Near critical interior dynamics in a model of state society interaction

Abstract

We study a normalized two dimensional competitive Lotka$-$Volterra system describing the interaction between state power and society power. Restricting attention to the positively invariant domain $[0,1]^2$, the analysis focuses on interior equilibrium dynamics where coexistence persists as the unique long run outcome. We show that when the interaction parameters approach the coexistence threshold $a_{12}a_{21}\to1^{-}$, convergence toward equilibrium becomes slow and trajectories exhibit prolonged transient dynamics. In this near critical regime, trajectories organize into a narrow corridor around the balance manifold despite the absence of bistability. The corridor structure can be characterized quantitatively through equilibrium gaps and interaction thresholds. Numerical simulations illustrate how asymmetric adjustment influences the geometry and persistence of these transient regimes.