The Lotka model

The Lotka model is based on the following reaction chain 

Here E is the infinite reservoir of matter. It is assumed that in an open system concentration ne is constant and E is linearly transformed into A followed by an autocatalytic transformation of A into B and its decay. The product P does not affect the reaction rate. The model (2.1.21) is described by a set of equations 

A biological interpretation of the model could be easily formulated in terms of prey animals A and predators B living on them. Let riA t) be a population density of prey animals who stimulate reproduction of predators 

The use of (2.1.14) gives the Lotka model a single solution for the stationary point 

The way in which the solution rii t) approaches its stationary value n = rij(oo) for a system with two degrees of freedom can be easily illustrated in a phase space after eliminating time t - Fig. 2.1. This type of 

The first theoretical model of a chemical reaction providing for oscillations in concentrations of reagents was the Lotka model from 1910. A mechanism of the hypothetical Lotka reaction has the following form  

The Lotka model also has an ecological intepretation. In such a formulation A is an inexhaustible source of food for animals X which, in turn, constitute feed for animals Y. The death rate of predators Y, denoted as P, is proportional to their number whereas the death rate of animals X can be neglected, since a decrease in the abundance of population X occurs mainly as a result of predators eating them. We must also assume that the considered populations are isolated from other predators and other animals being a prey. A model system may consist of hares (X) and lynxes (Y). Abundant populations of these animals, living in the same area, occur for example in Canada. 

We shall now proceed to giving a system of kinetic equations corresponding to the mechanism (6.52). Neglecting the effects associated with diffusion, the kinetic equations take the following form  

Stationary states of the system (6.55) satisfy for a 0 the equations deriving from condition (5.4) 

a catastrophe involving the change in a number of stationary solutions cannot occur. 

---

Fig. 1.29. The steady state in the Lotka model. Control parameter p = 0.2.

Fig. 2.1. A stable node. Four solutions of the Lotka model, equations (2.1.22)—(2.1.23) with the distinctive parameter Kp/401 = 2 are presented. The starting point of each trajectory is...

The Lotka model is an example of a rough system deviations of concentrations from their asymptotic values (2.1.24) occur independently on chosen parameters p, K, (3, i.e., small variations of these parameters cannot affect the way a system strives for the equilibrium state. 

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. 

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... 

In terms of the deterministic approach the Lotka model (2.1.21) is described by a set of equations... 

Results of the stochastic simulations for the Lotka model are presented in Fig. 2.16. 

Equations (2.2.76), (2.2.77) of the Lotka model are not analyzed so far. We suggest the readers to solve this problem as a home exercise. Despite... 

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). 

Following [21, 25], let us visualize the scheme (2.1.21) of the Lotka model in terms of a simple stochastic model as follows. 

Independent control parameters of the Lotka model are p and j3, describing reproduction of particles A and decay of B s, as well as the relative diffusion parameter k = DA/(DA+DB), and the space dimension d. Before discussing the solution of a complete set of the kinetic equations (8.3.20) to (8.3.24), let us formulate several statements. 

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. 

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. 

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). 

Statement 1) a stable stationary solution of a complete set of equations of the Lotka model holds. At long t the reaction rate K(t) strives for the stationary value. Time development of concentrations obeys standard chemical kinetics, Section 2.1.1. 

Figure 8.9 serves as a good illustration of different possible transient regimes arising as k is reduced. As stabilization time increases, r oc kT1, the Lotka model reveals a series of quasi-periodic motions, separated by chaotic transient phases. The main trend seen from the analysis of results, is emergence of the periodic motion with a minimal period. To get some important properties of the transient irregular regimes, such as the presence of main frequencies or a white noise, it is useful to analyze the Fourier spec-... 

The correlation functions for the Lotka model in the auto-oscillating regime are presented in Figs 8.14 and 8.15. The value of the parameter k = 0.02 corresponds to the curves plotted in Figs 8.7(c) and 8.8. The correlation functions motion is completely periodic, the results shown here correspond to the the minimum and maximum of K(t). 

---