196 K. E. F. WATT 



G. Biological productivity of a population can be governed by (a) environ- 

 mental factors extrinsic to the population, (b) the size and age distributions 

 of the population, which regulate competition pressure, or (c), both of {a) 

 and (b). Any species can, and probably will, be under the influence o£{a) in 

 the unfavourable or boundary part of its range, and {b) in the central part of 

 its range where cHmate is favourable for natality, growth and survival. Also, 

 particularly at the fringe of the species range, populations may be largely 

 regulated by {a) in some years and (b) in others. 



H. There are many different causal pathways by which population biomass 

 productivity can be governed. Each of intraspecific competition, interspecific 

 competition, food abundance and weather can regulate each of fecundity, 

 fertility, survival and growth. Parasites, predators and diseases can affect 

 growth and survival, and sometimes fecundity and fertility. A mathematical 

 model for a specific exploited population is only realistic and useful in so far 

 as it includes insight into all the relevant causal pathways. 



SYMBOLIC FORMULATION OF PRODUCTIVITY MODELS 



The foregoing considerations suggest how we must proceed in order to 

 develop a general mathematical model for use in showing how best to 

 exploit biological populations. It is clear that a model designed to show only 

 how to maximize productivity will not always be satisfactory, because it 

 will sometimes be impossible to increase productivity by means of a harvest- 

 ing regimen. On the other hand, it is also clear that a general model must 

 involve a term showing how all relevant factors interact to regulate pro- 

 ductivity, for the following two reasons. First, in those cases where a 

 population is not entirely, or at all, regulated by climate, we can in fact 

 increase productivity by judicious choice of harvesting regimen. A model is 

 not general unless it shows how to choose the harvesting procedure in such 

 cases. Second, where a population is entirely regulated by climate, the best 

 harvesting procedure is that which makes best utilization of the productivity 

 allowed by chmate. Hence an expression for the productivity which does 

 occur is required in equations for this category of cases also. 



Hence I have decided that the best possible statement of the optimum, or 

 maximum yield equation is as follows. 'The maximum biomass yield that 

 can possibly be obtained repeatedly, per unit time, from any biological 

 population is equal to the biomass present at the end of each unit time less 

 the minimum biomass of individuals that must be left behind per unit time 

 to guarantee replacement of the maximum possible yield by the next time 

 exploitation occurs.' It is understood that the age distribution of the 

 remainder is relevant and critical. 



