Feedback Stackelberg equilibrium strategies when the private label competes with the national brand

We consider a noncooperative differential game where a retailer sells her own private label in addition to the manufacturer's brand. We assume that each brand's goodwill evolves according to a modified Nerlove-Arrow dynamics, in such a way that the advertising effort of one brand hurts the competitor's goodwill stock. We characterize Feedback-Stackelberg pricing and advertising strategies and employ simulations to analyze their sensitivity to the main model parameters.