A note on positive non-constant steady-state solutions to a reaction–diffusion model

2010 ◽  
Vol 217 (8) ◽  
pp. 4234-4238
Author(s):  
Wei Dong ◽  
Zhongli Wei ◽  
Changci Pang
Keyword(s):  
Steady State ◽  
Diffusion Model ◽  
Reaction Diffusion ◽  
Reaction Diffusion Model ◽  
Steady State Solutions

scholarly journals The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime

2014 ◽  
Vol 19 (5) ◽  
pp. 1373-1410 ◽  
Author(s):  
Theodore Kolokolnikov ◽  
◽  
Michael J. Ward ◽  
Juncheng Wei ◽  
◽  
...  
Keyword(s):  
Steady State ◽  
Diffusion Model ◽  
Hot Spot ◽  
Reaction Diffusion ◽  
Urban Crime ◽  
Reaction Diffusion Model ◽  
The Stability

Stability and Hopf Bifurcation in a Reaction–Diffusion Model with Chemotaxis and Nonlocal Delay Effect

2018 ◽  
Vol 28 (04) ◽  
pp. 1850046 ◽  
Author(s):  
Dong Li ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Diffusion Model ◽  
Dirichlet Boundary ◽  
Reaction Diffusion ◽  
High Concentration ◽  
Delay Effect ◽  
One Dimensional ◽  
Nonlocal Delay ◽  
Reaction Diffusion Model ◽  
Steady State Solutions

Chemotaxis is an observed phenomenon in which a biological individual moves preferentially toward a relatively high concentration, which is contrary to the process of natural diffusion. In this paper, we study a reaction–diffusion model with chemotaxis and nonlocal delay effect under Dirichlet boundary condition by using Lyapunov–Schmidt reduction and the implicit function theorem. The existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady state solutions are investigated. Moreover, our results are illustrated by an application to the model with a logistic source, homogeneous kernel and one-dimensional spatial domain.

Bifurcations in Twinkling Patterns for the Lengyel–Epstein Reaction–Diffusion Model

2021 ◽  
Vol 31 (11) ◽  
pp. 2150164
Author(s):  
J. Sarría-González ◽  
Ivonne Sgura ◽  
M. R. Ricard
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Diffusion Model ◽  
Reaction System ◽  
Reaction Diffusion ◽  
Spatially Inhomogeneous ◽  
Reaction Diffusion Model ◽  
Spatially Homogeneous ◽  
Time Periodic ◽  
Theoretical Results

Conditions for the emergence of strong Turing–Hopf instabilities in the Lengyel–Epstein CIMA reaction–diffusion model are found. Under these conditions, time periodic spatially inhomogeneous solutions can be induced by diffusive instability of the spatially homogeneous limit cycle emerging at a supercritical Bautin–Hopf bifurcation about the unstable steady state of the reaction system. We report numerical simulations by an Alternating Directions Implicit (ADI) method that show the formation of twinkling patterns for a chosen parameter value, thus confirming our theoretical results.

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