Abstract
The aim of the present paper, how the people behave towards the offer of two products in two different patches. In this work, an innovation diffusion model with six-compartments for two different patches is proposed. There is a delay in the adoption of product-1 in patch-2 and delay of adoption of product-2 in patch-1. The entire population in both the patches is classified into three different groups (i) non-adopter (ii) adopter of product-1 (iii) adopter of product-2. Dynamical behavior of the proposed system is studied, and Basic influence numbers (BINs) of the model are calculated. Stability analysis is executed for all the possible equilibrium points with and without delays. Hopf bifurcation analysis is too carried out taking the delay of adoption to adopt the product-1 in patch-2, and product-2 in patch-1 are bifurcation parameter and obtained the threshold values. Moreover, sensitivity analysis is carried out for the system parameter used in the interior equilibrium. Finally, exhaustive numerical simulations have been carried out by utilizing MATLAB, to supports analytical results.
Similar content being viewed by others
References
Rogers, E.M.: Diffusion of Innovations, 1st edn. The Free Press, New York (1962)
Bass, F.M.: A new product growth model for consumer durable. Manag. Sci. 15(5), 215–227 (1969)
Fort, L.A., Woodlock, J.W.: Early perdiction of market succcess for new grocerry products. J. Mark. 25, 31–38 (1960)
Fisher, J.C., Pry, R.H.: A simple substitution model of technological change. Technol. Forecast. Soc. 3, 75–88 (1971)
Tenneriello, C., Fergola, P., Ma, Z., Wang, W.: Stability of competitive innovation diffusion model. Ric. Mat. 51(2), 185–199 (2002)
Modis, V.: Technological forecasting at the stock market. Technol. Forecast. Soc. 62, 173–202 (1999)
Modis, T.: Genetic re-engineering of corporations. Technol. Forecast. Soc. 56, 107–118 (1997)
Singh, H., Dhar, J., Bhatti, H.S., Chandok, S.: An epidemic model of childhood disease dynamics with maturation delay and latent period of infection. Model. Earth Syst. Environ. 2(2), 79 (2016)
Singh, H., Dhar, J., Bhatti, H.S.: Bifurcation in disease dynamics with latent period of infection and media awareness. Int. J. Bifurc. Chaos 26, 1650097 (2016)
Tuli, R., Dhar, J., Bhatti, H.S., Singh, H.: Dynamical response by the instant buyer and thinker buyer in an innovation diffusion marketing model with media coverage. J. Math. Comput. Sci. 7(6), 1022–1045 (2017)
Yumei, Y., Wang, W., Zhang, Y.: An innovation diffusion model for three competitive products. Comput. Math. Appl. 46, 1473–1481 (2003)
Rider, R.K., Weinberg, C.: Competitive dynamics and introduction of new products: the motion pricture timiing game. J. Mark. Res. 35(1), 1–15 (1998)
Krishan, T., Bass, F.M., Jain, D.: Optimal pricing strategy for new products. Manag. Sci. 45(12), 1650–1663 (1999)
Chintagunta, P.K., Rao, V.R.: Pricing strategies in a dynamic duopoly: a differential game model. Manag. Sci. 42, 1501–1513 (1996)
Eliashberg, J., Jeuland, A.: The impact of competitive entry in a developing market upon dynamic pricing strategies. Mark. Sci. 5, 20–36 (1986)
Horsky, D., Simon, L.S.: Advertising and the diffusion of new products. Mark. Sci. 2, 1–17 (1983)
Dockner, E., Jorgensen, S.: Optimal advertising policy for diffusion models of new product innovations in monopolistic situation. Manag. Sci. 34, 119–130 (1988)
Teng, J.T., Thompson, G.L.: Oligopoly models for optimal advertising. Manag. Sci. 29, 1087–1101 (1983)
Lee, S.J., Lee, D.J., Oh, H.S.: Technological forecasting at the Korean stock market: a dynamic competition analysis using Lotka–Volterra model. Technol. Forecast. Soc. Change 72, 1044–1057 (2005)
Rogers, E.M., Everett, M.: Diffusion of Innovation, 4th edn. Free Press, New York (1995)
Wendi, W., Fergola, P., Tenneriello, C.: An innovation diffusion model in patch environment. Appl. Math. Comput. 134, 51–67 (2003)
Sahu, G.P., Dhar, J.: Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate. Appl. Math. Model. 36(3), 908–923 (2012)
Driwssche, P.V., Watmough, J.: Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)
Ruan, S.: Absolute stabilty, conditional stability and bifurcation in Kolmogrov-type predator–prey systems with discrete delays. Q. Appl. Math. 59(1), 159–174 (2001)
Singh, H., Dhar, J., Bhatti, H.S.: Dynamics of prey generalized predator system with disease in prey and gestation delay for predator. Model. Earth Syst. Environ. 2(2), 52 (2016)
Lin, X., Wang, H.: Stability analysis of delay differential equations with two discrete delays. Can. Appl. Math. Q. 20(4), 519–533 (2012)
Acknowledgements
We are thankful to the managing editor and reviewers for there valuable suggestions to improve the manuscript. Also, I express the warm thanks to I.K.G. Punjab Technical University, Punjab for providing me the facilities for the research being required.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declare that there is no conflict of interest.
Ethical standard
It is ensure that principles of ethical and professional conduct have been followed.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tuli, R., Dhar, J. & Bhatti, H.S. Innovation diffusion model with interactions and delays in adoption for two competitive products in two different patches. Ricerche mat 68, 705–726 (2019). https://doi.org/10.1007/s11587-019-00435-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-019-00435-1