Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-24T03:04:53.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of an Innovation Diffusion Model with Time Delay

Part of: Qualitative theory Manifolds Stability theory Infinite-dimensional dissipative dynamical systems

Published online by Cambridge University Press:  07 September 2017

Rakesh Kumar ,
Anuj K. Sharma  and
Kulbhushan Agnihotri
Rakesh Kumar*
Affiliation:
Department of Applied Sciences, SBS State Technical Campus, Ferozepur-152004, Punjab, India
Anuj K. Sharma*
Affiliation:
Department of Mathematics, LRDAV College, Jagraon, Ludhiana-142026, Punjab, India
Kulbhushan Agnihotri*
Affiliation:
Department of Applied Sciences, SBS State Technical Campus, Ferozepur-152004, Punjab, India
*
*Corresponding author. Email addresses:[email protected] (R. Kumar), [email protected] (A. K. Sharma), [email protected] (K. Agnihotri)
*Corresponding author. Email addresses:[email protected] (R. Kumar), [email protected] (A. K. Sharma), [email protected] (K. Agnihotri)
*Corresponding author. Email addresses:[email protected] (R. Kumar), [email protected] (A. K. Sharma), [email protected] (K. Agnihotri)
Get access

Abstract

A nonlinear mathematical model for innovation diffusion is proposed. The system of ordinary differential equations incorporates variable external influences (the cumulative density of marketing efforts), variable internal influences (the cumulative density of word of mouth) and a logistically growing human population (the variable potential consumers). The change in population density is due to various demographic processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem involves two dynamic variables viz. a non-adopter population density and an adopter population density. The model is analysed qualitatively using the stability theory of differential equations, with the help of the corresponding characteristic equation of the system. The interior equilibrium point can be stable for all time delays to a critical value, beyond which the system becomes unstable and a Hopf bifurcation occurs at a second critical value. Employing normal form theory and a centre manifold theorem applicable to functional differential equations, we derive some explicit formulas determining the stability, the direction and other properties of the bifurcating periodic solutions. Our numerical simulations show that the system behaviour can become extremely complicated as the time delay increases, with a stable interior equilibrium point leading to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit cycles with more local maxima and minima per cycle, and finally chaotic solutions.

Keywords

Innovation diffusion modelevaluation periodstability analysisHopf bifurcationnormal form theorycentre manifold theorem

