Advertisement

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

  • Rishi TuliEmail author
  • Joydip Dhar
  • Harbax S. Bhatti
Original Research
  • 40 Downloads

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.

Keywords

Boundedness Positivity Basic influence number Delay Hopf bifurcation Sensitivity analysis 

Mathematics Subject Classification

34C23 34D20 92B05 92D30 

Notes

Acknowledgements

I express my warm thanks to I.K.G. Punjab Technical University, Punjab for providing me the facilities for the research being required.

References

  1. 1.
    Bass, F.M.: A new product growth model for consumer durable. Manag. Sci. 15(5), 215–227 (1969)CrossRefzbMATHGoogle Scholar
  2. 2.
    Rogers, E.M.: Diffusion of Innovation, 4th edn. Free Press, New York (1995)Google Scholar
  3. 3.
    Tenneriello, C., Fergola, P., Ma, Z., Wang, W.: Stability of competitive innovation diffusion model. Ric. Mat. 51(2), 185–199 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Mckeown, M.: The Truth About Innovation. Pearson Financial Times (2008)Google Scholar
  5. 5.
    Day, G.S.: Analysis for strategic market decisions. MN.West, St. Paul (1986)Google Scholar
  6. 6.
    Centrone, F., Goia, A., Salinelli, E.: Demographic process in a model of innovation diffusion with dynamic market. Technol. Forcasting Soc. Change 74(3), 27–266 (2007)Google Scholar
  7. 7.
    Sharma, A., Sharma, A.K., Agnihotri, K.: The dynamics of plankton-nutrient interaction with delay. Appl. Math. Comput. 231, 503–515 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Shukla, J.B., Kushwah, H., Agarwal, A., Shukla, A.: Modeling the effects of variable external influences and demographic processes on innovation diffusion. Nonlinear Anal. Real World Appl. 13, 186–196 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hale, J.K.: Ordinary Differential Equations. Wley, New York (1969)zbMATHGoogle Scholar
  10. 10.
    Wang, W., Fergola, P., Tenneriello, C.: Innovation diffusion model in patch environment. Appl. Math. Comput. 134(1), 51–67 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fergola, P., Tenneriello, C., Ma, Z., Petrillo, F.: Delayed innovation diffusion processes with positiveand negative word-of-mouth. Int. J. Differ. Equ. Appl. 1, 131–147 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yumei, Y., Wendi, W.: Global stability of an innovation diffusion model for n products. Appl. Math. Lett. 19, 1198–1201 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yu, Y., Wang, W.: Stability of innovation diffusion model with nonlinear acceptance. Acta Math. Sci. 27, 645–655 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kang, Y.: Deley Differential Equations with Applications in Population Dynamics. Academic Press, London (1993)Google Scholar
  15. 15.
    Fanelli, V., Maddalena, L.: A time delay model for the diffusion of a new technology. Nonlinear Anal. Real World Appl. 13(2), 643–649 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yumei, Y., Wendi, W., Yong, Z.: An innovation diffusion model for three competitive products. Comput. Math. Appl. 46, 1473–1481 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dhar, J., Tyagi, M., Sinha, P.: An innovation diffusion model for the survival of a product in a competitive market: basic influence numbers. Int. J. Pure Appl. Math. 89(4), 439–448 (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Singh, H., Dhar, J., Bhatti, H.S.: Bifurcation in disease dynamics with latent period of infection and media awareness. Int. J. Bifurc. Chaos 26(06), 1650097 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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
  20. 20.
    Kumar, R., Sharma, A.K., Agnihotri, K.: Stability and bifurcation analysis of a delayed innovation diffusion model. Acta Math. Sci. 38(2), 709–732 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, L., Xu, R., Feng, G.: A stage-structured predator-prey system with the delay. J. Appl. Math. Comput. 33, 267–281 (2010)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ruan, S.: Absolute stabilty, conditional stability and bifurcation in kolmogrov-type predator-prey systems with discrete delays. Q. App. Math. 59(1), 159–174 (2001)CrossRefzbMATHGoogle Scholar
  25. 25.
    Singh, H., Dhar, J., Bhatti, H.S.: Dynamics of a prey generalized predator system with disease in prey and gestation delay for predator. Model. Earth Syst. Environ. (2016)Google Scholar
  26. 26.
    Song, Y., Han, M., Wei, J.: Stability and hopf bifurcation analysis on a simplified bem neural network with delays. Phys. D 200(3), 184–204 (2005)Google Scholar
  27. 27.
    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

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Research ScholarIKG-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