'as 



Equation (2.4) in this case has a simple solution: 

 \ ^ \ ^ Q* h P 



n„ , "^^ P 



ofl. Pofl = U V'l s='r PTJ" • (2.7) 



^ul+i^al u=2 X^u+^^au "^' ^' ' «=1 s=l Pa+i 



complete solution of system (2.7) requires explicit assignment of the 

 variation of X^ , xt^ and u^ with p^ and the parameters of the 

 environment. In addition to the variations which are specific for each 

 community, at least the following two are common. Suppose the 

 conditional probabilities X\ of birth of cx-indi viduals of the first age 

 group from a-indi viduals of age groups (probabilities of fertility by 

 ages) are assigned. 



Then : 



xt = 



;|J' >^IpIs • (2.8) 



On the other hand, suppose the death of a-indi viduals , defined by 

 the probability u^g , depends on K independent events--both in the 

 environment and within the biocenosis (catastrophies , fishing, 

 predators, parasites, food shortages, natural death, etc.). The 

 probability of these fatal events is represented as M^gi^ (k=l,K) . Some 

 of them, in turn, may depend on the status of the system p'-. 



Then, as we can easily demonstrate: 



mIs = n + [^1^ (p'ik - 1)'^]"^}"^ . (2.9) 



We note that only where Pctsk^c<l does approximate additiveness 

 occur: 



K 



v\s = kli i^ask + (c) , (2.10) 



which, as a rule, is not considered in the ecologic literature. 



Let us study a few particular cases of the relationships we have 

 derived. Let us begin with the case of a single population (a=l), 

 divided into b-^ age groups. Using equations (2.1) and (2.6) in our 

 case, we produce, for the probability of the first age group 



t+1 t t t t t 

 p = p (1 - y - X ) + p X . (2.11) 



11 11 11 11 2 1 



We will consider that the reproductive capacity is equal only for 

 age groups beginning with group Sq. Mathematically, this can be 

 expressed as: 



345 



