__Introduction__

In fishery biology, the most useful manner of expressing the decay (=decrease) of an age group of fishes through time is by means of exponential rates. These rates, of which three are normally defined, are given in the following two expressions:

N _{T} = N_{o} · e^{-Zt} | (18) |

where N_{o} is the (initial) number of fishes at time t = o, and N_{t} is the number of remaining
fishes at the end of time t, Z being the instantaneous rate of total mortality. An
advantage of instantaneous rates is that they can be added or subtracted. Thus we have

Z = M + F | (19) |

where M is the instantaneous rate of natural mortality and F the instantaneous rate of fishing mortality (see Figure 11). Obviously, when

F = 0 then, Z = M | (20) |

which means that natural and total mortality have the same value when there is no fishing (in an unexploited stock).

Figure 10 Length-frequency data on the slipmouth *Leiognathus bindus* caught off Calicut, India in years 1956, 1957 and 1958,
with suggested “growth segments” for use with a Gulland and Holt Plot (see Table 6 and text). Based on data
of Balan 1967

Figure 11 Showing the decrease of a cohort of fishes, over time, starting with 100 fishes at age 0, with 3 levels of mortality

The fishery biologist, as far as mortalities are concerned, has two main jobs:

(a) To estimate the value of Z;

(b) To split - where appropriate - an estimated value of Z into its component parts M and F.

__Estimating Total Mortality__

Total mortality from the mean size in the catch

(a) When a large number of length-frequency data have been obtained from a given stock, by a given gear, Z can be estimated from the mean length () in the catch from a given population by means of

Where L_{∞} and K are parameters of the von Bertalanffy growth equations, is the mean length
in the catch, and “L' is the smallest length of animals that are fully represented in catch
samples” (Beverton and Holt 1956).

(b) Another equation which can be used to estimate Z from the mean length in the catch is

where L_{∞}, , L' and K are defined above while n is the number of fishes used for the estimation
of (based on Ssentengo and Larkin 1973). It will be noted that when n is large, the term
n/n+1 tends toward unity and hence can be neglected.^{1}

(c) The equation corresponding to Expression 22 is for weight (when growth is isometric):

where W_{∞} and K are parameters of the von Bertalanffy growth function for weight growth, which
has the form

W _{t} = W_{∞} (1-e^{-K (t-to)})^{3} | (24) |

with and W' being the weight corresponding to and L' as obtained (along with W_{∞}) from a
conversion from length to weight, by means of the appropriate length-weight relationship.

Equation 24 is particularly useful in that it is quite easy to weigh a catch and to divide it by the number of fish to obtain .

Figure 12 Nomogram for the estimation of the selection factors of fishes from measurements of their body proportions.
Based on data in Sinoda *et al.*, 1979 and Meemeskul (1979). Original. (Note: all selection factors estimated
by means of this nomogram refer to __total length__. Hence: SF . mesh size = L_{c} (total length))

Year | Effort | n | Z_{1} | Z_{2} | |
---|---|---|---|---|---|

1966 | 2.08 | 4 733 | 13.25 | 2.41 | 2.95 |

1967 | 2.08 | 11 902 | 13.01 | 2.69 | 3.24 |

1968 | 3.50 | 12 503 | 12.99 | 2.72 | 3.27 |

1969 | 3.60 | 9 060 | 13.07 | 2.62 | 3.16 |

1970 | 3.80 | 8 132 | 12.37 | 3.73 | 4.29 |

1972 | 7.19 | 3 635 | 12.30 | 3.88 | 4.44 |

1973 | 9.94 | 10 510 | 12.01 | 4.61 | 5.14 |

1974 | 6.06 | 7 960 | 12.60 | 3.30 | 3.85 |

4.87 | 12.70 | 3.25 | 3.79 | ||

Constants: L _{∞} = 20.0; K = 1.16; L_{c} = 10.0 (Z_{1} = equation 21; Z_{2} = equation 22) |

A parameter closely related to L' (see above for definition) is the mean length at first
capture (L_{c}), or the length at which 50% of the animals sampled are retained by the gear.

L_{c} is normally estimated from selection experiments (see Gulland 1969) which, however,
are rather time and resource consuming. To facilitate estimation of L_{c}, a figure is therefore
included which allows for values of L_{c} to be obtained from attributes of fish that are very
easy to obtain, such as the length/depth ratio, or the “girth factor” of fishes (Figure 12).

