Oscillations Lotka—Volterra mechanism

Autocatalysis occurs as in the Lotka-Volterra mechanism and the oregonator. If the concentrations of A and B are maintained constant, the concentrations of X and Y oscillate with time. A graph of the concentration of X against that of Y is a closed loop (the limit cycle of the reaction). The reaction settles down to this limit cycle whatever the initial concentrations of X and Y, i.e. the limit cycle is an attractor for the system. The reaction mechanism Is named after the city of Brussels, where the research group that discovered It Is based. 

Lotka-Volterra mechanism A simple chemical reaction mechanism proposed as a possible mechanism of oscillating reactions. The process involves a conversion of a reactant R into a product P. The reactant flows into the reaction chamber at a constant rate and the product is removed at a constant rate, i.e. the reaction is in a steady state (but not in chemical equilibrium). The mechanism involves three steps ... 

Autocatalysis is involved in the first two steps of this process. It appears that oscillating chemical reactions have mechanisms that are different from the Lotka-Volterra mechanism. This type of mechanism does occur in certain types of complex system such as predator-prey relationships in biology. It was in the biological context that the mechanism was investigated by the Italian mathematician Vito Volterra (1860-1940). 

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 ... 

Section we show that presence of two such intermediate stages is more than enough for the self-organization manifestation. Lotka [22] was the first to demonstrate theoretically that the concentration oscillations could be in principle described in terms of a simplest kinetic scheme based on the law of mass action [4], Its scheme given by (2.1.21) is similar to that of the Lotka-Volterra model, equation (2.1.27). The only difference is the mechanism of creation of particles A unlike the reproduction by division, E + A - 2A, due to the autocatalysis, a simpler reproduction law E —> A with a constant birth rate of A s holds here. Note that analogous mechanism was studied by us above for the A + B — B and A + B — 0 reactions (Chapter 7). 

How the autocatalytic regulation of PFK leads to glycolytic oscillations is a question that benefits from being put in theoretical terms. The knowledge of the mechanism of oscillations and the availabiUty of numerous experimental data early on prompted the construction of models for the PFK reaction. The first model for glycolytic oscillations, proposed by Higgins (1964), was based on the activation of the enzyme by its second product, FBP. This model, however, only admitted relatively unstable oscillations of the Lotka-Volterra type (see chapter 1, and Nicolis Prigogine, 1977). 

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. 

A to the first line, Rossler (1976) was the first to provide a chemical model of chaos. It was not a mass-action-type model, but a three-variable system with Michaelis-Menten-type kinetics. Next Schulmeister (1978) presented a three-variable Lotka-type mechanism with depot. This is a mass-action-type model. In the same year Rossler (1978) presented a combination of a Lotka-Volterra oscillator and a switch he calls the Cause switch showing chaos. This model was constructed upon the principles outlines by Rossler (1976a) and is a three-variable nonconservative model. Next Gilpin (1979) gave a complicated Lotka-Volterra-type example. Arneodo and his coworkers (1980, 1982) were able to construct simple Lotka-Volterra models in three as well as in four variables having a strange attractor. 

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. 

The mathematical background of the periodic oscillations around the steady state Xj, is a conservation law which plays quite an analogous role for the equations of motion (2.44) of the Volterra-Lotka model as the conservation of energy in the ordinary undamped harmonic oscillator in classical mechanics. The reader immediately verifies that the quantity... 

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