Bifurcation Analysis of an Advertising Diffusion Model

doi:10.1007/s42967-023-00353-y

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1540-1561.doi: 10.1007/s42967-023-00353-y

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Bifurcation Analysis of an Advertising Diffusion Model

Yong Wang, Yao Wang, Liangping Qi

Institute of Science and Technology, Tianjin University of Finance and Economics, Tianjin, 300222, China

收稿日期:2023-08-10 修回日期:2023-11-01 接受日期:2023-11-18 出版日期:2024-03-01 发布日期:2024-03-01
通讯作者: Yong Wang,E-mail:ywang@tjufe.edu.cn E-mail:ywang@tjufe.edu.cn
作者简介:Yao Wang, E-mail:yaomath@163.com
基金资助:
The authors would like to thank the editors and the anonymous reviewers for their valuable comments. This work is supported by the National Natural Science Foundation of China (Nos. 11701410 and 12001397) and the Natural Science Foundation of Tianjin City, China (No. 20JCQNJC00970).

Bifurcation Analysis of an Advertising Diffusion Model

Yong Wang, Yao Wang, Liangping Qi

Institute of Science and Technology, Tianjin University of Finance and Economics, Tianjin, 300222, China

Received:2023-08-10 Revised:2023-11-01 Accepted:2023-11-18 Online:2024-03-01 Published:2024-03-01
Supported by:
The authors would like to thank the editors and the anonymous reviewers for their valuable comments. This work is supported by the National Natural Science Foundation of China (Nos. 11701410 and 12001397) and the Natural Science Foundation of Tianjin City, China (No. 20JCQNJC00970).

摘要/Abstract

摘要： This research aims to investigate the impact of diffusion on the stability and bifurcation behavior of advertising diffusion systems. The study findings suggest that in the absence of diffusion, a higher proportion of crowd contact positively contributes to the stability of the system. Specifically, the study employs the interval partitioning method to discuss the k-mode Turing bifurcation and derives a more explicit Turing bifurcation line. Moreover, the study examines the k-mode Hopf bifurcation with the proportion of crowd contact acting as the bifurcation parameter. Furthermore, the weakly nonlinear analysis method is implemented to scrutinize the pattern formation in the Turing instability region. Finally, numerical simulation is utilized to validate the analytical findings obtained in this study.

关键词: Diffusive advertising model, Stability, Turing bifurcation, Pattern formation

Abstract: This research aims to investigate the impact of diffusion on the stability and bifurcation behavior of advertising diffusion systems. The study findings suggest that in the absence of diffusion, a higher proportion of crowd contact positively contributes to the stability of the system. Specifically, the study employs the interval partitioning method to discuss the k-mode Turing bifurcation and derives a more explicit Turing bifurcation line. Moreover, the study examines the k-mode Hopf bifurcation with the proportion of crowd contact acting as the bifurcation parameter. Furthermore, the weakly nonlinear analysis method is implemented to scrutinize the pattern formation in the Turing instability region. Finally, numerical simulation is utilized to validate the analytical findings obtained in this study.

Key words: Diffusive advertising model, Stability, Turing bifurcation, Pattern formation

中图分类号:

Yong Wang, Yao Wang, Liangping Qi. Bifurcation Analysis of an Advertising Diffusion Model[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1540-1561.

