PROBLEMS IN POPULATION INPUT-OUTPUT DYNAMICS 199 



-^min' ^i> ^2' ^3' ^4, ^5 and ^g are constants and t is the length of time over 

 which reproduction occurs. While the equation is unfamiliar, the family of 

 curves it describes is not. For example, equation (6) is intended to describe 

 stock-recruitment curves as discussed by Ricker (1954), after, of course, 

 suitable terms have been introduced in the pre-recruitmcnt phase. 



Equation (6) v^as derived from three differential equations, which, in turn 

 were developed after an exhaustive examination of entomological literature. 

 However, there is every reason to believe that the three phenomena described 

 by equation (6) operate in the same manner in all sexual animals. The 

 equation states that reproductive rate at any time and at any density of 

 potential reproducers is under the influence of three interacting factors. 



These factors are (i) the increased difficulty of potential mates encountering 

 each other as density decreases, (2) the fecundity-depression effect of inter- 

 ference, and (3) the fecundity-depressing effect of competition for oviposition 

 sites. Many experiments have been performed which demonstrate the 

 existence of all three of these factors, particularly by Japanese scientists. 



Equations such as equation (6) can be constructed for all the terms in the 

 yield equation. Once the yield equation is developed so that all growth and 

 survival coefficients can be expressed in terms of the factors that regulate 

 these coefficients, our position is as follows. First, since applied mathematics 

 is language, not science, our yield equation does no more than state in succinct 

 symbolic language the extent of our quantitative biological insight into the 

 productivity dynamics of the exploited population under study. The best 

 equation we can write at any time, therefore, is a measure of the precision 

 and accuracy of the data we have collected. Precision and accuracy, in turn, 

 are measures of the volume of rephcation and ingenuity of design in our 

 data collecting procedures. The yield equation we have written should 

 indicate the magnitude and types of effects on productivity of all the factors 

 which can govern productivity. Second, once we have the appropriate 

 equation, we can manipulate it to find out how to maximize productivity. 

 In general, the yield equation can be thought of as a hypersurface in w-space, 

 yield being the dependent variable, and such factors as water temperatures 

 and fishing pressures in various seasons, and size of spawning stock being 

 the independent variables. (This language can be translated into the appropri- 

 ate terms for problems in forestry, big game management, algae culture, etc. 

 In fact, the problem as stated and the solutions being outlined are completely 

 general.) 



The problem is to find the values of the independent variables over which 

 man has control which give the highest possible yield peak in our hyper- 

 surface, which may be thought of as the w-dimensional counterpart of a 

 3 -dimensional mountain range. 



