FULLENBAUM and BELL: AMERICAN LOBSTER FISHERY 



The dynamics of a fish stock may be depicted 

 by the logistic growth function (Lotka, 1956).^ 



X(t) = 



1 + Ce 



-KLt 



where L>0,O0,/e>0, (7) 



Kx = rKX 



(11) 



where r is a technological parameter.'- Finally, 

 the total revenue function for the industry may 

 take the following form: 



where L, C, and K are assumed to be environ- 

 mental constants. Differentiating (7) and sub- 

 stituting we obtain, 



X = ^ = kLX - /v'X2 = aX - 6X2 (g) 



at 



where 



a = kL, b = k. 



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

 nonzero steady-state biomass, alb (i.e., L). 

 Alternatively, the limit of X{t) as f ^ °° yields 

 identical results. The maximum of (8) occurs 

 when X is equal to al2b. Thus 



max 3^ = a^l4b 



(9) 



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

 is assumed to have no interactive effects, so that 

 the instataneous growth rate is reduced by the 

 amount harvested:'" 



^ = gX - 6X2 - Kx. 

 at 



(10) 



The economic component of the model re- 

 quires 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. 



"Graham (1935) was the first biologist to apply the 

 logistic growth model to exploited fish populations. 



'" Schaefer, (1954) was the first population dynamicist 

 to develop the function specified in equation ( 10). 



'• Alternatively, one could assume that the proportion 

 of the biomass caught declines as the number of vessels 

 increases: 



Kx = [\ - (\ - nf^]X. 



0<f<l 



With this specification, ; represents the proportion of the 

 biomass taken by the first vessel and also represents the 

 percentage taken by each succeeding vessel of the remain- 

 ing biomass. This form was first developed by E. W. Carl- 

 son (1970. An economic theory of common property re- 

 sources. Unpubl. manuscr. Econ. Res. Lab., Natl. Mar. 

 Fish. Serv., NOAA College Park, Md.). 



pKx = (a- (iKx)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 function where ex-vessel price 

 is inversely related to landings, holding all 

 other factors constant.'-'^ With total costs equal 

 to Ktt, the profit function becomes 



77 = (a - iiKx)Kx - Krf. 



(13) 



Given these formulations, the system in (10) - 

 (13) can be reduced to two steady-state func- 

 tions. The first, which condenses all relevant 

 biotechnological factors, is the ecological equilib- 

 rium equation. It plots the relationship between 

 the biomass and the number of vessels (or fish- 

 ing effort) needed to harvest the yield such that 

 the biomass is in equilibrium. We can derive 

 this equation by setting X equal to zero, sub- 

 stituting (11) into (10), and solving for K in 

 terms of X.- 



K = -{a- bX). 



(14) 



Similarly, the second equilibrium function plots 

 the relationship between X and A' under a zero 

 profit state, i.e., under conditions that K — 0, 

 or that there is no entry to or exit from the fish- 

 ery. Thus, by setting (13) equal to zero and 

 substituting (11) into (13), we obtain 



K = 



a 



drX ^V2X2 



(15) 



'•^ A reviewer ot this article has pointed out that ; 

 is not likely to be constant over any large number of 

 years. Since there are no time series observations on X, r 

 cannot be tested to see whether it varies over time or is 

 a constant. In this case, we are merely following the 

 simplified Schaefer model. 



13 Such complicating factors as per capita income and 

 its influence on ex-vessel prices can be introduced later 

 as changes in the parameter, q. 



15 



