Fish Population Dynamic and Stock Assessment

4.1.4.4.1. Estimation of Z by means of a catch curve

This method is usually referred to as the length converted catch curve or linearized length converted catch curve. In this method, Z is estimated by sampling a multiaged population of fishes. Plotting the natural logarithm ($$\log_e$$ ) of the number of fishes in the sample (N) against their respective age (t) is called catch curve, and this is of the following equation.

$$\log_e$$N = a + bt ............................................... (1)

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

Requirements to be met by ‘b’ to be a good estimator of Z

According to Pauly, 1984, the requirements to be met by ‘b’ to be a good estimator of Z should include the followings:

• The values of $$log_e$$N should include the age group of fishes fully vulnerable to the gear in question and this corresponds to using only the descending part of the catch curve and it should be larger than the L' . (Smallest length of animals that are fully represented in the catch samples).
• All age groups used in the plot are recruited with the same abundance or the variations in recruitment fluctuations should be small and of random character.
• All age groups used in the plot should be equally vulnerable to the gear used for sampling.
• The samples used should be large and it should cover all age groups and it should represent average population structure over the period considered.
• Inclusion of fish whose size close to that of their asymptotic size must be avoided as this will result in their age being closely overestimated.
• Since ‘Z’ is equal to slope (with sign changed) of the catch curve, the real age which requires an estimate of T₀ can be replaced by relative age i.e. by setting ‘t’ = T₀.

The equation - 1 can be rewritten as

$$log_e$$(N/D t) = a + bt

Where, $$\triangle$$t is the time needed to grow from lower ($$t_1$$ ) to the upper ($$t_2$$ ) limit of a given length class.

t = relative age corresponding to mid range of the length class in question.

Input data needed

a) LFD data and the catch in numbers in the respective length group.

b) Values of L $$\infty$$ and K needed for catch curve analysis, estimated from growth equations.