (violated for S $ 303) and for the mixture of competition and mutualism ffiffiffiffiffi SC p v2p=ð Þ pz2 (violated for S $ 15). Since Eð Þ j j X =sw0 for any distribution of X, the stability criteria form a strict hierarchy in which the mixture matrices are the least likely to be stable, the random matrices are intermediate, and the predator–prey matrices are the most likely to be stable (Fig. 2 and Table 1). Considerations based on qualitative stability2 and numerical simulations16 are consistent with this hierarchy. In the three cases above, the mean interaction strength is zero, and the coefficients come from the same distribution. In fact we can shuffle the interaction strengths, thereby transforming a network of one type into another: the difference in stability is driven exclusively by the arrangement of the coefficients in pairs with random, opposite and same signs, respectively. This feature allows us to further derive the stability criteria for all intermediate cases by using linear combinations of the three cases above (Supplementary Information). Two ecologically important cases, however, cannot produce a mean interaction strength of zero. In mutualistic networks all interactions are positive, whereas in competitive networks they are negative. In these cases, for large systems, all the eigenvalues except one (equal to the row sum) are contained in an ellipse (Fig. 3 and Supplementary Figs 1 and 2). In mutualistic networks in which all interaction pairs are positive and drawn from the distribution of jXj independently with probability C, the stability criterion becomes ð Þ S{1 CEð Þ j j X =svh (that is, row sum , 0; Supplementary Information). For competitive matrices, in which interaction pairs are drawn from the distribution of 2jXj with probability C, the criterion is ffiffiffiffiffi SC p 1zð Þ 1{2C E2 ð Þ j j X =s2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1{CE2 ð Þ j j X =s2 q zCEð Þ j j X =svh 1 Department of Ecology and Evolution, University of Chicago, 1101 East 57th Street, Chicago, Illinois 60637, USA. 2 Computation Institute, University of Chicago, 5735 South Ellis Avenue, Chicago, Illinois 60637, USA. 8 MARCH 2012 | VOL 483 | NATURE | 205 ©2012 Macmillan Publishers Limited. All rights reserved
RESEARCH LETTER Random Predator-prey Mixture 20 20 20 10 0 0 -10 -10 -20 20 20 20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 Real Real Real 1.0 米来来米米来米 Random 1.0 米来米米来来米军 Predator-prey 1.0 米米米米米米士 Mixture 0.8 0.8 0.8 。6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 米米亲米米米米 0 米米米米米米米米 0.6 0.8 1.0 1.2 1.4 1.5 2.0 2.5 3.0 3.5 4.0 0.3 0.4 0.50.60.7 0.80.9 aSC aSC SC Figure 1 Distributions of the eigenvalues and corresponding stability systematically varied C(crosses)or (plus signs)to obtain vSC spanning profiles.a,For X~N(0,),S=250,C=0.25 and =1,we plot the [0.5,...,1.0,....1.5]of the critical value for stability (indicated in red,1 in the eigenvalues of 10 matrices (colours)with-d=-1 on the diagonal and off- case of random matrices).The profiles were obtained by computing the diagonal elements,following the random,predator-prey or mixture probability of stability out of 1,000 matrices.The predator-prey case is as the prescriptions.The black ellipses are derived analytically in the text. random but with =0.5 and critical value /(-2).The mixture case is as the b,Numerical simulations for the corresponding stability profiles.For the random but with critical value r/(n+2).In all cases,the phase transition random case,starting from S=250,C=0.5,=0.1 and d=1,we between stability and instability is accurately predicted by our derivation. 0.25 有口 (Supplementary Information).In both cases,stability decreases rapidly o Predator-prey 女日 with higher complexity,and mutualistic matrices are less likely to be 口Random 年口 V Competition stable than their competitive counterpart(Fig 2 and Table 1). 