Barry and Tegner: A predictive model for size-frequency distribution 



15 



VON BERTALANFFY GROWTH 



AGE (t) 



Figure 1 



Brody-Bertalanffy growth for various values of the Brody growth 

 coefficient K. Values of K are listed adjacent to each curve. 



the maximum size, 6 is a scaling factor to account for 

 a size at recruitment larger than (for a recruitment 

 size of 0, 6 = 1), and K is the Brody growth coefficient 

 (Ricker 1975) which constrains the shape of the func- 

 tion. Higher values of A' result in a more rapid approach 

 to asymptotic size (Fig. 1). If 6 is considered to be unity 

 and age (t) rather than time is used, the equation 

 simplifies to 



S^a-hei^^). 



(5) 



Assuming these functions adequately describe the 

 mortality and growth schedules of the population, we 

 can combine equations (3) and (5) to derive an expres- 

 sion for the size-frequency distribution of the popula- 

 tion. By definition, the number of individuals alive in 

 an age interval tj to to in equation (3) is equal to the 

 number in the corresponding size interval Sj to S._. , as 

 determined by equation (5). 



Ns dr = N, dS, 



and rearranging, 



"-"■i- 



(6) 



Because we assume these relationships to be strictly 

 deterministic, we can solve equation (5) for t 



K 



In 1 





(7) 



Next, we form the first derivative of equation (7) with 

 respect to t, yielding 



d^ 



dS, 



1 



KS^ 1 





(8) 



Combining equations (6), (3), (7), and (8) yields an ex- 

 pression for number as a function of size. 



Ns = Noe 



z s. 



KS^ 1- 



S. 



which simplifies to: 



N. = 





-I: 



(i-O 



(9) 



This size distribution function (9) describes popula- 

 tion abundance as a function of size, rather than age, 

 for any conditions of constant mortality (Z) and growth 

 coefficient (A'). By evaluating its first derivative at 

 dNIdS = 0, we can identify conditions necessary for 

 the existence of a zero slope (modes or troughs) in the 

 size distribution. This derivative is 



dN 

 dS 



Nu 

 AS„ 



1 1 



S..P-1 



S. 



■i.^(-f 



g-) 



(10) 



The conditions where dN/ds = are: 



TV,, = : trivial 



S, = S^ : trivial 



S = oo : trivial 



Z = K : growth is balanced by mortality. 



The only non-trivial condition where the size-fre- 

 c}uency function has a slope of zero is when Z = K. In 

 this case all size classes are equally abundant, since the 

 solution to equation (9) indicates that when Z = K. Ng 

 is constant and independent of 5^. Therefore, there 

 are no conditions of growth and mortality that are 

 capable of producing a bimodal distribution when using 

 these simplifying assumptions. Thus, the hypothesis 

 that bimodality arises from rapid growth to adult size 