MSC classification

Secondary: 34C23: Bifurcation 34D20: Stability 37L10: Normal forms, center manifold theory, bifurcation theory 49Q12: Sensitivity analysis 92D25: Population dynamics (general)
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 455 - 481
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Fourt, L.A. and Woodlock, J.W., Early prediction of market success for new grocery products, J. Marketing 25, 31–38 (1960).Google Scholar
[2] Mansfield, E., Technical change and the rate of imitation, Econometrica: J. Econometric Soc. 29, 741–766 (1961).CrossRefGoogle Scholar
[3] Bass, F., A new product growth model for consumer durables, Management Sc. 15, 215–227 (1969).Google Scholar
[4] Hale, J.K., Ordinary Differential Equations, Wiley, New York (1969).Google Scholar
[5] Sharif, M.N. and Kabir, C., A generalized model for forecasting technological substitution, Technological Forecasting Social Change 8, 353–364 (1976).CrossRefGoogle Scholar
[6] Sharif, M.N. and Ramanathan, K., Binomial innovation diffusion models with dynamic potential adopter population, Technological Forecasting Social Change 20, 63–87 (1981).Google Scholar
[7] Easingwood, C., Mahajan, V. and Muller, E., A nonsymmetric responding logistic model for forecasting technological substitution, Technological Forecasting Social Change 20, 199–213 (1981).CrossRefGoogle Scholar
[8] Mahajan, V., Muller, E. and Kerin, R.A., Introduction strategy for new products with positive and negative word-of-mouth, Management Sc. 30, 1389–1404 (1984).Google Scholar
[9] Skiadas, C.H., Two generalized rational models for forecasting innovation diffusion, Technological Forecasting and Social Change, Elsevier 27, 39–61 (1985).Google Scholar
[10] Skiadas, C.H., Innovation diffusion models expressing asymmetry and/or positively or negatively influencing forces, Technological Forecasting Social Change 30, 313–330 (1986).Google Scholar
[11] Mahajan, V. and Peterson, R.A., Models for Innovation Diffusion, Vol. 48, SAGE (1985).Google Scholar
[12] Dockner, E. and Jorgensen, S., Optimal advertising policies for diffusion models of new product innovation in monopolistic situations, Management Sc. 34, 119–130 (1988).Google Scholar
[13] Mahajan, V., Muller, E. and Bass, F.M., New Product Diffusion Models in Marketing: A Review and Directions for Research, Springer (1991).Google Scholar
[14] Rogers, E.M., Diffusion of Innovations, The Free Press, New York (1995).Google Scholar
[15] Horsky, D. and Simon, L S., Advertising and the diffusion of new products, Marketing Sc. 2, 1–17 (1983).CrossRefGoogle Scholar
[16] Mahajan, V., Muller, E. and Bass, F.M., New-product diffusion models, Handbooks in Operations Research and Management Science 5, 349–408 (1993).CrossRefGoogle Scholar
[17] Dhar, J., Tyagi, M. and Sinha, P., Dynamical behavior in an innovation diffusion marketing model with thinker class of population, Int. J. Business Management Economics and Research 1, 79–84 (2010).Google Scholar
[18] Dhar, J., Tyagi, M. and Sinha, P., The impact of media on a new product innovation diffusion: A mathematical model, Boletim da Sociedade Paranaense de Matemática 33, 169–180 (2014).CrossRefGoogle Scholar
[19] Kuang, Y., Delay Differential Equations: with Applications in Population Dynamics, Academic Press (1993).Google Scholar
[20] Fergola, P., Tenneriello, C., Ma, Z. and Petrillo, F., Delayed innovation diffusion processes with positive and negative word-of-mouth, Int. J. Differential Equations Application 1, 131–147 (2000).Google Scholar
[21] Kot, M., Elements of Mathematical Ecology, Cambridge University Press (2001).Google Scholar
[22] Wang, W., Fergola, P., Lombardo, S. and Mulone, G., Mathematical models of innovation diffusion with stage structure, Appl. Math. Modelling 30, 129–146 (2006).Google Scholar
[23] Wang, W., Fergola, P. and Tenneriello, C., Innovation diffusion model in patch environment, Appl. Math. Comput. 134, 51–67 (2003).Google Scholar
[24] Tenneriello, C., Fergola, P., Zhien, M. and Wendi, W., Stability of competitive innovation diffusion model, Ricerche di Matematica 51, 185–200 (2002).Google Scholar
[25] Yu, Y., Wang, W. and Zhang, Y., An innovation diffusion model for three competitive products, Comput. Math. Appl. 46, 1473–1481 (2003).Google Scholar
[26] Yu, Y. and Wang, W., Global stability of an innovation diffusion model for n products, Appl. Math. Lett. 19, 1198–1201 (2006).Google Scholar
[27] Yu, Y. and Wang, W., Stability of innovation diffusion model with nonlinear acceptance, Acta Mathematica Scientia 27, 645–655 (2007).Google Scholar
[28] Centrone, F., Goia, A. and Salinelli, E., Demographic processes in a model of innovation diffusion with dynamic market, Technological Forecasting Social Change 74, 247–266 (2007).Google Scholar
[29] Shukla, J., Kushwah, H., Agrawal, K. and Shukla, A., Modeling the effects of variable external influences and demographic processes on innovation diffusion, Nonlinear Analysis: Real World Applications 13, 186–196 (2012).Google Scholar
[30] Hale, J.K., Functional Differential Equations, Springer (1971).Google Scholar
[31] Birkhoff, G. and Rota, G., Ordinary Differential Equations, Ginn, Boston (1989).Google Scholar
[32] Boonrangsiman, S., Bunwong, K. and Moore, E.J., A bifurcation path to chaos in a time-delay fisheries predator–prey model with prey consumption by immature and mature predators, Mathematics and Computers in Simulation 124, 16–29 (2016).Google Scholar
[33] Sharma, A., Sharma, A.K. and Agnihotri, K., The dynamic of plankton–nutrient interaction with delay, Appl. Math. Comp. 231, 503–515 (2014).Google Scholar
[34] Song, Y., Wei, J. and Han, M., Local and global hopf bifurcation in a delayed hematopoiesis model, Int. J. Bifurcation Chaos 14, 3909–3919 (2004).Google Scholar
[35] Li, F. and Li, H., Hopf bifurcation of a predator–prey model with time delay and stage structure for the prey, Mathematical and Computer Modelling 55, 672–679 (2012).Google Scholar
[36] Luenberger, D.G., Introduction to Dynamic Systems; Theory, Models, and Applications, John Wiley & Sons, Inc. (1979).Google Scholar
[37] Pal, D. and Mahapatra, G., Effect of toxic substance on delayed competitive allelopathic phytoplankton system with varying parameters through stability and bifurcation analysis, Chaos, Solitons & Fractals 87, 109–124 (2016).Google Scholar
[38] Freedman, H. and Rao, V.S.H., The trade-off between mutual interference and time lags in predator-prey systems, Bulletin Math. Biol. 45, 991–1004 (1983).Google Scholar
[39] Hassard, B.D., Kazarinoff, N.D. and Wan, Y.H., Theory and Applications of Hopf Bifurcation, Vol. 41, Cambridge University Press Archive (1981).Google Scholar
[40] Ruan, S. and Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous Discrete Impulsive Systems Series A 10, 863–874 (2003).Google Scholar
[41] Kar, T. and Pahari, U., Modelling and analysis of a prey–predator system with stage-structure and harvesting, Nonlinear Analysis: Real World Appl. 8, 601–609 (2007).Google Scholar
[42] Liu, W. and Jiang, Y., Dynamics of a modified predator-prey system to allow for a functional response and time delay, East Asian J. Appl. Math. 6, 384–399 (2016).Google Scholar
[43] Faria, T. and Magalhães, L.T., Normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation, J. Diff. Eqs. 122, 181–200 (1995).Google Scholar
[44] Diamond, D., The impact of government incentives for hybrid-electric vehicles: Evidence from us states, Energy Policy 37, 972–983 (2009).Google Scholar
[45] Kivi, A., Smura, T. and J. Töyli, Technology product evolution and the diffusion of new product features, Technological Forecasting Social Change 79, 107–126 (2012).Google Scholar