# Bifurcation Analysis of Time-Delay Model of Consumer with the Advertising Effect

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## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Model and Its Dynamics

**Proposition**

**1.**

- 1.
- Semi-trivial equilibrium point ${E}_{1}(1,1,0)$,
- 2.
- Nontrivial equilibrium ${E}_{2}({a}_{0},1,k)$, where ${a}_{0}=\frac{\gamma (a+k)}{\gamma (a+k)+bk}$.

**Lemma**

**1.**

- 1.
- $Re\left({\lambda}_{1,2}\right)>0$ iff $\gamma <1$;
- 2.
- $Re\left({\lambda}_{1,2}\right)<0$ iff $\gamma >1$;
- 3.
- $Re\left({\lambda}_{1,2}\right)=0$ and $Im\left({\lambda}_{1,2}\right)\ne 0$ iff $\gamma =1$;
- 4.
- ${\lambda}_{1}={\lambda}_{2}$ iff $\beta =\frac{1}{4}(6-\gamma -\frac{1}{\gamma})$.

**Lemma**

**2.**

- 1.
- $Re\left({\lambda}_{1,2}\right)>0$ iff ${\mathsf{\Psi}}_{1}<0$;
- 2.
- $Re\left({\lambda}_{1,2}\right)<0$ iff ${\mathsf{\Psi}}_{1}>0$;
- 3.
- $Re\left({\lambda}_{1,2}\right)=0$ and $Im\left({\lambda}_{1,2}\right)\ne 0$ iff ${\mathsf{\Psi}}_{1}=0$;
- 4.
- ${\lambda}_{1}={\lambda}_{2}$ iff $\mathsf{\Delta}={\mathsf{\Psi}}_{1}^{2}-4{\mathsf{\Psi}}_{2}=0$.

**Theorem**

**1.**

- 1.
- A hyperbolic saddle if $\gamma >1$;
- 2.
- An unstable Equilibrium Point if $\gamma <1$;
- 3.
- A non-hyperbolic point if the parameter satisfies one of the following conditions
- (a)
- $\gamma =1$;
- (b)
- $d=0$.

**Theorem**

**2.**

- 1.
- A hyperbolic saddle if ${\mathsf{\Psi}}_{1}<0$;
- 2.
- A stable Equilibrium Point if ${\mathsf{\Psi}}_{1}>0$;
- 3.
- A non-hyperbolic Equilibrium Point if ${\mathsf{\Psi}}_{1}=0$.

## 3. Stability and Bifurcation Analysis

#### 3.1. Hopf Bifurcation of Semi-Trivial Equilibrium Point ${E}_{1}$

**Proof.**

#### 3.2. Hopf Bifurcation of Non-Trivial Equilibrium Point ${E}_{2}$

**Theorem**

**4.**

**Proof.**

#### 3.3. Numerical Simulations

## 4. Delayed Model and Its Dynamics

#### 4.1. Stability of Equilibria and Bifurcations of Periodic Solutions

**Lemma**

**3.**

- $\left({L}_{3}\right)$
- If either ${A}_{0}>0$ and ${A}_{1}>0$ or ${A}_{1}^{2}<4{A}_{0},$ then Equation (26) does not have any positive roots.
- $\left({L}_{4}\right)$
- If either ${A}_{0}<0$ or ${A}_{1}<0$ and ${A}_{1}^{2}=4{A}_{0}$ then Equation (26) has one positive root.
- $\left({L}_{5}\right)$
- If ${A}_{0}>0$ and ${A}_{1}<0$ and ${A}_{1}^{2}>4{A}_{0}$ then Equation (26) has two positive roots.

- If the conditions $\left({L}_{1}\right)$ or $\left({L}_{2}\right)$ do not hold, the fixed point E is unstable for all $\tau \ge 0.$
- If the conditions $\left({L}_{1}\right),\left({L}_{2}\right)$ and $\left({L}_{3}\right)$ hold, the fixed point E is stable for all $\tau \ge 0.$
- If the conditions $\left({L}_{1}\right),\left({L}_{2}\right)$ and $\left({L}_{4}\right)$ hold, the equilibrium point E is stable for $\tau \in [0,{\tau}_{1,0})$ and unstable for $\tau >{\tau}_{1,0}.$ Moreover, the Hopf bifurcation occurs when $\tau ={\tau}_{1,0}.$
- If the conditions $\left({L}_{1}\right),\left({L}_{2}\right)$ and $\left({L}_{5}\right)$ hold, there is an integer $p\ge 1$, such that$$0<{\tau}_{1,0}<{\tau}_{2,0}<{\tau}_{1,1}<{\tau}_{2,1}<\cdots <{\tau}_{1,p-1}<{\tau}_{1,p}<{\tau}_{2,p-1}<\cdots .$$So the equilibrium point E is stable for $\tau \in [0,{\tau}_{1,0})\cup \left({\tau}_{2,j},{\tau}_{1,j+1}\right)$ for $j=0,1,\dots ,p-1$ and unstable for $\tau \in \left({\tau}_{1,j},{\tau}_{2,j}\right)\cup ({\tau}_{1,p},\infty )$ for $j=0,1,\dots ,p-1.$ Moreover, the Hopf bifurcation occurs when $\tau ={\tau}_{1,j}$ and $\tau ={\tau}_{2,j}$ for $j=0,1,\dots ,p.$

