Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

2016 ◽  
Vol 86 (1) ◽  
pp. 73-89 ◽  
Author(s):  
Xiaosong Tang ◽  
Yongli Song ◽  
Tonghua Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Herd Behavior ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

Stability and Hopf bifurcation analysis of a diffusive predator–prey model with Smith growth

2015 ◽  
Vol 08 (01) ◽  
pp. 1550013 ◽  
Author(s):  
M. Sivakumar ◽  
M. Sambath ◽  
K. Balachandran
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Neumann Boundary Condition ◽  
Local Stability ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

In this paper, we consider a diffusive Holling–Tanner predator–prey model with Smith growth subject to Neumann boundary condition. We analyze the local stability, existence of a Hopf bifurcation at the co-existence of the equilibrium and stability of bifurcating periodic solutions of the system in the absence of diffusion. Furthermore the Turing instability and Hopf bifurcation analysis of the system with diffusion are studied. Finally numerical simulations are given to demonstrate the effectiveness of the theoretical analysis.

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

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.

scholarly journals Dynamics of a diffusive predator–prey model with herd behavior

2020 ◽  
Vol 25 (1) ◽  
Author(s):  
Yan Li ◽  
Sanyun Li ◽  
Fengrong Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Positive Constant ◽  
Herd Behavior ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Prey Model ◽  
The Stability

This paper is devoted to considering a diffusive predator–prey model with Leslie–Gower term and herd behavior subject to the homogeneous Neumann boundary conditions. Concretely, by choosing the proper bifurcation parameter, the local stability of constant equilibria of this model without diffusion and the existence of Hopf bifurcation are investigated by analyzing the distribution of the eigenvalues. Furthermore, the explicit formula for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are also derived by applying the normal form theory. Next, we show the stability of positive constant equilibrium, the existence and stability of periodic solutions near positive constant equilibrium for the diffusive model. Finally, some numerical simulations are carried out to support the analytical results.

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