年口 △Mixture Having derived the stability criteria,we want to assess the effect of 0.20- 0 g口 ◇Mutualism imposing realistic food web structure within the predator-prey case.It is 每口 0 believed that realistic food web structures should improve stability7 In community matrices of food webs,producers have positive columns and negative rows,with the opposite for top predators.To test whether 0.15 these variations affect stability,we plotted the eigenvalues for predator- prey webs in which interactions are arranged,following the cascade20 0 and niche"models.Imposing realistic structures results in eigenvalues 0.10 with larger real parts than the corresponding unstructured predator- prey case (Supplementary Information and Supplementary Fig.3). 0 Thus,the cascade and niche models produce networks that arelesslikely to be stable than their unstructured predator-prey counterpart,with the 0.05 niche model having a larger discrepancy:imposing realistic food web structure hampers stability. Similarly,we measured the effect ofrealistic structures on mutualistic networks.Several published mutualistic networks are bipartite- there are two types of node(for example plants and pollinators),and 0 500 1,000 1.500 interactions occur exclusively between different types.In addition, bipartite mutualistic networks tend to be nested"':the interactions of Figure 2 Stability criteria for different types of interaction.We fixed the specialists form a subset of those of the generalists.Nestedness is 6=d/=4,and for a given connectance C we solved for the largest integer S believed to beget stability2.We plotted the eigenvalues for these two that satisfies the stability criterion for each type of interactions.Combinations types of structure and compared the results with those obtained for the of S and Cbelow each curve lead to stable matrices with a probability close to 1. unstructured mutualistic case(Fig.3,Supplementary Information and The interaction types form a strict hierarchy from mutualism(most unlikely to Supplementary Fig.4).Asstated above,stability in mutualistic networks be stable)to predator-prey(most likely to be stable). is determined by the row sum.The bipartite case yields row sums that, I VOL 483 I 8 MARCH 2012 2012 Macmillan Publishers Limited.All rights reserved
(Supplementary Information). In both cases, stability decreases rapidly with higher complexity, and mutualistic matrices are less likely to be stable than their competitive counterpart (Fig. 2 and Table 1). Having derived the stability criteria, we want to assess the effect of imposing realisticfood web structure within the predator–prey case. It is believed that realistic food web structures should improve stability7,8,17. In community matrices of food webs, producers have positive columns and negative rows, with the opposite for top predators. To test whether these variations affect stability, we plotted the eigenvalues for predator– prey webs in which interactions are arranged, following the cascade20 and niche21 models. Imposing realistic structures results in eigenvalues with larger real parts than the corresponding unstructured predator– prey case (Supplementary Information and Supplementary Fig. 3). Thus, the cascade and nichemodels produce networks that are less likely to be stable than their unstructured predator–prey counterpart, with the niche model having a larger discrepancy: imposing realistic food web structure hampers stability. Similarly, we measured the effect of realistic structures on mutualistic networks. Several published mutualistic networks are bipartite11–14: there are two types of node (for example plants and pollinators), and interactions occur exclusively between different types. In addition, bipartite mutualistic networks tend to be nested11: the interactions of the specialists form a subset of those of the generalists. Nestedness is believed to beget stability12–14. We plotted the eigenvalues for these two types of structure and compared the results with those obtained for the unstructured mutualistic case (Fig. 3, Supplementary Information and Supplementary Fig. 4).As stated above, stability inmutualistic networks is determined by the row sum. The bipartite case yields row sums that, −20 −10 0 10 20 −20 −10 0 10 20 −20 −10 0 10 20 −20 −10 0 10 20 Real −20 −10 0 10 20 Real −20 −10 0 10 20 Real Imaginary −1 a Random b 0.6 0.8 1.0 