Stochastic Lotka-Volterra model

Here X, Y are internal and A external components, andO is the zero complex. The deterministic kinetic equation is 

According to linear stability analysis the trivial stationary point is an unstable saddle point, while the nontrivial stationary point is a marginally stable centre. 

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 

The stochastic version of the Lotka-Volterra model leads to qualitatively different results from the deterministic approach. The master equation is 

Similar results were given by Keizer (1976). However, he also demonstrated that the fluctuations might be bounded, if fluctuations for the external components (including the zero complex) are also allowed. Possible behaviours of the realisations were demonstrated by simulation methods (Fig. 5.2). 

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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.

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. 

Irregular behaviour of concentrations and the correlation functions observed in the chaotic regime differ greatly from those predicted by law of mass action (Section 2.1.1). Following Nicolis and Prigogine [2], the stochastic Lotka-Volterra model discussed in this Section, could be considered as an example of generalized turbulence. 

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. 

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]. 

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). 

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],... 

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. 

As was mentioned in Subsection 5.6.2, stochastic versions of the Lotka-Volterra model lead to qualitatively different results from the deterministic model. The occurrence of a similar type of results is not too surprising. A simple model for random predator-prey interactions in a varying environment has been studied, staring from generalised Lotka-Volterra equations (De, 1984). The transition probability of extinction is to be determined. The standard procedure is to convert the problem to a Fokker-Planck equation (adopting continuous approximation) and to find an approximation procedure for evaluating the transition probabilities of extinction and of survival. 

Reddy, V. T. N. (1975). Existence of steady-state in stochastic Volterra-Lotka model. 

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