Non-Linear Bifurcation Analysis of Reaction-Diffusion Activator-Inhibator System (original) (raw)
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arXiv (Cornell University), 2018
A reaction-diffusion system exhibiting Turing's diffusion driven instability is considered. The equation for an activator is supplemented by unilateral terms of the type s − (x)u − , s + (x)u + describing sources and sinks active only if the concentration decreases below and increases above, respectively, the value of the basic spatially constant solution which is shifted to zero. We show that the domain of diffusion parameters in which spatially non-homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms. It is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources. The case of mixed (Dirichlet-Neumann) boundary conditions as well as that of pure Neumann conditions is described.
A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type
Interfaces and Free Boundaries
A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in R N , with N > 2. It is shown that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatio-temporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the considered class of reactiondiffusion equations.
Spot bifurcations in three-component reaction-diffusion systems: The onset of propagation
Physical Review E, 1998
We present an analytical investigation of the bifurcation from stationary to traveling localized solutions in a three-component reaction-diffusion system of arbitrary dimension with one activator and two inhibitors. We show that increasing one of the inhibitors' time constants leads to such a bifurcation. For a limit case, which comprises the full range of stationary two-component patterns, the bifurcation is supercritical and no other bifurcation precedes it. Bifurcation points and velocities close to the branching point are predicted from the shape of the stationary solution. Existence and stability of the traveling solution are checked by means of multiple scales perturbation theory. Numerical simulations agree with the analytical results.
Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink
Discrete & Continuous Dynamical Systems - B, 2017
In this paper, we concern about the dynamics of a diffusive enzyme-catalyzed system arising from glycolysis, describing a biochemical reaction in which a substrate is converted into a product with positive feedback and into a branched sink. The temporal and spatiotemporal dynamics of the system under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equation is presented. For the reaction-diffusion model, firstly the parameter regions for the stability or instability of the unique constant steady state are discussed. Finally, bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied. Numerical simulations are presented to verify and illustrate the theoretical results.
Journal of Mathematical Biology, 1981
Pattern formation in a unicomponent reaction-diffusion system with trigger type dynamics and combined boundary conditions is considered. The boundary permeabilities and reservoir concentrations as well as the dimension of the system are the control parameters. The whole assemblage of steady states, their bifurcations and changes under the variation of these parameters is described. Among all steady distributions possible for given values of the parameters, only the simplest ones prove to be asymptotically stable. The relation to catastrophe theory is discussed.
Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems
Journal of Differential Equations, 1990
We apply a general heteroclinic and homoclinic bifurcation theory to the study of bifurcations of travelling waves of b&able reaction diffusion systems. Using the notion of separation, we first prove the existence of a cusp point of the set of travelling front solutions in the parameter space. This as well as the symmetry of the system yields a coexisting pair of front and back solutions which undergoes the homoclinic bifurcation producing a pulse solution. All the hypotheses imposed on the general heteroclinic and homoclinic bifurcation theorem are rigorously verified for a system of bistable reaction diffusion equations containing a small parameter E by using singular perturbation techniques, especially the SLEP method. A relation between the stability of front (or back) solutions and the intersecting manner of the stable and unstable manifolds is also given by means of the separation.
Instabilities in reaction-diffusion systems
Applied mathematical sciences, 2014
In this paper we discuss the solution stability of two-component reaction-diffusion systems with constant diffusion coefficients. Linear stability analysis is performed near the steady state solution of the system discussing the dependence of the system stability on its parameters. A comprehensive linear stability analysis results in three types of instabilities: (1) Stationary periodic, (2) Oscillatory uniform and (3) Stationary uniform. Precise parameter regimes are identified for each. Mathematics Subject Classification: 35B36, 35K57, 70K50
A subcritical bifurcation for a nonlinear reaction–diffusion system
2010
In this paper the mechanism of pattern formation for a reaction-diffusion system with nonlinear diffusion terms is investigated. Through a linear stability analysis we show that the cross-diffusion term allows the pattern formation. To predict the form and the amplitude of the pattern we perform a weakly nonlinear analysis. In the supercritical case the Stuart-Landau equation is found, which rules the evolution of the amplitude of the most unstable mode. With the increasing distance from the bifurcation value of the cross-diffusion parameter, the weakly nonlinear analysis fails and a Fourier-Galerkin approach is adopted. In the subcritical case the weakly nonlinear analysis must be pushed up to the fifth order, recovering the quintic Stuart-Landau equation for the amplitude of the pattern. The bifurcation diagram of this equation shows a range of the bifurcation parameter in which two qualitatively different stable states coexist (the origin and two large amplitude branches). Therefore the evolution of the pattern corresponds to a hysteresis cycle.