1.2 1.4 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 σ SC P(stability) Random −1 Predator−prey 1.5 2.0 2.5 3.0 3.5 4.0 Predator−prey −1 Mixture 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mixture σ SC σ SC Figure 1 | Distributions of the eigenvalues and corresponding stability profiles. a, For X , N(0, s2 ), S 5 250, C 5 0.25 and s 5 1, we plot the eigenvalues of 10 matrices (colours) with 2d 5 21 on the diagonal and offdiagonal elements, following the random, predator–prey or mixture prescriptions. The black ellipses are derived analytically in the text. b, Numerical simulations for the corresponding stability profiles. For the random case, starting from S 5 250, C 5 0.5, s 5 0.1 and d 5 1, we systematically varied C (crosses) or s (plus signs) to obtain s ffiffiffiffiffi SC p spanning [0.5,…, 1.0, …, 1.5] of the critical value for stability (indicated in red, 1 in the case of random matrices). The profiles were obtained by computing the probability of stability out of 1,000 matrices. The predator–prey case is as the random but with s 5 0.5 and critical value p/(p 2 2). The mixture case is as the random but with critical value p/(p 1 2). In all cases, the phase transition between stability and instability is accurately predicted by our derivation. 0 500 1,000 1,500 0 0.05 0.10 0.15 0.20 0.25 S C Predator−prey Random Competition Mixture Mutualism Figure 2 | Stability criteria for different types of interaction. We fixed h 5 d/s 5 4, and for a given connectance C we solved for the largest integer S that satisfies the stability criterion for each type of interactions. Combinations of S and C below each curve lead to stable matrices with a probability close to 1. The interaction types form a strict hierarchy from mutualism (most unlikely to be stable) to predator–prey (most likely to be stable). RESEARCH LETTER 206 | NATURE | VOL 483 | 8 MARCH 2012 ©2012 Macmillan Publishers Limited. All rights reserved
LETTER RESEARCH Table 1|Stability criteria for different types of interaction and network structure Sma(C.0) Interaction Stability criterion (0.1.2.0) (0.1,4.0) (02,4.0 Nested mutualism 28 Mutualism s-1c 16(15) 41(51) 22(20) Bipartite mutualism 1 41 23 Mixture vc<2 7(14) 58(59) 33(29) Competition 17(15) 62(63) 38(33) Random VSC<0 50(40) 168(160) 88(80) Niche predator-prey 149 461 245 Cascade predator-prey 298 1.134 535 Predator-prey 314(302) 1,201(1,211) 603(605) π-2 In all cases,the criterion is derived for large SxS matrices with X~N(O)(and thus E(X))connectance Cand -d/.Numerical simulations report.for a given combination of Cand 0,the largest S (S)yielding a probability of stability0.5(computed using 1.000 matrices)In parenthesis are the analytical predictions for large S,are equal to the unstructured case.Accordingly,we did not hampers stability.We conclude that weak interactions,contrary to find a discrepancy in stability for the bipartite case.However,in nested current beliefs,can destabilize predator-prey systems.Weakening structures some rows and columns have sums that are larger than the interactions shifts E(X)closer to zero and therefore makes average(generalist plants and animals).Consequently,nested matrices predator-prey systems closer to their random counterpart.With the are inherently less likely to be stable than unstructured ones.These same argument,weak interactions can stabilize the mixture of competi- findings are confirmed by numerical simulations.Using the same tion and mutualism case and have no effect on random networks. method,we found that asymmetric coupling of interaction strengths Variability in interaction strengths was previously found to be (where each large Mi is coupled with a small M),contrary to current detrimental for stability in large food webs2 and competitive networks expectations2,does not influence stability in mutualistic networks For example,consider a uniform distribution X~U-v3,v3 (Supplementary Information and Supplementary Fig.5). and contrast it with the normal case X~N(0,)Both parameteriza- We have considered how the arrangement of the interactions affects tions lead to E(X)=0 and Var(X)=a2.In the