Near critical interior dynamics in a model of state society interaction

TL;DR

This paper analyzes a two-dimensional Lotka-Volterra model of state-society interaction, revealing slow convergence and prolonged transients near the coexistence threshold, with implications for understanding dynamic balance in political systems.

Contribution

It introduces a detailed analysis of interior equilibrium dynamics near critical thresholds in a state-society interaction model, highlighting transient behaviors and corridor structures.

Findings

Convergence slows as parameters approach the coexistence threshold.

Trajectories form a narrow corridor around the balance manifold.

Asymmetric adjustments affect transient regime geometry.

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.