FISHERY BULLETIN: VOL. 75, NO. 3 



(7) 



Using Equations (5), (6), and (7); G, F, and D 

 are calculated as follows for fish of any length: 



G=q  f 

 F=q f 



Pi 



D =q  f  P,  (1 - P 2 ). 



(8) 



(9) 



(10) 



Since G, F, and D vary with L and f, they are 

 time dependent functions. 



Aging 



The aging process of yellowtail flounder is 

 simulated by advancing individuals of each age- 

 size compartment to the next higher age-group 

 within the same size category. 



Growth 



The mechanism used in the model to simulate 

 growth was based on the von Bertalanffy growth 

 function. The von Bertalanffy function can be 

 expressed in many forms, but the following is 

 most applicable to this study: 



*-» ~ J->m + U'O L m ) 



-kt 



(11) 



where L m is the maximum length obtained by the 

 fish of the population, L is the length of a fish at 

 the beginning of a time interval of duration t, k 

 is the growth rate coefficient that applies during 

 the interval, and L is the length obtained by the 

 end of the interval. The derivative of Equation 

 (11) is identical to the growth equation deduced 

 by von Bertalanffy (1938). 



A single value of L m is usually assumed for an 

 entire population. In the model, differences in the 

 mean length of size categories are maintained by 

 assigning a unique maximum value to L for each 

 size category (L ml , L m2 , . . ., L ml ). Fish are distrib- 

 uted among the size categories in the following 

 manner. AssumeL^ is a normally distributed ran- 

 dom variable with mean L m4 and standard devia- 

 tion s m . For G lt G 2) . . ., G 7 , the portion of the 

 population in each size category respectively (in 

 the absence of fishing), the range of values of L m 

 included in each size category can be determined 

 from a standard normal table. The mean value of 



L m for the jth size category (L„ y ) is obtained by 

 integrating the product of the normal density 

 function and the random variable L m over the 

 range of values of L m included in the size category 

 and then dividing the result by Gj. 



Taylor (1962) showed that k of the von Berta- 

 lanffy function was related to water temperature 

 for a number of species, and there is evidence 

 (which is discussed later in this paper) that this 

 is also true for the Southern New England 

 yellowtail flounder. The influence of temperature 

 on k is simulated by adjusting k by a multipli- 

 cative growth-temperature factor, T g , defined as 



T g = \+c 



l-i 



(T - T) 



(12) 



where T is an index of temperature and f is the 

 average value of the index over the total period 

 for which data are available. T is an exogenous 

 variable of the model. 



Different values of k (k x , k 2 ) were necessary to 

 describe the growth of yellowtail flounder less 

 than and greater than 2 yr old (Lux and Nichy 

 1969). Seasonal variations of growth were incor- 

 porated into the model by multiplicative quarterly 

 growth factors K x , K 2 , K s , K 4 (with an average 

 value of 1.0). The length of age-size compartment 

 i,j after an interval of time t is calculated accord- 

 ing to Equation (11) using the length of the com- 

 partment at the beginning of the interval L„ y , 

 and k as follows: 



where n indicates the quarter of the year 

 indicates age less than or greater than 2 j 



(13) 

 and a 



yr. 



Spawning 



Spawning occurs during May or at 0.4 of each 

 year. The fecundity-length function of the yellow- 

 tail flounder was assumed to be of the usual form, 



Fe(L) 



(14) 



where Fe is the egg production of a mature female 

 fish of length L. Not all fish mature at the same 

 age or length. Royce et al. (1959) found that 

 maturation was more closely associated with 

 length than age. A relationship of the following 

 form, expressing the probability of a fish of specific 

 length being mature (P 4 ) was assumed. 



468 



