Skip to main content
SpringerLink
Log in

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

  • Special Feature: Review
  • Unravelling ecological networks
  • Published:
Population Ecology

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Allesina S, Pascual M (2008) Network structure, predator–prey modules, and stability in large food webs. Theor Ecol 1:55–64

    Article  Google Scholar 

  • Allesina S, Pascual M (2009) Food web models: a plea for groups. Ecol Lett 12:652–662

    Article  PubMed  Google Scholar 

  • Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483:205–208

    Article  CAS  PubMed  Google Scholar 

  • Allesina S, Alonso D, Pascual M (2008) A general model for food web structure. Science 320:658–661

    Article  CAS  PubMed  Google Scholar 

  • Anderson GW, Guionnet A, Zeitouni O (2010) An introduction to random matrices. Cambridge University Press, Cambridge

    Google Scholar 

  • 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 33–42

  • Bai Z (1997) Circular law. Ann Probab 25:494–529

    Article  Google Scholar 

  • Bai Z, Silverstein JW (2009) Spectral analysis of large dimensional random matrices. Springer, New York

    Google Scholar 

  • Dunne JA, Williams RJ, Martinez ND (2002) Food-web structure and network theory: the role of connectance and size. Proc Natl Acad Sci USA 99:12917–12922

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Gardner MR, Ashby WR (1970) Connectance of large dynamic (cybernetic) systems: critical values for stability. Nature 228:784

    Article  CAS  PubMed  Google Scholar 

  • Ginibre J (1965) Statistical ensembles of complex, quaternion, and real matrices. J Math Phys 6:440–449

    Article  Google Scholar 

  • Girko VL (1985) Circular law. Theor Probab Appl 29(4):694–706

    Article  Google Scholar 

  • Girko VL (1986) Elliptic law. Theor Probab Appl 30(4):677–690

    Article  Google Scholar 

  • Hanski I, Ovaskainen O (2000) The metapopulation capacity of a fragmented landscape. Nature 404:755–758

    Article  CAS  PubMed  Google Scholar 

  • Hastings A (2001) Transient dynamics and persistence of ecological systems. Ecol Lett 4:215–220

    Article  Google Scholar 

  • Hiai F, Petz D (2000) The semicircle law, free random variables and entropy, vol 77. American Mathematical Society, Providence

    Google Scholar 

  • Kondoh M (2003) Foraging adaptation and the relationship between food-web complexity and stability. Science 299:1388–1391

    Article  CAS  PubMed  Google Scholar 

  • Levins R (1968) Evolution in changing environments: some theoretical explorations. Princeton University Press, Princeton

    Google Scholar 

  • Magurran AE, Henderson PA (2003) Explaining the excess of rare species in natural species abundance distributions. Nature 422:714–716

    Article  CAS  PubMed  Google Scholar 

  • May RM (1972) Will a large complex system be stable? Nature 238:413–414

    Article  CAS  PubMed  Google Scholar 

  • May RM (2001) Stability and complexity in model ecosystems. Princeton University Press, Princeton

    Google Scholar 

  • McCann KS (2000) The diversity–stability debate. Nature 405:228–233

    Article  CAS  PubMed  Google Scholar 

  • McCann KS, Hastings A, Huxel GR (1998) Weak trophic interactions and the balance of nature. Nature 395:794–798

    Article  CAS  Google Scholar 

  • Metha M (1967) Random matrices and the statistical theory of energy levels. Academic, New York

    Google Scholar 

  • Moore JC, Hunt HW (1988) Resource compartmentation and the stability of real ecosystems. Nature 333:261–263

    Article  Google Scholar 

  • Naumov A (2012) Elliptic law for real random matrices. arXiv:1201.1639

  • Neubert MG, Caswell H (1997) Alternatives to resilience for measuring the responses of ecological systems to perturbations. Ecology 78:653–665

    Article  Google Scholar 

  • 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 449:599–602

    Article  CAS  PubMed  Google Scholar 

  • Nguyen H, O’Rourke S (2012) The elliptic law. arXiv:1208.5883

  • Pimm SL (1979) The structure of food webs. Theor Popul Biol 16:144–158

    Article  CAS  PubMed  Google Scholar 

  • Pimm SL (1984) The complexity and stability of ecosystems. Nature 307:321–326

    Article  Google Scholar 

  • Pimm SL, Lawton JH, Cohen JE (1991) Food web patterns and their consequences. Nature 350:669–674

    Article  Google Scholar 

  • Roberts A (1974) The stability of a feasible random ecosystem. Nature 251:607–608

    Article  Google Scholar 

  • 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):902

    Google Scholar 

  • Solé RV, Alonso D, McKane A (2002) Self-organized instability in complex ecosystems. Philos Trans R Soc B-Biol Sci 357:667–681

    Article  Google Scholar 

  • Sommers H, Crisanti A, Sompolinsky H, Stein Y (1988) Spectrum of large random asymmetric matrices. Phys Rev Lett 60:1895

    Article  PubMed  Google Scholar 

  • Stouffer DB, Bascompte J (2011) Compartmentalization increases food-web persistence. Proc Natl Acad Sci USA 108:3648–3652

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Stouffer DB, Camacho J, Amaral LAN (2006) A robust measure of food web intervality. Proc Natl Acad Sci USA 103:19015–19020

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Tang S, Allesina S (2014) Reactivity and stability of large ecosystems. Front Ecol Evol 2:21

    Article  Google Scholar 

  • Tang S, Pawar S, Allesina S (2014) Correlation between interaction strengths drives stability in large ecological networks. Ecol Lett 17:1094–1100

    Article  Google Scholar 

  • Tao T, Vu V, Krishnapur M (2010) Random matrices: universality of ESDs and the circular law. Ann Probab 38:2023–2065

    Article  Google Scholar 

  • Van Mieghem P, Cator E (2012) Epidemics in networks with nodal self-infection and the epidemic threshold. Phys Rev E 86(016):116

    Google Scholar 

  • 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 25–34

  • Wigner EP (1958) On the distribution of the roots of certain symmetric matrices. Ann Math 67:325–327

    Article  Google Scholar 

  • Williams RJ, Martinez ND (2000) Simple rules yield complex food webs. Nature 404:180–183

    Article  CAS  PubMed  Google Scholar 

  • Wood PM (2012) Universality and the circular law for sparse random matrices. Ann Appl Probab 22:1266–1300

    Article  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Department of Ecology and Evolution, University of Chicago, 1101 E. 57th, Chicago, IL, 60637, USA

    Stefano Allesina & Si Tang

  2. Computation Institute, University of Chicago, Chicago, USA

    Stefano Allesina

Authors
  1. Stefano Allesina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Si Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefano Allesina.

Additional information

This manuscript was submitted for the special feature based on a symposium in Osaka, Japan, held on 12 October 2013.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allesina, S., Tang, S. The stability–complexity relationship at age 40: a random matrix perspective. Popul Ecol 57, 63–75 (2015). https://doi.org/10.1007/s10144-014-0471-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10144-014-0471-0

Keywords

Search

Navigation