(^) Often input–output management schemes assume strong density dependence of
survival and growth. That is, if a stock is drawn down it is expected to respond with
better survival and growth, because more resources become available to support that.
Of course, some other animal may fill that carrying capacity. For example, in the case
of slow-growing fish, the substitute appears often to be fast growing, and thus
opportunistic, sometimes squid or jellyfish. In the following, “fish” will simply mean
the product of a fishery, which could be clams, shrimp, squid, or, of course, fish.
Stock Size (Measuring B)
(^) The first-order problem is simply to estimate the biomass and age composition of the
stock. As any wildlife manager will tell you, it is not simple to estimate accurately the
abundance of deer or geese. Stocks in the ocean are even more difficult, since it is not
easy to peer into the ocean. So, the size of stocks is not usually known explicitly
(tagging methods – see below – not withstanding). The most common approach is to
suppose that the harder it is to catch a fish, the fewer fish there are. That is, the yield,
Y, per unit fishing effort, X, should be less if the stock, B, is smaller (and vice versa):
(Eqn. 17.2)
(^) where q is the catchability coefficient, the proportion of the stock removed by one unit
of effort. The captured fish are weighed or counted at the dock, and the effort made to
catch them is estimated in some suitable units like hook-hours or number of days
fishing by standard boats. You get that measure of effort by urging or requiring fishers
to keep logs. A government fishery agency might require log submission as a
condition of sale. For fisheries that have, over the course of time, established several
approximate equilibria between fishing pressure and population dynamics, this can
work; the shape of the Y/X vs. X curve (X on the abscissa) declines monotonically. In
the simplest cases the relationship is roughly linear (Fig. 17.5a). If the ordinate
intercept is called qPnatural, an estimate of catch rate from one unit of effort for the
stock size without fishing, we can represent this relationship by:
(Eqn. 17.3)
(^) where k is the slope of the relationship. In a relative sense, this Y/X vs. X relation
gives us an idea how many fish there are in a stock.
Fig. 17.5 (a) CPUE (catch per unit effort) for yellowfin tuna in the eastern tropical
Pacific, 1934–1965. (b) Line in (a) converted to show catch vs. effort, a parabola.
(^) (After Schaefer 1967.)