364 



Fishery Bulletin 102(2) 



N, 



S-(S-k )e' 



k, =R t IN t = 



k^e** 



[S-(S-k f) )e- 2, "\' 



Mortality proportional to school size: m dk 

 Random aggregation: w N 



and Albert (1999). Dorogovtsev and Mendes (2000) and 

 Albert and Barabasi (2002). 



When the aggregation model is preferential attachment 

 (w ) (ignoring for the moment the nonstationarity of N 

 and P), then the partial differential of a school of size k lt 

 with respect to R t has been shown by Dorogovtsev and 

 Mendes (2000) to asymptotically be 



dk u im, = P,ik,,IR,), 



(AD 



where ft, is the net rate of decay per each mortality event, 

 i.e., 



R, 



l + 2pSi-^(S-Ml-e~ 2 °*] 

 a 



fit 



l-w pa lm. 



With specific-mortality models, fi, is 



(A2) 



N t = 



k,=R,/N,=- 



S-iS-k )e' 



S-iS-kpte 



-2at 



l + 2^t-^S-k )[l-e~ 

 a 



Random mortality encounters: m N 

 Random aggregation: w N 



V i„ \ S <J 



R l =S-£-]n\Z-(e""-l) + l\ 

 a [ko 



m dl : p, = l-ial ii)iN, -Dk, = l-ial /j)R, 

 m dN : p, = l-(a/ n)(N t -ljktl N t = l-ial n)k, 

 m dR : p, = l-ial f.t)iN, -DIN, =l-(a//i) 

 m dk : P, = l-(a/fi)(N t -l) = l-(a/p)P, Ik,, 



where the approximations on the right assume that the 

 number of schools is large. The first term of (A2) denotes 

 the removal of a fish proportional to school size for a mor- 

 tality event; the second term denotes aggregation events 

 proportional to school size. If ft, is independent of time 

 ift,=ft), then Dorogovtsev and Mendes (2000) showed that 

 under continuum conditions 



Pi k ) 



7 = 1 + 1/ /J. 



(A3) 



W,=- 



2ftf 



S-{S-ko)e 



k,=R,IN, 



Appendix 2 



Characteristics of school-size distribution under 

 preferential attachment 



Much of the recent literature on evolving complex net- 

 works has been directed at determining the degree distri- 

 bution, i.e., the probability P(k) of a node having k links 

 (Albert and Barabasi, 2002). When the network grows 

 or declines proportional to k or when links are rewired 

 to be proportional to k, then P(k> can be determined by 

 using continuum theory (Dorogovtsev and Mendes, 2000; 

 Albert and Barabasi, 2002) leading to scale-free degree 

 distributions. Therefore, when preferential attachment 

 and nonrandom mortality are used, then the model may be 

 couched as a scale-free network in the manner of Barabasi 



Equation A3 is equivalent to the results of Dorogovtsev 

 and Mendes (2000), Krapivsky et al. (2000), and Albert 

 and Barabasi (2002) and suggest that ft, may be a useful 

 approximation for determining the power-law tail of the 

 school-size distribution (Appendix Fig. 1). 



The simulation results showed the dynamics of P kt . 

 When the aggregated initial condition was imposed, at 

 the start of the simulations there were no schools with 

 only one fish in them (P,, = 0). Eventually, as the number 

 of schools and fish declined, P„ became positive. Finally, 

 as the distribution became scale-free, -i)P x ,l Hk became 

 negative and remained so throughout the remainder of the 

 simulation or until a single giant cluster was formed (Ap- 

 pendix Fig. 1). Conversely, if the initial conditions began 

 with schools being disaggregated, then dP u ldk began as 

 a negative number and remained so until either a giant 

 cluster formed or there were no more fish remaining. 



An approximation is suggested by the above results 

 for circumstances when the initial conditions are disag- 

 gregated and when there is preferential attachment: the 

 differential equation dP,. , Idt when k = 1 ( Eq. 6 ) is replaced 



by 



dP u ldt 



-irP y ,IN, + ;?Ml-P u )/JV,. 



(A4) 



