Theoretical Analysis of Stability and Bifurcation in Models of Political Violence Dynamics in North East Nigeria

The insurgency crisis in North East Nigeria, particularly in Borno, Yobe, and Adamawa States, exhibits complex dynamic patterns including cyclical intensification, spatial diffusion, and tipping point transitions between relative stability and open conflict that resist characterisation by linear statistical approaches. This dissertation develops an original theoretical framework using dynamical systems theory and bifurcation analysis to model the evolution of political violence dynamics in the North East. The mathematical model is formulated as a system of nonlinear ordinary differential equations with state variables representing insurgent strength, government counter-insurgency capacity, civilian support, and economic grievance level. Theoretical contributions include a proof of the existence of multiple equilibria corresponding to stable peace, unstable peace, and endemic conflict states, and a characterisation of the saddle-node bifurcation that governs the transition between peace and conflict regions of parameter space. Slow manifold reduction is used to reduce the four-dimensional system to a two-dimensional normal form near the bifurcation point, providing analytically tractable insight into transition dynamics. The model is parameterised using event data from the Armed Conflict Location and Event Data Project for the 2010 to 2023 period in the three states. Empirical estimation using approximate Bayesian computation identifies the grievance-to-violence coupling parameter as the dimension where the system is closest to the bifurcation threshold. Keywords: bifurcation theory, political violence dynamics, North East Nigeria, dynamical systems, insurgency modelling.