参考文献

[1] Abd-Rabo, M.A., Zakarya, M., Cesarano, C., Aly, S.: Bifurcation analysis of time-delay model of consumer with the advertising effect. Symmetry 13(3), 417 (2021)
[2] Barnett, W., Cymbalyuk, G.: Bifurcation Analysis, pp. 1-6. Springer, New York (2013)
[3] Bass, F.M.: A new product growth model for consumer durables. Manage. Sci. 15, 215-227 (1976)
[4] Botero, M.V.P., Ortiz, S.B.V.: The potential market for sustainable housing under the contingent valuation method. City of Palmira. Cuadernos de Administración 35(65), 45-59 (2019)
[5] Carvalho Braga, D., Mello, L.F., Rocsoreanu, C., Sterpu, M.: Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discret. Contin. Dyn. Syst. Ser. B 11, 785-803 (2009)
[6] Chen, H., Zhang, C.: Bifurcations and hydra effects in a reaction-diffusion predator-prey model with Holling II functional response. J. Appl. Anal. Comput. 13(1), 424-444 (2023)
[7] Dorfman, R., Steiner, P.O.: Optimal Advertising and Optimal Quality, pp. 165-166. Springer, Berlin, Heidelberg (1976)
[8] Ellis, J., Petrovskaya, N., Petrovskii, S.: Effect of density-dependent individual movement on emerging spatial population distribution: Brownian motion vs levy flights. J. Theor. Biol. 464, 159-178 (2019)
[9] Feichtinger, G.: Hopf bifurcation in an advertising diffusion model. J. Econ. Behav. Organ. 17(3), 401-411 (1992)
[10] Fu, X., Wu, R., Chen, M., Liu, H.: Spatiotemporal complexity in a diffusive Brusselator model. J. Math. Chem. 59, 2344-2367 (2021)
[11] Gambino, G., Giunta, V., Lombardo, M.C., Rubino, G.: Cross-diffusion effects on stationary pattern formation in the Fitzhugh-Nagumo model. Discret. Contin. Dyn. Syst. B 27(12), 7783 (2022)
[12] Glaister, S.M.: Advertising policy and returns to scale in markets where information is passed between individuals. Economica 41, 139-156 (1974)
[13] Golovin, A., Matkowsky, B.J., Volpert, V.A.: Turing pattern formation in the Brusselator model with superdiffusion. SIAM J. Appl. Math. 69(1), 251-272 (2008)
[14] Hu, D., Zhang, Y., Zheng, Z., Liu, M.: Dynamics of a delayed predator-prey model with constant-yield prey harvesting. J. Appl. Anal. Comput. 12(1), 302-335 (2022)
[15] Jacquemin, A.: Optimal control and advertising policy. Metroeconomica 25, 200-207 (1973)
[16] Jiang, W., Cao, X., Wang, C.: Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discret. Contin. Dyn. Syst. Ser. B 27(2), 1163-1178 (2022)
[17] Jin, D., Yang, R.: Hopf bifurcation in a predator-prey model with memory effect and intra-species competition in predator. J. Appl. Anal. Comput. 13(3), 1321-1335 (2023)
[18] Kondo, S., Miura, T.: Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329(5999), 1616-1620 (2010)
[19] Lu, M., Xiang, C., Huang, J., Wang, H.: Bifurcations in the diffusive Bazykin model. J. Differential Equations 323(25), 280-311 (2022)
[20] Mazzucato, M., Semmler, W.: The determinants of stock price volatility: an industry study. Nonlinear Dyn. Psychol. Life Sci. 6, 197-216 (2002)
[21] McGee, J.: The economics of advertising. Econ. J. 83(329), 295-297 (1973)
[22] Nerlove, M.L., Arrow, K.J.: Optimal advertising policy under dynamic conditions. Economica 29, 167-168 (1962)
[23] Peres, R., Muller, E., Mahajan, V.: Innovation diffusion and new product growth models: a critical review and research directions. Int. J. Res. Mark. 27, 91-106 (2010)
[24] Piccardi, C., Casagrandi, R.: Remarks on Epidemic Spreading in Scale-Free Networks, pp. 77-89. Springer, Berlin, Heidelberg (2009)
[25] Qu, M., Zhang, C.: Turing instability and patterns of the Fitzhugh-Nagumo model in square domain. J. Appl. Anal. Comput. 11(3), 1371-1390 (2021)
[26] Song, D., Song, Y., Li, C.: Stability and Turing patterns in a predator-prey model with hunting cooperation and Allee effect in prey population. J. Theor. Biol. 30(9), 2050137 (2020)
[27] Song, Y., Yang, R., Sun, G.: Pattern dynamics in a Gierer-Meinhardt model with a saturating term. Appl. Math. Model. 46, 476-491 (2017)
[28] Wang, Y., Zhou, X., Jiang, W.: Bifurcations in a diffusive predator-prey system with linear harvesting. Chaos Solitons Fractals 169, 113286 (2023)
[29] Wang, Y., Zhou, X., Jiang, W., Qi, L.: Turing instability and pattern formation in a diffusive Sel’kov-Schnakenberg system. J. Math. Chem. 61(5), 1036-1062 (2023)
[30] Yang, G., Tang, X.: Dynamics analysis of three-species reaction-diffusion system via the multiple scale perturbation method. J. Appl. Anal. Comput. 12(1), 206-229 (2022)

Bifurcation Analysis of an Advertising Diffusion Model

Bifurcation Analysis of an Advertising Diffusion Model

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