#### 4.2. Direction and Stability of the Hopf Bifurcation

- If ${\mu}_{2}>0(<0)$, the direction of bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for $\tau >{\tau}_{m}(\tau <{\tau}_{m})$,
- If ${\beta}_{2}>0(<0)$, the solutions of bifurcating periodic solutions are orbitally stable (unstable),
- If ${\tau}_{2}>0(<0)$, the periods of bifurcating periodic solutions increase (decrease).

**Remark**

**1.**

#### 4.3. Numerical Simulations

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Smith, R.E.; Chen, J.; Yang, X. The impact of advertising creativity on the hierarchy of effects. J. Advert.
**2008**, 37, 47–62. [Google Scholar] [CrossRef] - Blecher, E. The impact of tobacco advertising bans on consumption in developing countries. J. Health Econ.
**2008**, 27, 930–942. [Google Scholar] [CrossRef] - Aslam, S.; Jadoon, E.; Zaman, K.; Gondal, S. Effect of word of mouth on consumer buying behavior. Mediterr. J. Soc. Sci.
**2011**, 2, 497. [Google Scholar] - He, Q.; Qu, H. The impact of advertising appeals on purchase intention in social media environment analysis of intermediary effect based on brand attitude. J. Bus. Adm. Res.
**2018**, 7, 17. [Google Scholar] [CrossRef] [Green Version] - Jovanović, P.; Vlastelica, T.; Kostić, S.C. Impact of advertising appeals on purchase intention. Manag. J. Sustain. Bus. Manag. Solut. Emerg. Econ.
**2017**, 21, 35–45. [Google Scholar] - Bass, F.M. A new product growth model for consumer durables. Manag. Sci.
**1969**, 15, 215–322. [Google Scholar] [CrossRef] - Dodson, J.A., Jr.; Muller, E. Models of new product diffusion through advertising and word-of-mouth. Manag. Sci.
**1978**, 24, 1568–1578. [Google Scholar] [CrossRef] [Green Version] - Feichtinger, G. Hopf bifurcation in an advertising diffusion model. J. Econ. Behav. Organ.
**1992**, 17, 401–411. [Google Scholar] [CrossRef] - Landa, F.J.; Velasco, F. Dynamic analysis of the current market and potential of organizations. Eur. J. Manag. Econ. Co.
**2004**, 13, 131–140. [Google Scholar] - Nie, P.; Abd-Rabo, M.A.; Sun, Y.; Ren, J. A consumption behavior model with advertising and word-of-mouth effect. J. Nonlinear Model. Anal.
**2019**, 1, 461–489. [Google Scholar] - Feichtinger, G.; GHEZZI, L.L.; Piccardi, C. Chaotic behavior in an advertising diffusion model. Int. J. Bifurc. Chaos
**1995**, 5, 255–263. [Google Scholar] [CrossRef] - Sirghi, N.; Neamtu, M. Deterministic and stochastic advertising diffusion model with delay. WSEAS Trans. Syst. Control
**2013**, 4, 141–150. [Google Scholar] - Kuznetsov, Y.A. Elements of Applied Bifurcation Theory, 2nd ed.; Marsden, J.E., Sirovich, L., Eds.; Springer: New York, NY, USA, 1998. [Google Scholar]
- Ren, J.; Yuan, Q. Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate. Chaos Interdiscip. J. Nonlinear Sci.
**2017**, 27, 083124. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ren, J.; Yu, L.; Zhu, H. Dynamic analysis of discrete-time, continuous-time and delayed feedback jerky equations. Nonlinear Dyn.
**2016**, 86, 107–130. [Google Scholar] [CrossRef] - Charykov, N.A.; Charykova, M.V.; Semenov, K.N.; Keskinov, V.A.; Kurilenko, A.V.; Shaimardanov, Z.K.; Shaimardanova, B.K. Multiphase open phase processes differential equations. Processes
**2019**, 7, 148. [Google Scholar] [CrossRef] [Green Version] - Li, H.; Cheng, J.; Li, H.B.; Zhong, S.-M. Stability analysis of a fractional-order linear system described by the caputo–fabrizio derivative. Mathematics
**2019**, 7, 200. [Google Scholar] [CrossRef] [Green Version] - Mahmoud, G.M.; Arafa, A.A.; Mahmoud, E.E. Bifurcations and chaos of time delayed Lorenz system with dimension 2n + 1. Eur. Phys. J. Plus
**2017**, 132, 461. [Google Scholar] [CrossRef] - Li, L.; Shen, J. Bifurcations and dynamics of the Rb-E2F pathway involving miR449. Complexity
**2017**, 2017, 1409865. [Google Scholar] [CrossRef] [Green Version] - Rihana, F.A.; Lakshmananb, S.; Maurer, H. Optimal control of tumour-immune model with time-delay and immuno-chemotherapy. Appl. Math. Comput.
**2019**, 353, 147–165. [Google Scholar] [CrossRef] - Yin, Z.; Yu, Y.; Lu, Z. Stability analysis of an age-structured SEIRS model with time delay. Mathematics
**2020**, 8, 455. [Google Scholar] [CrossRef] [Green Version] - Sun, C.; Lin, Y.; Han, M. Stability and Hopf bifurcation for an epidemic disease model with delay. Chaos Solitons Fractals
**2006**, 30, 204–216. [Google Scholar] [CrossRef] - Wang, F.; Wang, H.; Xu, K. Diffusive logistic model towards predicting information diffusion in online social networks. In Proceedings of the 2012 32nd International Conference on Distributed Computing Systems Workshops, Macau, China, 18–21 June 2012; pp. 133–139. [Google Scholar]
- J<i>a</i>¨ntschi, L. The eigenproblem translated for alignment of molecules. Symmetry
**2019**, 11, 1027. [Google Scholar] [CrossRef] [Green Version] - Dimitrova, N.; Zlateva, P. Global stability analysis of a bioreactor model for phenol and Cresol Mixture Degradation. Processes
**2021**, 9, 124. [Google Scholar] [CrossRef] - Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.W. Theory and Applications of Hopf Bifurcation; Cambridge University Press: Cambridge, UK, 1981; Volume 41. [Google Scholar]
- Song, Y.; Wei, J. Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos. Chaos Solitons Fractals
**2004**, 22, 75–91. [Google Scholar] [CrossRef] - Yang, J.; Zhao, L. Bifurcation analysis and chaos control of the modified Chua’s circuit system. Chaos Solitons Fractals
**2015**, 77, 332–339. [Google Scholar] [CrossRef]

