{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T07:55:21Z","timestamp":1773734121995,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,4]],"date-time":"2021-03-04T00:00:00Z","timestamp":1614816000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"King Khalid University","doi-asserted-by":"publisher","award":["RGP. 2\/51\/42."],"award-info":[{"award-number":["RGP. 2\/51\/42."]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Given the economic importance of advertising and product promotions, we have developed a diffusion model to describe the impact of advertising on sales. The main message of this study is to show the effect of advertising diffusion to convert potential buyers into actual customers which may result in persistent alteration in marketing over time. This work is devoted to studying the dynamic behavior of a reaction-diffusion model and its delayed version with the advertising effect. For the non-delay model, it is proven the existence of Hopf bifurcation. Moreover, the stability and direction of bifurcation of periodic solutions are detected. On the other hand, we consider there is a lag for responding of potential buyers to the advertising. Therefore, the time delay \u03c4 is deemed as an additional factor in the diffusion model. We have determined the critical values for the delay parameter that yield periodic solutions. Furthermore, the direction and the stability of bifurcating periodic solutions is studied. For supporting the theoretical analysis and demonstrate complex dynamic behaviors, numerical simulations including families of periodic curves are given.<\/jats:p>","DOI":"10.3390\/sym13030417","type":"journal-article","created":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T00:39:07Z","timestamp":1614904747000},"page":"417","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Bifurcation Analysis of Time-Delay Model of Consumer with the Advertising Effect"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5063-5301","authenticated-orcid":false,"given":"Mahmoud A.","family":"Abd-Rabo","sequence":"first","affiliation":[{"name":"Zhengzhou Key Laboratory of Big Data Analysis and Application, Henan Academy of Big Data, Zhengzhou University, Zhengzhou 450052, China"},{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut branch, Assiut 71524, Egypt"}]},{"given":"Mohammed","family":"Zakarya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut branch, Assiut 71524, Egypt"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, Universit\u00e0 Telematica Internazionale Uninettuno, 00186 Rome, Italy"}]},{"given":"Shaban","family":"Aly","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut branch, Assiut 71524, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"47","DOI":"10.2753\/JOA0091-3367370404","article-title":"The impact of advertising creativity on the hierarchy of effects","volume":"37","author":"Smith","year":"2008","journal-title":"J. Advert."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1016\/j.jhealeco.2008.02.010","article-title":"The impact of tobacco advertising bans on consumption in developing countries","volume":"27","author":"Blecher","year":"2008","journal-title":"J. Health Econ."},{"key":"ref_3","first-page":"497","article-title":"Effect of word of mouth on consumer buying behavior","volume":"2","author":"Aslam","year":"2011","journal-title":"Mediterr. J. Soc. Sci."},{"key":"ref_4","first-page":"17","article-title":"The impact of advertising appeals on purchase intention in social media environment analysis of intermediary effect based on brand attitude","volume":"7","author":"He","year":"2018","journal-title":"J. Bus. Adm. Res."},{"key":"ref_5","first-page":"35","article-title":"Impact of advertising appeals on purchase intention","volume":"21","author":"Vlastelica","year":"2017","journal-title":"Manag. J. Sustain. Bus. Manag. Solut. Emerg. Econ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1287\/mnsc.15.5.215","article-title":"A new product growth model for consumer durables","volume":"15","author":"Bass","year":"1969","journal-title":"Manag. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1568","DOI":"10.1287\/mnsc.24.15.1568","article-title":"Models of new product diffusion through advertising and word-of-mouth","volume":"24","author":"Dodson","year":"1978","journal-title":"Manag. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/S0167-2681(95)90016-0","article-title":"Hopf bifurcation in an advertising diffusion model","volume":"17","author":"Feichtinger","year":"1992","journal-title":"J. Econ. Behav. Organ."},{"key":"ref_9","first-page":"131","article-title":"Dynamic analysis of the current market and potential of organizations","volume":"13","author":"Landa","year":"2004","journal-title":"Eur. J. Manag. Econ. Co."