PROBLEMS IN POPULATION INPUT-OUTPUT DYNAMICS 201 



would be well advised to study some of the more advanced literature in this 

 area, by Arrow et al (1958), Bellman (195?) and Kuhn & Tucker (1951)- 

 These treatises deal with problems very reminiscent of the actual problems 

 encountered by biologists. The equations may be non-linear and even 

 sequencing of operations in time is considered. 



Finally, a variety of electronic computer techniques are available for 

 finding extrema in hypersurfaces as quickly as possible (Box, 1954; Box & 

 Coutie, 1956; Box & Wilson, 1951; Box & Youle, 1955)- The computer 

 programme for such problems suggests a blind man trying to fmd the 

 highest point of the Himalayas. The blind climber would constantly go up 

 the direction of steepest ascent. Unlike the chmber, however, a routine can 

 be built into the computer to prevent it from becoming 'lost' on a local 

 maximum. 



SUMMARY 



Because of burgeoning human populations and limited ability of the earth's 

 surface to produce food, mankind is now confronted with two crucially 

 important problems. The maximum amount of renewable natural resources 

 that the world can produce each year must be estimated with a high degree 

 of precision and accuracy. Second, it must be determined how maximum 

 levels of production can be attained and maintained. 



It would be desirable to have some broad unifying theoretical model of 

 the mechanics of exploited populations that would suggest how a great 

 variety of cases could be handled mathematically. The aim of the empirical 

 and subsequent mathematical analysis in each case, for trees, wheat, algae, 

 fish or other exploited resource would be to show how to maximize yield 

 without impairing the abihty of the population to replace itself. 



A general mathematical model which aims to show how biomass produc- 

 tivity can be maximized is not a ubiquitously applicable theoretical tool. 

 This is because it will only be possible to increase productivity by adjusting 

 a fishing or harvesting regimen in those cases where productivity has been 

 depressed by intraspecific competition. In cases where a population or 

 organisms is existing in an area where the cUmate is only rarely favourable 

 for natahty, growth and survival, competition is not important as a regulator 

 of productivity. 



More general and insightful mathematical tools for analysing yield 

 problems are suggested if we think in terms of minimizing biomass wastage, 

 rather than maximizing biomass productivity. Such tools allow us to analyse 

 effects of wastage due to all factors, not just those caused by competition. 



Data from, and techniques for various types of cases are discussed. 