**Figure 1.**Hopf bifurcation Diagrams of model at ${E}_{1}(1,1,0)$ (3) for bifurcation values of $\gamma =1$.

**Figure 2.**Hopf bifurcation Diagrams of model at ${E}_{2}(\frac{5.1\gamma}{0.5+5.1\gamma},1,0.1)$ (3) for bifurcation values of ${\gamma}_{1}=0.169751,{\gamma}_{2}=0.634171$.

**Figure 6.**The phase portrait of the limit cycle at $E(\frac{5.1\gamma}{0.5+5.1\gamma},1,0.1)$, ${\gamma}_{1}=0.169751$.

**Figure 8.**The phase portrait of the limit cycle at $E(\frac{5.1\gamma}{0.5+5.1\gamma},1,0.1)$, ${\gamma}_{2}=0.634171$.

**Figure 9.**The critical time delays ${\tau}_{1,j}$ (blue curves) and ${\tau}_{2,j}$ (red dash curves) for system (14) against $\beta $.

The Symbol | The Meaning |
---|---|

${k}_{0}>0$ | individuals stream into the firms market, |

$\epsilon >0$ | the current customers leaving the market forever, |

$\beta \ge 0$ | current customers switching to a rival brand, |

$\alpha >0$ | proportionality measuring of the advertising effectiveness. |

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**MDPI and ACS Style**

Abd-Rabo, M.A.; Zakarya, M.; Cesarano, C.; Aly, S.
Bifurcation Analysis of Time-Delay Model of Consumer with the Advertising Effect. *Symmetry* **2021**, *13*, 417.
https://doi.org/10.3390/sym13030417

**AMA Style**

Abd-Rabo MA, Zakarya M, Cesarano C, Aly S.
Bifurcation Analysis of Time-Delay Model of Consumer with the Advertising Effect. *Symmetry*. 2021; 13(3):417.
https://doi.org/10.3390/sym13030417

**Chicago/Turabian Style**

Abd-Rabo, Mahmoud A., Mohammed Zakarya, Clemente Cesarano, and Shaban Aly.
2021. "Bifurcation Analysis of Time-Delay Model of Consumer with the Advertising Effect" *Symmetry* 13, no. 3: 417.
https://doi.org/10.3390/sym13030417