Nonlinearity-generated Resilience in Large Complex Systems

TL;DR

This paper extends May's model to include nonlinear terms, revealing a resilience gap around stable fixed points and showing how systems near tipping points lose resilience, with fixed points proliferating beyond a certain radius.

Contribution

It introduces a nonlinear extension of May's model with higher-order terms, uncovering a resilience gap and the exponential growth of fixed points near equilibrium.

Findings

Resilience gap exists around stable fixed points.

Radius r* vanishes at stability loss threshold.

Number of fixed points grows exponentially beyond r*.

Abstract

We consider a generic nonlinear extension of May's 1972 model by including all higher-order terms in the expansion around the chosen fixed point (placed at the origin) with random Gaussian coefficients. The ensuing analysis reveals that as long as the origin remains stable, it is surrounded by a "resilience gap": there are no other fixed points within a radius r*>0 and the system is therefore expected to be resilient to a typical initial displacement small in comparison to r*. The radius r* is shown to vanish at the same threshold where the origin loses local stability, revealing a mechanism by which systems close to the tipping point become less resilient. We also find that beyond the resilience radius the number of fixed points in a ball surrounding the original point of equilibrium grows exponentially with N, making systems dynamics highly sensitive to far enough displacements from the origin.