There may be circumstances when fisheries managers are able to specify an upper limit
to fishing intensity, beyond which an extremely undesirable state of the fishery is agreed to
exist. As noted, this may be referred to as a Limit Reference Point (LRP). For mathematical
simplicity we assume in the following, that F_{MSY} is a conventionally-accepted value, known
without error. In the following example we further assume that the LRP used in this case is
a pre-established value for the fishing mortality corresponding to MSY conditions.

The managers acknowledge that they are operating in an uncertain environment and
that the current ‘status quo’ for the fishery, and the F-value during the last season ( = F_{NOW}),
was not precisely known, but that some rough estimates of its standard deviation can be
made. In the hypothetical case in question, there is strong evidence that the fishing intensity
last year was below F_{MSY} , and it is assumed that if the same effort were to be exerted in the
next season, we could expect the probability distribution of fishing mortality rates to remain
the same. The managers feel however that it would be useful to define a target reference
point in such a way that this results in a small, prespecified risk that F_{MSY} is not exceeded.

Given this situation, the following illustrates one procedure for calculating appropriate
target values for F_{NOW} which result in a pre-specified probability of an agreed LRP being
respected. In this example, the LRP is assumed to be a pre-established value for F_{MSY} = 0.6.
Although there is no unambiguous evidence in the literature as to the most appropriate
distribution function to use for F, it is believed that the uncertainty in the relative position of
F_{NOW} < F_{MSY} can reasonably be expressed by a normal distribution (see Fig 18 below), although
similar calculations could readily be performed for other distribution functions.

Let us assume that the current level of the fishing mortality, F, is less than the target
reference point, F_{MSY}. For mathematical simplicity we assume that F_{MSY} is known without
uncertainty. We further assume that the uncertainty of F, is described by a lognormal
distribution where, with equal probability, i.e. with equal levels of confidence, the actual value
of F in the fishery may be twice or half the estimated parameter, F_{NOW}. We seek to find a
value for this limit reference point, F_{NOW}, that lies safely below F_{MSY}, leaving only an
acceptably small probability that the ‘true’ value of the current F is, in fact, greater than F_{MSY}.

Mathematically the suggested procedure is the following: The level of risk the fishery
can safely tolerate, (quantified in the appendix figure as the shaded area on the right hand tail
of the normal distribution), is equivalent to the integral of the probability that the current F
exceeds our limit reference point. Referring to this chosen level of acceptable risk as P
(F > F_{MSY}), we must solve for the value of F_{NOW} that corresponds to the target reference point
that provides this margin of safety. The cumulative probability for the right hand tail of the
normal distribution may then be written as:

From this we can obtain F_{NOW} using a mathematical package (e.g. MAPLE,
MATHEMATICA) designed for equation solving.

The standard deviation of F could then, in theory, be estimated from the historical record by comparing the predicted F's and the ‘true’ values obtained retrospectively from VPA, or by agreeing on a ‘likely’ margin of error given estimates of the variance in population size estimates. Furthermore, this type of historical comparison could enable fishery managers to describe more accurately the nature of the distribution in uncertainty of F.

[P(x)] | STANDARD σ = 0.25 | DEVIATION: = 0.5 | = 1.0 | |
---|---|---|---|---|

P(F_{NOW} > F_{MSY}) | 30% | 0.53 | 0.475 | 0.39 |

20% | 0.50 | 0.42 | 0.33 | |

10% | 0.45 | 0.365 | 0.26 |