Lotka-Volterra mechanism

Chemistry Basic Elements Lotka-volterra Mechanism ... 

A reaction may be periodic if its network provides for restoration of a reactant or intermediate that has been depleted, while conversion of main reactants to products continues. Periodic behavior often results from competition of two or more contending mechanisms. Predator-prey fluctuations in ecology (Lotka-Volterra mechanisms) provide an easily visualized example. The Belousov-Zhabotinsky reaction—catalyzed oxidation of malonic acid by bromate—involves a similar competition between two pathways. 

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 occurs as in the Lotka-Volterra mechanism and the brusselator. The mechanism was named after Oregon in America, where the research group that discovered it is based. 

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). 

We can illustrate oscillatory behavior with the Lotka-Volterra mechanism ... 

The known mechanisms that produce oscillatory behavior have two characteristics in common. The first is autocatalysis. The product of a step must catalyze that step, as in steps 1 and 2 of the Lotka-Volterra mechanism. The second is that nonlinear differential equations occur. That is, the variables must occur with powers greater than unity or as products. A mechanism has been proposed for the BZ reaction that has 18 steps and involves 21 different chemical species. A computer simulation of the 18 simultaneous rate differential equations for the mechanism has been carried out and does produce oscillatory behavior. It also exhibits the interesting behavior that all curves in phase space corresponding to different initial states eventually approach... 

As of 2006, the website http //tu-dresden.de/Members/thomas.petzoldt provides a program that you can run to carry out the solution to the Lotka—Volterra mechanism. 

The key feature of the Lotka-Volterra mechanism shown below... 

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). 

When we speak of mathematical models for biology, we usually refer to formulae (such as the Hardy-Weinberg theorem, or the Lotka-Volterra equations) that effectively describe some features of living systems. In our case, embryonic development is not described by integrals and deconvolutions, and the formulae of the reconstruction algorithms cannot be a direct description of what happens in embryos. There is however another type of mathematical model. The formulae of energy, entropy and information, for example, apply to all natural processes, irrespective of their mechanisms, and at this more general level there could indeed be a link between reconstruction methods and embryonic development. For our purposes, in fact, what really matters are not the formulae per se, but... 

The physical separation between genotype and phenotype has an extraordinary consequence, because mental genotypes can be directly instructed by mental phenotypes, and this means that cultural heredity is based on a transmission of acquired characters. Cultural inheritance, in other words, is transmitted with a Lamarckian mechanism, whereas biological inheritance relies on a Mendelian mechanism which is enormously slower. As a result, cultural evolution is much faster than biological evolution, and almost all differences between biology and culture can be traced back to the divide that exists in their hereditary mechanisms. The discovery that human artifacts (i.e. cultural phenotypes) obey the Lotka-Volterra equations has two outstanding consequences. The first is that selection accounts for all types of adaptive evolution natural selection is the mechanism hy which all phenotypes - biological as well as cultural - diffuse in the world. 

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... 

The pure Volterra-Lotka model leads to a mathematically marginal case The singular point (= stationary state) is a center encircled by closed orbits. The latter are non-stationary solutions depending on the initial conditions in analogy to the solutions of undamped heuniltonian systems in theoretical mechanics. Therefore the original model is sensitive to small perturbations which may have different causes [4.1, 23-26, 30, 31]. On the other hand, Volterra-Lotka cycles have been observed by biologists in nature for several entirely different species [4.21, 22, 27-29]. It has been stressed that the existence of such cycles could depend on the availabihty of appropriate facilities for migration [4.28, 31]. 

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