Ant System (AS) is the first algorithm in the Ant Colony Optimization (ACO) domain to have successfully implemented. But, a little have been put forward about the mathematical analysis of the stochastic model based AS. In this paper, a deterministic solution of the classical Ant dynamics is introduced. A transfer function model is developed and the system characterization is done in frequency domain. It is helpful to explore the system behavior that gives the supportive analysis on the stability of the Ant System. Also we deduce the necessary bounds of the trail persistence q which will control the ant dynamics to avoid over accumulation of pheromone and search for good optimal solution using Region of Convergence (ROC) criterion. Simulation results also present supportive evidence of the analysis. © Springer India 2015.