Bounds on Transient Instability for Complex Ecosystems

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.

Environmentally‐induced noise dampens and reddens with increasing trophic level in a complex food web

“…Our results illustrate how successive predator-prey interactions stabilize food webs by reducing both the magnitude and temporal variability of species. Increased reddening from the bottom to the top of the food web also implies increased predictability (positive autocorrelation) in the transient dynamics of the population (Caravelli and Staniczenko 2016) which, in the present study, emerge in response to yearly changes in the ecosystem productivity. This suggests that, besides maintaining single species of high conservation value, it is also important to maintain entire food webs including the long food chains on which such species depend.…”

“…Whilst our analysis focussed on only one – albeit empirically well founded – aquatic food web, it sets the stage for the potential generalization of our results by studying a broader range of food webs. Such generality may be expected given the low reactivity of the predator–prey interaction networks to perturbations compared to mutualistic and competitive networks (Caravelli and Staniczenko ).…”

“…Whilst our analysis focussed on only one -albeit empirically well founded -aquatic food web, it sets the stage for the potential generalization of our results by studying a broader range of food webs. Such generality may be expected given the low reactivity of the predator-prey interaction networks to perturbations compared to mutualistic and competitive networks (Caravelli and Staniczenko 2016). Most previous studies investigating the role of coloured environmental noise on species' dynamics have focussed on a single species and explored how environmental noise introduced into the demographic processes alters the fluctuations in the population dynamics (Kaitala et al 1997, Reuman et al 2008.…”

Degree heterogeneity and stability of ecological networks

“…Moreover, since we use a model-independent framework and linear stability analysis, our findings are not limited to ecological networks, but instead hold for any system of differential equations resting at an equilibrium point. For example, our results could be relevant in economic [57,58], organizational [59] and biological [60] systems where competition, cooperation and consumer -resource interactions occur.…”

Correction: Bounds on Transient Instability for Complex Ecosystems

“…Caravelli F, Staniczenko PPA (2016) Correction: Bounds on Transient Instability for Complex Ecosystems. PLoS ONE 11(9): e0162430.…”