Product - The Stability Of The Diffusion Model Analysis

The aim of this work is to construct some mathematical diffusion models about competitive products in the market, to analyze the asymptotic behavior of these models and to study the effect of advertisement on the competitive products. In the first part, innovation diffusion models with a few products are proposed. For innovation diffusion model with three products, the global stability of the unique positive equilibrium is proved, by using the generalized Poincare-Bendixson theory and the competitive system theory. The threshold between the extinctionand existence of the product without advertisement in the market is considered.For four products, the local stability of the unique positive equilibrium is proved through Hurtwiz theory. Lastly, distributed time delays, with adopter rejection, are introduced to the model for three products. Liapunov functional is constructed to find the sufficient condition of the positive equilibrium.The asymptotic behavior of two innovation diffusion models with nonlinear contact is investigated in the second part. First, model for two products with nonlinear contact is considered. By using monotonicity and Poincare-Bendixson theory, the global stability of the unique positive equilibrium is proved. Furthermore, model with three products, under the condition of equally probable contact, has globally stable positive equilibrium.In the last part, a mathematical model is proposed to describe the dynamics of users in three different patches. This model has an unique positive equilibrium which is globally stable. Then periodic advertisements are incorporated in the model and the existence and stability of positive periodic solutions are investigated.