Resilience of dynamical systems in nature: from definitions to indicators

Biological systems have evolved to maintain properties that are crucial for survival, even though they are subject to parametric uncertainty as well as continuously exposed to exogenous disturbances. Robustness and resilience describe a system’s ability to preserve its functions despite intrinsic uncertainties and extrinsic perturbations; however, numerous competing definitions of these concepts coexist and often lack a rigorous control-theoretic formulation.
We consider a family of systems obtained as stochastic perturbations of a nominal deterministic system and we introduce a new notion of resilience, formally defined as a quantification of the probability that the qualitative behaviour of the nominal deterministic system with respect to a prescribed attractor is preserved under stochastic perturbations. We show that this framework naturally generalises the notion of probabilistic robustness, and we demonstrate its efficacy when applied to widely used models in biology. In particular, we focus on a stochastic model of the positive gene autoregulating feedback loop and we probabilistically quantify its resilience, i.e., its ability to preserve the equilibrium associated with a prescribed concentration of transcription factor, and the corresponding basin of attraction, in the presence of noise; for this system, we also explore a complementary definition of resilience that relies on the Fokker-Planck equation.

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