Dynamics of an advertising competition model with sales promotion

• V–W models have received much attention to theoretical analysis and many applications in physics, biology, engineering and business. In the literature, very little has been known on periodic solution and its bifurcation in with advertising competition while sales promotion is rarely considered into a differential model.
• In this study we incorporate advertising competition with sales promotion into a differential model, and restrict our attention to the complex dynamics of this model and the effect of sales promotion on sales level.
• Numerical simulations on periodic solutions, bifurcation diagram, and sales promotion strategies are shown through an illustrative example. They agree well with our theoretical analysis.

In this paper, an advertising competition model with sales promotion is constructed and investigated. Conditions of the existence and stability of period-T solutions are obtained by means of the discrete map. Flip bifurcation is analyzed by using the center manifold theory and three sales promotion strategies are discussed. Example and numerical simulations are illustrated which agree well with our theoretical analysis.