Powers: Recruitment as an evolving random process of aggregation and mortality 



355 



School size 



J3 



-r 1.000,000 



Figure 2 



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

 model and w N as the aggregation model. This simulation started with disag- 

 gregated initial conditions (N = S), where S=10 6 . The aggregation parameter 

 was a=10~ 6 . The top panel shows school-size distributions (frequency in log) at 

 selected times ((). The lower panel gives the mean school size (kbar) and school 

 abundance (JV) versus time. 



them. At this point the reduction in the number of schools 

 is accelerated because of the mortality of fish that are in 

 "schools" in which they are the only member, and because 

 of the loss of schools attributed to aggregation. 



Model comparisons 



Numerical integration of Equations 3-5 matched the sim- 

 ulation results (Fig. 6, when P 1( =0), indicating that the 

 mathematical model describes the simulation dynamics. 

 The numerical techniques are sufficiently efficient to be 

 used in a curve-fitting context. Evaluations of the approxi- 

 mation (Appendix 2) indicate that the approximation may 

 be useful for predictions of recruitment, when compared 



with the simulations. However, the components of recruit- 

 ment, k t and N r were biased (Fig. 7). Further research is 

 needed to develop estimates of P 1 1 and, more generally, 

 P(k) under other models and initial conditions. 



Recruitment was compared between mortality models 

 and aggregation models (Fig. 8). If the mortality model 

 was either m dl or m dR , then the mortality rate was not af- 

 fected by the school-size distribution: random attachment 

 and preferential attachment perform equally as well in 

 terms of survival at a given time. But if mortality encoun- 

 ters proportional to school density (.m dN ) were imposed, 

 then there were better survival rates with preferential at- 

 tachment than with random attachment (Fig. 8, A and B). 

 Conversely, mortality encounters proportional to school 