Figure 12, however, does not allow for a direct estimation of L_{c}. Rather, once the
length/depth ratio or the girth factor have been estimated, Figure 12 is used to estimate a
selection factor (SF) such that

L _{c} = SF · mesh size (of cod end)^{1} | (25) |

The parameter L_{c} is used in stock assessment in a number of models, notably that of Beverton
and Holt (1966). In cases where “knife-edge” selection occurs (see Beverton and Holt 1966,
or Gulland 1969), L_{c} = L'; generally however, L'> L_{c}.

^{1} When the sampling gear is a trawl

__Estimating Z by means of a Catch Curve__

Another method of estimating Z consists of sampling a multi-aged population of fishes,
then plotting the __natural__ logarithm (log_{e}) of the number of fishes in the sample (N) against
their respective age (t) or

log _{e}N = a + bt | (26) |

where the value of b, with sign changed, provides an estimate of Z.

Several requirements must be met for the values of -b to be a good estimator of Z. Among these, we may mention:

(a) Only those values of log_{e}N must be included which pertain to age group of fishes
fully vulnerable to the gear in question (among other things, the fish must be
larger than L' as defined above): This corresponds to using only the “descending
part” of a catch curve (see Figure 13).

(b) Recruitment must have been constant within the period covered, or have varied in a random fashion only.

When suitable length-frequency samples are available, a catch curve may also be constructed
through previous conversion from length to age by means of a set of growth parameters.
Here, however, care must be taken not to include fish whose size is close to that
of their asymptotic size, as this may result in their age being grossly over-estimated. This
latter feature incidentally makes it imperative that a scatterdiagram be drawn in order to
properly identify the section of the catch curve which can be used to estimate Z (Figure 13).
It will be noted, also, that since Z is equal to the slope (with sign changed) of the catch
curve the real age - which requires an estimate of t_{o} - can be here replaced by relative age,
i.e., by setting t_{o} = 0 (Table 8, Figure 13).

Also, when converting a length-frequency sample to a catch curve, a problem must be considered which doesn't occur when fish have been aged individually. This problem is due to the fact that length growth not being linear, it takes an older fish longer than a younger fish to grow through a given size-group. Put another way, among bigger fish, a given magnitude of size interval (e.g., a l-cm length group) will contain more age groups than among small fish.

Compensating for this “piling-up” effect is, however, quite straightforward and can be achieved, e.g. by rewriting equation (26) as

log _{e} (N/∆t) = a + bt | …(26a) |

where ∆t is the time needed to grow from the lower (t_{1}) to the upper (t_{2}) limit of a given
length class, while t is the relative age corresponding to the midrange of the length class
in question. The procedure to convert a length-frequency to a length-structured catch curve
is illustrated in Table 8 and Fig. 13.

Class limits | Mid-range^{2} | N | t_{1}^{3} | t_{2}^{3} | ∆t | Adjusted number per length class log_{e}N/∆t | Mean relative age t ^{3} | Remarks | ||
---|---|---|---|---|---|---|---|---|---|---|

Lower^{2} | Upper^{2} | |||||||||

6.000 | 6.999 | 6.5 | 3 | 0.510 | 0.612 | 0.102 | 3.38 | 0.56 | Not used, ascending part of curve | |

7.000 | 7.999 | 7.5 | 143 | 0.612 | 0.720 | 0.109 | 7.18 | 0.67 | ||

8.000 | 8.999 | 8.5 | 271 | 0.721 | 0.837 | 0.116 | 7.76 | 0.78 | ||

9.000 | 9.999 | 9.5 | 318 | 0.837 | 0.961 | 0.125 | 7.84 | 0.90 | ||

10.000 | 10.999 | 10.5 | 416 | 0.961 | 1.096 | 0.134 | 8.04 | 1.03 | ||

11.000 | 11.999 | 11.5 | 488 | 1.096 | 1.242 | 0.146 | 8.11 | 1.17 | ||

12.000 | 12.999 | 12.5 | 614 | 1.242 | 1.402 | 0.160 | 8.25 | 1.32 | ||

13.000 | 13.999 | 13.5 | 613 | 1.402 | 1.579 | 1.777 | 8.15 | 1.49 | Portion used for estimating Z | |