uniform case, stability,and have found several counterintuitive results.These results E(XD)=Gv3/2=0.866g,whereas in the normal case E(X])= can be accounted for by the fact that we provide a very conservative test o√2/π≈0.798a.This means that the uniform distribution,on for the effects of structure on stability (Supplementary Information). average,leads to stronger interactions than the corresponding normal We now assess the role of the magnitude of interaction strengths.In case.In turn,this has a large effect on stability:the criterion for the fact,our findings extend to any distribution of coefficient strengths predator-prey case becomes VSC<40 for the uniform distri- (Supplementary Information). bution,whereas it is VSC<π/（π-2）0=2.750 for the normal case. Typically,ecologists have regarded as the 'average interaction The random case is unaffected by the choice of the distribution strength"2.However,o does not provide information on weak inter- (VSC<0),whereas in the mixture of competition and mutualism actions we can have the same for two distributions with distinct we have VSC<40/7=0.5710 for the uniform distribution and shapes,and thus different proportions of weak and strong interactions VSC<π0/（π+2）≈0.6l0 for the normal case..These considerations (Supplementary Information).We analyse how the shape of the dis- extend to any choice of distribution for the interaction strengths tribution affects stability for fixed S,C,d and o.If the distribution (Supplementary Information and Supplementary Figs 6 and 7):weak contains many weak interactions,the expected magnitude E(X)0. interactions,all other things being equal,are destabilizing for food In contrast,if weak interactions are rare,E(X)0.In the predator- webs,stabilizing for mutualistic and competitive networks(and their prey systems,lowering E(X)decreases0/(1-E2(X)/2)and thus mixture),and have no effect on random networks. a 0.5- 1.0- -0.5 b 0.5 1.0- 0.5 1.0 0.5 Rea Figure 3 Distribution of the eigenvalues for the three types of mutualism. red arrow)is equal to that of the unstructured mutualistic case.The nested a,Unstructured mutualism.b,Bipartite mutualism.c,Nested and bipartite matrices,in which generalist species yield (on average)larger row and column mutualism.In all cases,S=250,c=0.1,C=0.2 and d=1.Note that the sums,have larger rightmost eigenvalues.Thus,highly nested matrices are less bipartite case does produce extreme negative real eigenvalues (green arrow) likely than the other two cases to be stable. coupled with positive ones,but the row sum (and thus the rightmost eigenvalue, 8 MARCH 2012 VOL 483 2012 Macmillan Publishers Limited.All rights reserved
for large S, are equal to the unstructured case. Accordingly, we did not find a discrepancy in stability for the bipartite case. However, in nested structures some rows and columns have sums that are larger than average (generalist plants and animals). Consequently, nested matrices are inherently less likely to be stable than unstructured ones. These findings are confirmed by numerical simulations. Using the same method, we found that asymmetric coupling of interaction strengths (where each large Mij is coupled with a small Mji), contrary to current expectations22, does not influence stability in mutualistic networks (Supplementary Information and Supplementary Fig. 5). We have considered how the arrangement of the interactions affects stability, and have found several counterintuitive results. These results can be accounted for by the fact that we provide a very conservative test for the effects of structure on stability (Supplementary Information). We now assess the role of the magnitude of interaction strengths. In fact, our findings extend to any distribution of coefficient strengths (Supplementary Information). Typically, ecologists have regarded s as the ‘average interaction strength’1,2. However, s does not provide information on weak interactions9,10,17: we can have the same s for two distributions with distinct shapes, and thus different proportions of weak and strong interactions (Supplementary Information). We analyse how the shape of the