Multistationarity

Investigating all possible solutions of a differential equation it may happen that, although the solutions tend to an equilibrium point, the equilibrium point depends on the initial conditions. Here we do not think of the necessity to remain in the same stoichiometric compatibility class, we think of the nonuniqueness within a single class. This is the case of multistationarity. The same thing may happen with multiple periodic solutions and with multiple chaotic attractors, or even with several attractors with all the three types. 

In the previous section regular behaviour has been dealt with now we turn to multistationarity, oscillation and chaos. The other possible combinations seem to be more complicated, and they have not been regularly studied so far. (See, however, Rossler (1983a).) 

In these sections we remain within the area of homogeneous reaction kinetics. Questions regarding phenomena connected with spatial effects, like propagating fronts, will be touched on in Chapter 6. 

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Thomas, R., and Kaufman, M. (2001). Multistationarity, the basis of cell differentiation and memory. 

I. Structural conditions of multistationarity and other nontrivial behavior. Chaos Interdisciplinary... 

Traditional reaction kinetics has dealt with the large class of chemical reactions that are characterised by having a unique and stable stationary point (i.e. all reactions tend to the equilibrium ). The complementary class of reactions is characterised either by the existence of more than one stationary point, or by an unstable stationary point (which could possibly bifurcate to periodic solutions). Other extraordinarities such as chaotic solutions are also contained in the second class. The term exotic kinetics refers to different types of qualitative behaviour (in terms of deterministic models) to sustained oscillation, multistationarity and chaotic effects. Other irregular effects, e.g. hyperchaos (Rossler, 1979) can be expected in higher dimensions. 

Exotic chemical systems, mostly oscillatory reactions, but also systems exhibiting multistationarity and chaotic effects, have extensively been investigated. Phenomena in chemical, biological and industrial chemical systems are the experimental basis of the theoretical studies. 

It is often mentioned that the stationary distribution of chemical reactions is generally the Poisson distribution (Prigogine, 1978). However, the significance of the Poisson distribution is rather limited other unimodal distributions can also reflect the regular behaviour of the fluctuating chemical sytems. Multimodality of the stationary distribution might be associated, at least loosely speaking, with multistationarity of deterministic... 

Multistability is the more general notion we start with. It means that the system of differential equations has multiple attractors. Here we only deal with multistationarity, the case in which all the attractors are equilibrium points. 

The first experimental results on multistationarity seem to be those by Liljenroth (1918). 

Ganapathisubramian Showalter (1984b) measured the steady state iodide concentration in the iodate-arsenous acid system as a function of reciprocal residence time and found multistationarity. 

Multistationarity in kinetic models of continuous flow stirred tank reactors... 

In this field, as in any other part of reaction kinetics, there are two approaches investigation of simple, but possibly realistic models see, for example, Caram Scriven (1976), Othmer (1976), Luss (1980, 1981) Gray Scott (1983a, b) or the example of Horn Jackson (1972, cited here as Exercise 1) or a search for general criteria that ensure or exclude multistationarity in large classes of mechanisms. This second approach was initiated by Rumschitzky Feinberg (in preparation Rumschitzky, 1984). Here we present results of this second approach, and some of the Problems contain results on specific models derived using the first approach. 

The theorems by Feinberg, Horn, Jackson and Vol pert provide sufficient conditions to exclude multistationarity. These theorems can be applied in the case of homogeneous systems, and in the case of inhomogeneous systems, if the system can be modelled by formal elementary reactions as shown several times above. An especially important case of an inhomogeneous systems is the isothermal continuous (flow) stirred tank reactor (CSTR). By a CSTR we mean one in which there is perfect mixing and in which, at each instant, every component within the reaction vessel is also contained in the effiuent stream. 

It was mentioned (in Section 1.4) that multistationarity in deterministic models might be associated (however, only approximately) with multimodality of the stationary distribution. It is generally assumed that... 

The presence of multistationarity can be illustrated in other way. The last two equations allow one to conclude that ... 

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