Population Ecology

, Volume 57, Issue 1, pp 63–75 | Cite as

The stability–complexity relationship at age 40: a random matrix perspective

Special Feature: Review Unravelling ecological networks


Since the work of Robert May in 1972, the local asymptotic stability of large ecological systems has been a focus of theoretical ecology. Here we review May’s work in the light of random matrix theory, the field of mathematics devoted to the study of large matrices whose coefficients are randomly sampled from distributions with given characteristics. We show how May’s celebrated “stability criterion” can be derived using random matrix theory, and how extensions of the so-called circular law for the limiting distribution of the eigenvalues of large random matrix can further our understanding of ecological systems. Our goal is to present the more technical material in an accessible way, and to provide pointers to the primary mathematical literature on this subject. We conclude by enumerating a number of challenges, whose solution is going to greatly improve our ability to predict the stability of large ecological networks.


Complexity Eigenvalue Food web Random matrix Stability 



SA and ST funded by NSF #1148867. Thanks to G. Barabás for comments. D. Gravel and an anonymous reviewer provided valuable suggestions.


  1. Allesina S, Pascual M (2008) Network structure, predator–prey modules, and stability in large food webs. Theor Ecol 1:55–64CrossRefGoogle Scholar
  2. Allesina S, Pascual M (2009) Food web models: a plea for groups. Ecol Lett 12:652–662CrossRefPubMedGoogle Scholar
  3. Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483:205–208CrossRefPubMedGoogle Scholar
  4. Allesina S, Alonso D, Pascual M (2008) A general model for food web structure. Science 320:658–661CrossRefPubMedGoogle Scholar
  5. Anderson GW, Guionnet A, Zeitouni O (2010) An introduction to random matrices. Cambridge University Press, CambridgeGoogle Scholar
  6. Backstrom L, Boldi P, Rosa M, Ugander J, Vigna S (2012) Four degrees of separation. In: Proceedings of the 3rd annual ACM web science conference. ACM, New York, pp 33–42Google Scholar
  7. Bai Z (1997) Circular law. Ann Probab 25:494–529CrossRefGoogle Scholar
  8. Bai Z, Silverstein JW (2009) Spectral analysis of large dimensional random matrices. Springer, New YorkGoogle Scholar
  9. Dunne JA, Williams RJ, Martinez ND (2002) Food-web structure and network theory: the role of connectance and size. Proc Natl Acad Sci USA 99:12917–12922CrossRefPubMedCentralPubMedGoogle Scholar
  10. Gardner MR, Ashby WR (1970) Connectance of large dynamic (cybernetic) systems: critical values for stability. Nature 228:784CrossRefPubMedGoogle Scholar
  11. Ginibre J (1965) Statistical ensembles of complex, quaternion, and real matrices. J Math Phys 6:440–449CrossRefGoogle Scholar
  12. Girko VL (1985) Circular law. Theor Probab Appl 29(4):694–706CrossRefGoogle Scholar
  13. Girko VL (1986) Elliptic law. Theor Probab Appl 30(4):677–690CrossRefGoogle Scholar
  14. Hanski I, Ovaskainen O (2000) The metapopulation capacity of a fragmented landscape. Nature 404:755–758CrossRefPubMedGoogle Scholar
  15. Hastings A (2001) Transient dynamics and persistence of ecological systems. Ecol Lett 4:215–220CrossRefGoogle Scholar
  16. Hiai F, Petz D (2000) The semicircle law, free random variables and entropy, vol 77. American Mathematical Society, ProvidenceGoogle Scholar
  17. Kondoh M (2003) Foraging adaptation and the relationship between food-web complexity and stability. Science 299:1388–1391CrossRefPubMedGoogle Scholar
  18. Levins R (1968) Evolution in changing environments: some theoretical explorations. Princeton University Press, PrincetonGoogle Scholar
  19. Magurran AE, Henderson PA (2003) Explaining the excess of rare species in natural species abundance distributions. Nature 422:714–716CrossRefPubMedGoogle Scholar
  20. May RM (1972) Will a large complex system be stable? Nature 238:413–414CrossRefPubMedGoogle Scholar
