Unilateral sources and sinks of an activator in reaction-diffusion systems exhibiting diffusion-driven instability (original) (raw)

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

Diffusion-Driven Instability in Reaction–Diffusion Systems

Journal of Mathematical Analysis and Applications, 2001

For a stable matrix A with real entries, sufficient and necessary conditions for A y D to be stable for all non-negative diagonal matrices D are obtained. Implications of these conditions for the stability and instability of constant steadystate solutions to reaction᎐diffusion systems are discussed and an example is given to show applications.

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.

Positional differentiation as pattern formation in reaction-diffusion systems with permeable boundaries. Bifurcation analysis

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.

Stabilization strategies for some reaction–diffusion systems

Nonlinear Analysis: Real World Applications, 2009

A two-component reaction-diffusion system is considered. The question of stabilizing to zero one of the components of the solution via an internal control acting on a small subdomain and preserving nonnegativity of both components is investigated. Our results apply to predator-prey systems.

Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations

Journal of Mathematical Biology, 1996

Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.

Stationary multiple spots for reaction–diffusion systems

Journal of Mathematical Biology, 2008

In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reaction-diffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activator-substrate systems (such as the Gray-Scott system or the Schnakenberg model). The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature. We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator. Existence of multi-spots is proved using tools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis. Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots. Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis. The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations.

Non-Linear Bifurcation Analysis of Reaction-Diffusion Activator-Inhibator System

Journal of Biological Physics, 1999

The paper first deals with the linear stability analysis of an activator-inhibitor reaction diffusion system to determine the nature of the bifurcation point of the system. The non-linear bifurcation analysis determining the steady state solution beyond the critical point enables us to determine characteristic features of the spatial inhomogeneous pattern formation arising out of the bifurcation of the state of the system.