the quantities C* and a* can serve as experimental estimates of the 

 theoretical parameters 



1 - 2xi . xi - x, 

 C = ^ and A - ^ ^ 



1 - Xl (1 - Xl)(l - Xg) 



For numerical calculations, it is convenient to analyze the asymptotic 

 case of small values of a, when the expressions (2.27) and (2.28) are 

 simplified: 



l.i ."A l_i .^ _ . 2- - 2 



en 2 c ^2 „ 2 A(z. - zj a'^(z, - zy 



YA . ^ ^ (1 ^ L_ + 1 2_), (2.33) 



1 Azf 1 - A 2 24 



i _ 7 i c" " 1 ? 



C ^2 2 ^ ^ 2c^ 



a2(Zi - z„)2 



6A . 1 + ^ ^ . (2.34) 



24 



To estimate the mean probabilities of a-indivi duals of a 

 community p, we must estimate the mean birth rate l of its individuals, 

 limited to a great extent by the trophic nature of the area of 

 distribution. These estimates represent a special problem in 

 trophodynamics. The importance of the equations presented above for the 

 relative indices a and tt consists in that they can be calculated without 

 knowing the values of l, before they are determined. Let us analyze a 

 numerical example of the use of these equations. 



Suppose, within an a-component community (a = 10) with a mean 

 probability of a-indivi duals within it pi = 0.01 and assigned parameters 

 C = 0.1 and a = 0.15, there is a commercial population of interest to us 

 (component a) and we would like to exploit it annually (time step of 

 model one year) so that the corresponding parameter y^ = 0.4, while 

 leaving the other components alone (y = y^ = 0-^ =1-2). What, in 

 this case, is the maximum possible value of the relative mean 

 probability of a-indivi duals of the community ^2 = P2/Pi» and what is 

 the possible increase in v^ to the value 113 = Pt/P^, after some rather 

 long time of readaptation of the community to the new conditions of 

 stable (with characteristic y) exploitation? 



By definition, we have 



773 > 0.5 and 1x2 > 0.75, 



the quantities 0.5 and 0.75 were defined as follows: first, using C = 

 0.1 and a = 0.15 and the equations (2.33) and 2.34), we found the values 

 of X = 0.5 and <5 = 1 , then, using them and Fig. 9, we determined the 

 values of a^i = 0.5 and Ap = 0.25; then using them and equations 

 (2.31'), we found the values of the estimates M3 > 0.5 and P2 ^ ^.75 and 

 then, finally, using them, we found the desired values of the estimates 

 TT3 and tt2 (see (2.32)). 



354 



