Stability, Steady-State Bifurcations, and Turing Patterns in a Predator-Prey Model with Herd Behavior and Prey-taxis

2017 ◽  
Vol 139 (3) ◽  
pp. 371-404 ◽  
Author(s):  
Yongli Song ◽  
Xiaosong Tang
Keyword(s):  
Steady State ◽  
Turing Patterns ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

CROSS-DIFFUSION INDUCED TURING PATTERNS IN A SEX-STRUCTURED PREDATOR–PREY MODEL

2012 ◽  
Vol 05 (04) ◽  
pp. 1250016 ◽  
Author(s):  
JIA LIU ◽  
HUA ZHOU ◽  
LAI ZHANG
Keyword(s):  
Steady State ◽  
Degree Theory ◽  
Reaction Diffusion ◽  
Turing Patterns ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Existence And Stability

In this paper, we consider a sex-structured predator–prey model with strongly coupled nonlinear reaction diffusion. Using the Lyapunov functional and Leray–Schauder degree theory, the existence and stability of both homogenous and heterogenous steady-states are investigated. Our results demonstrate that the unique homogenous steady-state is locally asymptotically stable for the associated ODE system and PDE system with self-diffusion. With the presence of the cross-diffusion, the homogeneous equilibrium is destabilized, and a heterogenous steady-state emerges as a consequence. In addition, the conditions guaranteeing the emergence of Turing patterns are derived.

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.

Turing patterns in a predator–prey model on complex networks

2020 ◽  
Vol 99 (4) ◽  
pp. 3313-3322 ◽  
Author(s):  
Chen Liu ◽  
Lili Chang ◽  
Yue Huang ◽  
Zhen Wang
Keyword(s):  
Complex Networks ◽  
Turing Patterns ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model
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