Sustaining of two competing products under the impact of the media including the experience of adopters

Abstract

In the present study, we proposed an innovation diffusion model with four-compartments to investigate the interaction and diffusion of two competitive products in a particular region. Herein, Positivity, Boundedness and Basic influence numbers (BINs) are examined. Asymptotic stability analysis is carried out for all feasible steady-states. It is investigated that the adopter free steady-state is stable if BINs are less than one for both the competitive products. Hopf bifurcation analysis is also carried out by taking the adoption experience period of the adopters, i.e., $\tau _1, \tau _2$ as the bifurcation parameter and obtained the threshold values. Further, when $\tau _1>0, \tau _2>0$, the interior steady-state $E^*$ is stable for specific threshold parameters $\tau _1<\tau _{10^{*}}^{+},\tau _2>\tau _{20^{*}}^{+}$ or $\tau _1>\tau _{10^*}^{+},\tau _2<\tau _{20^{*}}^{+}$. If both $\tau _1, \tau _2$ crosses the threshold parameters, i.e., $\tau _1>\tau _{10^{*}}^{+},\tau _2>\tau _{20^{*}}^{+}$ system perceived oscillating behavior and Hopf bifurcation occurs. Moreover, sensitivity analysis is carried out for the system parameter used in the interior steady-state. Exhaustive numerical simulation supports analytical results. Finally, it exhibited that in light of the impact of media, non-adopter joins the adopter class rapidly as the effect of the media increases in the region.