MULLEN: FISH THROUGH VARIABLE DIFFUSIVITY 



of effort given that the effort and fish are in the 

 same area, so q has dimension of (1/area). The esti- 

 mated value of 0.000039 (lATTC in press) for the 

 entire fishery was therefore multiplied by the num- 

 ber of cells, 400, to obtain the q for each cell. 



The price of fish, p, was set at $1,200 per ton, 

 which approximated that of the last half of 1987 

 (Parks et al. 1988). The cost of a unit of effort, c, 

 was estimated by assuming that in 1987 there 

 existed an economic equilibrium, that is the fisher- 

 men just covered their costs in this year but made 

 no net profits. If that were the case, the cost of a 

 unit of effort would be simply the total catch times 

 the price divided by the total effort for that year. 

 This gives c to be approximately $24,000 per days 

 effort, which was used as the initial value for this 

 parameter when exploitation was included. To 

 evaluate the effect of different restraints upon fish- 

 ermen in different areas, c was in some cases made 

 position dependent. The proportion of profits re- 

 invested in effort, a, was set arbitrarily at 0.2. 



For the initial run all but two cells had carrying 

 capacities set at 1,000 tons; the two exceptions, posi- 

 tioned at (10,5) and (10,10) were given carrying 

 capacities of 10,000 tons. This range of a factor of 

 10 for the capacities was chosen because it corre- 

 sponds with the range shown by annual productivity 

 over the region (Berger et al. 1987). The total car- 

 rying capacity specified within the model, 418,000 

 tons, was close to that estimated for the fishery. The 

 abundance for each cell was initially set at the local 

 carrying capacity, except for the cells with the 

 higher capacities where the abundance was 0.8 of 



that, to speed convergence. Every run of the model 

 was continued until a steady state was evident; the 

 possibility of multiple steady states, implicit in such 

 a nonlinear model was not investigated. 



RESULTS 



The equilibrium distribution of abundance at zero 

 exploitation was calculated for constant a^. The 

 abundance of tunas was little higher in the cells of 

 high capacity than elsewhere. This was true even 

 when the "hot" cells were given capacities 100 times 

 that of the others, and when a^ was reduced by a 

 factor of 10, well below any observed value. 



It was believed that if the area of the region of 

 increased capacity were greater, then the leakage 

 of extra production of fish would be less from that 

 region, and a significant local increase in abundance 

 might be seen. By analogy with a coal fire, indivi- 

 dual coals cool quickly, but if they are grouped 

 together, they have proportionately less surface 

 area and so cool more slowly. The local increase in 

 abundance was greater when the area of higher 

 capacity was expanded, but most of the increased 

 production of the local area still diffused away. The 

 region of higher capacities was increased to 3 x 3 

 cells and even 5x5 cells, but there still was not 

 the sort of variation in abundance that one can infer 

 from catch records. Variations of the same order 

 as appear to occur in the ocean were not found until 

 a - was allowed to vary with the degree of satura- 

 tion of the local carrying capacity. 



Figure la shows the distribution of fish when 



Figure L— Distribution of abundance with constant diffusivity (a), 

 and with diffusivity a function of the local saturation of carrying 

 capacity (b). b 



fiBUNDflNCE 

 Constant Diffusn'ity 



ftBUNDflMCE 

 Dependant Dif -f us ii> 1 1 y 



355 