distribution affects stability for fixed S, C, d and s. If the distribution contains many weak interactions, the expected magnitude Eð Þ j j X <0. In contrast, if weak interactions are rare, Eð Þ j j X <s. In the predator– prey systems, lowering Eð Þ j j X decreases h 1{E2 ð Þ j j X =s2 and thus hampers stability. We conclude that weak interactions, contrary to current beliefs9,10,17, can destabilize predator–prey systems. Weakening the interactions shifts Eð Þ j j X closer to zero and therefore makes predator–prey systems closer to their random counterpart. With the same argument, weak interactions can stabilize the mixture of competition and mutualism case and have no effect on random networks. Variability in interaction strengths was previously found to be detrimental for stability in large food webs23 and competitive networks17. For example, consider a uniform distribution X , U {s ffiffi 3 p ,s ffiffi 3 p and contrast it with the normal case X , N(0, s2 ). Both parameterizations lead to Eð Þ X ~0 and Var(X) 5 s2 . In the uniform case, Eð Þ j j X ~s ffiffi 3 p 2<0:866 s, whereas in the normal case Eð Þ j j X ~ s ffiffiffiffiffiffiffiffi 2=p p <0:798 s. This means that the uniform distribution, on average, leads to stronger interactions than the corresponding normal case. In turn, this has a large effect on stability: the criterion for the predator–prey case becomes ffiffiffiffiffi SC p v4 h for the uniform distribution, whereas it is ffiffiffiffiffiffi SC p vp=ð Þ p{2 h<2:75 h for the normal case. The random case is unaffected by the choice of the distribution ( ffiffiffiffiffi SC p vh), whereas in the mixture of competition and mutualism we have ffiffiffiffiffi SC p v4h=7<0:571 h for the uniform distribution and ffiffiffiffiffi SC p vph=ð Þ pz2 <0:61 h for the normal case. These considerations extend to any choice of distribution for the interaction strengths (Supplementary Information and Supplementary Figs 6 and 7): weak interactions, all other things being equal, are destabilizing for food webs, stabilizing for mutualistic and competitive networks (and their mixture), and have no effect on random networks. Table 1 | Stability criteria for different types of interaction and network structure Smax(C, h) Interaction Stability criterion (0.1, 2.0) (0.1, 4.0) (0.2, 4.0) Nested mutualism 9 28 18 Mutualism ð Þ S{1 C ffiffiffi 2 p r vh 16 (15) 41 (51) 22 (20) Bipartite mutualism 17 41 23 Mixture ffiffiffiffiffiffi SC p v hp p z 2 17 (14) 58 (59) 33 (29) Competition ffiffiffiffiffiffi SC p 1z 2 { 2C p { 2C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p { 2C p r zC ffiffiffi 2 p r vh 17 (15) 62 (63) 38 (33) Random ffiffiffiffiffiffi SC p vh 50 (40) 168 (160) 88 (80) Niche predator–prey 149 461 245 Cascade predator–prey 298 1,134 535 Predator–prey ffiffiffiffiffiffi SC p v hp p { 2 314 (302) 1,201 (1,211) 603 (605) In all cases, the criterion is derived for large S 3 S matrices with X , N(0, s2 ) (and thus Eð Þ j j X ~s ffiffiffiffiffiffiffiffi 2=p p ), connectance C and h 5 d/s. Numerical simulations report, for a given combination of C and h, the largest S (Smax) yielding a probability of stability $ 0.5 (computed using 1,000 matrices). In parenthesis are the analytical predictions. Real Imaginary −7 −6 −5 −4 −3 −2 −10 1 2 3 4 5 −0.5 1.0 0.5 a b c −0.5 1.0 0.5 −0.5 1.0 0.5 Figure 3 | Distribution of the eigenvalues for the three types of mutualism. a, Unstructured mutualism. b, Bipartite mutualism. c, Nested and bipartite mutualism. In all cases, S 5 250, s 5 0.1, C 5 0.2 and d 5 1. Note that the bipartite case does produce extreme negative real eigenvalues (green arrow) coupled with positive ones, but the row sum (and thus the rightmost eigenvalue, red arrow) is equal to that of the unstructured mutualistic case. The nested matrices, in which generalist species yield (on average) larger row and column sums, have larger rightmost eigenvalues. Thus, highly nested matrices are less likely than the other two cases to be stable. LETTER RESEARCH 8 MARCH 2012 | VOL 483 | NATURE | 207 ©2012 Macmillan Publishers Limited. All rights reserved
