7 Production/biomass

Total mortality catch curves

Total mortalities can be estimated from catch curves, i.e., from catch composition data, either in terms of age-structured or of length-converted catch curves. The estimation can be carried out using appropriate software for analysis, but require careful consideration of how representative the samples are of the entire population, and the impact of related bias on the estimated mortality parameters.

Total mortality from sum of components

The P/B rate can be estimated as the sum of natural mortality (M = M0 + M2, assuming that the M1 term is included in M0) and fishing mortality (F), i.e., Z = M + F ignoring here potential migration and biomass accumulation. In the absence of catch-at-age data from an unexploited population, natural mortality for finfish can be estimated from an empirical relationship (Pauly, 1980^[2]) linking M, two parameters of the von Bertalanffy Growth Function (VBGF) and mean environmental temperature, i.e.,

[latex]M = K^{0.65} \cdot L_{\infty}^{-0.279} \cdot T_c^{0.462}\tag{1}[/latex]

where, M is the natural mortality (year^-1), K is the curvature parameter of the VBGF (year^-1), L_∞ is the asymptotic length (total length, cm), and T_c is the mean habitat (water) temperature, in °C .

In equilibrium situations, fishing mortality (F, year^-1) can be estimated directly from the catch (C, including discards, t km^-2 year^-1) and biomass (B, t km^-2)

[latex]F = C/B \tag{2}[/latex]

Total mortality from average length

Beverton and Holt (1957^[3]) showed that total mortality (Z = P/B, year-1), in fish population whose individuals grow according to the von Bertalanffy Growth Function (VBGF), can be expressed as

[latex]Z=P/B=\frac{K \cdot (L_{\infty}-\bar L)}{\bar L-L^{'}}\tag{3}[/latex]

where K is the VBGF curvature parameter (year^-1, expressing the rate at which L_∞ is approached), L_∞ is the asymptotic length, i.e., the mean size the individuals in the population would reach if they were to live and grow indefinitely, L̅ is the mean length in catches, and L’ represents the mean length at entry into the fishery, assuming knife-edge selection. Note that the L̅–L’ denominator must be positive.

Total mortality and longevity

Mortality rates (P/B = Z) are not just nuisance parameters, they really mean something that one can relate to. How much does a population produce relative to its biomass?  A lot for plankton and not very much for whales, right?

Mortality rates thus relates to size, e.g., for a small (1-2 mm) zooplankton like Acartia tonsa, the P/B is up around 45 year^-1. The much larger Calanus finmarchicus can live for several years and may have a P/B closer to 7 or 8 year^-1. Whales? P/B will be below 0.1 year^-1.

A good way to relate to such numbers is to turn them on their head. That is, think of the B/P ratio (year) to get a sense for the P/B (year^-1) ratio.  So if a blue whale has a P/B of 0.025 year^-1, the inverse B/P is 40 years – that’s the average longevity of blue whales (if P/B indeed is 0.025 year^-1).  Seals with P/B of 0.14 year-1 would have an average longevity of 7 years, cod with a P/B of 0.25 year^-1 an average longevity of 4 years, and anchovy with P/B of 2.0 year^-1 would on average live half a year.

It makes sense, and longevity provides a good handle for evaluating what reasonable estimates of P/B may be.