Volterra model

Diffusive instability can appear in simple predator-prey models. Bartumeus et al. (2001) used the linear stability analysis and demonstrated that a simple reaction-diffusion predator-prey model with a ratio-dependent functional response for the predator can lead to Turing structures due to diffusion-driven instabilities. 

Example 13.10 Lotka-Volterra model Solve the following equations and prepare a state-space plot where X is plotted against 7 using the solution. 

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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. 

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

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. 

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. 

Fig. 2.13. The random trajectory in the stochastic Lotka-Volterra model, equation (2.2.64). Parameters are a/k = /3/k = 20, the initial values NA = NB = 20. When the trajectory coincides with the NB axis, prey animals A are dying out first and predators second.

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. 

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). 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

Let us reformulate the standard Lotka-Volterra model [23, 24] described by the set (2.1.27) in terms of the diffusion-controlled process as it was suggested for the first time by Kuzovkov [21, 25-27], Its basic elements are as follows. 

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. 

Lotka-Volterra model reveals different kind of autowave processes with the non-monotonous behaviour of the correlation functions accompanied by their great spatial gradients and rapid change in time. Due to this fact the space increment Ar time increment At was variable to ensure that the relative change of any variable in the kinetic equations does not exceed a given small value. The difference schemes described above were absolutely stable and a choice of coordinate and time mesh was controlled by additional calculations with reduced mesh. 

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. 

The performed calculations demonstrate that a type of the asymptotic solution of a complete set of the kinetic equations is independent of the initial particle concentrations, iVa(0) and 7Vb(0). Variation of parameters a and (3 does not also result in new asymptotical regimes but just modifies there boundaries (in t and k). In the calculations presented below the parameters 7Va(0) = 7Vb(0) =0.1 and a = ft = 0.1 were chosen. The basic parameters of the diffusion-controlled Lotka-Volterra model are space dimension d and the ratio of diffusion coefficients k. The basic results of the developed stochastic model were presented in [21, 25-27],... 

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.

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... 

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 ... 

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