RESEARCH LETTER We have derived stability criteria for unstructured networks in 3.MacArthur,R.Fluctuations of animal populations and a measure of community which species interact at random,in predator-prey,mutualistic,and stability.Ecology 36,533-536(1955). 4 Elton,C.S.Animal Ecology (Univ.of Chicago Press,2001). competitive pairs.These results hold for arbitrary diagonal values and 5. McCann,K.S.The diversity-stability debate.Nature 405,228-233(2000). arbitrary distribution of interaction strengths (Supplementary 6 Levins R.Evolution in Changing Environments:Some Theoretical Explorations Information).Our analysis shows that,all other things being equal, (Princeton Univ.Press,1968). McNaughton,S.J.Stability and diversity of ecological communities.Nature 274, weak interactions can be either stabilizing or destabilizing depending 251-253(1978). on the type of interactions between species.In predator-prey systems, 8. Yodzis.P.The stability of real ecosystems.Nature 289,674-676(1981). realistic structure and weak interactions are detrimental for stability 9. McCann,K.S.,Hastings,A.Huxel,G.R.Weak trophic interactions and the balance However,in natural food webs,which seem to persist in time,weak of nature.Nature 395,794-798(1998). 10.Emmerson,M.Yearsley,J.M.Weak interactions,omnivory and emergent food- interactions are preponderant24.The persistence of these networks web properties.Proc.R.Soc.Lond.B 271,397-405 (2004). might be explained by the interplay between their structure and weak 11.Bascompte,J.Jordano,P.Melian,C.J.&Olesen,J.M.The nested ass eTmbyof interactions,even though each would be destabilizing if taken in plant-animal mutualistic networks.Proc.Natl Acad.Sci.USA 100,9383-9387 2003). isolation.For example,as suggested previously,generalist predators 12.Okuyama,T.Holland,J.N.Network structural properties mediate the stability of could have weak interactions with their numerous prey,reducing the mutualistic communities.Ecol Lett 11,208-216(2008). effect of the realistic structure and driving the system closer to the 13.Bastolla,U.et al.The architecture of mutualistic networks minimizes competition and increases biodiversity.Nature 458,1018-1020(2009). unstructured case. 14.Thebault,E Fontaine,C.Stability of ecological communities and the architecture Predator-prey systems differ markedly from the other cases studied of mutualistic and trophic networks.Science 329,853-856(2010). here.Suppose that a network is unstable.The system can be stabilized 15.DeAngelis,D.L.Waterhouse,J.C.Equilibrium and nonequilibrium concepts in either by lowering C,S or(decreasing its complexity),or by increas- ecological models.Ecol.Monogr.57,1-21 (1987). 16.Allesina,S.Pascual,M.Network structure,predator-prey modules,and stability ing the self-regulation d.This is in line with May's argument:large and in large food webs.Theor.Ecol.1,55-64 (2008). highly interconnected systems are difficult to stabilize.For random 17.Gross,T.Rudolf,L,Levin,S.A.Dieckmann,U.Generalized models reveal networks,reducing complexity is the only way to stabilize the system. stabilizing factors in food webs.Science 325.747-750(2009). 18.Tao,T.,Vu,V.Krishnapur,M.Random matrices:universality of ESDs and the However,in the other cases,networks can be stabilized by altering the circular law.Ann.Probab.38,2023-2065(2010). distribution of interaction strengths;by modifying the parameters of Sommers,H.J.,Crisanti,A,Sompolinsky,H.Stein,Y.Spectrum of large random the system we can typically change the distribution of the off-diagonal asymmetric matrices.Phys.Rev.Lett 60,1895-1898(1988). elements without altering the diagonal ones(Supplementary Informa- 20. Cohen,J.E..Briand,F.Newman,C.M.&Palka,Z.J.Community FoodWebs:Dataand Theory(Springer,1990). tion).For competition,mutualism and their mixture,stability is 21. Williams,R.J.Martinez,N.D.Simple rules yield complex food webs.Nature 404 achievable by decreasing the average interaction strength E(X), 180-183(2000). which is akin to lowering complexity.On the contrary,predator-prey 22. Bascompte,J.Jordano,P.Olesen,J.M.Asymmetric coevolutionary networks facilitate biodiversity maintenance.Science 312,431-433 (2006). networks can be stabilized by increasing the strength of interaction 23. Kokkoris,G.D.Jansen,V.AA.,Loreau,M.