Hanusse theorem

From the point of view of the Hanusse theorem just discussed, a system with two intermediate stages and mono- and bimolecular reactions are not capable to reveal any temporary, spatial and spatio-temporal structures. However, results obtained in the past few years permit reconsideration of such an absolute point of view. 

The Hanusse theorem [23] discussed in Section 2.1.1 was later generalized for the case of diffusion by Tyson and Light [32], Therefore, the mono- and bimolecular reactions with one or two intermediate products are expected to strive asymptotically, as t —> oo, for the stationary spatially-homogeneous solution Ci(r, oo) = nt(oo) corresponding to equations (2.1.2) for a system with the complete particle mixing. 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

As it was mentioned in Section 2.1.1, the concentration oscillations could be simulated quite well by a set of even two ordinary differential equations of the first order but paying the price of giving up the rigid condition imposed on interpretation of mechanisms of chemical reactions namely that they are based on mono- and bimolecular stages only (remember the Hanusse theorem [19]) An example of what Smoes [7] called the heuristic-topological model is the well-known Brusselator [2], Its scheme was discussed in Section 2.1.1 see equations (2.1.33) to (2.1.35). 

For example, the standard synergetic approach [52-54] denies the possibility of any self-organization in a system with with two intermediate products if only the mono- and bimolecular reaction stages occur [49] it is known as the Hanusse, Tyson and Light theorem. We will question this conclusion, which in fact comes from the qualitative theory of non-linear differential equations where coefficients (reaction rates) are considered as constant values and show that these simplest reactions turn out to be complex enough to serve as a basic models for future studies of non-equilibrium processes, similar to the famous Ising model in statistical physics. Different kinds of auto-wave processes in the Lotka and Lotka-Volterra models which serve as the two simplest examples of chemical reactions will be analyzed in detail. We demonstrate the universal character of cooperative phenomena in the bimolecular reactions under study and show that it is reaction itself which produces all these effects. 

The two-species-competing system (7.25) cannot show oscillatory behaviour (in consequence of the Hanusse-Tyson-Light-Pota theorem it is different from the Lotka-Volterra model). The conditions of stability of equilibria were studied by Takeuchi Adachi (1983b). Accordingly, the ( ++) is globally stable if and only if a < I, 6 < 1. ( +0) and ( 0+) are globally stable if and only if a <, I and, b < respectively. For the case of a >, b > ( +0) and ( 00) are locally stable. 

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