PROBLEMS IN POPULATION INPUT-OUTPUT DYNAMICS 197 



Writing the above in mathematical shorthand, and again trying to use 

 notation consistent with that of Holt et ah (i959)» we have 



Max y^+i = Pf+i - P(min)«+i (2) 



where P^+i represents the biomass present at the end of the interval ^ to ( + i 

 and P(min)t+i represents the minimum biomass of individuals that must be 

 left behind at any harvesting time, ^+1, to guarantee replacement of 

 Max y^+i- 



While a great deal is known now about how to develop expanded versions 

 of equation (2), application of this equation to any particular biological 

 population is still a relatively tricky proposition, I will mention here only a 

 few of the types of technical difficulties that can be encountered. 



It may be possible to maximize Y^+i, for example, by removing some 

 individuals prior to their reproductive age. Suppose the optimum yield 

 procedure is to remove all reproductive individuals each harvest, and leave 

 behind individuals that will reproduce between harvests. How many such 

 immature individuals should be left behind; We cannot leave the bare 

 minimum required for reproduction, because natural mortality will reduce 

 this bare minimum between harvest time and the time reproduction takes 

 place. Hence enough immature individuals must be left behind so that even 

 after losses due to natural mortahty, there will be sufficient reproducers by 

 the time when reproduction occurs. However, the 'cushion', or extra number 

 of individuals left behind to absorb the effect of natural mortahty from one 

 year to the next should vary from year to year, because climate varies from 

 year to year and hence percentage natural mortality differs from one year 

 to another. Since we cannot know what climate will be like in advance, the 

 size of the 'cushion' in such instances should be sufficient to absorb the effect 

 of natural mortality under the worst possible conditions for survival that 

 could occur. 



Before proceeding to elaboration of equation (2), let us consider in a 

 general way the strategies of harvest dictated in different situations by this 

 equation. 



In cases where the environment is favourable and competition regulates 

 productivity, the equation tells us to fish, harvest, hunt or otherwise exploit 

 the population so that P^+i is a maximum, and this in turn maximizes Y^+i, 

 if exactly P(unn)t+i is left behind. However, where P^+i is entirely regulated 

 by a rigorous environment, and nothing man can do will increase it, the 

 equation tells us to maximize Y^+i by fishing so that no more biomass than 

 Pimin)t+i is left behind at any harvesting time. It follows, of course, that 

 where P^+i is partly determined by competition, and partly by the environ- 

 ment, a mixture of the forgoing two strategies will be required. 



