The stochastic Lotka model

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 -I- B B and A -1- B 0 reactions (Chapter 7). 

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

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 po,i and pop remain the same as in Section 8.2. Other kinetic equations are slightly simplified, a number and multiplicity of integrals are reduced. 

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Fig. 2.16. The random trajectory in the stochastic Lotka model, equation (2.2.76). Parameters are fco//3 = fijk = 10, the initial values Na = Nb = 10. When the trajectory touches the Na axis, the predators B are dying out and the population of the prey animals A infinitely...

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