Fish Population Dynamic and Stock Assessment

4.1.4.2. Estimation of ‘Z’ by age composition data

(a) Heinchke’s method

This method is the oldest method. This method could be applied for short lived species (fish or shrimp or exploited aquatic animal) whose life span is more or less 2 years. Further ‘0’ year class should constitute the fully recruited group.

$$A={ N_0 \over {\sum N}}$$………………………….. (1)

N₀ is the number of 0 year old fish and $$\sum$$N is the total number of fish comprising all age groups.

From the annual mortality rate, S could be calculated as S = 1 – A. .................... (2)

Z could be estimated by the following equation,

S = e^-Z ............................................................ (3)

(– Z =$$\log_e$$ S ; Z = - $$\log _e$$ S)

(b) Jackson method

In this method, ‘Z’, could be estimated between two adjacent age groups in a given year or between the abundance of the same year class in two adjacent years using the following formula.

  S =$$Nt+1 \over Nt$$ ………………………………. (4)

Since the annual mortality rate (A) is the complement of S i.e., A = 1 – S, ‘Z’ could be estimated by passing equations 3 and 4.

 Z=$$-\log_e {Nt+1 \over Nt}$$ or Z=$$\log_e {Nt \over Nt+1}$$ …………………….. (5)

(c)  Estimation of Z (if more than 3 age groups are represented in the fishery).

If in a species more than 5 age groups (year classes) are represented in the fishery, the following formula is used. 

        S= $${{N_1 + N_2 + N_3 + N_4 + N_5} \over {N_0 + N_1 + N_2 + N_3 + N_4 + N_5 }}$$........………………… (6)

     Z = -$$\log_e$$ S

(d)  Robson and Chapman’s method

            The ‘Z’ could be estimated from age composition data i.e., the numbers caught per age group.

       Z = - ln$${A \over {B+A-1}}$$ ……………………………………. (7)

         Where,

            A  =   N1 + 2*N2 + 3*N3 + 4*N4 + 5*N5+………………

            B  =   N0 + N1 + N2 + N3 + N4 + N5+………………….

(e)       Cushing method

When incomplete age composition data are available for a fish stock, for example only 2nd and 4th year age groups are available, Z could be calculated from the following relation.

$$Z$$ = $${1 \over 3}\times \log_e {{N_2} \over {N_4}$$........................................................................................ (8)