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



361 



A Mortality encounters proportional to school 

 r 



density; aggregated initial conditons 



Mortality encounters proportional to school 



C Mortality encounters proportional to school 

 size; aggregated initial conditons 



o if 







D Mortality encounters proportional to school 

 size; disaggregated initial conditons 



2,000.000 



Stock size (no. of fish) 



Random attachment 



Preferential attachment 



Figure 8 



Stock-recruitment results with preferential attachment comparer] with random attach- 

 ment (w versus w N ). (A) Recruitment at r=l with mortality encounters proportional to 

 school density im dN ) at a=l(H' and ,«=5xl0 4 with an aggregated initial condition; (Bl 

 recruitment at r=10~ 5 with mortality encounters proportional to school density lm dN ) at 

 a=0.2 and ji=1 with a disaggregated initial condition; (C) recruitment at t=l with mortal- 

 ity encounters proportional to school size (.m dk ) at a=10~ 9 and jt<=5xl0~ 41 with an aggre- 

 gated initial condition; (Dl recruitment at t=l with mortality encounters proportional to 

 school size l»! iM .) at a=5x 10~ 7 and ti=l with a disaggregated initial condition. 



history strategies may evolve. Perhaps, the random ag- 

 gregation model would be most effective for solitary preda- 

 tory fish when their mortality is imposed by a m dk -type 

 model. For fish, this may be more likely to occur at later 

 life stages than at recruitment. If mortality encounters 

 are proportional to fish (m rlR ), then results are intermedi- 

 ate and preferential attachment and random attachment 

 perform equally as well. 



The density-dependent mortality models implicitly in- 

 corporate a predator-prey interaction. Alternative preda- 

 tor-prey interactions examined were those in which preda- 

 tor density was proportional to fish, to schools, or to the 

 number offish within a school (school size) throughout a 

 recruitment period. In reality mortality is perpetrated by 

 a variety of agents at many different scales. Some agents 

 act at the scale of the population (Nk ), some at the scale of 



schools (N ), some at the scale of mean school size (k ), and 

 some at the scale of a local school (£■). The mixture of preda- 

 tory agents and their densities can cause various kinds of 

 dynamics including oscillatory, chaotic, and stable behav- 

 ior (Wilson 1996, Pascual and Levin 1999). Therefore, it is 

 unlikely that the models in this study, in which predator- 

 prey ratios are constant, would be predictive of anything 

 other than average behavior during recruitment. However, 

 the analytical approach allows changes in the scale of 

 predator-prey interaction over time. We can model this as 

 m l =2uN t a k t b , where a and b are dynamic (time-dependent) 

 and, perhaps, correlated. Although we may wish to use 

 the Beverton-Holt model (a=b=2) or the Ricker-like model 

 (a=2, b=l) as a representation of average dynamics, it 

 remains that recruitment variability will be influenced by 

 the dynamics of the exponents, a and b. Numerical evalua- 



