AN UNSOLVED PROBLEM OF BIOLOGY 



average contribution of offspring to the test-tube population 

 of the future. Each test-tube may lay claim to an equal share 

 of the ancestry of future generations, and its reproductive 

 value is invariant with its age.^ 



The next step in the argument is vital. Although each indi- 

 vidual test-tube takes an equal share of the ancestry of the 

 future population, each age-group most certainly does not. The 

 older the age-group, the smaller is its overall reproductive 

 value. The group of test-tubes 2-3 months old, for example, 

 makes a very much greater contribution than the group 11-12 

 months old. This is not because the test-tubes of the senior 

 group are individually less fertile — their fertility is ex hypothesi 

 unchanged — but merely because there are fewer of them; and 

 there are fewer of them not because they have become more 

 fragile — their vulnerability is likewise unaltered — but simply 

 because, being older, they have been exposed more often to the 

 hazard of being broken. It is simply the old story of the pitcher 

 and the well. 



Some of the consequences of this decline in the reproductive 



1 The actuarial characteristics of a 'potentially immortal population' are 



particularly simple: the life table is defined by the relation l^. = /^e"'^^, 



where /^ is the size of the original cohort, l^. is the number of them that 



/' 1 dl^\ 

 survive to the age of x, and a is the force of mortality [ {^ = — ) , in- 



\ Ij. dx J 



dependent ex hypothesi of age. The probability Pj. of surviving from birth to 



age X is simply IxJIq ^ 6~'**. If the number of offspring born to each member 



of the population in each unit of age remains constant, as we have supposed, 



at the value h, then the reproductive value remains constant throughout 



1 /'GC h 



life at the value J?~. = — I hpx .dx = — \ and this will also be its value at 



PxJx ^ ^ 



birth (the net reproduction ratio R^. If the regimen of constant mortality 



and fertility has been in progress long enough, and numbers are not 



declining (6> jLt),then a stable age-distribution will be reached in which the 



fraction of the population falling within the age interval a? to a; -H Dx is 



rx+Dx 

 given by Ca; = I he~^^ dx; the proportion of the population aged x 



J X 



and upwards is thus simply e~^^. 



61 