14.000 | 14.999 | 14.5 | 493 | 1.579 | 0.176 | 1.999 | 7.13 | 1.67 | ||

15.000 | 15.999 | 15.5 | 278 | 1.776 | 1.999 | 0.223 | 7.13 | 1.88 | ||

16.000 | 16.999 | 16.5 | 93 | 2.000 | 2.257 | 0.257 | 5.89 | 2.12 | ||

17.000 | 17.999 | 17.5 | 73 | 2.257 | 2.560 | 0.303 | 5.48 | 2.40 | ||

18.000 | 18.999 | 18.5 | 7 | 2.260 | 2.930 | 0.370 | 2.94 | 2.74 | ||

19.000 | 19.999 | 19.5 | 2 | 2.930 | 3.404 | 0.473 | 1.44 | 3.15 | ||

20.000 | 20.999 | 20.5 | 2 | 3.404 | 4.063 | 0.659 | 1.11 | 3.70 | Not used, too close to L_{∞} | |

21.000 | 21.999 | 21.5 | 0 | 4.064 | 5.159 | 1.094 | - | 4.53 | ||

22.000 | 22.999 | 22.5 | 1 | 5.160 | 9.208 | 4.047 | -1.40 | 6.19 | ||

23.000 | 23.999 | 23.5 | 1 | - | - | - | - |

^{1} Form Ziegler (1979) who also gives: L_{∞} = 23.1 and K = 0.59 and T = 28°C

^{2} Total length, in cm

^{3} Computed by means of equation 32, with t_{o} = 0

Figure 13. A catch curve based on length converted to age, and corrected for the time
needed for the fish to grow through the size classes (see next). Based on
the data of Table 8, the regression equation is log_{e}(N_{∆t}) = 14.8 - 4.19t,
(r = 0.988), which provides an estimate of Z = 4.2.

__Splitting Z into M and F__

Splitting Z into M and F by means of a plot of Z on effort

When values of Z are available for several years pertaining to different annual values of effort (f), the value of M can be calculated from

Z = M + qf | (27) |

where q is the “catchability coefficient”, which relates f and fishing mortality (F) through

F = q · f | (28) |

Thus, a series of Z (mean annual) values can be plotted against their corresponding values of f and a straight line fitted to the points by means of linear regression technique. This results in a regression line with the equation

y = a + bx | (29) |

where Z = y and x = f, the slope (b) of which provides an estimate of the catchability coefficient
q while the intercept value (a) is an estimate of M (see Figure 14) in which values
of Z based on Equations 21 and 22 have been plotted.^{1}

Splitting Z into M and F by means of an independent estimate of M

When only one value of Z is available, or when the available values of Z and f cover too small a range of Z and f values for reasonable values of M and q to be obtained, the catchability coefficient (q) may be estimated through

where is the mean of the available values of Z (or a single value of Z) and is the mean of the values of f (or a single value of f), M being an independent estimate of natural mortality (see Table 7 and Figure 14).

__Method for Obtaining Independent Estimate of M__

It has been demonstrated by various authors that the value of the parameter K of the VBGF in fishes is closely linked with their longevity. This can be demonstrated on the basis of the observation that, generally, in nature, the oldest fishes of a stock grow to reach about 95 percent of their asymptotic length (Taylor, 1962; Beverton, 1963). Thus when we have

L _{t} = L_{∞} (1-e^{-K(t-to)}) | (31) |

we also have

^{1} It will be noted that equation 22 produces estimates of Z which are higher than those
obtained using equation 21; Mr. P. Sparre (Danish Institute for Fishery and Marine
Research) suggests (pers.comm.) that equation 22 is biased upward.

Figure 14 Plots of total mortality (z) on effort (f) to estimate natural mortality and catchability coefficient in
*Selaroides leptolepis* from the Gulf of Thailand. Based on data in Table 7. The equations are:

Z_{1} = 1.96 + 0.263f (squares)

Z_{2} = 2.52 + 0.263f (dots)

or, if we insert 95 percent of L_{∞} for the oldest fish

thus

where t_{max} is the longevity of the fishes in question.

That natural mortality should, in fishes, be correlated with longevity, hence with K, seems obvious. Natural mortality should, in fishes, also correlate with size since large fishes should have, as a rule, fewer predators than small fishes.

