Many complex systems in Nature, from metabolic networks to ecosystems, appear to be poised at the edge of stability, hence displaying enormous responses to external perturbations. This feature, also known in physics as marginal stability, is often the consequence of a complex underlying interaction network, which can induce large-scale collective dynamics and critical behaviours.
In this seminar, I will discuss a reference model in theoretical ecology, the high-dimensional Lotka-Volterra model with random interactions and finite demographic noise.
By using techniques rooted in mean-field spin-glass theory, I will show how to obtain a complete characterisation of the phase diagrams. Notably, I will relate emergent collective behaviours and slow relaxation dynamics to the appearance of different disordered phases akin to those occurring in glassy systems in the low-temperature regime .
Finally, I will discuss the wide applicability of this framework to obtain predictions both in the case of weakly asymmetric interactions and with higher-order potentials in the dynamics of the species abundances , which turn out to be useful to model cooperative effects in ecological and biological communities.
 A. Altieri, F. Roy, C. Cammarota, G. Biroli, arXiv: 2009.10565 (2020);
 Altieri, G. Biroli, to be submitted (2021).
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