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Fishery Bulletin 102(2) 



(the number of nodes is increasing) and if the new nodes 

 were linked to existing nodes by preferential attachment. 

 Preferential attachment (or the "rich-get-richer" phenom- 

 enon) occurs when a new node is linked to an existing node 

 with a probability proportional to the number of links al- 

 ready attached to that node. More formally, the Barabasi 

 and Albert model is created by adding a new node at each 

 time step and by randomly linking it to m existing nodes 

 proportional to the number of links at the existing nodes. 

 After a large number of time steps, the probability of a node 

 having k links (the degree distribution) scales as a power- 

 law P(k)~k '-', where y = 3, independent of m. The Barabasi 

 and Albert result differs from the classic random network 

 model of Erdos and Renyi ( 1960 ) in which nodes are linked 

 randomly to existing nodes, leading to P(k)~exp(-Xk ). Sub- 

 sequent research has expanded on the Barabasi and Albert 

 model to examine aging, removal and rewiring of nodes, 

 removal of links, fitness and attractiveness of nodes, and 

 local modifications to preferential attachment (see Albert 

 and Barabasi, 2002, for a review of these developments) 



The generic occurrence of scale-free school-size distribu- 

 tions suggest that modeling of aggregation and mortality 

 processes using the analogy of random networks may be 

 fruitful. The approach may provide insight into recruit- 

 ment dynamics and a theoretical basis for further inves- 

 tigation. This study attempts to do that and is organized 

 in the following manner. First, a simulation model of the 

 recruitment process is developed in which aggregation and 

 mortality occur based upon some simple rules of prefer- 

 ential attachment and random attachment. Attachment 

 rules are presented as metaphors for more complex behav- 

 iors. Next, analytical models are created that mimic the 

 simulations, and results of the simulations and analytical 

 models are compared. Finally, the implications for existing 

 stock-recruitment models and investigation of recruitment 

 processes are discussed. 



ity is the removal of nodes (fish) and, if there are no more 

 fish in the school, then the removal of schools. A simulation 

 model with simple rules of mortality and aggregation was 

 created to examine the dynamics of these processes. 



The simulation model followed individual fish and 

 schools through a recruitment period, i.e., the passage 

 of time until an arbitrary time of recruitment. During a 

 recruitment period fish and schools undergo encounters 

 of mortality and aggregation. Starting at time r=0 with S 

 fish, iV,_ schools and k t t=0 fish in school i U=l,2, . . . , N ), 

 simulations were conducted by randomly generating an 

 encounter event (mortality or aggregation!. If the event 

 was a mortality, then a school was randomly selected by 

 using the appropriate mortality rate model (;?*. discussed 

 below). If the size of that school was greater than one, 

 then that size was reduced by one. If the school size was 

 equal to one, then the number of schools was reduced by 

 one and this school was eliminated from the list. 



If the event was an aggregation, then two distinct 

 schools were randomly selected by using the appropriate 

 aggregation rate model (w, also discussed below). The two 

 schools were combined, leaving one school whose size was 

 the sum of the two original ones and one fewer total num- 

 ber of schools. The probability of an event being a mortality 

 was ml(m+w) and the converse probability of an aggrega- 

 tion was l-m/(m+w). Time increments of each event were 

 computed using At=m~ l for mortality events and (mw)~ l 

 for aggregation events. Results at time t were collated into 

 the number offish surviving to time / (denoted by R, >, the 

 number of schools, N r the school size distribution, P t (k), 

 and the average school size, k r Note thati?, =N t k r Simula- 

 tions were run until there were no fish left. 



Encounter rates The encounter rates, m and w, were 

 based upon random movements in statistical mechanics 

 (Tolman, 1979) in which the encounter rate (£/) of objects 

 of type i with objects of type 7 is described by 



Methods 



U=(G, + GO Dp, {v? + v/)" 3 , 



(1) 



Simulation of individuals in ecology and population 

 dynamics (individual-based models) have become increas- 

 ingly popular (McCauley et al., 1993). However, it is often 

 difficult to understand the dynamics of large individu- 

 ally based models (Pascual and Levin, 1999). Thus, it is 

 important to obtain models that describe dynamics of 

 groups that incorporate individual behavior (Flierl et 

 al., 1999). The models that are developed here include an 

 individually based model (simulation model) and an ana- 

 lytical model that describes "mean-field" dynamics of the 

 individuals behavior. 



Simulation model 



The recruiting fish of a year class may be modeled as a 

 network offish in which a fish "links" to other fish to form 

 schools. (Note that in this context it is assumed that a 

 "school" includes aggregations consisting of a single fish). 

 Thus, the process of aggregation is a process of adding 

 links to nodes (aggregation of schools). Similarly, mortal- 



where G, = the size of the detection space at which object 

 detects object type./'; 

 Z), = the density of objects of type i; and 

 u, = the velocity (in three-dimensional space) at 

 which object i moves in the environment. 



For these simulations the G parameters were scaled to 

 one and the velocity parameters (v's) were collapsed into 

 two encounter rates: ,11 for mortality encounters (scaled to 

 unity) and a for aggregation encounters. 



Mortality rate In the simulations, mortality of fish is 

 perpetrated by mortality agents. If the mortality agents 

 randomly encounter schools of fish, then the probability 

 of a successful mortality (the removal of a fish from the 

 system) is proportional to the school size k. Under these 

 conditions Equation 1 reduces to Equation 2 with 



G=G, 



■UV? +V 3 . )<={!■■ 



(2) 



m = 2^ENk, 



