TURING-HOPF BIFURCATION IN THE REACTION-DIFFUSION SYSTEM WITH DELAY AND APPLICATION TO A DIFFUSIVE PREDATOR-PREY MODEL

2019 ◽  
Vol 9 (3) ◽  
pp. 1132-1164 ◽  
Author(s):  
Yongli Song ◽  
◽  
Heping Jiang ◽  
Yuan Yuan ◽  
◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Predator Prey ◽  
System With Delay ◽  
Predator Prey Model ◽  
Prey Model

scholarly journals Global stability of solutions in a reaction-diffusion system of predator-prey model

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4665-4672
Author(s):  
Demou Luo ◽  
Hailin Liu
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

In this article, we investigate the global asymptotic stability of a reaction-diffusion system of predator-prey model. By applying the comparison principle and iteration method, we prove the global asymptotic stability of the unique positive equilibrium solution of (1.1).

scholarly journals Effect of Diffusion and Cross-Diffusion in a Predator-Prey Model with a Transmissible Disease in the Predator Species

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Guohong Zhang ◽  
Xiaoli Wang
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Predator Population ◽  
Predator Species ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Transmissible Disease

We study a Lotka-Volterra type predator-prey model with a transmissible disease in the predator population. We concentrate on the effect of diffusion and cross-diffusion on the emergence of stationary patterns. We first show that both self-diffusion and cross-diffusion can not cause Turing instability from the disease-free equilibria. Then we find that the endemic equilibrium remains linearly stable for the reaction diffusion system without cross-diffusion, while it becomes linearly unstable when cross-diffusion also plays a role in the reaction-diffusion system; hence, the instability is driven solely from the effect of cross-diffusion. Furthermore, we derive some results for the existence and nonexistence of nonconstant stationary solutions when the diffusion rate of a certain species is small or large.

Bifurcation Analysis and Spatiotemporal Patterns in a Diffusive Predator–Prey Model

2014 ◽  
Vol 24 (06) ◽  
pp. 1450081 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Shiping Lu ◽  
Yuepeng Wang
Keyword(s):  
Pattern Formation ◽  
Spiral Wave ◽  
Reaction Diffusion ◽  
Spatiotemporal Chaos ◽  
Diffusion System ◽  
Target Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Impact

In this paper, we consider a species predator–prey model given a reaction–diffusion system. It incorporates the Holling type II functional response and a quadratic intra-predator interaction term. We focus on the qualitative analysis, bifurcation mechanisms and pattern formation. We present the results of numerical experiments in two space dimensions and illustrate the impact of the diffusion on the Turing pattern formation. For this diffusion system, we also observe non-Turing structures such as spiral wave, target pattern and spatiotemporal chaos resulting from the time evolution of these structures.

scholarly journals Stability and Hopf bifurcation of a delayed reaction–diffusion predator–prey model with anti-predator behaviour

2019 ◽  
Vol 24 (3) ◽  
pp. 387-406
Author(s):  
Jia Liu ◽  
Xuebing Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Discrete Time Delay ◽  
Predator Behaviour

In this paper, we study the dynamics of a delayed reaction–diffusion predator–prey model with anti-predator behaviour. By using the theory of partial functional differential equations, Hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated. It reveals that the discrete time delay has a destabilizing effect in the model, and a phenomenon of Hopf bifurcation occurs as the delay increases through a certain threshold. By utilizing upperlower solution method, the global asymptotic stability of the interior equilibrium is studied. Finally, numerical simulation results are presented to validate the theoretical analysis.

scholarly journals Complexity in a prey-predator model

2008 ◽  
Vol Volume 9, 2007 Conference in... ◽  
Author(s):  
Baba I. Camara ◽  
Moulay A. Aziz Alaoui
Keyword(s):  
Functional Response ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Type Ii ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Réponse Fonctionnelle ◽  
International Audience ◽  
Predator Model

International audience In this paper we consider a predator-prey model given by a reaction-diffusion system. It incorporates the Holling-type-II and a modified Leslie-Gower functional response. We focus on qualitaive analysis, bifurcation mecanisms and patterns formation. Nous considérons un modèle proie-prédateur exprimé sous forme de système de réaction diffusion. En absence de diffusion, le système étudié est de type Holling-type-II et la réponse fonctionnelle une forme modifiée du terme de Leslie-Gower. Dans cet article, nous nous intéressons à l’analyse qualitative des solutions , l’étude des bifurcations et la formation de motifs spatio-temporels.

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