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



353 



and the estimates may provide a theoretical framework for 

 empirical research on recruitment processes. 



Noting that R,=N t k t , the recruitment dynamics depicted 

 in the simulations may be modeled by using Equations 

 3-6 in which recruitment is dependent on the particular 

 mortality and aggregation models that are chosen (m and 

 w; Table 1): 



dR,l = -m = d(N,k, "/dt=(dk l Idf'N, +(dN, I dt"k, (3) 



dNJdt 



-2aN t {N t -l)\ l 



dN, I dt = -w - m x P u 



dk,/dt = mft, [( k, - 1)1 ( N, - 1) 

 -< m - m^P v )l N,+ wk t /<JV,_i ) 



(4) 



(5) 



dP kl Idt = ( m k+l P k+u - m k P Kt )IN t - w k P, !t I N, k > 1 



i-l 



+ wY J P,P k -,IN l (6) 



= m 2 P u IN t - mi P u a-P u >/< AT, -1) k = l 



-w 1 P u IN l , 



where P h , = the proportion of schools with k fish in them 

 at time t . 



Also, m k and w k denote encounter rates appropriate to 

 schools of size k, whereas unsubscripted m and w denote 

 mean field dynamics and, thus, the k t t 's are replaced by 

 k t 's (see Table 1). 



The first term in Equation 4 denotes the reduction in 

 number of schools due to aggregation events; the second 

 term denotes a reduction due to mortality events on 

 schools with one fish in them. Similarly, the first term in 

 the mean school-size equation ( Eq. 5 ) describes the change 

 in mean school size due to mortality events on schools 

 of size equal to one; the second term is due to mortality 

 events on schools of size greater than one; and the third 

 term is due to aggregation events. Finally, the first term 

 in Equation 6 describes the change in probability of school 

 size k due to mortality; the second term describes loss due 

 to aggregation; and the third describes gain due to aggre- 

 gation. Of particular importance is P l t : when P x , is zero, 

 the loss of schools occurs only due to aggregation. When 

 Pj , is positive, then loss of schools is accelerated due to 

 mortality (Eq. 4). 



The goal is to obtain solutions to Equations 3-6 as 

 functions of a, ju, and the initial conditions. If one can be 

 assured that there will not be a school composed of one 

 fish during a particular recruitment period (Pj ,=0), then 

 Equation 6 is eliminated, the Pj ( terms drop out of Equa- 

 tions 4 and 5, and a numerical or analytical solution to 

 the differential equations can be obtained, which is com- 

 putationally feasible for use in fitting to stock-recruitment 

 data. For example, when there is preferential aggregation 

 (w ) and mortality agents are proportional to schools 

 (m dN ), the equations reduce to 



dk t ldt-- 



l dN 



I N, + w k,( Nj_-1)= -2/jNtk, + 2aN t k t 3 



Analytical solutions were obtained for some of the mor- 

 tality and aggregation models when P 1( =0 throughout 

 the recruitment process (Appendix 1). In particular for 



m„ 



!R andw pa- 



R t = SHl+2iitS) (7) 



N t =N Q +(a/n"[S-S/e%* %itS"] (8) 



^=^/[l+2^S + 2ctfS^]. 



(9) 



which is the Beverton-Holt stock-recruitment model 

 expanded to include equations for the number of schools 

 and the mean school size. Interestingly, Equation 9 indi- 

 cates that monitoring the school-size distribution two or 

 more times during a recruitment procession would yield 

 estimates of the stock-recruitment parameters without 

 having direct measures of the number of surviving fish. 

 Equation 7 predicts recruitment by using one parameter. 

 j.1. , the rate of mortality encounters during the recruitment 

 period. However, spawning stock biomass is often used as a 

 surrogate for the number of initial stock, S. Thus, another 

 parameter is needed to convert spawning stock biomass 

 to S in Equation 7. In that case the recruitment model 

 becomes R t = aS/(l+2utaS), where a is another parameter 

 related to fecundity. The additional parameter will be 

 needed for all the models discussed here, if spawning stock 

 biomass is the measure of initial stock. 



The assumption that P lt =0 for all t of a recruitment 

 period may not be justified in all situations. An approxi- 

 mation was developed (Appendix 2) to be applied when 

 the initial conditions are disaggregated and when there 

 is preferential attachment. In this circumstance, the dif- 

 ferential equation (Eq. 6) when k = l is replaced by 



dP u /dt = -wP u /N, + m ( 1 - P u )IN t . 



(10) 



Results 



Simulations 



Several hundred simulations were conducted under vari- 

 ous initial stock sizes (S), aggregation parameters (o), ini- 

 tial aggregation conditions, and mortality and aggregation 

 models (m and w). An example set of results are presented 

 in Figures 1-5 (a key to figures is in Table 1). 



A typical example of the evolution of the school-size 

 distribution is given in Figure 1 for the disaggregated 

 initial condition, a=10" 6 , S=10 6 , mortality model m di and 

 aggregation model w . In this example both the mortality 

 and aggregation models exhibit preferential attachment, 

 and the school-size distribution approaches scale-free be- 

 havior P(k)~k->, although y evolves over time. Eventually, 

 a so-called "giant cluster" is formed by the aggregation 

 process, in which all the fish attach to one school. This has 



