310 Appendix 



may be set up. This has the solution 



If = k e^'l(l +De^'y+°'+'- (29) 



If death occurs when the units fall below a, we have 



We also have 



= I D{\ +Z))^+«/(l -{-De^y^'' (30) 



dt ^ \+De^' ^ ' 



and 



I"' - ^^Td^ (^^) 



We have now found a difference equation model which leads to a 

 Perks (logistic) formula for /x^. In formula (31) the upper limit of /x, is 

 _p(/S + a); p ^ 0-1 and the limit ^ 0-7 so that (jS + a) ~ 7. 



25. The distribution of a in the population at age implied by 

 equation (29) is a decreasing geometrical progression, i.e. 



D D D 



1+D' {l+Df"'(l-{-D)'^ 



For human mortality D is small (of the order of 10~^) so that the dis- 

 tribution is very slowly decreasing with increasing a. 



26. The significant result which emerges from the experiments 

 made along these lines is that to provide results which have some 

 reasonable semblance to observed human mortality the backward 

 type of model has to be adopted. This is consistent with death 

 being regarded as the culmination of a degenerative process such 

 that death occurs when the organism reaches a certain level of 

 degeneration. The mathematical models are based on numerical 

 results for adult ages and interpolation back to birth is possibly a 

 questionable process, a more suitable approach being to regard the 

 life and death process as a period during which the organism is 

 building up to a complex situation with a subsequent degeneration. 

 This would lead to models in which the whole of life process would 

 be looked upon as the resultant effect of two opposing forces. 



