Oscillation Volterra-Lotka type

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. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

This type of a pattern of singular points is called a centre - Fig. 2.3. A centre arises in a conservative system indeed, eliminating time from (2.1.28), (2.1.29), one arrives at an equation on the phase plane with separable variables which can be easily integrated. The relevant phase trajectories are closed the model describes the undamped concentration oscillations. Every trajectory has its own period T > 2-k/ujq defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum oj < u>o, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since... 

The Lotka-Volterra type of equations provides a model for sustained oscillations in chemical systems with an overall affinity approaching infinity. Perturbations at finite distances from the steady state are also periodic in time. Within the phase space (Xvs. Y), the system produces an infinite number of continuous closed orbits surrounding the steady state... 

Now we keep the prey growth rate linear g R) = aR but instead use a more realistic type-II functional response, f R) = kRl(KM + R), to describe the food-uptake. As shown in Fig.15.3b again the A -isocline is a vertical line, but the ii-isocline N = Km + R) is now a straight line with increasing slope. This has the consequence that the rotating trajectory is increasing in amplitude. Thus, the effect of a functional response is to destabilize the Lotka-Volterra oscillations. 

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. 

Example 13.10 Sustained oscillations of the Lotka—Volterra type... 

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). 

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