## N

removing fish (the catchability) so that F = qE. We already know that MSY is rK/4, but essentially all other population sizes will produce sustainable harvests (Figure 6.4b): as long as the harvest equals the biological production, the stock size will remain the same and the harvest will be sustainable. This is most easily seen by considering the steady state of Eqs. (6.8) for which rN [1 -(N/K)] = qEN. This equation has the solution N = 0, which we reject because it corresponds to extinction of the stock or solutionN=K [1 -(qE/r)] . We conclude that the steady state yield is

Figure 6.4. (a) Atlantic cod, Gadus morhua, perhaps a poster-child for poor fishery management (Hutchings and Myers 1994, Myers et al. 1997a, b). (b) Steady state analysis of the Schaefer model. I have plotted the biological production rN (1 - (N/K)) and the harvest on the same graph. The point of intersection is steady state population size. (c) As either effort or catchability increases, the line y = qEN rotates counterclockwise and may ultimately lead to a steady state that is less than MNP, in which case the stock is considered to be overfished, in the sense that a larger stock size can lead to the same sustainable harvest. If qE is larger still, the only intersection point of the line and the parabola is the origin, in which case the stock can be fished to extinction.

which we recognize as another parabola (Figure 6.5) with maximum occurring at E* = r/2q.

Verify that, if E = E*, then the steady state yield is the MSY value we determined from consideration of the biological growth function (as it must be).

Furthermore, note from Eqs. (6.8) that catch is FN (=qEN), regardless of whether the stock is at steady state or not. Hence, in the Schaefer

Steady state yield qEK 1- f

Figure 6.5. The steady state yield Y = qEK[1 -(qE/r)] is a parabolic function of fishing effort E.

Steady state yield qEK 1- f

Figure 6.5. The steady state yield Y = qEK[1 -(qE/r)] is a parabolic function of fishing effort E.

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