Chemical reaction Lotka model

First model for oscillating system was proposed by Volterra for prey-predator interactions in biological systems and by Lotka for autocatalytic chemical reactions. Lotka s model can be represented as... 

Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

Staying within a class of mono- and bimolecular reactions, we thus can apply to them safely the technique of many-point densities developed in Chapter 5. To establish a new criterion insuring the self-organisation, we consider below the autowave processes (if any) occurring in the simplest systems -the Lotka and Lotka-Volterra models [22-24] (Section 2.1.1). It should be reminded only that standard chemical kinetics denies their ability to selforganisation either due to the absence of undamped oscillations (the Lotka model) or since these oscillations are unstable (the Lotka-Volterra model). 

Statement 1) a stable stationary solution of a complete set of equations of the Lotka model holds. At long t the reaction rate K(t) strives for the stationary value. Time development of concentrations obeys standard chemical kinetics, Section 2.1.1. 

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. 

At approximately the same time, Lotka proposed his famous models of oscillating chemical reactions based on irreversible autocatalytic processes. The first model included one autocatalytic step and gave damped oscillations. The second model became a paradigm in oscillating chemistry. It consists of two consecutive autocatalytic steps, resulting in undamped oscillations. The Lotka models attracted great attention from theoretical biologists, because... 

In our example we will consider the earliest model of an oscillating homogeneous chemical reaction (A. J. Lotka, /. Am. Chem. Soc. 42 (1920) 1595 Proc. Natl. Acad. Sci. USA6 (1920) 410), which is based on the reaction sequence... 

It should be emphasized that there have been exceptions to this attitude. In 1910 and 1920 Lotka published his theory of chemical reactions in which the oscillations of reagent concentrations could appear. An essential feature of the Lotka models was nonlinearity. In mathematics and physics a trend has long persisted to examine linear systems and phenomena and to replace non-linear models by (approximate) linear models. The trend, originating from insufficient mathematical means, has turned into specific philosophy. The non-linear Lotka models thus constituted a deviation from a canon. Hence, general arguments of thermodynamic nature, lack of interest in non-linear models and commonness of observations of a monotonic attainment of the equilibrium in chemical reactions were the reasons for skepticism and disbelief which the results of Belousov have met with. 

The first theoretical model of a chemical reaction providing for oscillations in concentrations of reagents was the Lotka model from 1910. A mechanism of the hypothetical Lotka reaction has the following form ... 

This feedback-type behavior has been first considered in the domain of mathematics, with explicit targeting chemistry. In 1910. Alfred Lotka proposed some differential equations that corresponded to the kinetics of an autocatalytic chemical reaction, and then with lto Volterra derived a differential equation that describes a general feedback mechanism (oscillations) known as the Lotka-Volterra model. However, chemistry has not been ready yet for this link. 

The examination of competitive interactions among different species has been one of the main topics of mathematical biology. The most often used mathematical model is still a generalisation of the Lotka-Volterra model systems of polynomial ordinary differential equations expressible in terms of formal chemical reactions have also been investigated. The main problem is to find criteria for the coexistence of species. All species in the communities... 

Theoretical discussion of oscillatory reaction began since the discovery of the most famous Lotka s model in 1921. After the non-equilibrium theories of Prigogine (1968), many theories to explain the oscillatory chemical reactions have been proposed. A brief summary of some important oscillator models have been described below. 

A second requirement is that the system contain some sort of feedback, i.e., that the product of some reaction exert an influence upon its own rate of production. Probably the simplest type of feedback and the one most commonly found in chemical oscillators is autocatalysis. The simple model (Lotka, [16]) containing two coupled autocatalytic reactions shown in Table 1 was probably the first chemical" mechanism to give sustained oscillations. While autocatalysis is far more prelavent in biological than in chemical systems, autocatalytic chemical reactions do exist many of them are listed as "clock reactions in collections of lecture demonstrations. 

Autocatalysis can cause sustained oscillations in batch systems. This idea originally met with skepticism. Some chemists believed that sustained oscillations would violate the second law of thermodynamics, but this is not true. Oscillating batch systems certainly exist, although they must have some external energy source or else the oscillations will eventually subside. An important example of an oscillating system is the circadian rhythm in animals. A simple model of a chemical oscillator, called the Lotka-Volterra reaction, has the assumed mechanism ... 

The Lotka-Volterra model [23, 24] considered in the preceding Section 8.2 involves two autocatalytic reaction stages. Their importance in the self-organized chemical systems was demonstrated more than once [2], In this... 

Now the question is how to construct the simplest model of a chemical oscillator, in particular, a catalytic oscillator. It is quite easy to include an autocatalytic reaction in the adsorption mechanism, for example A+B—> 2 A. The presence of an autocatalytic reaction is a typical feature of the known Bmsselator and Oregonator models that have been studied since the 1970s. Autocatalytic processes can be compared with biological processes, in which species are able to give birth to similar species. Autocatalytic models resemble the famous Lotka-Volterra equations (Berryman, 1992 Valentinuzzi and Kohen, 2013), also known as the predator-prey or parasite-host equations. 

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