Abstract
The study of local stability has a long tradition in community ecology. Stability describes whether an ecological system will eventually return to its original steady state after being perturbed. More recently, the study of the transient dynamics of ecological systems has been recognized as crucial, given that continuously disturbed systems might never reach a steady state, and thus the instantaneous response to perturbations could largely determine species persistence. A stable equilibrium can be nonreactive -- all perturbations decay immediately, or reactive -- some perturbations are initially amplified before decaying. Here we derive analytical criteria for the reactivity of large ecological systems in which species interact at random. We find that in large ecological systems both stability and reactivity are governed by the same quantities: number of species, means of the intra- and inter-specific interaction strengths, variance of inter-specific interactions, and the correlation of pairwise interactions. We identify two phase transitions, one from nonreactivity to reactivity and one from stability to instability. As reactivity is an intermediate state between nonreactivity and instability, it could be used to develop an early-warning signal for systems approaching instability.
Highlights
The relationship between complexity and stability of ecological communities has been an important driver of theoretical ecology (e.g., Yodzis, 1981; Pimm, 1984; McCann, 2000; May, 2001; Martinez et al, 2005)
Similar to the derivation for the stability criterion, in which the goal is to estimate the largest real part of the eigenvalues of M, the derivation of the reactivity criterion relies on estimating the largest eigenvalue of H when S is sufficiently large
The stability and reactivity criteria are derived for ecological networks with random (i.e., Erdos-Rényi) structure
Summary
The relationship between complexity and stability of ecological communities has been an important driver of theoretical ecology (e.g., Yodzis, 1981; Pimm, 1984; McCann, 2000; May, 2001; Martinez et al, 2005). This interest was sparked by the work of Robert May, who in 1972 showed that large random communities are inevitably unstable (May, 1972). Starting—as May did— directly from the “community matrix” (Levins, 1968) describing the effects of species on each other around an equilibrium point, we derived stability criteria for large, complex ecosystems, and showed that a large number of species can coexist at a locally stable equilibrium if predator-prey interactions are preponderant
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