SISSENWINE: COMPARTMENTALIZED SIMl I.ATION MODEL 



landing). The equilibrium catch is the sum of 

 recruitment and growth (of the individual fish of 

 the nominal stock) minus loss due to natural mor- 

 tality. Based on this relationship using earlier 

 estimates of equilibrium catch and recruitment 

 (Sissenwine 1974) and assuming annual natural 

 mortality of 0.1, Sissenwine (1975) estimated the 

 average annual weight gain per fish of the South- 

 ern New England yellowtail flounder fishery for 

 1944-65. These estimates ranged from 72 to 331 

 g/fish per year and are significantly correlated 

 (Kendall rank correlation coefficient (t) of —0.60) 

 with annual average air temperature at Block 

 Island. Estimates of k of the von Bertalanffy func- 

 tion derived from growth increments of age- 

 classes for 1962-71 were also significantly cor- 

 related (r = -0.42) with temperature at Block 

 Island. Thus, the model was designed to simulate 

 the effect of temperature on growth. 



The instantaneous growth rate of a fish is 

 related to k by the following equation: 



dw 

 dt 



kc t c 2 {L m — L) L 



<v 



(19) 



The proportion of yellowtail flounder recruits 

 entering each size category of age-group 1 was 

 assumed as follows: G l = G 7 = 0.05, G 2 = G 6 = 

 0.10, G 3 = G 5 = 0.20, and G 4 == 0.30. 



Lux and Nichy (1969) reported a value of 500 

 mm for parameter L,„ of the von Bertalanffy 

 growth function for the yellowtail flounder. They 

 selected this value since it was the maximum 

 length observed. The model described in this 

 paper requires values of L m for each of the seven 

 size categories. Considering the magnitude of s„, 

 (33.9 mm, see Table 2) a value of 500 mm for L m4 

 may yield fish far in excess of the maximum 

 length observed. Therefore, a more conservative 

 value was used: L m4 = 480 mm. 



The probability density function of L m was used 

 to calculate values of L mi for i = 1, 2, 3, 5, 6, 7. 

 The range of values of L m represented by each size 

 category (Z u to Z 2l ) was calculated based on G, 

 and the normal density table and found to be as in 

 Table 3. The mean value of L m for each size cate- 

 gory equals the integral of L m times its density 

 function divided by the integral of the density 

 function (results also shown in Table 3). 



Equation (19) was derived by substituting Equa- 

 tion (11) into Equation (4) and differentiating 

 with respect to t. For the values of k, c 1; c 2 , and L m 

 reported by Lux (1969b) and Lux and Nichy 

 (1969), dwldt is 143, 172, 182, and 163 g/yr for a 

 length of 250, 300, 350, and 400 mm, respectively. 

 Most of the fish in the catch are within this range 

 of length. Therefore, only a minor proportion of 

 the estimated range in annual growth per fish can 

 be accounted for by changes in size composition of 

 the stock. Thus, within the constraints of the 

 model described here (c 1; c 2 , L m do not vary with 

 time), k must be nearly proportional to the rate 

 of weight gain. 



During the period 1944-65 there were 4 yr in 

 which the estimated average annual air tempera- 

 ture was greater than 11°C and 7 yr in which it 

 was less than 10°C. For the four warmer years, 

 temperature averaged 11.2°C and growth per fish 

 averaged 88 g. For the seven colder years, tem- 

 perature averaged 9.5°C and growth 222 g. 

 Assuming k proportional to annual average 

 weight gain per year, c 14 was estimated as 

 -0.466 by solving: 



{1 + c 14 (lL2 - f)}l 

 {l + c 14 (9.5 - T)} = 88/222. 



TABLE 3. — Range and mean for L m , the maximum length pa- 

 rameter of the von Bertalanffy growth function, representing 

 each of the size categories of the yellowtail flounder model. 



Size Range of L m Mean of L m 

 category (mm) (mm) 



Size Range of Lm Mean of L m 

 category (mm) (mm) 



0.0-425.1 

 425.1-^45.4 

 445.4-467.2 

 467.2-4928 



410.9 

 436 3 

 457.0 

 480 



492.8-514.6 

 514.6-534.9 

 534.9-x 



503.0 

 523.6 

 549.1 



Lux and Nichy (1969) estimated the growth rate 

 coefficient (k of the von Bertalanffy growth func- 

 tion) for yellowtail flounder older than 2 yr of age 

 as 0.335. For the period during which Lux and 

 Nichy collected their data, the average annual 

 temperature at Block Island was about 9.8°C. This 

 temperature results in a growth-temperature 

 factor (T g ) of 1.175. Lux and Nichy's estimate 

 was divided by T g resulting in an estimate of k 2 = 

 0.285. An estimate of k x (=0.56) was determined 

 using the model so that fish would grow to a 

 realistic length by age 2. 



The seasonal nature of yellowtail flounder 

 growth was exhibited when the average lengths 

 of age-groups were determined quarterly (Lux 

 and Nichy 1969). In general, the mean size of 

 an age-group changes little from the first to the 

 second quarter. Thus, most growth apparently 

 occurs during the second half of the year. In order 



471 



