Delay Induced Hopf Bifurcation in a Nonlinear Innovation Diffusion Model: External Influences Effect

A nonlinear innovation diffusion model which incorporates the evaluation stage (time delay) is proposed to describe the dynamics of three population classes for non-adopter and adopter densities. The local stability of the various equilibrium points is investigated. It is observed that the system is locally asymptotically stable for a delay limit and produces periodic orbits via a Hopf bifurcation when evaluation period crosses a critical value. Applying normal form theory and center manifold theorem, we study the properties of the bifurcating periodic solutions. The model shows that the adopter population density achieves its maturity stage faster if the cumulative density of external influences increases. Several numerical examples confirm our theoretical results.