scholarly journals Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

2018 ◽  
pp. 1-22
Author(s):  
Xiuli Sun ◽  
Luan Wang ◽  
Baochuan Tian
Keyword(s):  
Hopf Bifurcation ◽  
Population Model ◽  
Delay Effect ◽  
Nonlocal Delay

Stability and Hopf Bifurcation in a Reaction–Diffusion Model with Chemotaxis and Nonlocal Delay Effect

2018 ◽  
Vol 28 (04) ◽  
pp. 1850046 ◽  
Author(s):  
Dong Li ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Diffusion Model ◽  
Dirichlet Boundary ◽  
Reaction Diffusion ◽  
High Concentration ◽  
Delay Effect ◽  
One Dimensional ◽  
Nonlocal Delay ◽  
Reaction Diffusion Model ◽  
Steady State Solutions

Chemotaxis is an observed phenomenon in which a biological individual moves preferentially toward a relatively high concentration, which is contrary to the process of natural diffusion. In this paper, we study a reaction–diffusion model with chemotaxis and nonlocal delay effect under Dirichlet boundary condition by using Lyapunov–Schmidt reduction and the implicit function theorem. The existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady state solutions are investigated. Moreover, our results are illustrated by an application to the model with a logistic source, homogeneous kernel and one-dimensional spatial domain.

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

2018 ◽  
Vol 28 (11) ◽  
pp. 1850136 ◽  
Author(s):  
Ben Niu ◽  
Yuxiao Guo ◽  
Yanfei Du
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Immune Cell ◽  
Tumor Treatment ◽  
Delay Effect ◽  
Stable Periodic ◽  
The Stability ◽  
Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

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