The deterministic model with time delay for a new product diffusion in a market

Nicoleta SIRGHI; Mihaela NEAMTU; Gabriela MIRCEA; Doriana RAMESCU

Nicoleta SIRGHI West University of Timisoara
Mihaela NEAMTU West University of Timisoara
Gabriela MIRCEA West University of Timisoara
Doriana RAMESCU West University of Timisoara

Keywords: Consumers, Market, Advertising model, Equilibrium point, Stability

Abstract

The aim of this paper is to present and analyze a market through a nonlinear deterministic model. A firm launches a new product and devotes a fixed proportion of sales to advertising, while customers go through a three stage adoption process with some delay on the effect of advertisement. The mathematical model is described by three nonlinear differential equations with time delay, where the word-of-mouth and advertising effectiveness are taken into account. The variables consist of the number of non-adopters (unaware of the existence of the product or the number of people who have not repurchased it), the number of thinkers (the number of people who know about the product, but they have not yet purchased it) and the number of adopters (the number of people who have purchased the product). The time delays are introduced in both purchase decisions of the thinkers and repurchase decisions of the adopters as well. The positive equilibrium point is determined and the conditions for the asymptotic stability are provided, when there is no delay. When the delay is taken as bifurcation parameter the conditions for the existence of a Hopf bifurcation are given. The critical value of the delay is found where the asymptotic stability is lost. Numerical simulations and conclusions can be found in the last part of the paper.

Author Biography

Mihaela NEAMTU, West University of Timisoara

Faculty of Economics and Business Administration

Professor Phd.

References

[1] Bass, F.M. (1969). A new product growth model for consumer durables, Management Sci., 15, 215-227.

[2] Benito, C.C., González-Parra, G., Arenas, A. (2016). Fractional order financial models for awareness and trial advertising decisions, Computational Economics 48.4 (2016): 555-568.

[3] Ciurdariu A.L., Neamţu, M. (2012). The analysis of the deterministic and stochastic models with delays which describe the financial crises contagions, Int. Journal of mathematical models and methods in applied sciences, 4(6): 583-591.

[4] Dhar, J., Tyagi, M., P. Sinha, P. (2010). Dynamical Behavior in an Innovation Diffusion Marketing Model with Thinker Class of Population, Int. J. Buss. Mgt. Eco. Res., 1(1): 79-84.

[5] Dodson J. A., Muller, E. (1978). Models of new product diffusion through advertising and word-of-mouth, Management Sci., 24, 1978, pp. 1568--1578.

[6] Feichtinger, G., Ghezzi, L.L., Piccardi, C. (1995). Chaotic behavior in an advertising diffusion model, International Journal of bifurcation and chaos, 5(1): 255-263.

[7] Hassard, B.D., Kazarinoff, N.D., Wan, Y.H. (1981). Theory and applications of Hopf bifurcation, Cambridge University Press, Cambridge.

[8] Kutznetsov, Y.A. (1995). Elements of applied bifurcation theory, Springer Verlag.

[9] Lekvall, P., Wahlbin, C. (1973). A study of some assumptions underlying innovation diffusion functions, Swedish J. Economics, 75, 362-377.

[10] Mircea, G., Neamţu, M. Opriş, D., (2011). Uncertain, stochastic and fractional dynamical systems with delay, LAP LAMBERT Academic Publishing.

[11] Sirghi, N., Neamţu, M. (2013). Stability and Hopf bifurcation Analysis in an Advertising Diffusion Model with Delay, Mathematical Applications in Science and Mathematics, Proceedings of AMATHI'13, 4th European Conference for the Applied Mathematics and Informatics, Dubrovnik, Croatia, June 25-27, 205-211.

[12] Simon, H., Sebastian, K. H. (1987). Diffusion and advertising: The German telephone company, Managemnet Sci., 33: 451-466.

[13] Sîrghi, N., Neamţu M. (2013).Deterministic and stochastic advertising diffusion model with delay, WSEAS Transaction on system and control, 4.8 (2013): 141-150.