X = 



dX 

 dt 



= kLX - kX'^ = aX - bX- 



(8) 



where a = kL, h = k. 



If (8) is set equal to zero, we may solve for the 

 nonzero steady state biomass, a/h (i.e.. L). Alterna- 

 tively, the limit of X(t) as / ^ ac yields identical re- 

 sults. The maximum of (8) occurs when X is equal to 

 a/2b. Thus 



max ^ = a- 1 Ah. 

 dt 



(9) 



The introduction of fishing (i.e., harvest or Kx) is as- 

 sumed to have no interactive effects, so that the in- 

 stantaneous growth rate is reduced by the amount har- 

 vested: 



A^ = aX 

 dt 



hX^ - Kx. 



(10) 



The economic component of the model requires the 

 exact specification of an industry production function 

 and an industry revenue relationship. One hypothesis 

 regarding the fish catch is that the proportion of the 

 biomass caught is a direct function of the number of 

 vessels (or equivalent fishing effort) exploiting a given 

 ground.'" Thus, the total harvest rate is given as. 



Kx = iKX 



(11) 



where r is a technological parameter. Finally, the total 

 revenue function for the industry may take the follow- 

 ing form: 



pKx = (a-Bkx)Kx. (12) 



Equation (12) merely stipulates that the total revenue 

 is a quadratic function of total landings, Kx. Dividing 

 through by Kx will give us the familiar demand func- 

 tion where ex-vessel price is inversely related to land- 

 ings, holding all other factors constant.-" With total 

 costs equal to K-n. the profit function becomes 



TT = (a-BKx) Kx-Ktt. 



(13) 



Given these formulations the system in ( 10)-(13) can 

 be reduced to two steady state functions. The first, 

 which condenses all relevant biotechnological factors, 

 is the ecological equilibrium equation. It plots the rela- 

 tionship between the biomass and the number of ves- 

 sels (or fishing effort) needed to harvest the yield such 

 that the biomass is in equilibrium. We can derive this 

 equation by setting -Y equal to zero, substituting (11) 

 into (10), and solving for K in terms of A": 



"Alternatively, one could assume that the proportion of the 

 biomass caught declines as the number of vessels increases: 

 K.x = [\-{\ - l)k]X .()<t<\. With this specification./ represents the 

 proportion of the biomass taken by each succeeding vessel of the 

 remaining biomass. This form was first developed by Carlson 

 (1970). 



^°Such complicating factors as per capita income and its influence 

 on ex-vessel prices can be introduced later as changes in the 

 parameter, a . 



K =±(a - bX). 

 r 



(14) 



Similarly, the second equilibrium function plots the 

 relationship between X and K under a zero profit 

 state, i.e., under conditions that K = 0, or that there 

 is no entry to or exit from the fishery. Thus, by setting 

 (13) equal to zero and substituting (11) into (13), we 

 obtain. 



K = 



15) 



BrX Br^X"" 



These two curves are plotted in Figure 18. Their inter- 

 section at {X*. K*) denotes bioeconomic equilibrium. 

 The direction of the arrows describe the qualitative 

 dynamic changes of a point in phase space. Figure 18 

 represents the general case of exploitation. When (15) 

 is combined with (14), however, we can simulate 

 either non-exploitation (Fig. 19) or extinction as a pos- 

 sible dynamic result (Fig. 20). The state of the 

 fishery — exploited, unexploited, or extinct — depends 

 upon the parameters a. h. r, B, ir, and a and their 

 interrelationships. This completes ourgeneral model of 

 how a fishery functions. Now let us turn to a specific 

 application of the model. 



Figure 18. — Exploitation. 



Figure 19. — Non-exploitation. 



39 



