An algorithm for Hopf bifurcation analysis of a delayed reaction–diffusion model

2017 ◽  
Vol 89 (1) ◽  
pp. 345-366 ◽  
Author(s):  
Ş. Kayan ◽  
H. Merdan
Keyword(s):  
Hopf Bifurcation ◽  
Diffusion Model ◽  
Bifurcation Analysis ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Reaction Diffusion Model

scholarly journals TRAVELING WAVE SOLUTIONS IN A STAGE-STRUCTURED DELAYED REACTION-DIFFUSION MODEL WITH ADVECTION

2015 ◽  
Vol 20 (2) ◽  
pp. 168-187
Author(s):  
Liang Zhang ◽  
Huiyan Zhao
Keyword(s):  
Diffusion Model ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Stage Structure ◽  
Delayed Reaction ◽  
Wave Solutions ◽  
Reaction Diffusion Model ◽  
Travelling Fronts ◽  
Appl Math

We investigate a stage-structured delayed reaction-diffusion model with advection that describes competition between two mature species in water flow. Time delays are incorporated to measure the time lengths from birth to maturity of the populations. We show there exists a finite positive number c∗ that can be characterized as the slowest spreading speed of traveling wave solutions connecting two mono-culture equilibria or connecting a mono-culture with the coexistence equilibrium. The model and mathematical result in [J.F.M. Al-Omari, S.A. Gourley, Stability and travelling fronts in Lotka–Volterra competition models with stage structure, SIAM J. Appl. Math. 63 (2003) 2063–2086] are generalized.

scholarly journals Hopf Bifurcation in an Oscillatory-Excitable Reaction–Diffusion Model with Spatial Heterogeneity

2017 ◽  
Vol 27 (05) ◽  
pp. 1750065
Author(s):  
Benjamin Ambrosio
Keyword(s):  
Hopf Bifurcation ◽  
Qualitative Analysis ◽  
Spatial Heterogeneity ◽  
Numerical Simulations ◽  
Diffusion Model ◽  
Center Manifold ◽  
Reaction Diffusion ◽  
Reaction Diffusion Model

We focus on the qualitative analysis of a reaction–diffusion model with spatial heterogeneity. The system is a generalization of the well-known FitzHugh–Nagumo system, in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of a Hopf bifurcation and determine an equation of the center-manifold in which the solution asymptotically evolves. Numerical simulations illustrate the phenomenon.

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