349 



Abstract— The dynamics of the sur- 

 vival of recruiting fish are analyzed as 

 evolving random processes of aggrega- 

 tion and mortality. The analyses draw 

 on recent advances in the physics of 

 complex networks and, in particular, 

 the scale-free degree distribution aris- 

 ing from growing random networks 

 with preferential attachment of links 

 to nodes. In this study simulations 

 were conducted in which recruiting 

 fish 1) were subjected to mortality by 

 using alternative mortality encounter 

 models and 2) aggregated according 

 to random encounters (two schools 

 randomly encountering one another 

 join into a single school ) or preferential 

 attachment (the probability of a suc- 

 cessful aggregation of two schools is 

 proportional to the school sizes). The 

 simulations started from either a "dis- 

 aggregated" (all schools comprised a 

 single fish) or an aggregated initial con- 

 dition. Results showed the transition of 

 the school-size distribution with pref- 

 erential attachment evolying toward 

 a scale-free school size distribution, 

 whereas random attachment evolved 

 toward an exponential distribution. 

 Preferential attachment strategies 

 performed better than random attach- 

 ment strategies in terms of recruit- 

 ment survival at time when mortal- 

 ity encounters were weighted toward 

 schools rather than to individual fish. 

 Mathematical models were developed 

 whose solutions (either analytic or 

 numerical) mimicked the simulation 

 results. The resulting models included 

 both Beverton-Holt and Ricker-like 

 recruitment, which predict recruitment 

 as a function of initial mean school size 

 as well as initial stock size. Results 

 suggest that school-size distributions 

 during recruitment may provide infor- 

 mation on recruitment processes. The 

 models also provide a template for 

 expanding both theoretical and empiri- 

 cal recruitment research. 



Recruitment as an evolving random process 

 of aggregation and mortality 



Joseph E. Powers 



Southeast Fisheries Science Center 



National Marine Fisheries Service, NOAA 



75 Virginia Beach Drive 



Miami, FL 33149 



E-mail address: loseph powers@noaa.gov 



Manuscript approved for publication 

 10 December 2003 by Scientific Editor. 



Manuscript received 20 January 2004 

 at NMFS Scientific Publications Office. 



Fish. Bull. 102:349-365 (2004). 



The study of recruitment processes 

 has traditionally addressed mortal- 

 ity (predation and starvation) and 

 the effects of patchiness on mortality 

 (Vlymen, 1977; Beyer and Laurence, 

 1980; Hunter, 1984; Rothschild, 1986); 

 hence the importance of aggregation 

 and mortality in recruitment processes 

 of marine fish populations has long 

 been noted. Ecological processes of 

 starvation, growth, and predation of 

 larval fish, coupled with oceanographic 

 factors show the inherent variability in 

 these processes (Koslow, 1992; Mertz 

 and Myers, 1994, 1995; Pepin, 1991; 

 Rickman et al., 2000; Comyns et al., 

 2003). In particular Rickman et al. 

 (2000) have indicated the importance 

 of the magnitude of fecundity in the 

 variability of egg and larval mortal- 

 ity. Indeed, Koslow ( 1992 ) argued that 

 fecundity and the associated variability 

 in egg and larval mortality will limit 

 our ability to determine stock-recruit- 

 ment relationships. 



Stock-recruitment models have gen- 

 erally emphasized the static results 

 of recruitment processes rather than 

 the dynamics themselves. Indeed, al- 

 though the classic stock-recruitment 

 models such as the Beverton-Holt and 

 Ricker have been related to microscale 

 processes (Beverton and Holt, 1957; 

 Ricker, 1958; Paulik, 1973; Harris, 

 1975), the dynamics at those scales 

 were not explored, primarily because 

 there was not a theoretical basis for do- 

 ing so ( Rothschild, 1986 ). Nevertheless, 

 there is a need to develop a theoretical 

 understanding of small-scale inter- 

 action processes during recruitment, 

 particularly as they relate to group 

 formation. 



Group-formation ( aggregation of fish 

 into schools), schooling (shoaling) be- 

 havior, and the evolutionary motivations 

 for formation of schools continue to be 

 important research topics (Pitcher and 

 Parrish, 1993; Landa, 1998). Schooling 

 behavior has variously been attributed 

 to predator-avoidance, predator-attack 

 dilution, and hydrodynamic and forag- 

 ing advantages (see Pitcher and Par- 

 rish, 1993, for a review). One of the first 

 models for school formation was that of 

 Anderson ( 19S1 ) in which he empirically 

 observed skewed distributions in which 

 small schools were more prevalent than 

 larger ones. Subsequently, Bonabeau 

 and Dagorn (1995), Gueron and Levin 

 (1995), Niwa (1998), and Bonabeau et 

 al. (1999), developed group-size distri- 

 bution models. In particular, Bonabeau 

 et al. (1999) in comparing group-size 

 distributions of tunas, sardinella, and 

 buffalo suggested that power-law dis- 

 tributions may be quite generic. Niwa 

 (1998) noted that Anderson's original 

 model allowed for power-law distribu- 

 tions. Power-laws are termed scale-free 

 because they exhibit no intrinsic scale. 

 Similarly, existence of a power-law is 

 often referred to as "scaling." 



Recently, power-law distributions 

 have arisen in studies of the physics 

 of small-world and evolving networks 

 (for example the world wide web, ac- 

 tor collaborations, scientific citations 

 [Barabasi and Albert, 1999], biological 

 cellular networks [Fell and Wagner, 

 2000], and ecosystem structure [Sole 

 and Montoya, 2001]). In particular, 

 Barabasi and Albert (1999) demon- 

 strated that a randomly evolving net- 

 work would result in a scale-free degree 

 distribution if the network is growing 



