322 George A. Sacher 



This restriction is met if we have X > 3a (as is the case in the appHcations 

 considered). The normahzing integral in equation (15) is then not appreciably 

 less than unity, so equation (15) for the mortality rate reduces to 



(A>3a) (1-a) 



Equation (15) gives the dependence of the mortality rate on the parameters 

 /?, A and a (or /i, A and D) in the stationary state of a system specified by equation 

 (6) and subject to a stationary random force function with spectral density 4D. 

 Although this model is too simple and artificial to be an adequate description of 

 an actual mortality process, it should be noted that equations equivalent to 

 equation (4) give an approximate description of a number of different physiologic 

 mechanisms. 



Equation (15) can be extended to the case of time-dependent mortality rates, 

 as they are observed in animal populations, if the parameters are sufficiently 

 slowly changing functions of time, so that stationariness of the fluctuation 

 process is preserved. This is a reasonable assumption with regard to the life 

 tables of animal populations in their normal environments. It is also considered 

 for the purpose of this discussion that the fixed and the random components 

 of environmental forces are stationary throughout life. 



Experimental data on homeostatic capacity for a variety of mechanisms 

 as a function of age indicate that this capacity diminishes during adult life (5). 

 We therefore expect a steady decrease in the value of />. Since a^ = Djft, the 

 value of a will be increased by a decrease in /?. The observed dispersion of 

 physiologic variables does not increase markedly with age. This may imply 

 that the recovery constant does not diminish much during the life span, but it 

 may also be due to the eff'ect of the distribution of parameters in the population, 

 since it can be estimated that about half the total variance in a typical outbred 

 population is variance between members, and this variance is reduced by selec- 

 tion, for as mortality proceeds in a heterogeneous population the subpopulations 

 with the more disadvantageous parameter values will experience heavier 

 mortality and thus be preferentially eliminated from the surviving population. 



We have also examined (6) one simple mathematical model of a homeostatic 

 mechanism that introduces a plausible type of non-linearity of recovery. In 

 this model there arises a relation between the location of the mean state and the 

 value of the recovery constant. In the notation used here, 1 and /9 would 

 decrease concomitantly. 



The methodological difficulty in the study of mortality processes is that 

 mortality data are not by themselves sufficient for the unique determination of 

 their parameters, even in the simplest cases. In earlier treatments (7) the 

 expedient was therefore adopted of assuming that the mean state, X, is the 

 only parameter that changes with age. There is abundant evidence that the 

 mean values of physiologic variables change with age (8). Advantage was also 

 taken of the fact that changes in mean physiologic state with age are usually 

 small in degree. This justified taking the linear term of the expansion of A^ 

 about the initial value Aq, 



A2 = V + 2AoAA (16) 



