2. Stochastic Models of Communities . (B. S. Fleishman) 



A community is a complex open stochastic system (Fleishman, 1976), 

 but its deterministic models are justified in the first approximation by 

 the law of large numbers. 



There is an extensive literature on deterministic dynamic models of 

 community, based on differential, finite-difference and matrix 

 equations. 



A community consists of components--populations or groups of 

 populations, distinguished according to some principle. The individuals 

 of component a, in turn, may form a stable formation of n^ 

 individuals. This formation will be called an a-individual . Their 

 number in a certain biotope of volume V will be represented by H^. The 

 density of the a-individuals (density of the biomass) will be denoted by 

 the expression p^ = N^/V (B^ = Pa^a). where 6^ = "a^a is the mean 

 biomass of an a-individual, B^ is the mean biomass of an individual of 

 the a-component. 



Theoretically, we can divide a biotope of volume V into N 

 elementary cells of volume aV, occupied by only one a-individual or 

 empty. Suppose V^ = aVN^^ is the volume occupied by a-individuals, while 

 ^a+l ~ '^VNa+i is an empty volume with N^+i elementary cells 



( Z N« = N). 

 a=l 



If we analyze a stochastic model of a community without considering 

 the aggregation of a-individuals, i.e., if we consider all placements of 

 the Nc( a-individuals in the N cells equally possible, then the 

 probability of presence of an a-individual in any cell (probability of 

 an a-individual) is equal to p^ = HJH, while the probability of an 

 empty cell is equal to Pa+i = Ng+i/N. It is easy to show that the 

 probabilities are proportional to the densities p^ = PqAV. However, the 

 introduction of probabilities is important, due to the possibility of 

 working with them and using the apparatus of the theory of 

 probabilities. 



Thus, the status of a community is described by a probabilistic 

 vector p = (pc(). If the ath component of the community consists of one 

 population, it can be divided into b^, age groups, considering p^ - (p^s) 

 to be a vector, where p^^s is the probability of a-individuals of the sth 

 age group. 



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