  This paper introduces a novel approach to quantifying ecological resilience
in biological systems, particularly focusing on noisy systems responding to
episodic disturbances with sudden adaptations. Incorporating concepts from
non-equilibrium statistical mechanics, we propose a measure termed `ecological
resilience through adaptation,' specifically tailored to noisy, forced systems
that undergo physiological adaptation in the face of stressful environmental
changes. Randomness plays a key role, accounting for model uncertainty and the
inherent variability in the dynamical response among components of biological
systems. Our measure of resilience is rooted in the probabilistic description
of states within these systems, and is defined in terms of the dynamics of the
ensemble average of a model-specific observable quantifying success or
well-being. Our approach utilizes stochastic linear response theory to compute
how the expected success of a system, originally in statistical equilibrium,
dynamically changes in response to a environmental perturbation and a
subsequent adaptation. The resulting mathematical derivations allow for the
estimation of resilience in terms of ensemble averages of simulated or
experimental data. Finally, through a simple but clear conceptual example, we
illustrate how our resilience measure can be interpreted and compared to other
existing frameworks in the literature. The methodology is general but inspired
by applications in plant systems, with the potential for broader application to
complex biological processes.
