Appendix 309 



the solution of which leads to 



If = l^ ^pe^-*) {p{oj - 1) Yjoi ! (23) 



from which 



dt ^ co-t 



if the deaths occur at the ath hit. In this formula p'~0-3,a^ll 

 and oj --^ 110 for human mortality. 



22. No obvious physical model applies to equation (22), but the 

 relationship can be written in the backward form 



f = - ^ Z,« + «-±l r^ (24) 



at to — I co — t 



in which the rate at which a unit is lost is proportional to the number 

 of units remaining divided by the years of life remaining to the final 

 age CO. From a biological point of view the concept of a final age by 

 which the organism must be dead is unsatisfactory, but the fact that 

 satisfactory numerical results arise only from a backward formula 

 suggests that a closer study of this type of model might be more 

 profitable. 



23. The simplest backward model arises from the relationship 



^= -plf+pir (25) 



where the organism is assumed to lose a unit at rate p. 

 This has a solution 



If = l^ e-^* {pt)^°'l(n - a) ! (26) 



where n is a maximum number of units. If death is assumed to occur 

 when the number of units faUs below ?•, we have 



dCioQ 11.1) n — r ,„^. 



This is of similar form to equation (16) and is not suitable for human 

 data. 



24. By assuming that the rate of loss of a imit is proportional to 

 the number of units remaining the relation 



^ = -piP + c.) lf+piP + o^ + 1) r^ (28) 



