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



351 



where E = the density of mortality agents; and 



k = the encounter rate of fish with mortality 

 agents. 



Note that on average Equation 2 reduces to m = 2iiE l N l k l = 

 2iiE t R t = -dRIdt. Hence, if the density of mortality agents 

 is constant throughout the recruitment period, then mor- 

 tality is density independent and mortality is proportional 

 to abundance. An alternative interpretation of Equation 2 

 is that the mortality agents randomly encounter fish and 

 that all encounters result in a successful mortality. The 

 mortality model (Eq. 2) will be referred to as m dl (for den- 

 sity-independent). It is not expected that m di is the most 

 realistic, but rather it provides a basis for comparison. 



A second mortality alternative is where mortality agents 

 randomly encounter schools, whereupon they always per- 

 petrate a successful mortality: m N = 2iiE t N t . This model, 

 like m di , assumes that the density of mortality agents are 

 constant throughout the recruitment period. 



For purposes of simulation, the density of mortality 

 agents at the onset of the recruitment process was speci- 

 fied to be unity (£ =1). For the two mortality models, m di 

 and m N , this meant that £=1 throughout a simulation. 



More realistic density-dependent mortality models are 

 immediately suggested. The first is a density-dependent 

 model in which the ratio of mortality-agent density to the 

 number of schools remains constant throughout the re- 

 cruitment period, i.e., E t IN t remains constant throughout 

 the recruitment period. This leads to m dN = 2f.iN z k, where 

 E t IN t was set equal to one. In this model the ratio of mor- 

 tality agents to schools is constant, agents and schools 

 randomly encounter one another, and the probability of a 

 successful mortality (given there is an encounter) is pro- 

 portional to the number offish that are in the school that 

 is encountered (mortality success is related preferentially 

 toward larger schools). 



A second density-dependent model is where the mortal- 

 ity agent density is proportional to the number offish (E t l 

 R t is a constant set equal to one, m dR =2iiR 2 =2iiN 2 k 2 ). In 

 this model the ratio of mortality agents to the number 

 of fish in the population is constant; agents and schools 

 randomly encounter one another; and the probability of a 

 successful mortality (given there is an encounter) is pro- 

 portional to the number offish that are in the school that 

 is encountered (mortality success related preferentially 

 toward larger schools). Another interpretation of this 

 model is that agents randomly encounter fish, at which 

 time the fish suffers mortality at a probability independent 

 of school-size characteristics. 



A third density-dependent model depicts mortality- 

 agent density proportional to school size (E t lk t is a con- 

 stant set equal to one, m dk =2\.i Nk' 1 ). In this model the 

 ratio of mortality agents to mean school size is constant, 

 agents and schools randomly encounter one another, and 

 the probability of a successful mortality (given that there 

 is an encounter) is proportional to the number offish that 

 are in the school that is encountered. Another interpreta- 

 tion of this model is that agent density is proportional to 

 the number of schools, agents encounter schools prefer- 

 entially according to school size, and the probability of a 



successful mortality (given that there is an encounter) is 

 proportional to the number of fish that are in the school 

 that is encountered. 



Subsequently it will be shown that the first density- 

 dependent model is related to a Ricker-like stock-recruit- 

 ment model and the second model is exactly equivalent 

 to a Beverton-Holt model. Definitions of the mortality 

 models are summarized in Table 1. Note that in the ini- 

 tial applications of these mortality models, it is assumed 

 that a mortality encounter results in mortality of one 

 fish. More detailed mortality models in which a number 

 of fish greater than one are removed by mortality may be 

 implemented in the future. Clearly, these would be more 

 biologically realistic in many instances. However, the 

 emphasis of this study is on the possible scaling behavior 

 of school-size distributions. Barabasi and Albert (1999) 

 showed that the scaling behavior of a growing random 

 network is independent of the number of randomly se- 

 lected links at each time step. With this analogy, simple 

 increases in mortality per encounter are not expected to 

 alter the scaling behavior of the school-size distributions. 

 Therefore, the one-fish-per-mortality-encounter approach 

 was used in these initial simulations. 



Aggregation rate 



Similar to mortality-rate encounters, aggregations were 

 investigated as 1) random attachment of two unique 

 schools (w N =2aNiN-D) and 2) preferential attachment of 

 two unique schools i and./' (w =2aN(N-\)k l k J ; [Table 1]). 

 Note, the trivial alternative where there was no attach- 

 ment, (a=0), results in equivalence between the mortal- 

 ity models m dN , and m dR \ whereas m dl becomes a simple 

 proportional mortality rate. Thus, results of models with 

 o=0 are uninteresting in the context of this study and are 

 not presented. 



Initial conditions Each simulation was conducted with 

 one of two alternative initial conditions. The first alterna- 

 tive was one of complete disaggregation in which simula- 

 tions were initiated with S fish, S schools, and one fish in 

 each school (N Q =S, k Q =l). The second alternative initial 

 condition was constructed from the population dynamics 

 of a typical fish population. The main assertion of this 

 alternative is that the eggs or larval fish produced by 

 one female during spawning constitutes one school at the 

 onset of the recruitment process. Thus, the fecundity per 

 female at age is a measure of initial school size and the 

 abundance of females at age is a measure of the frequency 

 of schools of that size. More precisely, the initial condition 

 was constructed for a population of females greater than 

 five years of age (age of maturity), where their fecundity 

 at age, F ' is proportional to weight at age determined 

 from a von Bertalanffy growth equation with parameters 

 K=0.2 and L^ = 10, and an allometric parameter of 3: 

 (F =1000 [(l-exp(-age( 0.2)))] 3 ). Abundance at age, A age , 

 was computed with an instantaneous mortality rate of 0.2: 

 [A =Zexp(-0.2(age-5))]. The scalarXwas obtained from 

 the approximate solution to S =2F age A age , where F and A 

 were integer values and S was the initial number of fish 



