Fish Population Dynamic and Stock Assessment

 4.1.4.1. Estimation of Z by exponential equation

            In fishery biology, the most useful manner of expressing the decrease of the age group of fishes through time is by means of exponential equation. For example, if we put a group of fish in a pond and sampled the fish over a long period of time, the number of survivors of the original group decline over time as shown in the following figure.

            This can be modeled in the following equation.

            N_t = N₀ e^-Zt  .........................................................  (1)

Where            N_t        = Number of survivors after ‘t’ units of time

                        N₀        = Original number of fish

                        e          = Constant equal to 2.7182818

                                       (e is the base of natural logarithm)

                        Z          = Total instantaneous mortality rate in units of time -1 (eg 1/yr.)

N_t        = the number of fishes remaining at the end of time ‘t’. 

‘Z’ is most often reported in units of (yr) -1.  It is also reported in other units such as 1/month.  To convert ‘Z’ to units of yr-1, the equation (1) must be in units of months if ‘Z’ is in units of 1/month.  If ‘t’ is used as units of one year, then the number of time units is one twelfth (1/12) as large.  (12 units of 1 month would become 1 unit of 1 year).  So one should multiply Z by 12 in order for the answer (Nt) to stay the same.  For example Z is 0.2 / month, the Z in units of Yr-1 will be 2.4 yr-1 (12 x 0.2 = 2.4 yr-1).  

a) Annual survival rate (S)

            The proportion of the population that will survive after one unit of time has passed is known as annual survival rate (if time is expressed in years).  The annual survival rate is denoted by ‘S’.  The relationship between annual survival rate (S) and the total instantaneous mortality rate (Z) is described by the following notion.

            S =e^-Z  ……………………………………………………… (2)

b) Annual mortality rate (A)

            The proportion of the population that does not survive is known as ‘annual mortality rate’ if time is expressed in years.  It is designated by ‘A’.

            A = 1 – S ..........................................................................  ( 3)

            The annual mortality rate (A) and total instantaneous mortality rate (Z) will be more or the less the same to begin within a stock, but they progressively diverge as the mortality rate increases.