FISHERY BULLETIN: VOL. 87, NO. 2, 1989 



sity dependence of the relevant organism, but only 

 indirectly because the density is moderated by the 

 local environment. 



For apex predators, such as yellowfin and skip- 

 jack tuna, the quality of their environment is deter- 

 mined primarily by the availability of prey; the local 

 degree of saturation by predators is determined by 

 the availability of prey. Saturation is defined as the 

 biomass of predators divided by the maximum that 

 the locality could sustain; thus it is dependent upon 

 the intrinsic richness of the locality and the number 

 competing for those riches. For any particular area, 

 an increase in predators will decrease the availabil- 

 ity of prey; this will, in turn, reduce the quality of 

 that area for those predators. 



can be checked by multiplying through the above ex- 

 pression for F(A(x,y)) by r', and substituting for 

 K-(x,y). 



The term a- may be a constant, D, or propor- 

 tional to the local abundance divided by the local 

 carrying capacity. That is to say, unless constant, 



a^ix,y) = DA(x,y)IK(x,y). 



Effort was determined by a simple bioeconomic 

 equation taken from Clark (1985): 



dE{x,y) 

 dt 



= a(pqA - c) E(x,y), if E > 0; (4) 



THE MODEL 



The biomass of a particular species at any point 

 {x,y) may be modelled: 



dA{x,y) „,,, ,, 32 /a2 ^, ^ 

 , = Fi.A{x,y)) + — - \-A{x,y) 

 dt dx'- 4 



_9i 



A(x,y)\ 



- qE(e,y) A{x,y). 



(1) 



That is to say, the rate of change of the local biomass 

 with time, t, is determined by the production func- 

 tion, F{A(x,y)); the catch equation, where q is the 

 catchability coefficient and E(x,y) is the fishing ef- 

 fort expended; plus the diffusion of fish into or out 

 of the locality in both the x and y directions. The 

 key parameter governing diffusion is a-. It is 

 either constant, or a function of local biomass and 

 carrying capacity; variables that were also in the 

 domain of F. 



The production of biomass is modelled using a 

 modification of the Schaefer (1954) model: 



F(A(x,y)) = A(x,y) \r\l - 



Aix,y) 

 K-(x,y) 



M\ (2) 



where r' = r + M and K'(x,y) = ((r + M)lr) 

 K(x,y). The modification simply separates natural 

 mortality, M, from the intrinsic growth rate: the 

 modified form uses the gross, rather than net in- 

 trinsic growth rate, r. The form oiF(A(x,y)) is un- 

 changed; the function was rewritten so that birth 

 and death processes would be more explicit. In 

 particular, the carrying capacity is unchanged, as 



= otherwise. 



p represents the price per ton received by fishermen, 

 c is the cost to the fishermen of each unit of effort, 

 and a is the proportion of profits reinvested. An im- 

 plicit assumption is perfect liquidity, i.e., that a loss 

 immediately causes a reduction in effort of the same 

 magnitude as the increase created by a profit. 



Equation (1), which represents the kernel of the 

 model, is a nonlinear partial differential equation; 

 it might be possible to solve it analytically, but it 

 is difficult. Solutions were found numerically by 

 iterating explicitly using finite differences. 



The 5 million square nautical miles of the range 

 of yellowfin tuna within the eastern Pacific is repre- 

 sented by a grid of 20 x 20 cells, each cell repre- 

 senting an area of approximately 2 degrees in both 

 latitude and longitude. The northern and southern 

 edges of the model grid were joined, as were the 

 western and eastern edges; forming a torus and cir- 

 cumventing any boundary problems. Parameter D 

 was set at 0.08, equivalent to 1,000 square nautical 

 miles per day, a number within the mid range of 

 those found by Bayliff (1984). 



The carrjring capacity for the region as a whole, 

 and the intrinsic natural rate of increase, were ob- 

 tained by fitting a Schaefer (1954) model to catch 

 and effort data for this entire region. ^ This gave an 

 annual value for r of 1.61, and an estimate for the 

 carrying capacity of the entire region of 431,000 

 tons (lATTC in press). The annual rate of natural 

 mortality was estimated at 0.8 by Hennemuth 

 (1961). The catchability coefficient, q, is the prob- 

 ability that a particular fish will be caught by a unit 



^Patrick Tomlinson. Inter-American Tropical Tuna Commission, 

 c/o Scripps Institution of Oceanography, La Jolla, CA 92037, pers. 

 commun. May 1988. 



354 



