FISHERY BULLETIN: VOL. 73, NO. 4 



its ability to accommodate multivariable system 

 models such as multispecies models; models 

 describing economic as well as biological 

 phenomena; and detailed population models in- 

 corporating size, age, temperature, food supply, 

 etc. Once a three-variable example is presented, 

 generalization of the technique to models with 

 more than three variables is straightforward. The 

 following model of a single species population was 

 developed by Timin and Collier (1971) and contains 

 three state variables: A'', the population density; 

 W, the mean biomass per organism; and E, the 

 food density. The model is given in dimensionless 

 form, and thus the values of the model variables 

 are relative to reference values. The system's 

 dynamics are described by the following equa- 

 tions: 



^ = {h-d)N - f 

 dt 



dE 



dt 



— = a -qN-dE 



(9) 



(10) 



Static optimization can be used to determine the 

 maximum equilibrium yield condition. For./gq = 

 0.005, the equilibrium values are: A'^^eq = 0.16, fi'eq = 

 20.3, W^ = Wh^ = 0.8. Following the procedures 

 outlined in the Appendix (Equations (A-2) through 

 (A-4)), Equations (9), (10), and (11) were linearized 

 around this equilibrium to obtain: 



0.03 -0.56 



= -5.73 -0.12 -0.81 



0.14 0.02 -2.02 



u (12) 



= gq - {W + c)b - fiW - 



dt 



N 



(11) 



where: x^ = N - N_ 



eq 



where: t = time measured in a dimensionless unit 

 equal to the time required for the or- 

 ganism to metabolize an amount of 

 food equal to its own dry weight 

 (usually between two and four weeks 

 for commercial fish species) 

 b,d= birth and death rates per individual 

 /= fishing rate 



g = the ratio of the quantity (energy in- 

 gested minus energy not assimilated, 

 minus energy expended to catch, in- 

 gest and assimilate the ingested food) 

 to the amount of energy ingested 



q= food ingestion rate per individual 



c= coeflficient of energy loss associated 

 with births 



/A = metabolic heat loss coefficient 



a = rate of food supply 



$= proportionality constant for the rate 

 of food leaving the system through 

 decay or flushing 

 Wf, = mean organism biomass of harvested 

 individuals. 



Functional forms and parameters given as typical 

 by Timin and Collier are: 



834 



eq 



The following performance index J describes our 

 desire to keep the system near the desired 

 equilibrium (Appendix, Equation (A-6)): 



CO 



■J= J (Qll.*'? + Q22■vl+Q33•»■i + ^"')^^ (13) 







Here the weighting matrix [Q] becomes: 



[Q] = 



and the matrix [R] becomes a scalar R. A sub- 

 stantial difference between the single-variable and 

 multivariable cases is that in the latter case we can 

 no longer easily determine the feedback gains by 

 specifying the desired values of the time cons- 

 tants. Instead, the gains are calculated by 



I 



