Fish Population Dynamic and Stock Assessment

3.2.4.4. Bagenels’ least square method

Growth in length

The VBGF equation for growth in length is

$$Lt = L_\infty [1 - e^{-K(t-t_0)]$$ ........................................... (1)

The equation (1) can be rewritten

$$Lt+1 = L_\infty [1-e^{-K}+ e^{-K}Lt]$$ ..................................... (2)

The equation 2 gives a linear regression of Lt+1 on Lt of the type

Lt+1 = a + b Lt ...................................................... (3)

Where $$a = L_\infty (1- e^{-K})$$ and $$b = e^{-K}$$

'a' and 'b' are estimated using the linear regression equation ( y = a + bx)

Step (a): To estimate $$L_\infty$$ and Kn

Transformation of length at age data into Lt and Lt + 1

(Lt as x and Lt +1 as y)

Apply linear regression analysis ( y = a + bx)

Step (b): To estimate t_o

Take age in years as 'X' and $$log_{e} L_\infty - Lt$$ as ‘Y’

Apply linear regression analysis (y = a + bx)

In the graphic method, ‘ t_o’ is written as

t_o =$${ a - log_{e} L_\infty \over -b}$$

Which is of the simple linear form (y = a + bx)

‘t_o’ could be estimated algebraically.

t_o = $${1 \over k}\{(log e { L_\infty - Lt \over L_\infty}+ Kt\}$$

Growth in weight

The VBGF equation for growth in weight is

Wt = W $$_\infty$$ [(1- e ^{-K (t – t₀)} ]³……………………………… (1)

The equation (1) can be rewritten as

$$w_t^{1/3}+1$$ =W$$_\infty^{1/3}\left(1-e^{-k}\right)+e^{-k}w_t^{1/3}$$ …………………………….. (2)

The equation 2 gives a linear regression of Wt + 1 on Wt of the type.

$$w_t^{1/3}$$= a + b $$w_t^{1/3}$$………………………………………………. (3)

Where, a =$$w_\infty^{1/3}\left(1-e^{-k}\right)$$  and

b = e ^-K

Step (1): To find out ‘a’ & ‘b’ to arrive at $$w_\infty$$ & K.

Applying simple linear regression to obtain ‘a’ and ‘b’ for the values of Wt (x) on Wt+1 (y) in the data.

$$w_\infty^{1/3}$$=$$a\over{1-b}$$

K = - log e^b

Step (2): To find out ‘t_o’ graphically, take ‘t’ as ‘x’ and ‘$$\log_{e}^{{w_\infty}-w_t}$$ ’ as ‘y’

t_{o =}$${a-log_e^{w_\infty^{1/3}}}\over{-b}$$

To calculate growth parameters for weight based age data using von Bertalanffy equation, Wt and Wt +1 are converted to respective cube root equivalents. Once this is accomplished, the procedure mentioned in the different methods for the growth in length could be used for estimation of $$w_\infty$$ , K and ‘t_o‘. (This method is given as problem in Practical Section)

Estimation of t_o

            The Gulland and Holt plot does not allow for estimation of the third parameter of the VBGF, ‘t_o’.  This parameter is necessary. A rough estimate of ‘t_o’ may be obtained from the empirical relationship.

$$\log_{10}\left(-t_{0}\right)$$= - 0.3922 - 0.2752 $$\log_{10}$$ $$L_{\infty}-1.038 \log_{10}k$$