Appendix A. Optimal distribution of individual effort in a competitive fishery.
Here we show that provided , the catchability coefficient, is constant, the optimal spatial distribution of effort in a competitive fishery involves some harvesters fishing-the-line, with the remaining effort spread uniformly outside the reserve boundary.
First, we note that CPUE, fish density, and effort must be homogeneous in the unprotected area. The CPUE at any point (away from the boundary) in the unprotected area equals . Individuals in a competitive fishery locate themselves so as to maximize CPUE; if all individuals achieve this, CPUE must be constant across space. Thus if is constant, the distribution of fish, , must also be uniform for . In order for fish density to be uniform outside the reserve, the distribution of fishing effort must also be uniform, therefore .
We now consider individuals located at the reserve boundary. Total fishing rate on the boundary can be expressed as
Thus CPUE on the boundary = .
Harvesters away from the boundary will relocate their effort and fish on the boundary if , i.e., for . Therefore, equalizing CPUE on, and away from, the boundary requires that fish density is equal on the reserve boundary and in the unprotected area (i.e., for ).
Finally, we show that the proportion of effort at the boundary is non-zero. At steady state, the density outside the reserve boundary is obtained from the equation implying for . Inserting this condition in the flux boundary condition in Table 1, and remembering that has zero slope outside the reserve, yields
Thus the proportion of effort allocated at the boundary, , is non-zero, unless , i.e., unless there is zero spillover from the reserve.