&Troumbis,A.Y.Variability ininteraction E(X),and thus the coupling between predators and prey.Predator- strength and implications for biodiversity.J.Anim.Ecol 71,362-371 (2002). prey systems are therefore the only ones that can potentially elude 24. Wootton,J.T.&Emmerson,M.Measurement of interaction strength in nature. Annu.Rev.Ecol Evol.Syst 36,419-444(2005). May's conclusionsand support an arbitrarily large,complex and stable ecological network. Supplementary Information is linked to the online version of the paper at Our results show that the ubiquity of consumer-resource relation- www.nature.com/nature. ships in nature could be due to their intrinsic dynamical properties. Acknowledgements We thank J.Bergelson,L-F.Bersier,A.M.de Roos,A.Eklof,C.A. These findings are not limited to ecological networks,but instead hold Klausmeier,S.P.Lalley,R.M.May,K.S.McCann,M.Novak,P.P.A.Staniczenko and J.D. for any system of differential equations resting at an equilibrium point. Yeakel for comments and discussion.This research was supported by National Science Foundation grant EF0827493. Received 18 May 2011;accepted 6 January 2012. Author Contributions All authors contributed equally Published online 19 February 2012. Author Information Reprints and permissions information is available at www.nature.com/reprints.The authors declare no competing financial interests. 1.May,R.M.Will a large complex system be stable?Nature 238,413-414(1972). Readers are welcome to comment on the online version of this article at 2.May,R.M.Stability and Complexity in Model Ecosystems (Princeton Univ.Press, www.nature.com/nature.Correspondence and requests for materials should be 2001). addressed to S.A.(sallesina@uchicago.edu). 208 NATURE I VOL 483 I 8 MARCH 2012 2012 Macmillan Publishers Limited.All rights reserved
We have derived stability criteria for unstructured networks in which species interact at random, in predator–prey, mutualistic, and competitive pairs. These results hold for arbitrary diagonal values and arbitrary distribution of interaction strengths (Supplementary Information). Our analysis shows that, all other things being equal, weak interactions can be either stabilizing or destabilizing depending on the type of interactions between species. In predator–prey systems, realistic structure and weak interactions are detrimental for stability. However, in natural food webs, which seem to persist in time, weak interactions are preponderant24. The persistence of these networks might be explained by the interplay between their structure and weak interactions, even though each would be destabilizing if taken in isolation. For example, as suggested previously2 , generalist predators could have weak interactions with their numerous prey, reducing the effect of the realistic structure and driving the system closer to the unstructured case. Predator–prey systems differ markedly from the other cases studied here. Suppose that a network is unstable. The system can be stabilized either by lowering C, S or s (decreasing its complexity), or by increasing the self-regulation d. This is in line with May’s argument: large and highly interconnected systems are difficult to stabilize. For random networks, reducing complexity is the only way to stabilize the system. However, in the other cases, networks can be stabilized by altering the distribution of interaction strengths; by modifying the parameters of the system we can typically change the distribution of the off-diagonal elements without altering the diagonal ones (Supplementary Information). For competition, mutualism and their mixture, stability is achievable by decreasing the average interaction strength Eð Þ j j X , which is akin to lowering complexity. On the contrary, predator–prey networks can be stabilized by increasing the strength of interaction Eð Þ j j X , and thus the coupling between predators and prey. Predator– prey systems are therefore the only ones that can potentially elude May’s conclusions1,2 and support an arbitrarily large, complex and stable ecological network. Our results show that the ubiquity of consumer–resource relationships in nature could be due to their intrinsic dynamical properties. These findings are not limited to ecological networks, but instead hold for any system of differential equations resting at an equilibrium point. Received 18 May 2011; accepted 6 January 2012. Published online 19 February 2012. 