Turing–Hopf bifurcations in a predator–prey model with herd behavior, quadratic mortality and prey-taxis

2018 ◽  
Vol 496 ◽  
pp. 446-460 ◽  
Author(s):  
Xia Liu ◽  
Tonghua Zhang ◽  
Xinzhu Meng ◽  
Tongqian Zhang
Keyword(s):  
Hopf Bifurcations ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

Spatial Periodic Solutions in a Delayed Diffusive Predator–Prey Model with Herd Behavior

2015 ◽  
Vol 25 (11) ◽  
pp. 1550155 ◽  
Author(s):  
Chaoqun Xu ◽  
Sanling Yuan
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Critical Conditions ◽  
Hopf Bifurcations ◽  
Herd Behavior ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

A delayed diffusive predator–prey model with herd behavior subject to Neumann boundary conditions is studied both theoretically and numerically. Applying Hopf bifurcation analysis, we obtain the critical conditions under which the model generates spatially nonhomogeneous bifurcating periodic solutions. It is shown that the spatially homogeneous Hopf bifurcations always exist and that the spatially nonhomogeneous Hopf bifurcations will arise when the diffusion coefficients are suitably small. The explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorems for partial functional differential equations.

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.

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.

Spatiotemporal Model of a Predator–Prey System with Herd Behavior and Quadratic Mortality

2019 ◽  
Vol 29 (04) ◽  
pp. 1950049 ◽  
Author(s):  
Teekam Singh ◽  
Sandip Banerjee
Keyword(s):  
Herd Behavior ◽  
Spatiotemporal Model ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pattern Dynamics ◽  
Prey System ◽  
Prey Model ◽  
Prey Interaction ◽  
Mixed Pattern ◽  
Spotted Pattern

In this paper, we have investigated a diffusive predator–prey model with herd behavior. Also, we considered that the mortality of predators is linear as well as quadratic. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of control parameters. Using extensive numerical simulations, we obtain non-Turing spatiotemporal patterns in the model with linear mortality of predators, and Turing pattern formation, namely, spotted pattern and mixed pattern (spots-stripes) in model with quadratic mortality of predators. The results focus on the effect of the changing mortality rates of predator in pattern dynamics of a diffusive predator–prey model and help us in the better understanding of the dynamics of the predator–prey interaction in real environment.

Sign in / Sign up

Export Citation Format

Share Document