Bulletin of Mathematical Biology

, Volume 80, Issue 6, pp 1630–1654 | Cite as

How Stochasticity Influences Leading Indicators of Critical Transitions

  • Suzanne M. O’ReganEmail author
  • Danielle L. Burton
Original Article


Many complex systems exhibit critical transitions. Of considerable interest are bifurcations, small smooth changes in underlying drivers that produce abrupt shifts in system state. Before reaching the bifurcation point, the system gradually loses stability (‘critical slowing down’). Signals of critical slowing down may be detected through measurement of summary statistics, but how extrinsic and intrinsic noises influence statistical patterns prior to a transition is unclear. Here, we consider a range of stochastic models that exhibit transcritical, saddle-node and pitchfork bifurcations. Noise was assumed to be either intrinsic or extrinsic. We derived expressions for the stationary variance, autocorrelation and power spectrum for all cases. Trends in summary statistics signaling the approach of each bifurcation depend on the form of noise. For example, models with intrinsic stochasticity may predict an increase in or a decline in variance as the bifurcation parameter changes, whereas models with extrinsic noise applied additively predict an increase in variance. The ability to classify trends of summary statistics for a broad class of models enhances our understanding of how critical slowing down manifests in complex systems approaching a transition.


Critical transitions Bifurcations Demographic stochasticity Environmental stochasticity Early warning signals Additive noise Multiplicative noise 



Much of this work was conducted while SMO was a Postdoctoral Fellow and DLB was a Graduate Research Assistant at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation through NSF Award # DBI-1300426, with additional support from The University of Tennessee, Knoxville. Additional support was obtained from North Carolina A&T State University. The authors would like to thank John Drake, Eamon O’Dea, Suzanne Lenhart and an anonymous reviewer for thoughtful comments on the manuscript.


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorth Carolina A&T State UniversityGreensboroUSA
  2. 2.National Institute for Mathematical and Biological SynthesisUniversity of TennesseeKnoxvilleUSA
  3. 3.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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