STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL

doi:10.1016/S0252-9602(18)30776-8

Abstract

Abstract:

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage (time delay) is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period (time delay, τ) passes through a critical value. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Key words: Innovation diffusion model, stability analysis, Hopf-bifurcation, normal form theory, center manifold theorem

Rakesh KUMAR, Anuj Kumar SHARMA, Kulbhushan AGNIHOTRI. STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL[J].Acta mathematica scientia,Series B, 2018, 38(2): 709-732.

Trendmd

References

[1] Fourt L A, Woodlock J W. Early prediction of market success for new grocery products. The Journal of Marketing, 1960:31-38
[2] Mansfield E. Technical change and the rate of imitation. Econometrica:Journal of the Econometric Society, 1961:741-766
[3] Bass F. A new product growth model for consumer durables. Management Science, 1969, 15:215-227
[4] Hale J K. Ordinary Differential Equations. New York:Wiley, 1969
[5] Sharif M N, Kabir C. A generalized model for forecasting technological substitution. Technological Forecasting and Social Change, 1976, 8(4):353-364
[6] Sharif M N, Ramanathan K. Binomial innovation diffusion models with dynamic potential adopter population. Technological Forecasting and Social Change, 1981, 20(1):63-87
[7] Easingwood C, Mahajan V, Muller E. A nonsymmetric responding logistic model for forecasting technological substitution. Technological forecasting and Social change, 1981, 20(3):199-213
[8] Mahajan V, Muller E, Kerin R A. Introduction strategy for new products with positive and negative word-of-mouth. Management Science, 1984, 30(12):1389-1404
[9] Skiadas C. Two generalized rational models for forecasting innovation diffusion. Vol 27. Elsevier, 1985
[10] Skiadas C H. Innovation diffusion models expressing asymmetry and/or positively or negatively influencing forces. Technological Forecasting and Social Change, 1986, 30(4):313-330
[11] Mahajan V, Peterson R A. Models for innovation diffusion. Vol 48. Sage, 1985
[12] Dockner E, Jorgensen S. Optimal advertising policies for diffusion models of new product innovation in monopolistic situations. Management Science, 1988, 34(1):119-130
[13] Mahajan V, Muller E, Bass F M. New product diffusion models in marketing:A review and directions for research//Diffusion of technologies and social behavior. Springer, 1991:125-177
[14] Rogers E M. Diffusion of innovations. New York:The Free Press, 1995
[15] Horsky D, Simon L S. Advertising and the diffusion of new products. Marketing Science, 1983, 2(1):1-17
[16] Mahajan V, Muller E, Bass F M. New-product diffusion models. Handbooks in Operations Research and Management Science, 1993, 5:349-408
[17] Dhar J, Tyagi M, Sinha P. Dynamical behavior in an innovation diffusion marketing model with thinker class of population. International Journal of Business Management Economics and Research, 2010, 1:79-84
[18] Dhar J, Tyagi M, Sinha P. The impact of media on a new product innovation diffusion:A mathematical model. Boletim da Sociedade Paranaense de Matemática, 2014, 33(1):169-180
[19] Kuang Y. Delay differential equations:with applications in population dynamics. New York:Academic Press, 1993
[20] Fergola P, Tenneriello C, Ma Z, Petrillo F. Delayed Innovation Diffusion Processes with Positive and Negative Word-of-Mouth. International Journal Differential Equations Application, 2000, 1:131-147
[21] Kot M. Elements of mathematical ecology. Cambridge University Press, 2001
[22] Wang W, Fergola P, Lombardo S, Mulone G. Mathematical models of innovation diffusion with stage structure. Applied Mathematical Modelling, 2006, 30(1):129-146
[23] Wendi W, Fergola P, Tenneriello C. Innovation diffusion model in patch environment. Applied Mathematics and Computation, 2003, 134(1):51-67
[24] Tenneriello C, Fergola P, Zhien M, Wendi W. Stability of competitive innovation diffusion model. Ricerche di Matematica, 2002, 51(2):185-200
[25] Yu Y, Wang W, Zhang Y. An innovation diffusion model for three competitive products. Computers & Mathematics with Applications, 2003, 46(10):1473-1481
[26] Yu Y, Wang W. Global stability of an innovation diffusion model for n products. Applied Mathematics Letters, 2006, 19(11):1198-1201
[27] Yu Y, Wang W. Stability of innovation diffusion model with nonlinear acceptance. Acta Mathematica Scientia, 2007, 27(3):645-655
[28] Centrone F, Goia A, Salinelli E. Demographic processes in a model of innovation diffusion with dynamic market. Technological Forecasting and Social Change, 2007, 74(3):247-266
[29] Shukla J, Kushwah H, Agrawal K, Shukla A. Modeling the effects of variable external influences and demographic processes on innovation diffusion. Nonlinear Analysis:Real World Applications, 2012, 13(1):186-196
[30] Hale J K. Functional differential equations. Berlin, Heidelberg:Springer, 1971
[31] Luenberger D G D G. Introduction to dynamic systems; theory, models, and applications. 1979
[32] Song Y, Wei J, Han M. Local and global hopf bifurcation in a delayed hematopoiesis model. International Journal of Bifurcation and Chaos, 2004, 14(11):3909-3919
[33] Li F, Li H. Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey. Mathematical and Computer Modelling, 2012, 55(3):672-679
[34] Song Y, Yuan S. Bifurcation analysis in a predator-prey system with time delay. Nonlinear analysis:real world applications, 2006, 7(2):265-284
[35] Liu L, Li X, Zhuang K. Bifurcation analysis on a delayed sis epidemic model with stage structure. Electronic Journal of Differential Equations, 2007, 77:1-17
[36] Boonrangsiman S, Bunwong K, 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, 2016, 124:16-29
[37] Sharma A, Sharma A K, Agnihotri K. The dynamic of plankton-nutrient interaction with delay. Applied Mathematics and Computation, 2014, 231:503-515
[38] Segel L A, Edelstein-Keshet L. A primer on Mathematical models in biology. Siam, 2013
[39] Kuznetsov Y A. Elements of Applied Bifurcation Theory. Third Edition. Applied Mathematical Sciences. Vol 112. New York:Springer Verlag, 2004
[40] Hassard B D, Kazarinoff N D, Wan Y H. Theory and applications of Hopf bifurcation. Vol 41. CUP Archive, 1981
[41] Ruan S, Wei J. On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynamics of Continuous Discrete and Impulsive Systems Series A, 2003, 10:863-874
[42] Kar T, Pahari U. Modelling and analysis of a prey-predator system with stage-structure and harvesting. Nonlinear Analysis:Real World Applications, 2007, 8(2):601-609
[43] Faria T, Magalhães L T. Normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation. Journal of Differential Equations, 1995, 122(2):181-200
[44] Sharma A, Sharma A K, Agnihotri K. Analysis of a toxin producing phytoplankton-zooplankton interaction with Holling IV type scheme and time delay. Nonlinear Dynamics, 2015, 81(1/2):13-25
[45] Sharma A K, Sharma A, Agnihotri K. Bifurcation behaviors analysis of a plankton model with multiple delays. International Journal of Biomathematics, 2016, 9(6):1650086