Resilience (mathematics)

In mathematical modeling, resilience refers to the ability of a dynamical system to recover from perturbations and return to its original stable steady state.

The concept of resilience is particularly useful in systems that exhibit tipping points, whose study has a long history that can be traced back to catastrophe theory.

According to Holling, resilience is "a measure of the persistence of systems and of their ability to absorb change and disturbance and still maintain the same relationships between populations or state variables".

[4] Engineering resilience refers to the ability of a system to return to its original state after a disturbance, such as a bridge that can be repaired after an earthquake.

With time, the once well-defined and unambiguous concept of resilience has experienced a gradual erosion of its clarity, becoming more vague and closer to an umbrella term than a specific concrete measure.

[9] In ecology, resilience might refer to the ability of the ecosystem to recover from disturbances such as fires, droughts, or the introduction of invasive species.

[10] The exact definition of resilience has remained vague for practical matters, which has led to a slow and proper application of its insights for management of ecosystems.

Some stable systems exhibit critical slowing down where, as they approach a basic reproduction number of 1, their resilience decreases, hence taking a longer time to return to the disease-free steady state.

Ball-and-valley analogy of resilience. A resilient steady state can be thought of as a ball in a deep valley, and it takes a large perturbation to move the ball away to an alternative steady state. A steady state that is less resilient can be thought of as a ball in a shallow valley, where smaller perturbations can be enough to make the system unstable.