A hybrid differential game in renewable resources with sliding modes and crossing limit cycles

This paper investigates how ecological thresholds affect strategic exploitation in renewable resource games. We extend the standard single-agent optimal control harvesting model to a non-cooperative differential game with two agents, where the resource follows piecewise-smooth (PWS) dynamics and growth rates change abruptly at a stock threshold. The resulting three-dimensional (3D) state-costate system generates new strategic behaviour absent in smooth models. We identify sliding regions and hybrid crossing limit cycles (HCLCs) as potential open-loop Nash equilibria (OLNE), depending on parameter values. These outcomes correspond to distinct economic regimes: sliding equilibria capture threshold exploitation at fragile steady states, while HCLCs reflect endogenous boom-bust harvesting cycles. In contrast to two-dimensional (2D) optimal-control models where HCLCs cannot be optimal, our results show that strategic interaction in multi-agent settings can endogenously sustain such cycles. The analysis reveals how ecological discontinuities and strategic competition jointly shape resource dynamics, offering new insights into the stability and management of renewable resources.