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Mortality proportional to fish: m dR 

 Preferential aggregation: w 



Rt 



l + 2,uSt 



N t =£ + ^S 



k t =R,/N, 



ko fi{ 1 + 2[iSt 



l + 2/iSr + 2osSV 



Mortality proportional to fish: m dR 

 Random aggregation: w N 



Density-independent: m dl 

 Preferential aggregation: w pa 



R, = Se- 2 '" 



Ar t= J. _.£_£(!_ e -4/« 



^ 2/1 

 k,=R,/N,-- 



v- 2 "' 



1 _S*o «q_ c -4/i/) 



2 n 



Density-independent: m di 

 Random aggregation: w N 



N t = 



R, = Se" 2 '* 

 S 



k t =R t /N t = e~ 2f " (S-iS-^e-' 2 *) 



Appendix 1 



Analytical solutions to Equations 3-5 for selected mortal- 

 ity and aggregation models. Solutions assume that P lt = 

 for all t evaluated and that the number of schools is large. 

 No analytical solutions were found for (m dN ,w pa ), (m dN , 



W pa>' OT(m o 



• w pa>- 



Mortality proportional to schools: m dN 

 Random aggregation: w N 



R t =Se 



-2(1/ J feo_ 



|s-(S- v? 



-2a* 



