FISHERY BULLETIN: VOL. 73, NO. 4 



Note that a static optimization method, calculus, 

 has been used to find an optimal equilibrium. To 

 analyze the model's behavior in the vicinity of the 

 equilibrium point. Equation (1) is linearized by 

 expanding the right-hand side in a two-variable 

 Taylor series about the point (A^eq-Zeq)' ^.nd keep- 

 ing only the first-order terms (see Appendix). This 

 gives: 



d£ 



dt 



(«I'*(t" 



where: g = aN - bN^ 



x = N- N\ 



qfN 



" =f-fe^- 



eq 



(2) 

 (3) 



The new variables .r and u are the deviations in 

 population density and fishing effort from their 

 equilibrium values. After evaluating the deriva- 

 tives of g at the equilibrium point, we obtain: 



d£ 

 dt 



aq 

 2h 



(4) 



If the fishing effort is kept constant at its 

 equilibrium value, then n = Oand 



dx _ a 

 It ~ "2'^' 



This system is stable for all positive values of a, 

 which means that if disturbed from equilibrium, 

 the population will eventually return to it. The 

 solution is: 



x(0 = .r(ge" 2 "-'"', 



where x(f„) is the deviation at time f„ . Since the 

 time constant for this system is 2/o, it will take 

 that amount of time for the deviation to decay by 

 6S9r and for four time constants to decay by 98%. If 

 the constant a is small, this time can be very large. 

 Also, by keeping the fishing effort constant, we 

 cannot take advantage of higher yields obtainable 

 when x{to)>0, and risk overexploiting when x{t„)<0. 

 We will show that by making the fishing effort a 

 function of population level, we can change the 

 system's time constant and also avoid the above 

 difficulties. 



For example, the results of Schaefer (1954) give 

 the following values for the Pacific halibut: 



a 



h 



0.67 



3.05 X 10-9 



3.95 X 10-5 



where A^ is in pounds, t in years, and./Mn number of 

 skates. A standard skate of halibut gear consists 

 of eight lines of 300 feet each in length, with 

 shorter lines with hooks attached at 10-foot inter- 

 vals (Carrothers 1941). The time constant is 3 yr, 

 and thus 12 yr are required for a deviation in 

 population to disappear, assuming no other dis- 

 turbances act during that time. 



If we now specify, by means of a "performance 

 index," that we wish to keep A'^ near Aeq while 

 minimizing the variation in./' required to do so, we 

 can design a regulatory procedure which will keep 

 the fishery near the maximum equilibrium yield 

 condition. The performance index J which 

 specifies this desire is the so-called quadratic 

 index: 



J = J (Q.r2 + Rn^ dt. 



The squared terms indicate that we make no dis- 

 tinction between positive and negative deviations 

 from equilibrium. The positive constants Q and R 

 are the weighting factors which indicate the rela- 

 tive importance placed on keeping N near Agq (.r 

 near 0) versus keeping./' near ./'gq {n near 0). The 

 infinite upper limit indicates that we are interest- 

 ed in long-term as well as short-term effects of our 

 fishing effort regulation. 



The problem of determining the function u, 

 which minimizes the performance index, is solved 

 by the application of optimal control theory. Since 

 the system. Equation (4), is linear, and the index is 

 quadratic, the problem formulated above is 

 referred to as the Linear-Quadratic Optimal Con- 

 trol Problem. 



The solution for the control function is (see 

 Appendix): 



-Kx 



(5) 



K = 



1 



R 26 



117 p 



where Pis the positive steady-state solution of the 

 so-called Riccati equation: 



dP 



dt 



= -aP 



R 



W^' 



+ Q 



832 



