Activator inhibitor system

If a second variable participates in an additional feedback loop with a negative regulation, oscillations become possible. The mutual dependencies of the two variables, which have been coined activator and inhibitor, are depicted in Fig. 1. A, the autocatalytic species is the activator it activates the production of /, and I is the inhibitor because it slows down or inhibits the growth of A [13, 14]. Oscillations arise in activator-inhibitor systems if characteristic changes of the activator occur on a faster time-scale than the ones of the inhibitor. In other words, the inhibitor must respond to a variation of the activator variable with some delay. In fields as diverse as semiconductor physics, chemistry, biochemistry or astrophysics, and also in electrochemistry, most simple periodic oscillations can be traced back to such an activator-inhibitor scheme. 

The first corresponds to the classical activator-inhibitor system, where the elements fy<0 and g > 0 represent, respectively, Y (the inhibitor) inhibiting the formation of X (the activator), and X promoting the formation of Y. The second, with the opposite sign pattern for these off-diagonal elements, corresponds to a positive-feedback system such as the Gray-Scott model, where X is the autocatalyst and Y is typically a consumable reactant. In this case, both the autocatalyst and the reactant promote the formation of the autocatalyst, and, in turn, both species participate in the consumption of the reactant. In either case, a Turing instability can exist. 

Note that the type can depend on the stationary state (Pu, py). If the specific type does not matter, we will refer to (1.19) and (1.21) simply as activator-inhibitor systems. In our convention, the species U is the activator and V the inhibitor. Some... 

The Brusselator is a cross activator-inhibitor system if b > 1. The trace T and determinant A of the Jacobian are given by... 

The Schnakenberg model is a cross activator-inhibitor system iib > a. The deter-minant of the Jacobian is also always positive, h. = a + b), and no stationary bifurcation can occur. The Hopf threshold b is given by the cubic equation... 

The expressions for the trace T and the determinant A in terms of e, q, and h are somewhat lengthy and not enlightening. They are best evaluated for specific values of the parameters. The Oregonator also belongs to the class of pure activator-inhibitor systems. The autocatalytic species HBr02 is the activator and the oxidized catalyst the inhibitor. 

The derivation of the LE model in Sect. 1.4.9 is based on two assumptions. The formation and dissociation of the iodide ion-substrate complex is in a fast equilibrium and the substrate concentration [S] can be considered constant. Then the three-variable activator-inhibitor-substrate system reduces to a two-variable activator-inhibitor system, where the kinetics and the diffusion coefficient of the activator are rescaled by a factor /a [214, 246,247]. We now explore these assumptions in greater detail. Consider the following well-stirred two-variable activator-inhibitor system... 

Pearson has analyzed the effect of an immobile species on the Turing instability in two-variable activator-inhibitor systems for more general conditions [346]. Consider the 2+1 species system described by the following reaction-diffusion equations ... 

Note that A (AT) does not depend on and that G factors out the complex-ation reaction has no effect on the Turing condition. The (2+1)-variable activator-inhibitor-substrate system has the same Turing threshold as the two-variable activator-inhibitor system without substrate. Equation (10.32),... 

Spontaneous oscillations are a widespread phenomenon in nature. They have been studied for a large number of experiments, including electrochemical systems such as the oxidation of metals and organic materials [Miller and Chen (2006)]. Electrochemical systems exhibiting instabilities often behave like activator-inhibitor systems. In these systems the electrode potential is an essential variable and takes on the role of either activator or the inhibitor. If certain conditions are met, an activator-inhibitor system generates oscillations [Krischer (2001)]. In this section we present experimental data of electric potential self-oscillations on the electrode of IPMC which results in the oscillating actuation of the material. Furthermore, we also present a physical model to predict these oscillations. 

Intuitively, we can see the answer immediately. Formation of Turing patterns requires that concentrations of all reactants lie within ranges that allow the system to satisfy a set of conditions in the case of a two-variable activator-inhibitor system, eqs. (14.5), (14.6), (14.17), (14.19), and (14.21). Because of the way the experiment is done (recall Figure 14.2), each reactant concentration is position-dependent, ranging from its input feed value at the end of the gel where it enters to essentially zero at the other end. Clearly, the conditions for Turing pattern formation can be satisfied only in a portion of the gel, if at all. 

Some algebraic manipulation (Lengyel et al., 1992a), reduces the set of conditions for our prototype activator-inhibitor system in the presence of a complexing agent to the pair of inequalities... 

In contrast to the instabihty of uniform oscillations, the physical origin of the wavefront instability seems to have some relation to the conventional diffusion instability. To see this, we first give a brief quaUtative interpretation of the conventional diffusion instabihty. Here the notions activator and inhibitor seem to be helpful, and for simpUcity we imagine a two-component activator-inhibitor system. The instability then turns out to be due to relatively rapid diffusion of the inhibiting substance. Consider the activator-inhibitor kinetics (first, without diffusion) ... 

Note that our piecewise linear model corresponds qualitatively to this situation. The condition (7.3.8) is automatically satisfied for all k, and the steady state is stable irrespective of the presence or absence of diffusion. Still, the system may exhibit pulses or kinks under suitable initial conditions. Although such systems are not usually called activator-inhibitor systems, they still retain some similarity to activator-inhibitor systems if the flow in the XY phase space is seen globally beyond the linear regime about the steady state. In fact this similarity to activator-inhibitor systems has some connection with the similarity of the front instability to the conventional diffusion instability. Suppose that a<. If the equilibrium value of X (i.e., the zero value) is perturbed slightly but beyond the small threshold value a, then we have so that X starts to grow... 

In this section we analyze this phenomenon in the framework of the kinematical theory. We restrict our analysis to the activator-inhibitor systems which are described by the equations... 

Hence, in the activator-inhibitor systems with equal diffusion constant of both species the scroll rings always collapse and do not drift along the symmetry axis. This result has been derived directly from the reaction-diffusion equations in [52]. 

The Differential Flow Induced Chemical Instability (DIFICI) of Activator-Inhibitor Systems... 

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