6.  TWO FORMULAE FOR THE RAPID ESTIMATION OF POTENTIAL YIELDS IN (MORE OR LESS) VIRGIN STOCKS

Often, when the available data on the stocks of a certain area are insufficient for detailed assessment, approximate formulae must be used to estimate potential yields (=Py ≃ MSY).

The simplest and most commonly used of this type of formulae has been proposed by Gulland (1971) and has the form

where B_v is the virgin standing stock, e.g., as estimated in a trawling or acoustic survey and M the (exponential) coefficient of natural mortality in the fish stock in question.

This equation was derived by Gulland (1971) from two different models:

A. From Beverton and Holt's yield-per-recruit model (1966) based on the assumptions that:

1. Recruitment remains more or less constant even under high levels of fishing mortality.
2. That no growth overfishing occurs, even at high levels of fishing mortality.

B. From the simple Schaefer model (1954), based on the assumptions that:

1. Virgin biomass ( = B_v) is equal to “carrying capacity” (= B_∞) of the environment for the stock in question. (This is a tacit assumption, not explicitly stated by Gulland, 1971.)
2. That maximum sustainable yield (MSY) is extracted when the virgin stock is halved (at B_v/2).
3. That, when harvesting at the MSY level, fishing mortality (F) is roughly equal to the natural mortality (M):

In Southeast Asian demersal trawl fisheries, Equation 42 is generally used in conjuction with values of M set equal to unity (1), because Southeast Asian fishes are relatively small and short-lived. This value which is essentially a guess has been uncritically applied to a wide variety of stocks, including the multispecies stocks off the Northwest Australian Shelf. The paper by Sainsbury (1979) should be consulted for a reassessment of this value and some negative consequences of its use.

For the Western Indian Ocean south of the equator, a more conservative estimate of M = 0.5 has been suggested and used for assessing various stocks (Gulland, 1979).

A modification of Equation 42 has also been proposed (Gulland, 1979) which may be applied to lightly exploited stocks, namely

where Z_t is the exponential rate of total mortality (= F + M) in the year t, B_t being the standing stock size in that year. Needless to say, this equation is even more approximate than Equation 42 and its use is justified only as a very first approximation, when absolutely no other information on the fishery is available, and the fishery has been operating at a fairly constant level for several years.

A rather different approach may be proposed here:

In his derivation of the simple (parabolic) Schaefer model, Ricker (1975, page 315) arrives at equations, which, slightly rearranged give

where B_∞ is the carrying capacity of the environment for a given stock (and which m \?\ assumed to correspond to B_v, the virgin stock size) while r_m (“k” of Ricker, 1975) \?\ “intrinsic rate of increase” of the population.

Thus, to obtain as estimate of MSY (or Py) all that is needed actually is an estimate of B_∞ (or B_v) and an independent estimate of r_m.

Blueweiss et al. (1978), using data published by various authors demonstrated that, for a wide variety of animals (including fishes).

where r_m is expressed on a daily basis, and where W is the mean weight (in g) of the adult animals^munder consideration. Combining Equations 44 and 45 and converting to the year as a time unit, we obtain

which can be used to estimate potential yields when virgin stock size and the mean weight (in g) of the adults in that stock are known. This equation, it will be noted, requires no estimate of M and may, therefore, be used for double-checking potential yield estimates based on Equation 42.