A dynamic oligopoly Stackelberg–Cournot model with time delays

A discrete-time Stackelberg–Cournot competition model is analyzed, in which $n$ leader firms and $m$ follower firms interact in an oligopolistic market subject to time delays. Two equilibrium states are identified: a boundary equilibrium and a positive (interior) equilibrium, whose existence depends on the relative cost structures of the firms. For the non-delayed case, we derive conditions for the local stability of both equilibria and demonstrate that a flip (period-doubling) bifurcation occurs as the adjustment speed increases. When time delays are introduced, the stability properties of the boundary equilibrium remain unaffected, while the dynamics of the interior equilibrium become sensitive to the overall delay. In particular, when the leader and follower updates are effectively synchronized, reflecting a collective timing of reactions, the stability region of the positive equilibrium expands, and a Neimark–Sacker bifurcation is observed; in contrast, asynchronous updates yield dynamics similar to those of the non-delayed model. Numerical simulations validate our theoretical findings and illustrate the various routes to chaotic behavior. These results emphasize the role of reaction timing and delay interactions in shaping market dynamics in oligopolistic settings.