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 reservedWe 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 increas￾ing 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 Informa￾tion). 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 relation￾ships 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 food￾web 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