192 K. E. F. WATT 



parameters in a productivity equation. Such analysis leads to the formulation 

 of a specific management routine which will produce maximum sustained 

 yield of the studied resource. A small, but increasing number of workers in 

 this area are attempting comprehensive studies of population productivity in 

 the laboratory and field. Hence the time seems appropriate for considerable 

 effort to formulate a general theoretical model of population productivity. 



Many types of mathematical models of productivity have been formulated 

 to date. Four proposed by fisheries workers have been reviewed (Watt, 

 1956) and applied (Watt, 19596). Many others, often radically different in 

 conception, have been used by speciahsts in forestry and a variety of other 

 biological fields. 



A recent review of optimum yield theories being applied by foresters is 

 found in the Forestry Handbook (Forbes, 1955). Analogous thinking applied 

 to pasture grasses is discussed in the treatise by Voisin (19 59)- However, this 

 author is not aware of any models of exploited biological populations which 

 compare in sophistication with the most advanced models of fisheries. In 

 turn, by far the most comprehensive, realistic and thoroughly tested model 

 put forth by fish population ecologists is that of Beverton & Holt (1957) • 

 However, Beverton and Holt are both scheduled to give papers at this 

 Symposium, so I shall not discuss their model further, except to note that 

 this thinking is essentially directed towards analysis of population steady 

 states in a stable environment. 



The present paper is intended as a first step in attempting to synthesize a 

 productivity model that will be ubiquitously apphcable to renewable natural 

 resources. Various types of biological productive systems will be discussed 

 in order to point out the categories of productivity problems that can arise, 

 and the factors that must be considered in their solution. This will lead to a 

 general statement of the population productivity problem. 



TYPES OF EXPLOITED POPULATIONS 



The simplest conceivable type of productive system is a regularly exploited 

 population maintained under controlled conditions. Such laboratory studies 

 have been conducted on unicellular algae (Ketchum et al, I949)> blow-fly 

 larvae (Nicholson, 1955), guppies (Silliman & Outsell, 1957, 1958), Daphnia 

 (Slobodkin & Richman, 1956) and flour beetles (Watt, 1955)- 



In such cases, perhaps the best way to set up a mathematical formulation 

 is to fix attention on maximization of biological productivity, defmed as net 

 amount of biomass produced per unit time. Productivity determines the 

 maximum sustained yield per unit time we can remove. The ideal harvesting 

 regimen is that in which the sustained yield per unit time is exactly equal to 



