Sublinear Hierarchical Dynamical System with Cyclic Coupling

We introduce a mathematical model describing a dynamical system made up of interdependent compartments subject to intrinsic fragility and cross-support. The dynamics are driven by the competition between hierarchical sublinear dissipation and cyclic coupling, enabling a natural interpretation of transitions between distinct regimes (e.g., resilience versus degradation, vulnerability and crisis/extinction in finance or biology). We establish the existence, uniqueness, positivity, and global continuation of solutions. We also investigate the qualitative behavior of the system by studying the stability of equilibrium points and deriving threshold conditions characterizing increasing, decreasing, and stationary regimes. Numerical simulations are provided to illustrate the theoretical results and the transition phenomena between extinction and self-sustained dynamics.