Stability and Hopf Bifurcation of Steady State Solutions of a Singularly Perturbed Reaction-Diffusion System

1992 ◽  
Vol 23 (1) ◽  
pp. 99-149 ◽  
Author(s):  
Robert Gardner
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Singularly Perturbed ◽  
Diffusion System ◽  
Steady State Solutions

scholarly journals Existence of global solution and nontrivial steady states for a system modeling chemotaxis

2006 ◽  
Vol 2006 ◽  
pp. 1-23 ◽  
Author(s):  
Zhenbu Zhang
Keyword(s):  
Steady State ◽  
Global Existence ◽  
Global Solution ◽  
System Modeling ◽  
Reaction Diffusion ◽  
Steady States ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Steady State Solutions

We consider a reaction-diffusion system modeling chemotaxis, which describes the situation of two species of bacteria competing for the same nutrient. We use Moser-Alikakos iteration to prove the global existence of the solution. We also study the existence of nontrivial steady state solutions and their stability.

Stability and Hopf Bifurcation of a Reaction–Diffusion System with Weak Allee Effect

2021 ◽  
Vol 31 (07) ◽  
pp. 2150098
Author(s):  
Jia-Long Yue ◽  
Zhan-Ping Ma
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Allee Effect ◽  
Reaction Diffusion ◽  
Steady State Solution ◽  
State Solution ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Three Component Reaction ◽  
Species Population

A delayed three-component reaction–diffusion system with weak Allee effect and Dirichlet boundary condition is considered. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained via the implicit function theorem. Moreover, taking delay as the bifurcation parameter, the Hopf bifurcation near the spatially nonhomogeneous steady-state solution is proved to occur at a critical value. Especially, the direction of Hopf bifurcation is forward and the bifurcated periodic solutions are unstable. Finally, the general results are applied to four types of three-species population models with weak Allee effect in growth.

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