Lotka-Volterra system

The study of mathematical models of competition has led to the discovery of some very beautiful mathematics. This mathematics, often referred to as monotone dynamical systems theory, was largely developed by M. W. Hirsch [Hil Hi3], although others have made substantial contributions as well. In this section we describe a result that was first obtained in a now classical paper of DeMottoni and Schiaffino [DS] for the special case of periodic Lotka-Volterra systems. Later, it was recognized by Hale and Somolinos [HaS] and Smith [S4 S5] that the arguments in [DS] hold for general competitive and cooperative planar periodic systems. The result says that every bounded solution of such a system converges to a periodic solution that has the same period as the differential equation. 

Competition among two species means that the increase in one of the populations decreases the net growth rate of the second one, and vice versa. This happens when they feed on the same resources, or if they produce substances (toxins) that are toxic for the other species. A classical competition model was also introduced in Volterra (1926), and considered in a more general parameter range by Lotka (1932). It is known as the competitive Lotka-Volterra system ... 

Fig. 15.2. Predator-prey cycles in Lotka-Volterra system (15.1). Left neutrally stable center in the phase plane. Oscillations are seen as closed loops of orbits around the center. Different initial values lead to different trajectories. Right typical time series of predator N t) (solid line) and of prey R(t) (dotted line).

Fig. 5.2 The character of the behaviour of the Lotka-Volterra system depends on the value of rate constants, initial values, and even on the starting point of the pseudorandom number generator.

The Lotka-Volterra system exhibits undamped oscillation, and the amplitude of the oscillation is determined by the initial values (and not by the structure of the system). The equation of the trajectory is... 

A system of coupled autocatalytic reactions that can be used to illustrate the generation of stable oscillations under non-equilibrium conditions is the Lotka-Volterra system ... 

Fig. 3.14 Phase portrait of the Lotka-Volterra system with a critical point (on-line calculation http //twt.mpei.ac.ru/MCSAVorksheets/Chem/ChemKin-3-13.xmcd)...

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 ... 

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. 

Lastly, non-elementary several-stage reactions are considered in Chapters 8 and 9. We start with the Lotka and Lotka-Volterra reactions as simple model systems. An existence of the undamped density oscillations is established here. The complementary reactions treated in Chapter 9 are catalytic surface oxidation of CO and NH3 formation. These reactions also reveal undamped concentration oscillations and kinetic phase transitions. Their adequate treatment need a generalization of the fluctuation-controlled theory for the discrete (lattice) systems in order to take correctly into account the geometry of both lattice and absorbed molecules. As another illustration of the formalism developed by the authors, the kinetics of reactions upon disorded surfaces is considered. 

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... 

More interesting aspects of stochastic problems are observed when passing to systems with unstable stationary points. Since we restrict ourselves to mono- and bimolecular reactions with a maximum of two intermediate products (freedom degrees), s — 2, only the Lotka-Volterra model by reasons discussed in Section 2.1.1 can serve as the analog of unstable systems. 

Systems Under Birth and Death Conditions Lotka and Lotka-Volterra Models... 

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). 

A role of other parameters of the model is investigated by Kuzovkov [26], It is demonstrated that an increase of the ratio a//3 for a fixed lj0 = (cr/3)1/2 and the control parameter k acts to accelerate a change of the focal regime for chaotic. Simultaneously, the amplitudes of oscillations in concentration for particles of different kinds are no longer close. A study of the stochastic Lotka-Volterra model performed here shows that irregular concentration motion observed experimentally in the Belousov-Zhabotinsky systems [8] indeed could take place in a system with mono- and bimolecular stages and two intermediate products only. 

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... 

Here the first term arises from the diffusive approach of reactants A into trapping spheres around B s it is nothing but the standard expression (8.2.14). The second term arises due to the direct production of particles A inside the reaction spheres (the forbidden for A s fraction of the system s volume). Unlike the Lotka-Volterra model, the reaction rate is defined by an approximate expression (due to use of the Kirkwood superposition approximation), therefore first equations (8.3.9) and (8.3.10) of a set are also approximate. 

In a system with strong damping of the concentration motion the concentration oscillations are constrained they follow oscillations in the correlation motion. As compared to the Lotka-Volterra model, where the concentration motion defines essentially the autowave phenomena, in the Lotka model it is less important being the result of the correlation motion. This is why when plotting the results obtained, we focus our main attention on the correlation motion in particular, we discuss in detail oscillations in the reaction rate K(t). 

Therefore, the study of the stochastic Lotka and Lotka-Volterra models carried out in Chapter 8, has demonstrated that the traditional estimates of the complexity of the system necessary for its self-organisation are not correct. Incorporation of the fluctuation effects and thus introduction of a continuous number of degrees of freedom prove their ability for self-organisation and thus put them into a class of the basic models for the study of the autowave processes. 

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. 

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... 

This model resembles a Lotka-Volterra model, which may be used in studying the evolution of systems in time, such as biological clocks or the time-dependent properties of neural networks. 

In biological dissipative structures, self-oiganization may be related to the attractors in the phase space, which correspond to ordered motions of the involved biological elements (De la Fuenta, 1999). When the system is far from equilibrium, ordering in time or spontaneous rhythmic behavior may occur. The Lotka-Volterra model of the predator-prey interactions is a simple example of the rhythmic behavior. The interactions are described by the following kinetics... 

Example 13.8 Prey-predator system Lotka-Volterra model The Lotka-Volterra predator and prey model provides one of the earliest analyses of population dynamics. In the model s original form, neither equilibrium point is stable the populations of predator and prey seem to cycle endlessly without settling down quickly. The Lotka-Volterra equations are... 

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... 

Example 11 The simplest system with stable oscillation the Lotka Volterra model... 

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