2 .1 Dynamic Model of a Community Ignoring Aggregation 



The succession of a community (Odum, 1975) can be described by a 

 stochastic model with discrete time t = 1,2,.... The time step taken as 

 unity corresponds to the characteristic time interval for the community 

 in question: a day, a month, a year. The status of the community at 

 moment in time t+lp^"''-'- in the model in question can be represented as 

 fol lows: 



pt+1 = ptPt, (2.1) 



where Pt = ||P^3|| ( »» 3 = l,a)--is a stochastic matrix. In order to 

 consider the effect of delay in the reaction of the community, we can 

 require that the probabilities p^ag depend on the k states p^'l*^ , . . .p^-l , 

 preceeding state t. 



pW = pW(p^-^---.P^-M or Pt = Pt(pt-k,....pt-l). (2.2) 



The specifics of each community, related to the trophic and 

 tropical structure of interactions of its components, as well as its 

 interaction with the environment, is described by the unsteady equations 

 (2.2). Recording of these data, together with the k first states 

 p ,...,p , unambiguously defines all subsequent states of the 

 community. However, certain modifications of these interactions are 

 always possible, which ecologists consider permissible within a given 

 community (not leading, from their point of view, to a new community). 

 In our model, this is formalized by the assignment of a certain set Wg 

 of permissible functional transforms (2.2), describing the very same 

 community. 



A climax community (Odum, 1975) corresponds to the limiting 

 behavior of the model as t->". The limiting state p = lyn pt exists if 

 there is a limit P (.,...,.) = li^ Pt( .,...,.) , corresponding to steady 

 influence of the environment and relationships within the community in 

 the climax state. The limiting state is independent of the k initial 

 states and can be found from the equation 



p = pP(p,...,p) (2.3) 



It is convenient to find the solution of this equation in two 

 stages. In the first stage, we seek out the solution of the linear, 

 homogeneous equation p = p||pag|| to express p through p^g 



P = FLp^g] (2.4) 



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