360 



Fishery Bulletin 102(2) 



1 ,000,000 2,000,000 



2.000,000 



Stock size (no. offish) 



Mathematical model • Simulation model 



Figure 7 



Stock-recruitment relationships determined from the mathematical models (Eqs. 

 4, 5, and 10, disaggregated initial conditions) compared with simulation models. 

 (A and Bl Recruitment at r=l with mortality encounters proportional to school size 

 (m dk ) at a=5xl0~" and ,n = l; A is recruitment and B is the mean school size. iC and 

 D) Recruitment at t=10~ 5 with mortality encounters proportional to school density 

 im /v i at o=0.2 and ii=l; C is recruitment and D is mean school size. 



occur due to secondary effects of mortality encounters, 

 as well as other factors such as starvation. For example, 

 Sogard and Olla (1997) have shown predation-risk and 

 hunger to be related to group cohesion. 



The formation of a giant cluster (a single school en- 

 compassing all the fish) is an important feature of the 

 attachment process. The simulations showed that with 

 preferential attachment the recruitment process passes 

 through a phase where the size distribution is scale free, 

 then a critical point is reached where a giant cluster is 

 being formed, i.e. a single school begins to attract all the 

 fish. Research on complex networks has shown the condi- 

 tions for formation of the giant cluster (Aiello et al., 2000; 

 Albert and Barabasi, 2002). This should be investigated 

 for the school aggregation models because it is likely that 

 the mortality models used in the present study would no 

 longer be appropriate once the giant cluster is formed. In- 

 deed in some fish stocks, schools may aggregate into giant 

 clusters on a local scale and then aggregation may stop for 



reasons such as juveniles entering a benthic phase. The 

 resulting distribution of school sizes may be the cluster 

 distribution across benthic habitats. Spatial limitations 

 of aggregation are an important feature of individually 

 based models (Pascual and Levin, 1999). Again, this may 

 be an important area for research. 



What is the benefit of preferential attachment? If mor- 

 tality encounters are proportional to school density, then 

 recruitment survival rates are improved when there are 

 fewer schools for a given number of fish, i.e. when prefer- 

 ential attachment is employed rather than random attach- 

 ment (Fig. 8). Perhaps, preferential attachment strategies 

 are a useful evolutionary hedge against uncertainty in 

 the nature of the mortality dynamics. Conversely, when 

 mortality encounters are proportional to school size, then 

 better survival is achieved when schools are smaller, i.e. 

 with random attachment (Fig. 8). If mortality by preda- 

 tors is related to larger schools, or if attainment of prey is 

 inversely related to larger schools, then more solitary life 



