This paper proposes definitions of implementation and adoption delays, arising from firm and client behaviors, in the context of market share dynamics. Information delay refers to the existence of lags in the information used for the advertising policy, while adoption delay refers to the existence of a lag in the effect of the advertising policy. These are natural lags in the flow of information and have not been considered in several models proposed in the literature. In this paper, these delays are introduced into recently proposed extensions of the Vidale-Wolfe-Deal and Lanchester models of market share dynamics subjected to affine advertising control policies. Conditions for stability of the equilibrium market share are derived. In addition, it is shown that Hopf bifurcations leading to oscillatory behavior exist for certain parameter values, and corresponding conditions for these are given. The main results are: the equilibrium market shares of the extended Vidale-Wolfe-Deal and Lanchester models are both robust to implementation delays, but, in the case of adoption delays, for both models, numerical results show that there is a critical value such that if the sum of the adoption delays exceeds this value, there is an onset of oscillations of market shares, through a Hopf bifurcation.
Nota bibliográficaFunding Information:
The work of Walter Aliaga was supported in part by the Universidad del Pacifico and in part by the Fellowship from the Brazilian Government Agency Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)-Brasil under Grant 001. The work of Amit Bhaya was supported in part by a Research Grant from the Brazilian Government Agency CNPq under Grant BPP/PQ 310646/2016-2, and in part by the Grant Universal 40616/2016-9.
© 2013 IEEE.