Intuition tells us (and you will confirm in an exercise below) that yield as a function of F will look like Figure 6.10. When Fis small, we expect that yield will be an increasing function of fishing effort (from a Taylor expansion of the exponential). As F increases, fewer individuals reach high age (and large weight), so that yield declines. The slope of the yield versus effort curve will be largest at the origin and very often you will cc

Figure 6.10. The yield from a cohort as a function of fishing effort.

a — a encounter rules for setting fishing mortality that are called F0.x, which means to choose F so that the slope of the tangent line of the yield versus effort curve is 0. x times the value of the slope at the origin.

Since pm(a) is the probability that an individual of age a is mature, the number of spawners when the fishing mortality is F and the age of recruitment to the fishery is ar is S(ar,F) = ^aPm(a)N(a) and the spawning stock biomass produced by this cohort is SSB(ar,F) =^2apm(a)W(a)N(a) (note that F and ar are actually ''buried'' in N(a)). The number of spawners and the spawning stock biomass that we have just constructed will depend upon the initial size of the cohort. Consequently, it is common to divide these values by the initial size of the cohort and refer to the spawners per recruit or spawning stock biomass per recruit.

In the early 1990s, W. G. Clark (Clark 1991, 2002) noted that some of the biggest uncertainty in fishery management arises in the spawner recruit relationship. Clark proceeded to simulate a number of different stock recruitment relationships and studied how the long term yield was related to the fishing mortality F. In the course of this work, he used the spawning potential ratio, which is the value of F that makes SSB(F) a specified fraction of SSB(0). Formany fast growing stocks, a SSB(F) of 0.35 or 0.40 (that is, 35% or 40%) is predicted to produce maximum long term yields while for slower growing stocks the value is closer to 55% or 60% (MacCall 2002).

Imagine a stock with von Bertalanffy growth with parameters k = 0.25 yr-1, Lo = 50 cm, t0 = 0, M = 0.1 yr-1, and a length weight allometry W = 0.01 L3, where W is measured in grams. Assume that no fish lives past age 10. With knife-edge dynamics for recruitment to the fishery, the dynamics of the cohort are

Assume that N0 = 500 000 individuals. Compute the total yield (in metric tons = 1000 kg) per recruit assuming that fish are recruited to the fishery at age 2, 3, or 4. Make three separate plots of yield vs fishing effort for the three different ages of recruitment to the fishery. Pick one of these ages and construct a table of age vs number of individuals in the presence or absence of fishing. Next compute the number of spawners per recruit and spawning stock biomass per recruit, assuming that all individuals mature at age 3. Now convert your code to a time dependent problem for the number of fish of age a at time t, N(a, t), by assuming that recruitment N(0, t) is a Beverton-Holt function of spawning stock biomass S(t — 1) according to N(0, t) = 3S(t - 1)/[1 + 0.002S(t - 1)] and repeat the previous calculations.

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