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3.2.4.4. Bagenels’ least square method
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 to
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, ‘ to’ is written as
to =$${ a - log_{e} L_\infty \over -b}$$
Which is of the simple linear form (y = a + bx)
‘to’ could be estimated algebraically.
to = $${1 \over k}\{(log e { L_\infty - Lt \over L_\infty}+ Kt\}$$
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 eb
Step (2): To find out ‘to’ graphically, take ‘t’ as ‘x’ and ‘$$\log_{e}^{{w_\infty}-w_t}$$ ’ as ‘y’
to =$${a-log_e^{w_\infty^{1/3}}}\over{-b}$$
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 ‘to‘. (This method is given as problem in Practical Section)
Estimation of to
The Gulland and Holt plot does not allow for estimation of the third parameter of the VBGF, ‘to’. This parameter is necessary. A rough estimate of ‘to’ may be obtained from the empirical relationship.
$$\log_{10}\left(-t_{0}\right)$$= - 0.3922 - 0.2752 $$\log_{10}$$ $$L_{\infty}-1.038 \log_{10}k$$