354 



Fishery Bulletin 102(2) 



1 ,000.000 



o f = 0.1 1 



+ f=0.40 

 » f=0.95 



A O 



V 



\ 



sw 



l um i n al ii h i 



School size 



-r 1,000,000 



-- 5,000,00 



( 



Figure 1 



Simulated dynamics of school-size distributions with m dl as the mortality model 

 and w as the aggregation model. This simulation started with disaggregated 

 initial conditions (JV n = Sl, where S=10 6 . The aggregation parameter was oc=10 -6 . 

 The top panel shows school-size distributions (in log-log scale) at selected times 

 (/). The lower panel gives the mean school size (kbar) and school abundance (AM 

 versus time. 



been shown to be an analog of Bose-Einstein condensation 

 (Bianconi and Barabasi, 2001; Dorogovtsev and Mendes, 

 2002) and gelation (Krapivsky et al., 2000). Greater mix- 

 ing rates Cot's) and larger densities (N's) accelerate the 

 aggregation process and the formation of the giant cluster. 

 The average size, k, increases over time from the disag- 

 gregated initial condition until a giant cluster is formed. 

 The number of schools declines over time because of both 

 aggregation and the mortality of fish in schools that only 

 have one fish in them. 



When there is random aggregation beginning from a 

 disaggregated initial condition (a=10 -6 , S=10 K , m di , w N ; 



Fig. 2), the school-size distribution exhibits exponential 

 behavior P(k)~exp(-/Jt), with A evolving over time. This 

 is equivalent to the Erdos and Renyi (1960) results for 

 random graphs. A comparison of Figure 2 with Figure 1 

 shows the difference between preferential attachment and 

 random attachment, i.e., the difference between scale-free 

 and exponential models. 



Aggregated initial conditions (Figs. 3-5) result in a 

 transition from the initial distribution to either scale- 

 free or exponential distribution. During the transition, 

 the size of the smallest school gradually becomes smaller 

 until there is a finite probability of schools with one fish in 



