## Natural Mortality Rate

The natural mortality rate of a fish stock is notoriously difficult to estimate. Tagging studies can be used, but in large or open systems, such as the ocean, tagging studies are infeasible or provide hopelessly imprecise estimates. If the total mortality rate and fishing mortality rate can each be estimated, M is known by subtraction in eqn [18]. More commonly, a natural mortality rate is chosen indirectly by drawing on empirical relationships between M and observable life-history characteristics.

Many ofthese relationships have been derived through meta-analysis. For example, Hoenig estimated a linear relationship giving the total mortality rate Z as a function of maximum age A. The work was based on data from 134 stocks comprising 79 species of lightly exploited fish, mollusks, and cetaceans:

For small sample sizes (n <200), Hoenig recommended this variant:

where ar is the earliest age fully represented in the sample. When using either equation, the result is an estimate of the total mortality rate Z, from which an estimate of M can be obtained by subtracting F, if known. Another possibility is applying the method to an unfished or lightly fished stock and considering the result an estimate of M.

Alverson and Carney developed an empirical estimator of M based on the age at which an unfished cohort reaches its maximum weight, sometimes called the critical age a*:

where k is the growth coefficient of VBGF (eqn [2]). Their analysis further suggested a* « 0.38A.

Pauly examined data from 84 species of marine and freshwater fishes, linking M to average water temperature (T) in celsius and the VGBF growth parameters of eqn [2] or [4]:

In the above, Lœ is in centimeters; W1, in grams.

### Natural mortality at age

In many stocks, natural mortality appears to decrease with age. Several studies have suggested that Ma can be estimated from weight at age Wa. Lorenzen concluded that the relationship differs among ecosystems. For oceanic ecosystems, his model is

Despite questions about estimating Ma, the assumption of M decreasing with age is increasingly preferred to the alternative of M constant with age. To estimate Ma, eqn [23] can be used directly or its estimates scaled to other information on mortality over the lifespan.

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