scholarly journals Existence of Solutions for a Quasilinear Reaction Diffusion System

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Canrong Tian
Keyword(s):  
Existence Of Solutions ◽  
Upper And Lower Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Reaction Functions ◽  
Engineering Problems ◽  
Activator Inhibitor

The degenerate reaction diffusion system has been applied to a variety of physical and engineering problems. This paper is extended the existence of solutions from the quasimonotone reaction functions (e.g., inhibitor-inhibitor mechanism) to the mixed quasimonotone reaction functions (e.g., activator-inhibitor mechanism). By Schauder fixed point theorem, it is shown that the system admits at least one positive solution if there exist a coupled of upper and lower solutions. This result is applied to a Lotka-Volterra predator-prey model.

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.

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.

Analysis of a Delayed Predator-Prey Reaction-Diffusion System with Beddington-DeAngelis Functional Response and Two Stage Structure

2011 ◽  
Vol 204-210 ◽  
pp. 2065-2070
Author(s):  
Fang Wei Zhang ◽  
Shi He Xu ◽  
Bao Shi
Keyword(s):  
Functional Response ◽  
Upper And Lower Solutions ◽  
Reaction Diffusion ◽  
Stage Structure ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Mature Stage ◽  
Two Stage ◽  
Predator Prey ◽  
Positive Steady States

In this paper, we study a delayed predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and two stage structure on prey, an immature and a mature stage. The time delays are the time lengths between the immature’s birth and maturity prey species. Under some simple assumptions, using upper and lower solutions method and a comparison principle we prove that the unique constant positive steady states solution is global asymptotic stability. The results shows that the dynamic behavior of solutions to this model is similar to that of solutions to corresponding model without diffusion.

scholarly journals Stabilization for a Periodic Predator-Prey System

2007 ◽  
Vol 2007 ◽  
pp. 1-17
Author(s):  
Sebastian Aniţa ◽  
Carmen Oana Tarniceriu
Keyword(s):  
Internal Control ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Predator Prey ◽  
Periodic Environment ◽  
Prey System

A reaction-diffusion system modelling a predator-prey system in a periodic environment is considered. We are concerned in stabilization to zero of one of the components of the solution, via an internal control acting on a small subdomain, and in the preservation of the nonnegativity of both components.

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