  21. May RM (2001) Stability and complexity in model ecosystems. Princeton University Press, PrincetonGoogle Scholar
  22. McCann KS (2000) The diversity–stability debate. Nature 405:228–233CrossRefPubMedGoogle Scholar
  23. McCann KS, Hastings A, Huxel GR (1998) Weak trophic interactions and the balance of nature. Nature 395:794–798CrossRefGoogle Scholar
  24. Metha M (1967) Random matrices and the statistical theory of energy levels. Academic, New YorkGoogle Scholar
  25. Moore JC, Hunt HW (1988) Resource compartmentation and the stability of real ecosystems. Nature 333:261–263CrossRefGoogle Scholar
  26. Naumov A (2012) Elliptic law for real random matrices. arXiv:1201.1639
  27. Neubert MG, Caswell H (1997) Alternatives to resilience for measuring the responses of ecological systems to perturbations. Ecology 78:653–665CrossRefGoogle Scholar
  28. Neutel AM, Heesterbeek JA, van de Koppel J, Hoenderboom G, Vos A, Kaldeway C, Berendse F, de Ruiter PC (2007) Reconciling complexity with stability in naturally assembling food webs. Nature 449:599–602CrossRefPubMedGoogle Scholar
  29. Nguyen H, O’Rourke S (2012) The elliptic law. arXiv:1208.5883
  30. Pimm SL (1979) The structure of food webs. Theor Popul Biol 16:144–158CrossRefPubMedGoogle Scholar
  31. Pimm SL (1984) The complexity and stability of ecosystems. Nature 307:321–326CrossRefGoogle Scholar
  32. Pimm SL, Lawton JH, Cohen JE (1991) Food web patterns and their consequences. Nature 350:669–674CrossRefGoogle Scholar
  33. Roberts A (1974) The stability of a feasible random ecosystem. Nature 251:607–608CrossRefGoogle Scholar
  34. Sinha S, Sinha S (2005) Evidence of universality for the May–Wigner stability theorem for random networks with local dynamics. Phys Rev E 71(020):902Google Scholar
  35. Solé RV, Alonso D, McKane A (2002) Self-organized instability in complex ecosystems. Philos Trans R Soc B-Biol Sci 357:667–681CrossRefGoogle Scholar
  36. Sommers H, Crisanti A, Sompolinsky H, Stein Y (1988) Spectrum of large random asymmetric matrices. Phys Rev Lett 60:1895CrossRefPubMedGoogle Scholar
  37. Stouffer DB, Bascompte J (2011) Compartmentalization increases food-web persistence. Proc Natl Acad Sci USA 108:3648–3652CrossRefPubMedCentralPubMedGoogle Scholar
  38. Stouffer DB, Camacho J, Amaral LAN (2006) A robust measure of food web intervality. Proc Natl Acad Sci USA 103:19015–19020CrossRefPubMedCentralPubMedGoogle Scholar
  39. Tang S, Allesina S (2014) Reactivity and stability of large ecosystems. Front Ecol Evol 2:21CrossRefGoogle Scholar
  40. Tang S, Pawar S, Allesina S (2014) Correlation between interaction strengths drives stability in large ecological networks. Ecol Lett 17:1094–1100CrossRefGoogle Scholar
  41. Tao T, Vu V, Krishnapur M (2010) Random matrices: universality of ESDs and the circular law. Ann Probab 38:2023–2065CrossRefGoogle Scholar
  42. Van Mieghem P, Cator E (2012) Epidemics in networks with nodal self-infection and the epidemic threshold. Phys Rev E 86(016):116Google Scholar
  43. Wang Y, Chakrabarti D, Wang C, Faloutsos C (2003) Epidemic spreading in real networks: an eigenvalue viewpoint. In: Proceedings of 22nd international symposium on reliable distributed systems. IEEE, New York, pp 25–34Google Scholar
  44. Wigner EP (1958) On the distribution of the roots of certain symmetric matrices. Ann Math 67:325–327CrossRefGoogle Scholar
  45. Williams RJ, Martinez ND (2000) Simple rules yield complex food webs. Nature 404:180–183CrossRefPubMedGoogle Scholar
  46. Wood PM (2012) Universality and the circular law for sparse random matrices. Ann Appl Probab 22:1266–1300CrossRefGoogle Scholar

Copyright information

© The Society of Population Ecology and Springer Japan 2015

Authors and Affiliations

  1. 1.Department of Ecology and EvolutionUniversity of ChicagoChicagoUSA
  2. 2.Computation InstituteUniversity of ChicagoChicagoUSA

Personalised recommendations