Advertisement

Innovation diffusion model with interactions and delays in adoption for two competitive products in two different patches

  • Rishi TuliEmail author
  • Joydip Dhar
  • Harbax S. Bhatti
Article
  • 6 Downloads

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.

Keywords

Boundedness Positivity Delay Hopf bifurcation Sensitivity analysis 

Mathematics Subject Classification

34C23 34D05 34D020 

Notes

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.

Compliance with ethical standards

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.

References

  1. 1.
    Rogers, E.M.: Diffusion of Innovations, 1st edn. The Free Press, New York (1962)Google Scholar
  2. 2.
    Bass, F.M.: A new product growth model for consumer durable. Manag. Sci. 15(5), 215–227 (1969)zbMATHGoogle Scholar
  3. 3.
    Fort, L.A., Woodlock, J.W.: Early perdiction of market succcess for new grocerry products. J. Mark. 25, 31–38 (1960)Google Scholar
  4. 4.
    Fisher, J.C., Pry, R.H.: A simple substitution model of technological change. Technol. Forecast. Soc. 3, 75–88 (1971)Google Scholar
  5. 5.
    Tenneriello, C., Fergola, P., Ma, Z., Wang, W.: Stability of competitive innovation diffusion model. Ric. Mat. 51(2), 185–199 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Modis, V.: Technological forecasting at the stock market. Technol. Forecast. Soc. 62, 173–202 (1999)Google Scholar
  7. 7.
    Modis, T.: Genetic re-engineering of corporations. Technol. Forecast. Soc. 56, 107–118 (1997)Google Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    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)MathSciNetzbMATHGoogle Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    Yumei, Y., Wang, W., Zhang, Y.: An innovation diffusion model for three competitive products. Comput. Math. Appl. 46, 1473–1481 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Krishan, T., Bass, F.M., Jain, D.: Optimal pricing strategy for new products. Manag. Sci. 45(12), 1650–1663 (1999)zbMATHGoogle Scholar
  14. 14.
    Chintagunta, P.K., Rao, V.R.: Pricing strategies in a dynamic duopoly: a differential game model. Manag. Sci. 42, 1501–1513 (1996)zbMATHGoogle Scholar
  15. 15.
    Eliashberg, J., Jeuland, A.: The impact of competitive entry in a developing market upon dynamic pricing strategies. Mark. Sci. 5, 20–36 (1986)Google Scholar
  16. 16.
    Horsky, D., Simon, L.S.: Advertising and the diffusion of new products. Mark. Sci. 2, 1–17 (1983)Google Scholar
  17. 17.
    Dockner, E., Jorgensen, S.: Optimal advertising policy for diffusion models of new product innovations in monopolistic situation. Manag. Sci. 34, 119–130 (1988)Google Scholar
  18. 18.
    Teng, J.T., Thompson, G.L.: Oligopoly models for optimal advertising. Manag. Sci. 29, 1087–1101 (1983)zbMATHGoogle Scholar
  19. 19.
    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)Google Scholar
  20. 20.
    Rogers, E.M., Everett, M.: Diffusion of Innovation, 4th edn. Free Press, New York (1995)Google Scholar
  21. 21.
    Wendi, W., Fergola, P., Tenneriello, C.: An innovation diffusion model in patch environment. Appl. Math. Comput. 134, 51–67 (2003)MathSciNetzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Driwssche, P.V., Watmough, J.: Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetzbMATHGoogle Scholar
  24. 24.
    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)zbMATHGoogle Scholar
  25. 25.
    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)Google Scholar
  26. 26.
    Lin, X., Wang, H.: Stability analysis of delay differential equations with two discrete delays. Can. Appl. Math. Q. 20(4), 519–533 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.IKG-Punjab Technical UniversityKapurthalaIndia
  2. 2.Beant College of Engineering and TechnologyGurdaspurIndia
  3. 3.ABV-Indian Institute of Information Technology and ManagementGwaliorIndia
  4. 4.Department of Applied SciencesB.B.S.B.E.CFatehgarh SahibIndia

Personalised recommendations