The Volterra-Lotka Model

The better known Lotka-Volterra model [18, 19] unlike (2.1.21) is based on two autocatalytic stages 

Here the infinite food E supply is assumed. Its biological interpretation is similar to the Lotka model predators B live on prey animals A, both are reproduced by division. 

The singular point seen in the first quadrant of the phase plane (nA,nB) 

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 lit/ujo defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum w wq, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since 

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FIGURE 2.6 Population dynamics predicted by the Lotka-Volterra model for an initial population of 100 rabbits and 10 lynx. 

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. 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Analysis of equations for second momenta like (SNA5NB), (5Na)2) and (5NB)2) shows that all their solutions are time-dependent. In the Lotka-Volterra model second momenta are oscillating with frequencies larger than that of macroscopic motion without fluctuations (2.2.59), (2.2.60). Oscillations of k produce respectively noise in (2.2.68), (2.2.69). Fluctuations in the Lotka-Volterra model are anomalous second momenta are not expressed through mean values. Since this situation reminds the turbulence in hydrodynamics, the fluctuation regime in this model is called also generalized turbulence [68]. The above noted increase in fluctuations makes doubtful the standard procedure of the cut off of a set of equations for random values momenta. 

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). 

Since our principal aim in studying the Lotka-Volterra model is to clarify whether the limit cycle or chaotic regime could arise for this model, let us specify now the functions /za(r), /zb(r) and a(r) in a way simplifying the integral terms in (8.2.1) to (8.2.5). 

A set of equations (8.2.12) and (8.2.13) for the concentration dynamics is formally similar to the standard statement of the Lotka-Volterra model given... 

In other words, K(t) is afunctional of the joint correlation function of similar particles. In this respect, a set of equations (8.2.12) and (8.2.13) is similar to the stochastic treatment of the Lotka-Volterra model (equations (2.2.68) and (2.2.69)) considered in Section 2.3.1 using the similar time-dependent reaction rate (2.2.67). 

These two kinds of dynamics - for particle correlations and concentrations -become coupled through the reaction rate. The functionals J[Z] in (8.2.15) to (8.2.17) were defined in Chapter 5 (5.1.36) to (5.1.38) for different space dimensions d = 1,2,3. They emerge in those terms of (8.2.9) to (8.2.11) which are affected by the superposition approximation. It should be stressed that in the case of the Lotka-Volterra model it is the only approximation used for deriving the equations of the basic model. 

As it was said above, there is no stationary solution of the Lotka-Volterra model for d = 1 (i.e., the parameter k does not exist), whereas for d = 2 we can speak of the quasi-steady state. If the calculation time fmax is not too long, the marginal value of k = K.(a, ft, Na,N, max) could be also defined. Depending on k, at t < fmax both oscillatory and monotonous solutions of the correlation dynamics are observed. At long t the solutions of nonsteady-state equations for correlation dynamics for d = 1 and d = 2 are qualitatively similar the correlation functions reveal oscillations in time, with the oscillation amplitudes slowly increasing in time. 

Let us consider a projection of the complex many-dimensional motion (which variables are both concentrations and the correlation functions) onto the phase plane (iVa, iVb). It should be reminded that in its classical formulation the trajectory of the Lotka-Volterra model is a closed curve - Fig. 2.3. In Fig. 8.1 a change of the phase trajectories is presented for d = 3 when varying the diffusion parameter k. (For better understanding logarithms of concentrations are plotted there.)... 

Fig. 8.1. Phase portraits of the Lotka-Volterra model for d = 3 (a) Unstable focus (re = 0.9) (b) Stable focus (re = 0.2) (c) Concentration oscillations during the steady-state formation (re = 0.1) (d) Chaotic regime (re = 0.05). The values of the distinctive parameter are shown.

Therefore, oscillations of K (t) result in the transition of the concentration motion from one stable trajectory into another, having also another oscillation period. That is, the concentration dynamics in the Lotka-Volterra model acts as a noise. Since along with the particular time dependence K — K(t) related to the standing wave regime, it depends also effectively on the current concentrations (which introduces the damping into the concentration motion), the concentration passages from one trajectory onto another have the deterministic character. It results in the limited amplitudes of concentration oscillations. The phase portrait demonstrates existence of the distinctive range of the allowed periods of the concentration oscillations. 

Fig. 8.2. Chaotic oscillations in the Lotka-Volterra model. Parameter n = 0.05, d = 3.

Fig. 8.4. The reaction rate in the Lotka-Volterra model. Parameters k = 0.5, d = 2.

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... 

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). 

Analogously the Lotka-Volterra model, let us write down the fundamental equation of the Markov process in a form of the infinite hierarchy of equations for the many-point densities. Thus equations for the single densities (m + m ) — 1 read ... 

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. 

This statement comes from analytical and topological studies [4], Unlike the Lotka-Volterra model where due to the dependence of the reaction rate K(t) on concentrations NA and NB, the nature of the critical point varied, in the Lotka model the concentration motion is always decaying. Autowave regimes in the Lotka model can arise under quite rigid conditions. It is easy to show that not any time dependence of K(t) emerging due to the correlation motion is able to lead to the principally new results. For example, the reaction rate of the A + B -> 0 reaction considered in Chapter 6 was also time dependent, K(t) oc t1 d/4 but its monotonous change accompanied by a strong decay in the concentration motion has resulted only in a monotonous variation of the quasi-steady solutions of (8.3.20) and (8.3.21) jVa(t) (3/K(t) and N, (t) p/f3 = const. 

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). 

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... 

Figure 13.10. State-space plots with various initial values of X0 tor the Lotka-Volterra model.

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