ON TURING-HOPF INSTABILITIES IN REACTION-DIFFUSION SYSTEMS

2008 ◽  
Vol 03 (01n02) ◽  
pp. 257-274 ◽  
Author(s):  
MARIANO RODRÍGUEZ RICARD
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Hopf Bifurcations ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Turing Instabilities ◽  
Spatially Homogeneous

We examine the appearance of Turing instabilities of spatially homogeneous periodic solutions in reaction-diffusion equations when such periodic solutions are consequence of Hopf bifurcations. First, we asymptotically develop limit cycle solutions associated to the appearance of Hopf bifurcations in reaction systems. Particularly, we will show conditions to the appearance of multiple limit cycles after Hopf bifurcation. Then, we propose expansions to normal modes associated with Turing instabilities from spatially homogeneous periodic solutions associated to limit cycles which appear as a consequence of a Hopf bifurcation. Finally, we discuss examples of reaction-diffusion systems arising in biology and chemistry in which can be observed spatial and time-periodic patterning.

scholarly journals Studies on the Existence of Unstable Oscillatory Patterns Bifurcating from Hopf Bifurcations in a Turing Model

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Yan Zhang ◽  
Zhenhua Bao
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Hopf Bifurcations ◽  
Hopf Bifurcation Point ◽  
The One ◽  
Model Subject ◽  
Turing Model

We revisit a homogeneous reaction-diffusion Turing model subject to the Neumann boundary conditions in the one-dimensional spatial domain. With the help of the Hopf bifurcation theory applicable to the reaction-diffusion equations, we are capable of proving the existence of Hopf bifurcations, which suggests the existence of spatially homogeneous and nonhomogeneous periodic solutions of this particular system. In particular, we also prove that the spatial homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point of the system are always unstable. This together with the instability results of the spatially nonhomogeneous periodic solutions by Yi et al., 2009, indicates that, in this model, all the oscillatory patterns from Hopf bifurcations are unstable.

Bounds on the critical times for the Fisher-KPP equation

2021 ◽  
Vol 63 ◽  
pp. 448-468
Author(s):  
Marianito Rodrigo
Keyword(s):  
Upper And Lower Solutions ◽  
Critical Time ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Kpp Equation ◽  
Diffusive Process ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Comparison Functions

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

A New Mechanical Algorithm for Calculating the Amplitude Equation of the Reaction-Diffusion Systems

2011 ◽  
pp. 205-213
Author(s):  
Houye Liu ◽  
Weiming Wang
Keyword(s):  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Normal Form ◽  
Reaction Diffusion ◽  
Amplitude Equation ◽  
Diffusion Reaction ◽  
Future Studies ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Form Approach

Amplitude equation may be used to study pattern formatio. In this chapter, we establish a new mechanical algorithm AE_Hopf for calculating the amplitude equation near Hopf bifurcation based on the method of normal form approach in Maple. The normal form approach needs a large number of variables and intricate calculations. As a result, deriving the amplitude equation from diffusion-reaction is a difficult task. Making use of our mechanical algorithm, we derived the amplitude equations from several biology and physics models. The results indicate that the algorithm is easy to apply and effective. This algorithm may be useful for learning the dynamics of pattern formation of reaction-diffusion systems in future studies.

Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems

2010 ◽  
Vol 466 (2119) ◽  
pp. 1903-1917 ◽  
Author(s):  
Michael Sieber ◽  
Horst Malchow ◽  
Sergei V. Petrovskii
Keyword(s):  
Travelling Waves ◽  
Natural Populations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Oscillatory Reaction ◽  
Stochastic Forcing ◽  
Spatial Direction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

Ecological field data suggest that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reaction–diffusion equations have helped to identify possible scenarios, by which such spatio-temporal patterns may arise. In this paper, such solutions are tested for their robustness against an irregular temporal forcing, since most natural populations can be expected to be subject to erratic fluctuations imposed by the environment. It is found that small environmental noise is able to suppress periodic travelling waves in stochastic variants of oscillatory reaction–diffusion systems. Irregular spatio-temporal oscillations, however, appear to be more robust and persist under the same stochastic forcing.

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