Abstract
In this paper, an innovation diffusion model with the delay in adoption is planned to study the dynamics of the adopter of the same product within two different patches. It is determined that solutions are positive and bounded for the proposed system. Asymptotic stability analysis is carried out for all possible equilibrium points. The critical value of the delays \(\tau _1, \tau _2\) are determined. It is observed that for the interior equilibrium remains stable if either (or both) the adoption delays is (are) less than the threshold values, i.e., \(\tau _1<\tau _{10}^+,\tau _2>\tau _{20}^+\) or \(\tau _1>\tau _{10}^+,\tau _2<\tau _{20}^+\). If both \(\tau _1\) and \(\tau _2\) cross its thresholds, system perceived oscillating behavior, and Hopf bifurcation occurs. Sensitivity analysis for the basic influence number of the model has been examined. Subsequently, numerical simulations have been carried out to support our analytical findings with the different set of parameters.
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References
Rogers, E.M.: Diffusion of Innovation, 4th edn. Free Press, New York (1995)
Bass, F.M.: A new product growth model for consumer durable. Manag. Sci. 15(5), 215–227 (1969)
Lekvall, P., Wahlbin, C.: A study of some assumptions underlying innovation diffusion functions. Swed. J. Econ. 75, 362–377 (1973)
Muller, E.: Trial/Awareness advertising decisions, a control problem with phase diagrams with non-stationary boundaries. J. Econ. Dyn. Control 6, 333–350 (1983)
Singh, H., Dhar, J., Bhatti, H.S., Chandok, S.: An epedemic model of childhood disease dynamics with maturation delay and latent period of infection. Model. Earth Syst. Environ. 2(2), 1–8 (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)
Kalish, S.: Monopolist pricing with dynamic demand and production cost. Mark. Sci. 2, 135–159 (1983)
Sethi, S.P.: Optimal advertising policy with the conagion model. J. Optim. Theory. Appl. 29, 615–626 (1979)
Horsky, D., Simon, L.S.: Advertising and the diffusion of new products. Mark. Sci. 2, 1–17 (1983)
Maleknejad, k, Mirzaee, F.: Numerical solution of stochastic linear heat conduction problem by using new algorithims. Appl. Math. Comput. 163(1), 97–106 (2005)
Mirzaee, F., Bimesl, S., Tohidi, E.: A numerical framework for solving high-order pantography-delay Volterra integro-differential equations. Kuwait J. Sci. Eng. 43(1), 69–83 (2016)
Mirzaee, F., Bimesl, S.: A uniformly convergent Euler matrix method for telegraph equations having constant coefficients. Mediterr. J. Math. 13(1), 497–515 (2016)
Maurer, S.M., Huberman, B.A.: Competitive dynamics of websites. J. Econ. Dyn. Control 27, 2195–2206 (2003)
Kim, J., Lee, D.J., Ahn, J.: A dynamic competition analysis on the Korean mobile phone market using competitive diffusion model. Comput. Ind. Eng. 51, 174–182 (2006)
Lopez, L., Sanjuan, M.F.A.: Defining strategies to win in the internet market. Physica A 301, 512–534 (2001)
Mahajan, V., Peterson, R.A.: Innovation diffusion in a dynamic potential adopter population. Manag. Sci. 24, 1589–1597 (1978)
Wendi, W., Fergola, P., Tenneriello, C.: An innovation diffusion model in patch environment. Appl. Math. Comput. 134, 51–67 (2003)
Sisodiya, O.S., Mishra, O.P., Dhar, J.: Pathogen induced infection and Its control by vaccination: a mathematical model for cholera disease. Int. J. Appl. Comput. Math. 4, 74 (2018)
Giovangis, A.N., Skiadas, C.H.: A stochastic logistic innovation diffusion model studying the electricity consumption in Greece and the United States. Technol. Forecast. Soc. Change 61, 235–246 (1999)
Gruber, H.: Competition and innovation: the diffusion of mobile telecommunication in central and eastern Europe. Inf. Econ. Policy 3, 19–34 (2001)
Jun, D.B., Kim, S.K.: Forecasting telecommunication service subscribers in substitutive and competitive environments. Int. J. Forecast. 18, 561–581 (2002)
Dhar, J., Tyagi, M., Sinha, P.: An innovation diffusion model for the survival of a product in a competitive market: basic influence number. Int. J. Pure Appl. Math. 89(4), 439–448 (2013)
Dhar, J., Tyagi, M., Sinha, P.: The impact of media on a new product innovation diffusion: a mathematical model. Bol. Soc. Parana. Mat. 33(1), 169–180 (2015)
Kalish, S., Mahajan, V., Muller, E.: Waterfall and sprinkler new-product strategies in competitve global markets. Int. J. Res. Mark. 2, 105–119 (1995)
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.Vanden, 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 a prey generalized predator system with disease in prey and gestation delay for predator. Model. Earth Syst. Environ. 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)
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I express my warm thanks to I.K.G. Punjab Technical University, Punjab for providing me the facilities for the research being required.
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Tuli, R., Dhar, J. & Bhatti, H.S. Innovation Diffusion Model for the Marketing of a Product with Interactions and Delay in Adoption for Two Different Patches. Int. J. Appl. Comput. Math 4, 149 (2018). https://doi.org/10.1007/s40819-018-0583-x
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DOI: https://doi.org/10.1007/s40819-018-0583-x