1. May, R. M. Will a large complex system be stable? Nature 238, 413–414 (1972). 2. May, R. M. Stability and Complexity in Model Ecosystems (Princeton Univ. Press, 2001). 3. MacArthur, R. Fluctuations of animal populations and a measure of community stability. Ecology 36, 533–536 (1955). 4. Elton, C. S. Animal Ecology (Univ. of Chicago Press, 2001). 5. McCann, K. S. The diversity–stability debate. Nature 405, 228–233 (2000). 6. Levins, R. Evolution in Changing Environments: Some Theoretical Explorations (Princeton Univ. Press, 1968). 7. McNaughton, S. J. Stability and diversity of ecological communities. Nature 274, 251–253 (1978). 8. Yodzis, P. The stability of real ecosystems. Nature 289, 674–676 (1981). 9. McCann, K. S., Hastings, A. & Huxel, G. R.Weak trophic interactions and the balance of nature. Nature 395, 794–798 (1998). 10. Emmerson, M. & Yearsley, J. M. Weak interactions, omnivory and emergent foodweb properties. Proc. R. Soc. Lond. B 271, 397–405 (2004). 11. Bascompte, J., Jordano, P., Melia´n, C. J. & Olesen, J. M. The nested assembly of plant–animal mutualistic networks. Proc. Natl Acad. Sci. USA 100, 9383–9387 (2003). 12. Okuyama, T. & Holland, J. N. Network structural properties mediate the stability of mutualistic communities. Ecol. Lett. 11, 208–216 (2008). 13. Bastolla, U. et al. The architecture of mutualistic networks minimizes competition and increases biodiversity. Nature 458, 1018–1020 (2009). 14. The´bault, E. & Fontaine, C. Stability of ecological communities and the architecture of mutualistic and trophic networks. Science 329, 853–856 (2010). 15. DeAngelis, D. L. & Waterhouse, J. C. Equilibrium and nonequilibrium concepts in ecological models. Ecol. Monogr. 57, 1–21 (1987). 16. Allesina, S. & Pascual, M. Network structure, predator–prey modules, and stability in large food webs. Theor. Ecol. 1, 55–64 (2008). 17. Gross, T., Rudolf, L., Levin, S. A. & Dieckmann, U. Generalized models reveal stabilizing factors in food webs. Science 325, 747–750 (2009). 18. Tao, T., Vu, V. & Krishnapur, M. Random matrices: universality of ESDs and the circular law. Ann. Probab. 38, 2023–2065 (2010). 19. Sommers, H. J., Crisanti, A., Sompolinsky, H. & Stein, Y. Spectrum of large random asymmetric matrices. Phys. Rev. Lett. 60, 1895–1898 (1988). 20. Cohen, J. E., Briand, F., Newman, C.M. & Palka, Z. J.Community Food Webs: Data and Theory (Springer, 1990). 21. Williams, R. J. & Martinez, N. D. Simple rules yield complex food webs. Nature 404, 180–183 (2000). 22. Bascompte, J., Jordano, P. & Olesen, J. M. Asymmetric coevolutionary networks facilitate biodiversity maintenance. Science 312, 431–433 (2006). 23. Kokkoris, G. D., Jansen, V. A. A., Loreau, M. & Troumbis, A. Y. Variability in interaction strength and implications for biodiversity. J. Anim. Ecol. 71, 362–371 (2002). 24. Wootton, J. T. & Emmerson, M. Measurement of interaction strength in nature. Annu. Rev. Ecol. Evol. Syst. 36, 419–444 (2005). Supplementary Information is linked to the online version of the paper at www.nature.com/nature. Acknowledgements We thank J. Bergelson, L.-F. Bersier, A. M. de Roos, A. Eklof, C. A. Klausmeier, S. P. Lalley, R. M. May, K. S. McCann, M. Novak, P. P. A. Staniczenko and J. D. Yeakel for comments and discussion. This research was supported by National Science Foundation grant EF0827493. Author Contributions All authors contributed equally. Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of this article at www.nature.com/nature. Correspondence and requests for materials should be addressed to S.A. (sallesina@uchicago.edu). RESEARCH LETTER 208 | NATURE | VOL 483 | 8 MARCH 2012 ©2012 Macmillan Publishers Limited. All rights reserved