FISHERY BULLETIN: VOL. 84, NO. 3 



ing segments of the various ecosystem populations. 

 Population variables will now be uniquely assigned 

 for each box and expressed in density units, such 

 as numbers or kilograms per hectare. Spatial re- 

 distributions are assumed to occur during a given 

 time period via migration, net drift, or turbulent 

 dispersion. The resultant redistribution process is 

 expressed by defining population transfers between 

 boxes. 



Spatial redistribution is applied to the surviving 

 populations determined from Equation (4) and is 

 described by 



PUt + 1,2) 



M 



= I 



m= 1 



gr k Pf(t + 1,1) 



(7) 



where 



P\(t + 1,2) = density of surviving population i 

 in compartment k after spatial 

 redistributions 



Pf(t + 1,1) = density of surviving population i 

 in compartment m before spatial 

 redistributions 



M = total number of spatial compart- 



ments 

 = population i transport coefficient 

 for the exchange from compart- 

 ment m to compartment k. 



9i 



ink 



The g coefficient defines the population fraction in- 

 volved in the exchange with an adjustment to ac- 

 count for the difference in area or volume between 

 compartments. If no transit occurs between com- 

 partments, the value of the respective coefficient 

 is zero. 



Birth and Aging Processes 



The larvae and juvenile age classes of fish popula- 

 tions have markedly different survival rates and 

 behavioral characteristics than do adult populations. 

 These differences have potentially important first- 

 order ecological consequences and are, therefore, 

 of concern in the present model development. 



A modified version of the Leslie matrix as pre- 

 sented by Lefkovitch (1965) is adopted here. Popula- 

 tions are grouped by stages which can be of unequal 

 duration with no restriction to single year classes. 

 The birth and aging matrix transform for N such 

 stages is given by 



n («+i,s) 



P&0 + 1.S) 

 Pfc(*+l f 8) 



n^+u) 



Oil Ji2 j%  JiN 



a n b a . 



a l2 b l3 . 







P!i (t + 1,2) 

 Pfc(t+1,2) 



P% (t + 1,2) 



PL (t+1,2) 



(8) 



Pl(t + 



where 



P|- (t + 1,3) = density of population %, age class 

 j after accounting for births and 

 aging in compartment k 

 1,2) = density of population i, age class 

 j before accounting for births and 

 aging, but after accounting for 

 spatial redistributions to compart- 

 ment k 



a {j = fraction of population i, age class 



j advancing to age class j + 1 



6 y = fraction of population i, age class 



j remaining in age class j 



flj = fecundity function for population 



i, age class j in compartment k. 



The coefficients a and 6 are functions of the size 

 of the time step and the division of ages within the 

 population. Equation (8) also implies a fixed age 

 distribution within an age class, such as a uniform 

 distribution. 



The fecundity term,/, is a function of the popula- 

 tion age class, as well as being time and space depen- 

 dent. Explicit population crowding effects are 

 neglected here because they would be comingled 

 with the other density-dependent terms in Equation 

 (4). 



Composite Ecosystem Dynamics 

 Equations 



The above equations are combined and expressed 

 by the general ecosystem dynamics model given 

 below. The final surviving, redistributed, and aged 

 population vector at the end of the time period has 

 been redefined as P(t +1) = P(t +1,3). 



538 



