Mapping the distinct origins of bimodality in a classic model with alternative stable states


Stochastic ecological dynamics result from both transient and asymptotic features of the underlying system, yet explanations for observed patterns often emphasize asymptotics. For example, an ecological state (e.g., a particular population size or community composition) that occurs frequently and/or persists for a meaningful duration might be assumed to be a stable equilibrium, even though transients can also persist for a long time and may recur. In this paper, we consider one particular pattern—a bimodal distribution of states as a system is observed through time—and consider alternative causes for this pattern. First, we consider the “asymptotic” explanation that each mode corresponds to a distinct stable state. Second, we consider that one mode might correspond to a long transient. We explore the dynamics that result from each of these causes in a classic bistable model, focusing particularly on the degree of environmental stochasticity needed to generate a bimodal distribution of states in each case. Our results highlight that observations of a system’s dynamics do not provide enough information to determine the number and location of stable states. We conclude that a more serious and systematic consideration for the possible role of transients in driving observed dynamics will lead to stronger insights and understanding.

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We thank members of the Abbott lab and the CWRU E&E journal club for helpful feedback on an early drafts. Two anonymous reviewers made excellent recommendations that improved this article. We thank K. Cuddington, G. Gellner, T. Gross, A. Hastings, K. McCann, T. Rogers, and S. Schreiber for thought-provoking discussions about the existence of a “sweet spot” for stochastic dynamics.


KCA was supported by James S. McDonnell Foundation grant 220020364 and NSF grant DMS-1840221. The authors conceived and discussed the ideas in this paper during workshops funded by the Mathematical Biosciences Institute at Ohio State University (NSF-DMS 1440386) and the Banff International Research Station for Mathematical Innovation and Discovery.

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Correspondence to Karen C. Abbott.

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Abbott, K.C., Dakos, V. Mapping the distinct origins of bimodality in a classic model with alternative stable states. Theor Ecol (2020).

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  • Alternative stable states
  • Environmental stochasticity
  • Lotka-Volterra competition model
  • Quasi-potential
  • Unstable equilibrium