Bridging Robustness and Resilience for Dynamical Systems in Nature

Biological systems have evolved to maintain properties that are crucial for survival. Robustness and resilience are associated with a system's ability to preserve its functions despite uncertainties, fluctuations and perturbations, both intrinsic and extrinsic. However, due to the multidisciplinary nature of the research topic, numerous competing definitions of these concepts coexist and often lack a rigorous control-theoretic formulation. Here, we consider a family of ODE systems consisting of stochastic perturbations of a nominal deterministic system and we introduce possible formal definitions of resilience of such a family of systems aimed at probabilistically quantifying its ability to preserve a prescribed attractor. We show that our proposed definitions generalise the notion of probabilistic robustness, and we demonstrate their efficacy when applied to widely used models in biology.