Natural mortality in fishes can also be demonstrated to be correlated to mean environmental temperature (Pauly, 1978b, 1980a.) These various interrelationships can be expressed for length-growth data by the multiple regression

log _{10}M = -0.0066-0.279 log_{10}L_{∞} + 0.6543 log_{10}K + 0.4634 log_{10}T | (35) |

and for weight-growth data by

log M = -0.2107-0.0824 log _{10}W_{∞} + 0.6757 log_{10}K + 0.4687 log_{10}T | (36) |

Where M is the natural mortality in a given stock, L_{∞} (total length in cm) and W_{∞} (live
weight in g) being the asymptotic size of the fishes of that stock, and K their growth
coefficient. The value of T, finally, is the annual mean temperature (in °C) of the water
in which the stock in question lives, as obtained, e.g., from an oceanographic atlas.

These two regressions were derived from 175 sets of L_{∞} (or W_{∞}), K, T, and M values,
obtained from a survey of the pertinent literature (Pauly, 1980a) and apply, in this form,
to temperatures ranging from 5° to 30°C, and provide useful fish estimates of M, which
should, of course, be confirmed where possible by analyses on the individual species in
question.

Although Equations 35 and 36 generally give reasonable estimates of M for about any set of growth parameters and temperature value, there is a group of tropical fishes in which the estimates may be biased, namely the strongly schooling pelagic fishes - especially the Clupeidae - the natural mortality of which is generally overestimated by these equations. Thus, in their case, it might be appropriate to reduce the estimate of M somehow, e.g., by multiplying it by 0.8.

__Assessing the State of a Stock from its Mortality Rates Alone__

Once values of F and M are available, an exploitation ratio (E) may be computed from

which allows one to (roughly) assess if a stock is overfished or not, on the assumption that
the optimal value of E (E_{opt}) is about equal to 0.5, the use of E ≈ 0.5 as optimal value for
the exploitation ratio itself resting on the assumption that sustainable yield is optimized
when F ≈ M (Gulland, 1971).

Exercise: | (a) | Depict the decay of a cohort of fishes (N_{o} = 100 000) with M = 0.5 and
F = 1.0 with exploitation starting at age 0.5 year. |

(b) | Calculate values of Z from the data in Table 10 using Equations 21 and 22, and use these estimates to obtain a value of M and q by means of Expression 27. | |

(c) | Compare the value of M obtained in b with a value of M obtained from
Expression 35 using L_{∞} = 29, K = 1.2 and T = 28° C. | |

(d) | Use the data of Table 10 to draw a catch curve using Equations 26a and 32, with
t_{o} = 0, to convert the midlength values to estimates of relative age, and
estimate Z. Estimate M by means of Expression 35 and the appropriate growth
parameters. Compute F and E and assess whether the stock in question is
optimally exploited or not. |

Year | Effort^{2} | L^{3} | n |
---|---|---|---|

1966 | 2.08 | 15.7 | 12 370 |

1967 | 2.80 | 15.5 | 14 231 |

1968 | 3.50 | 16.1 | 10 956 |

1969 | 3.60 | 14.9 | 9 738 |

1970 | 3.80 | 14.4 | 12 631 |

1973 | 9.94 | 12.8 | 9 091 |

1974 | 6.06 | 12.8 | 15 229 |

^{1} Based on data in Boonyubol and Hongskul (1978) and SCS (1978)

^{2} In million trawling hours

^{3} To be used in conjunction with L_{∞}=29, K=1.2 and L_{c}=7.6

Class midlength^{2} | N | Class midlength^{2} | N | |
---|---|---|---|---|

7.5 | 11 | 17.5 | 428 | |

8.5 | 69 | 18.5 | 338 | |

9.5 | 187 | 19.5 | 184 | |

10.5 | 133 | 20.5 | 73 | |

11.5 | 114 | 21.5 | 37 | |

12.5 | 261 | 22.5 | 21 | |

13.5 | 386 | 23.5 | 19 | |

14.5 | 445 | 24.5 | 8 | |

15.5 | 535 | 25.5 | do not use | 7 |

16.5 | 407 | 26.5 | 2 |

^{1} Based on Ziegler (1979)

^{2} To be used in conjunction with L_{∞} = 29.2, K = 0.607 and T = 28°C; length = total length in