4. Acceptance or rejection of the hypothesis. If 

 the agreement of simulated with observed 

 jnelds is good, the hypothesis is accepted. If 

 it is rejected, a new hypothesis is erected by 

 adjustment of parameters in the analog model, 

 and steps 2 to 4 are repeated until satisfactorj' 

 fit of simulated to actual yields is either at- 

 tained or found unattainable. 



In practice, the process was never repeated more 

 than a few times, since improvement fell off rapidlj'. 

 Also, indefinite repetition would be out of keeping 

 with the scientific method. 



BASIC FORMULATIONS 



For the initial trials of the analog technique, I 

 adopted what seemed the simplest useful model of 

 a fish population. This model includes rate of 

 growth, rates of fishing and natural mortality, and a 

 recruitment-stock relation. It does not take ac- 

 count of immigration, emigration, or environmental 

 effects. Symbols used have been adapted from 

 Holt, Gulland, Taylor and Kurita (1959) in further- 

 ance of their admirable attempt to secure uniformity 

 in the terminology of fishery dynamics. Definitions 

 are as follows: 



A', 

 R 



f 

 F 



1 



M 



Z 



t 



tr 

 tc 



P, 



P,o 



W, 

 U'oj 

 K'r 



E 



= Xumber of recruits surviving at time /. 



= Initial number of recruits to fishable stock for a 



single year class. 

 = Fishing effort. 



= Instantaneous rate of fishing mortality. 

 = F f. 



= Instantaneous rate of natural mortality. 

 = F + M. 



= Age of fish in years. 



= .\ge of fish at recruitment to fishable stock. 

 = .\ge of fish when first vulnerable to capture by gear 



in use. 

 = Weight of all fish of a given year class surviving at 



time /. 

 = Weight of all fish present at beginning of season. 

 = Weight of individual fish at time (. 

 = Upper asymptotic limit of Wt. 

 = Weight of individual recruit at time tr. 

 = Estimated yield of fishable stock in weight, per year. 

 = .\ctual yield of fishable stock in weight, per year, 



from official statistics. 



= Rate of e.xploitation, = ^ ^^ f 1 - «-<'' + '^'j 

 = Subscript referring to individual year classes. 



Ifw = Initial weight of recruits to fishable stock for a single 



year class. 

 0,g = Constants of Gompertz growth curve. 



Because interest in this study is centered on the 

 commercial catch, the model is limited to the fishable 

 sizes and ages of fish. For a year class of fish passing 

 through the fishable stock, numbers of fish surviving 

 may be expressed according to the declining expo- 

 nential formula, as set forth in Beverton and Holt 

 (1957): 



iV, = /?p-(F+-w) c-g 



(1) 



In addition to the above, the following symbols 

 have been adopted for the formulations here: 



To take account of the growth of individual fish, 

 and to obtain yields in weight for comparison with 

 commercial catches, it is necessary to introduce a 

 formula for weight-at-age. Beverton and Holt em- 

 ployed the von Bertalanffy equation for length-at- 

 age, converting to weight-at-age by means of a cubic 

 length-weight relation. Use of the cubic relation 

 has been shown to lead to considerable error when 

 the real relation between length and weight involves 

 a power of length other than 3 (Paulik and Gales, 

 1964). Although this difficulty can be overcome by 

 use of the Incomplete Beta Function (Wilimovsky 

 and Wicklund, 1963) in the yield equation, the 

 formulation still is not well adapted to analog 

 computation. 



As an alternative to the von Bei'talanffy equation, 

 I investigated the characteristics of the equation 

 developed by Benjamin Gompertz. He applied it 

 as an expression of human mortalitj' rates, but 

 various forms of it have since been used as growth 

 curves for both length and weight of animals. Its 

 applicability in this respect was thoroughly dis- 

 cussed b}' Winsor (1932). In the form used by 

 Weymouth and Mc^NIillin (1931), it is seen to be an 

 exponential curve in which the slope declines expo- 

 nentially. They point out that the relative (as 

 oppo.sed to absolute) growth of an animal declines 

 with age because of an increasing proportion of inac- 

 tive material, and other causes. The declining slope 

 of the Gompertz curve is in accord with this phe- 

 nomenon. Also, it provided a good fit to the 

 empirical data of weight-at-age for several fishes. 



Beverton and Holt (1957) rejected the Gompertz 

 curve on the basis that it deals with growth as an 

 additive process only, ignoring the breakdo\\'n of 

 protoplasm. The net effect, however, of anabolism 

 and catabolism may well be the kind of declining 

 relative growth described by the Gompertz curve. 

 This curve thus did not appear to be rejected on 



ANALOG COMPUTER MODELS OF FISH POPULATIOXS 



33 



