Bounds on Transient Instability for Complex Ecosystems

PLoS One. 2016 Jun 21;11(6):e0157876. doi: 10.1371/journal.pone.0157876. eCollection 2016.

Abstract

Stability is a desirable property of complex ecosystems. If a community of interacting species is at a stable equilibrium point then it is able to withstand small perturbations to component species' abundances without suffering adverse effects. In ecology, the Jacobian matrix evaluated at an equilibrium point is known as the community matrix, which describes the population dynamics of interacting species. A system's asymptotic short- and long-term behaviour can be determined from eigenvalues derived from the community matrix. Here we use results from the theory of pseudospectra to describe intermediate, transient dynamics. We first recover the established result that the transition from stable to unstable dynamics includes a region of 'transient instability', where the effect of a small perturbation to species' abundances-to the population vector-is amplified before ultimately decaying. Then we show that the shift from stability to transient instability can be affected by uncertainty in, or small changes to, entries in the community matrix, and determine lower and upper bounds to the maximum amplitude of perturbations to the population vector. Of five different types of community matrix, we find that amplification is least severe when predator-prey interactions dominate. This analysis is relevant to other systems whose dynamics can be expressed in terms of the Jacobian matrix.

MeSH terms

  • Ecosystem*
  • Models, Theoretical*

Grants and funding

FC was supported by Invenia Labs Limited. PPAS was supported by an AXA Research Fellowship and British Ecological Society grant 4785/5824. The funder provided support in the form of salary for author FC, but did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. The specific roles of these authors are articulated in the “author contributions” section.