PROBLEMS IN POPULATION INPUT-OUTPUT DYNAMICS 193 



the sustained productivity, less the remainder that must be left behind to 

 sustain the yield. Where conditions are controlled, intraspecies competition 

 is the only variable regulating productivity. Up to a certain 'fishing' level, 

 increased yield results in increased productivity in the following time interval 

 due to decreased biomass wastage caused by competition. However, fishing 

 beyond the optimum level causes extinction of the harvested population. 

 How high the optimum is for any species depends on the age structure of the 

 population left behind after harvest, the numbers of individuals left behind 

 and the frequency of harvest. From species to species, the productivity for 

 given treatment depends on the biotic potential of the species. 



The effect of biotic potential on optimum yield level shows up clearly 

 when we examine the results from the aforementioned five laboratory studies: 



Where intraspecies competition is present, my earlier (1955, 1956) formu- 

 lation of the productivity equation is suitable. That is, we develop an 

 expression in which productivity from time t to time ^ + i is equated to 

 the difference between biomass found at the later and earlier times. Then, in 

 turn, we do enough research so that this difference can be expressed as a 

 function of the factors which can regulate the magnitude of the difference, 

 through their effect on growth, survival and recruitment. Finally, we use 

 calculus or other mathematical technique to fmd the values of the independent 

 variables which maximize biomass productivity. In extremely succinct 

 mathematical shorthand, avoiding symbols already defined otherwise by 

 Holt et al. (1959), 



Pb (t : t+1) represents biomass productivity from time t to 

 time ^-h I. 

 Bt represents biomass at time t. 

 Bt+i (Xj, Xg, . . . X^) represents biomass at time ^ + i as a function of 



the variables X^ ... X^ which governed 

 biomass production over the interval ^ to ^ + i. 



We may write the forgoing problem statement as that of seeking 



MaxP5(,:,+i) = ^m(Xi,X2,X3 ... Xj - B, (i) 



