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

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 τ 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.

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The influence of density in population dynamics with strong and weak Allee effect

Journal of the Egyptian Mathematical Society ◽

10.1186/s42787-021-00114-x ◽

2021 ◽

Vol 29 (1) ◽

Author(s):

Kamrun Nahar Keya ◽

Md. Kamrujjaman ◽

Md. Shafiqul Islam

Keyword(s):

Population Dynamics ◽

Allee Effect ◽

Global Bifurcation ◽

Reaction Diffusion ◽

Logistic Growth ◽

Allee Effects ◽

Reaction Diffusion Model ◽

Positive Steady States ◽

The Stability ◽

The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

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On the stability of a “reaction-diffusion” model in ecological systems

Electrical Engineering in Japan ◽

10.1002/eej.4390980617 ◽

1978 ◽

Vol 98 (6) ◽

pp. 121-126

Author(s):

Koichi Harada ◽

Takeshi Fukao

Keyword(s):

Diffusion Model ◽

Reaction Diffusion ◽

Ecological Systems ◽

Reaction Diffusion Model ◽

The Stability

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Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law

Entropy ◽

10.3390/e23111516 ◽

2021 ◽

Vol 23 (11) ◽

pp. 1516

Author(s):

Adel Ouannas ◽

Iqbal M. Batiha ◽

Stelios Bekiros ◽

Jinping Liu ◽

Hadi Jahanshahi ◽

...

Keyword(s):

Diffusion Model ◽

Reaction Diffusion ◽

Linear Control ◽

Control Law ◽

Synchronization Problem ◽

Reaction Diffusion Model ◽

Spatial Dimensions ◽

Error System ◽

The Stability ◽

Complex Formations

The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion model is handled and examined. In addition, a novel convenient control law is designed in a linear form and, on the other hand, the stability of the associated error system is demonstrated through utilizing a suitable Lyapunov function. To illustrate the applicability of the proposed schemes, several numerical simulations are performed in one- and two-spatial dimensions.

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Effect of delay on pattern formation of a Rosenzweig–MacArthur type reaction–diffusion model with spatiotemporal delay

International Journal of Biomathematics ◽

10.1142/s1793524522500085 ◽

2021 ◽

Author(s):

Gao-Xiang Yang ◽

Xiao-Yu Li

Keyword(s):

Time Delay ◽

Diffusion Model ◽

Reaction Diffusion ◽

Multiple Scale ◽

Predator Prey ◽

Turing Bifurcation ◽

Reaction Diffusion Model ◽

Scale Perturbation ◽

Spatiotemporal Delay ◽

Theoretical Results

In this paper, a predator–prey reaction–diffusion model with Rosenzweig–MacArthur type functional response and spatiotemporal delay is investigated through using the tool of Turing bifurcation theories. First, by taking the average time delay as a bifurcation parameter, conditions of occurrence of Turing bifurcation are obtained through employing the Routh–Hurwitz criteria. Second, as the average time delay varies the amplitude equations of Turing bifurcation patterns including spots pattern and stripes pattern are also obtained through the multiple scale perturbation method. Finally, the two kinds of spatiotemporal evolution distributions of species such as spots pattern and stripes pattern are shown to illustrate theoretical results.

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Hopf Bifurcation in a Delayed Diffusive Leslie–Gower Predator–Prey Model with Herd Behavior

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741950055x ◽

2019 ◽

Vol 29 (04) ◽

pp. 1950055

Author(s):

Fengrong Zhang ◽

Yan Li ◽

Changpin Li

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Differential Equations ◽

Periodic Solutions ◽

Herd Behavior ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

The Stability ◽

The Impact

In this paper, we consider a delayed diffusive predator–prey model with Leslie–Gower term and herd behavior subject to Neumann boundary conditions. We are mainly concerned with the impact of time delay on the stability of this model. First, for delayed differential equations and delayed-diffusive differential equations, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated respectively. It is observed that when time delay continues to increase and crosses through some critical values, a family of homogeneous and inhomogeneous periodic solutions emerge. Then, the explicit formula for determining the stability and direction of bifurcating periodic solutions are also derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are shown to support the analytical results.

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