POLITICALLY INSTRUMENTALIZED TERRORISM DYNAMICS (PITD): A STOCHASTIC SPATIOTEMPORAL DIFFERENTIAL GAME FRAMEWORK FOR MODELING ELECTORAL VIOLENCE AND SOCIOECONOMIC DISRUPTION IN NIGERIA

Abstract

The metamorphosis of terrorism into an instrument of political sabotage introduces profound nonlinearities and exogenous stochastic forcing into regional security dynamics. This paper introduces the Politically Instrumentalized Terrorism Dynamics (PITD) model, a complex nonlinear framework formulated as a system of stochastic differential equations (SDEs) coupled with a meta-population spatial network. We investigate the complex dynamical behaviour of the system, proving the existence of a backward (subcritical) bifurcation driven by a nonlinear saturation term in state capacity, which mathematically explains the hysteresis and failure of localized kinetic interventions. Furthermore, we formulate a non-cooperative Nash differential game between an incumbent government and opposing political elites, deriving optimal control strategies via Pontryagin’s Maximum Principle. To capture the chaotic sensitivity of the system to electoral shocks, we introduce a Wiener process representing unpredictable exogenous volatility. Computational validation using an explicit Euler-Maruyama Monte Carlo scheme, calibrated with synthetic geospatial data, demonstrates that the spatial displacement of violence exhibits fractal-like diffusion across a multi-regional network. The stochastic simulations confirm that the Nash equilibrium control strategies remain robust, maintaining bounded 95% confidence trajectories despite high-variance environmental shocks. Our findings establish a rigorous mathematical paradigm for understanding the chaotic transition of organic grievances into politically engineered complex systems.