Volterra-Lotka equation

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

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... 

Fig. 2.3. A centre. Three solutions of the Lotka-Volterra equations (2.1.28)-(2.1.29) are presented the distinctive parameter a//3 = 1. The starting point of each trajectory is shown by...

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. 

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. 

The Lotka-Volterra equations written in the dimensionless parameters contain only several control parameters birth and death rates a, (3 and the ratio of diffusion coefficients k = Da/(Da + Z)B), 0 k < 1, i.e., Da = 2k, Db = 2(1 - k) whereas their sum is constant, DA + DB = 2. Lastly, it is also the space dimension d determining the functionals J[Z], equations (5.1.36) to (5.1.38), the Laplace operator (3.2.8) as well as the boundary condition (8.2.21) for the correlation functions of similar particles. Before discussing the results of the joint solution of the complete set of the kinetic equations, let us consider first the following statements. 

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],... 

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 ... 

Due to a similarity of reaction stages in the Lotka and Lotka-Volterra models the equations for the pop and pop remain the same as in Section 8.2. Other kinetic equations are slightly simplified, a number and multiplicity of integrals are reduced. 

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 could be proved in the manner similar to that used in Section 8.2. It is important to note that the correlation dynamics of the Lotka and Lotka-Volterra model do not differ qualitatively. A stationary solution exists for d = 3 only. Depending on the parameter k, different regimes are observed. For k kq the correlation functions are changing monotonously (a stable solution) but as k < o> the spatial oscillations of the correlation functions (unstable solution) are observed. In the latter case a solution of non-steady-state equations of the correlation dynamics has a form of the non-linear standing waves. In one- and two-dimensional cases there are no stationary solutions of the Lotka model. 

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... 

The physical separation between genotype and phenotype has an extraordinary consequence, because mental genotypes can be directly instructed by mental phenotypes, and this means that cultural heredity is based on a transmission of acquired characters. Cultural inheritance, in other words, is transmitted with a Lamarckian mechanism, whereas biological inheritance relies on a Mendelian mechanism which is enormously slower. As a result, cultural evolution is much faster than biological evolution, and almost all differences between biology and culture can be traced back to the divide that exists in their hereditary mechanisms. The discovery that human artifacts (i.e. cultural phenotypes) obey the Lotka-Volterra equations has two outstanding consequences. The first is that selection accounts for all types of adaptive evolution natural selection is the mechanism hy which all phenotypes - biological as well as cultural - diffuse in the world. 

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... 

To investigate the stability properties of the Lotka-Volterra equation in the vicinity of the equilibrium point (X, Y) = (0, 0), we linearize the equations ofXand Yappearing on the right side of Eqs. (13.63) and (13.64). These functions are already in the form of Taylor series in the vicinity of the origin. Therefore, the linearization requires only that we neglect the quadratic terms in XT, and the Lotka-Volterra equations become... 

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

Statement 1. Provided K t) = K = const, i.e., neglecting change in time of the correlation functions, equations (8.2.12) and (8.2.13) of the concentration dynamics describe undamped concentration oscillations with the frequencies uj < uq = /a, dependent on the initial conditions. The dependence u = uj K) is weak. This statement is based on the analysis of the Lotka-Volterra model by both topological and analytical methods (see Section 2.1.1). 

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