2 .2 Model Study of O ptimal Strategies of a Community . 



The structure of a community in our model is defined by the vector 

 x-(Xfy) of the death of an a-individual from causes within the community, 

 particularly natural death related to the trophic structure, etc. 

 Obviously, it is less variable with time than is the vector X = (Xot) of 

 birth of a-individuals. Therefore, vector x will be considered a fixed 

 parameter of the model while vector x, defining the behavior of the 

 community, will be considered variable. 



As an indication of the limiting effect of the medium on the 

 community, we can report the mean value X*, while we can use the mean 

 value y of the mortality vector y=(ya) due to extrabiocenotic causes 

 (behavior of the environment) as an indicator of the intensity of the 

 action of death-causing factors of the environment on the community. 

 The set of vectors y and X, with nonnegative components, not exceeding 

 unity, with fixed values of y and X, will be represented 

 by W- and W- . Furthermore, additional conditions are placed on 

 vect'^r X, related to equation (2.2). Therefore, the set W^ of its 

 possible values is included in the set W-(W^ W-) . We note that 

 analogous limitations on the vector p=(Ma) ^" ^^^ case extend only to 

 the fixed vector x and do not effect the vector y. 



Empirical analysis of succession processes in a community reveals a 

 tendency toward S- shaped growth of its summary biomass right up to the 

 limiting climax value, limited by the steady conditions of the 

 environment. This tendency, accompanied by fluctuations in the biomass 

 of the individual components, is considered by Yu. Odum as a strategy of 

 the community (Odum, 1975, p. 345). 



Let us formalize these general ecologic concepts within the 

 framework of the model for quantitative analysis which we have studied. 



Grouping populations within components, we can achieve relatively 

 small differences in mean densities 3a of the biomasses of a- 

 individuals. If this, for any reason, cannot be done, then we must 

 limit ourselves to analysis of only the higher trophic levels of 

 multi component communities, for which B^ is not divided into orders. 

 The assumptions we have made allow us to extend the tendency toward 

 growth of the summary (or mean) biomass of the community to the mean 

 population density p, which is proportional to the mean 

 probability p (see above). Thus, we can consider the following 

 function, monotonically increasing p, to be the "goal funct onal" of the 

 community: 



M = M(X,p) = p/X(l - ap) = J^ E WWa=i(-)- (2-15) 



Xci a=l XV 



la 

 *In what follows, the averaging operator x represents x= — S-iXq 



see (2.13) . " ""^ 



347 



