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Theoretical Ecology

, Volume 9, Issue 4, pp 477–485 | Cite as

The duality of stability: towards a stochastic theory of species interactions

  • Gabriel GellnerEmail author
  • Kevin S. McCann
  • Alan Hastings
ORIGINAL PAPER

Abstract

Understanding the dynamics of ecological systems using stability concepts has been a key driver in ecological research from the inception of the field. Despite the tremendous effort put into this area, progress has been limited due to the bewildering number of metrics used to describe ecological stability. Here, we seek to resolve some of the confusion by unfolding the dynamics of a simple consumer-resource interaction module. In what follows, we first review common dynamical metrics of stability (CV, eigenvalues). We argue using the classical type II consumer-resource model as an example where the empirical stability metric, CV, hides two different, but important, aspects of stability: (i) stability due to mean population density processes and (ii) stability due to population density variance processes. We then employ a simple stochastic consumer-resource framework in order to elucidate (i) when we expect these two different aspects of stability to arise in ecological systems and, importantly, highlight (ii) the fact that these two different aspects of stability respond differentially, but predictably, to changes in fundamental parameters that govern biomass flux and loss in any consumer-resource interaction (e.g., attack rates, carrying capacity, mortality).

Keywords

Stability Stochasticity Transients Consumer-Resource Dynamics 

References

  1. Arnold L (1998) Random dynamical systems. Springer, BerlinCrossRefGoogle Scholar
  2. Arnold L (2001) Recent progress in stochastic bifurcation theory. IUTAM Symposium Nonlinearity Stochastic Structural Dynam Solid Mech Appl 85:15–27CrossRefGoogle Scholar
  3. Boettiger C, Ross N, Hastings A (2013) Early warning signals: the charted and uncharted territories. Theor Ecol 6:255–264CrossRefGoogle Scholar
  4. Carpenter SR (2006) Rising variance: a leading indicator of ecological transition. Ecol Lett 9:308–315Google Scholar
  5. Chesson PL, Ellner S (1989) Invasibility and stochastic boundedness in monotonic competition models. J Math Biol 27:117–138CrossRefGoogle Scholar
  6. Chisholm RA, Filotas E (2009) Critical slowing down as an indicator of transitions in two-species models. J Theor Biol 257:142–149CrossRefPubMedGoogle Scholar
  7. Cohen JE (1995) Unexpected dominance of high frequencies in chaotic nonlinear population models. Nature 378:610–612CrossRefPubMedGoogle Scholar
  8. Drake JM, Griffen BD (2010) Early warning signals of extinction in deteriorating environments. Nature 467:456–459CrossRefPubMedGoogle Scholar
  9. Emmerson MC, Yearsley JM (2004) Weak interactions, omnivory and emergent food-web properties. Proceedings. Biol Sci / Royal Soc 271:397–405CrossRefGoogle Scholar
  10. Gellner G, McCann KS (2016) Consistent Role of Weak and Strong Interactions in High- and Low-Diversity Trophic Food Webs. Nat Commun 7(11180). doi: 10.1038/ncomms11180
  11. Goodman D (1975) The theory of diversity-stability relationships in ecology. Q Rev Biol 50:237–266CrossRefGoogle Scholar
  12. Grimm V, Wissel C (1997) Babel, or the ecological stability discussions: an inventory and analysis of terminology and a guide for avoiding confusion. Oecologia 109:323–334CrossRefGoogle Scholar
  13. Hildenbrandt H, Müller MS, Grimm V (2006) How to detect and visualize extinction thresholds for structured PVA models. Ecol Model 191:545–550CrossRefGoogle Scholar
  14. Ives AR, Carpenter SR (2007) Stability and diversity of ecosystems. Science 317:58–62CrossRefPubMedGoogle Scholar
  15. King AA, Schaffer WM (1999) The rainbow bridge: Hamiltonian limits and resonance in predator-prey dynamics. J Math Biol 39:439–469CrossRefPubMedGoogle Scholar
  16. Lande R (1994) Risk of population extinction from fixation of new deleterious mutations. Evolution 48:1460–1469CrossRefGoogle Scholar
  17. McCann KS (2000) The diversity-stability debate. Nature 405:228–233CrossRefPubMedGoogle Scholar
  18. McCann KS (2011) Food webs. Princeton University Press, PrincetonGoogle Scholar
  19. McCann KS, Hastings A, Huxel GR (1998) Weak trophic interactions and the balance of nature. Nature 395:794–798CrossRefGoogle Scholar
  20. Murdoch WM, Briggs CJ, Nisbet RM (2003) Consumer-resource dynamics. Princeton University Press, PrincetonGoogle Scholar
  21. Neutel A-M, Heesterbeek JAP, De Ruiter PC (2002) Stability in real food webs: weak links in long loops. Science (New York, NY) 296:1120–1123CrossRefGoogle Scholar
  22. Neutel A-M, Heesterbeek JAP, 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
  23. Nunney L, Campbell KA (1993) Assessing minimum viable population size: demography meets population genetics. Trends Ecol Evol 8:234–239CrossRefPubMedGoogle Scholar
  24. Olarrea J, de la Rubia FJ (1996) Stochastic Hopf bifurcation: the effect of colored noise on the bifurcaton interval. Phys Rev 53:268–271CrossRefGoogle Scholar
  25. Pimm SL (1984) The complexity and stability of ecosystems. Nature 307:321–326CrossRefGoogle Scholar
  26. Pineda-Krch M, Blok HJ, Dieckmann U, Doebeli M (2007) A tale of two cycles—distinguishing quasi-cycles and limit cycles in finite predator-prey populations. Oikos 116:53–64CrossRefGoogle Scholar
  27. Veraart AJ, Faassen EJ, Dakos V, van Nes EH, Lürling M, Scheffer M (2012) Corrigendum: recovery rates reflect distance to a tipping point in a living system. Nature 484:404–404CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Gabriel Gellner
    • 1
    Email author
  • Kevin S. McCann
    • 2
  • Alan Hastings
    • 1
  1. 1.Department of Environmental Science and PolicyUniversity of California DavisDavisUSA
  2. 2.Department of Integrative BiologyUniversity of GuelphGuelphCanada

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