},{"key":"ref_10","first-page":"461","article-title":"A consumption behavior model with advertising and word-of-mouth effect","volume":"1","author":"Nie","year":"2019","journal-title":"J. Nonlinear Model. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1142\/S021812749500020X","article-title":"Chaotic behavior in an advertising diffusion model","volume":"5","author":"Feichtinger","year":"1995","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_12","first-page":"141","article-title":"Deterministic and stochastic advertising diffusion model with delay","volume":"4","author":"Sirghi","year":"2013","journal-title":"WSEAS Trans. Syst. Control"},{"key":"ref_13","unstructured":"Marsden, J.E., and Sirovich, L. (1998). Elements of Applied Bifurcation Theory, Springer. [2nd ed.]."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"083124","DOI":"10.1063\/1.5000152","article-title":"Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate","volume":"27","author":"Ren","year":"2017","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s11071-016-2875-1","article-title":"Dynamic analysis of discrete-time, continuous-time and delayed feedback jerky equations","volume":"86","author":"Ren","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Charykov, N.A., Charykova, M.V., Semenov, K.N., Keskinov, V.A., Kurilenko, A.V., Shaimardanov, Z.K., and Shaimardanova, B.K. (2019). Multiphase open phase processes differential equations. Processes, 7.","DOI":"10.3390\/pr7030148"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Li, H., Cheng, J., Li, H.B., and Zhong, S.-M. (2019). Stability analysis of a fractional-order linear system described by the caputo\u2013fabrizio derivative. Mathematics, 7.","DOI":"10.3390\/math7020200"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1140\/epjp\/i2017-11739-6","article-title":"Bifurcations and chaos of time delayed Lorenz system with dimension 2n + 1","volume":"132","author":"Mahmoud","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1409865","DOI":"10.1155\/2017\/1409865","article-title":"Bifurcations and dynamics of the Rb-E2F pathway involving miR449","volume":"2017","author":"Li","year":"2017","journal-title":"Complexity"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.amc.2019.02.002","article-title":"Optimal control of tumour-immune model with time-delay and immuno-chemotherapy","volume":"353","author":"Rihana","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yin, Z., Yu, Y., and Lu, Z. (2020). Stability analysis of an age-structured SEIRS model with time delay. Mathematics, 8.","DOI":"10.3390\/math8030455"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.chaos.2005.08.167","article-title":"Stability and Hopf bifurcation for an epidemic disease model with delay","volume":"30","author":"Sun","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Wang, F., Wang, H., and Xu, K. (2012, January 18\u201321). Diffusive logistic model towards predicting information diffusion in online social networks. Proceedings of the 2012 32nd International Conference on Distributed Computing Systems Workshops, Macau, China.","DOI":"10.1109\/ICDCSW.2012.16"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"J<i>a<\/i>\u00a8ntschi, L. (2019). The eigenproblem translated for alignment of molecules. Symmetry, 11.","DOI":"10.3390\/sym11081027"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Dimitrova, N., and Zlateva, P. (2021). Global stability analysis of a bioreactor model for phenol and Cresol Mixture Degradation. Processes, 9.","DOI":"10.3390\/pr9010124"},{"key":"ref_26","unstructured":"Hassard, B.D., Kazarinoff, N.D., and Wan, Y.W. (1981). Theory and Applications of Hopf Bifurcation, Cambridge University Press."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.chaos.2003.12.075","article-title":"Bifurcation analysis for Chen\u2019s system with delayed feedback and its application to control of chaos","volume":"22","author":"Song","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.chaos.2015.05.028","article-title":"Bifurcation analysis and chaos control of the modified Chua\u2019s circuit system","volume":"77","author":"Yang","year":"2015","journal-title":"Chaos Solitons Fractals"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/417\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:32:35Z","timestamp":1760160755000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/417"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,4]]},"references-count":28,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["sym13030417"],"URL":"https:\/\/doi.org\/10.3390\/sym13030417","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,4]]}}}