The feasibility of equilibria in large ecosystems: A primary but neglected concept in the complexity-stability debate

Abstract

The consensus that complexity begets stability in ecosystems was challenged in the seventies, a result recently extended to ecologically-inspired networks. The approaches assume the existence of a feasible equilibrium, i.e. with positive abundances. However, this key assumption has not been tested. We provide analytical results complemented by simulations which show that equilibrium feasibility vanishes in species rich systems. This result leaves us in the uncomfortable situation in which the existence of a feasible equilibrium assumed in local stability criteria is far from granted. We extend our analyses by changing interaction structure and intensity, and find that feasibility and stability is warranted irrespective of species richness with weak interactions. Interestingly, we find that the dynamical behaviour of ecologically inspired architectures is very different and richer than that of unstructured systems. Our results suggest that a general understanding of ecosystem dynamics requires focusing on the interplay between interaction strength and network architecture.

Metacommunity‐scale biodiversity regulation and the self‐organised emergence of macroecological patterns

“…Community assembly models, therefore, converge on dynamic steady states defined by the onset of structural instability. By applying alternative mathematical approaches to studying this phenomenon (Yodzis 1988;Tokita 2004;Rossberg 2013;Dougoud et al 2018;Barbier et al 2018;Galla 2018), structural instability can be interpreted as resulting from the amplification of perturbations through complex indirect interactions in large communities.…”

Predicting Collapse of Complex Ecological Systems: Quantifying the Stability-Complexity Continuum

“…In addition, we derived a metric that uses only measurable quantities and that quantifies the proximity to collapse and level of structural stability. It has been noted in various studies that both feasibility and local stability are important concepts for ecosystems [39,40,43,47] and that real systems tend to grow in size to the point at which they are structurally unstable [48]. By combining resilience, structural stability, and persistence, we show that there exists parameter ranges for which a system can be resilient but structurally unstable, an observation that introduces a previously overlooked phase in parameter space.…”

“…7) might indicate. Inherently it also elucidates system response behaviour beyond the feasibility region, in addition to the feasibility investigation in [39], with extinctions acting as a stabilising mechanism. In [45] a different way of capturing structural stability is proposed, measured as the size and shape of the feasibility region of a system when varying intrinsic growth rates.…”

“…1 in mind it is clear that assuming feasibility [16,26,31,36,44] amounts to incorrect predictions of collapse since it ignores single-species extinctions as a stabilising mechanism (upholding resilience). Or stated differently, for random systems the parameter region of feasibility is a subset of the region of resilience as noted by [43] for competitive/mutualistic systems, [39] including cascade and niche structured food-webs and [45] showing the universality of the feasibility region. The boundary May introduced states that a random system is stable when σ < 1/ √ cN which implies that fixed-points are unstable for σ > 1/ √ cN .…”

Positive Solutions for Large Random Linear Systems

“…At the equilibrium dxn dt = 0, the abundance vector x n is solution of (1.1) and a key issue is the existence of a feasible solution, that is a solution x n where all the x k 's are positive. Dougoud et al [5], based on Geman and Hwang [8], proved that a feasible solution is very unlikely to exist if α n ≡ α is a constant. In fact, the CLT proved in [8] asserts that for any fixed number M of components…”