Adaptive Capacity, Structural Compression, and Collapse: A Minimal Dynamical Formalization of Tainter, Holling, and Scheffer

## Description The collapse of complex adaptive systems has been widely studied across disciplines, including anthropology, ecology, and nonlinear dynamics. Influential frameworks such as Joseph Tainter’s theory of diminishing returns to complexity, C. S. Holling’s adaptive cycle (Panarchy), and Marten Scheffer’s work on tipping points all converge on a shared structural intuition: systems become increasingly fragile when adaptive capacity is eroded relative to accumulated structural constraints. Despite this conceptual convergence, these traditions have largely developed in parallel, without a shared low-dimensional dynamical formalization that allows direct comparison, parameterization, and falsifiability. This paper proposes a minimal phenomenological model that formalizes this shared intuition as a coupled system of ordinary differential equations (ODEs). The model describes the interaction between adaptive capacity (R) and structural compression (Φ), and introduces a viability ratio T = R/Φ as a scalar indicator of systemic resilience. The resulting dynamics exhibit bistability and separatrix geometry, consistent with the qualitative structure of critical transitions observed in empirical systems. Rather than presenting a new theory, this work positions itself as a dynamical bridge between established collapse frameworks. It provides a compact analytical representation that is compatible with:- Tainter’s diminishing returns to complexity (increasing Φ)- Holling’s conservation phase in the adaptive cycle (high Φ, low R)- Scheffer’s tipping points (separatrix crossing and critical thresholds) The model is intentionally minimal and phenomenological. It does not aim to capture domain-specific mechanisms in full detail, but instead offers a normal-form representation of a recurring dynamical structure in complex systems. Potential empirical operationalizations are outlined, including proxies for structural compression (e.g., network coupling, correlation structure, effective dimensionality) and adaptive capacity (e.g., recovery rates, information flow, response diversity). The framework is designed to be testable using established early-warning signal methodologies. This preprint represents an initial formalization stage. Empirical validation, parameter estimation, and cross-domain testing are proposed as future work. collapse dynamicscomplex systemsadaptive capacitystructural compressionbistabilitycritical transitionstipping pointspanarchydynamical systemsearly warning signalsresiliencenonlinear dynamicsphenomenological modelODE systemssystems theory