PALM: FISHERY REGULATION VIA CONTROL THEORY 



with initial condition P(0) 

 is: 



0. Tlie solution for P 



P = 



aR + a '^R'^ + qhi-RQ/h- 



q^a^2b" 



Thus P and K are functions of the weighting fac- 

 tors R and Q, which must be specified. 



Note that this method yields three results: 1) 

 that the optimal control function for u is a linear 

 function of x (a linear feedback law); 2) the means 

 to calculate the feedback gain K, once R and Q are 

 specified; and 3) that A' is negative in this example 

 (we assume that a, h, and q are positive). The third 

 result indicates that the control law, Equation (5), 

 opportunely calls for an increase in fishing effort 

 when the population increases (.r > 0), and con- 

 servatively calls for a decrease in effort when the 

 population decreases (,r<0). 



In this simple single-variable case we can utilize 

 the first result and avoid specifying R and Q by 

 substituting wfrom Equation (5) into Equation (4). 

 The result is: 





t \2b Zj 



26 2 



The time constant for this system is: 



T = 



a 

 2" 



2b ^ 



(6) 



Using this approach it is possible to choose Kso as 

 to give a desired value of the time constant. 



Alternately, K may be chosen by specifying the 

 magnitude of the deviation in fishing effort we 

 will allow in order to counteract an expected 

 deviation in population level. Written in terms of 

 magnitudes, Equation (5) becomes: 



■r. 



where: .f,„ = maximum magnitude expected for x 

 u„, — maximum magnitude specified for u. 



Once K has been determined,./' as a function of A^ 

 can be found by substituting .r and u from Equa- 

 tions (2) and (3) into Equation (5) to obtain: 



/ = f - K{N 



^eq). 



(7) 



To evaluate the effects of the above regulation 

 scheme under various conditions, the above 

 expression is substituted into Equation (1), which 

 can then be solved by computer for A^ and / as 

 functions of time. 



As an example with the previously mentioned 

 results of Schaefer (1954) for the Pacific halibut, a 

 maximum deviation in N of 5% from A^eq was pos- 

 tulated, and a maximum deviation in /of 5% from 

 ./'eq was specified. Thus: 



A^ 



eq 



a/26 = 1.098 x lO^ 



./;„ = a/2q = 8.48 x 10^ 



eq 



x„, = O.OSA^eq 



u„, = 0.05/ 



eq- 



Using the second method for computing K, we 

 obtain: 



K = - 



0.05/-, 



eq 



0.05A^ 



= -0.772 X 101 



eq 



From Equation (6) the new time constant is found 

 to be 1.5 yr, which is one-half the value for the case 

 without feedback control. The fishing effort found 

 from Equation (7) is: 



'■^ 



(»-fe) 



=-A^= 0.772 



q 



X 10 -4 A^. (8) 



In view of the impossibility of continuously and 

 instantly measuring population size and varying 

 fishing effort, /as given by Equation (8) was in- 

 terpreted as follows. It was assumed that a limit is 

 imposed on fishing effort at the beginning of each 

 year and held constant during that year, and its 

 value / is calculated from Equation (8), with A^ 

 being the average population over a yearly inter- 

 val terminating three-tenths of a year before the 

 imposition of the new limit. That is, three-tenths 

 of a year is allowed for collecting and analyzing 

 the population data used to calculate the next 

 year's limit. With this discretized version of / 

 computer simulation results show that the system 

 time constant is 1.8 yr, which is reasonably close to 

 the 1.5 yr predicted by the continuous model. Thus 

 it is possible to use the analysis based on the con- 

 tinuous model in the realistic situation involving 

 data-collection limitations and limit-imposition 

 constraints. 



THREE-VARIABLE EXAMPLE 



An advantage of the optimal control method is 



833 



