The deterministic model with time delay for a new product diffusion in a market
The aim of this paper is to present and analyze a market through a nonlinear deterministic model. A firm launches a new product and devotes a fixed proportion of sales to advertising, while customers go through a three stage adoption process with some delay on the effect of advertisement. The mathematical model is described by three nonlinear differential equations with time delay, where the word-of-mouth and advertising effectiveness are taken into account. The variables consist of the number of non-adopters (unaware of the existence of the product or the number of people who have not repurchased it), the number of thinkers (the number of people who know about the product, but they have not yet purchased it) and the number of adopters (the number of people who have purchased the product). The time delays are introduced in both purchase decisions of the thinkers and repurchase decisions of the adopters as well. The positive equilibrium point is determined and the conditions for the asymptotic stability are provided, when there is no delay. When the delay is taken as bifurcation parameter the conditions for the existence of a Hopf bifurcation are given. The critical value of the delay is found where the asymptotic stability is lost. Numerical simulations and conclusions can be found in the last part of the paper.
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