Sudden collapse in complex systems can arise without prolonged visible deterioration, challenging traditional monitoring approaches based on gradual trends. We develop minimal stochastic and network-based models to investigate how rare shocks, finite capacity constraints, and structural interdependence interact to produce abrupt population decline. Simulations show that systems may remain stable for extended periods before experiencing rapid, nonlinear collapse once capacity thresholds are exceeded. To characterize proximity to such transitions, we introduce a response-based metric, the Shock Sensitivity Indicator (SSI), which quantifies system damage per unit applied shock. Normalization relative to a stable reference regime yields the Normalized Shock Sensitivity Ratio (NSSR), a dimensionless measure of fragility. In stochastic simulations, persistent elevation of NSSR precedes collapse even when aggregate state variables remain approximately stable, demonstrating the limitations of trend-based early-warning indicators in shock-dominated regimes. Phase diagrams reveal sharp boundaries separating stable and high-risk regimes, governed by the balance between expected shock load and effective system capacity. These results provide a general framework for diagnosing fragility in capacity-limited systems subject to stochastic stress. Although motivated by population and health-system contexts, the framework applies broadly to ecological, infrastructural, and other complex adaptive systems.
Roshan (Sat,) studied this question.