Estimation of growth The study of growth means the determination of the body size as a - PowerPoint Presentation

1. Estimation of growth

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3. The study of growth means the determination of the body size as a function of age. Therefore, all stock assessment methods in fisheries work essentially with the age composition data.

4. Individual growth mode1Growth in length can be modeled in a number of ways. The model most frequently used in fisheries was developed by von Bertalanffy (1957). The growth equation is referred as von-Bertalanffy’s equation (VBGF).Lt = L [ 1 – e –K(t – to)] Where, Lt = length of the fish at time ‘t’ L = asymptotic length. – this is the theoretical length beyond which fish does not grow. K = catabolic growth coefficient –rate at which length approaches L t = Age of the fish. t0 = Theoretical age at which length of the fish would have been zero.In the above equation L , K and t0 are called growth parameters.

5. Methods for estimating growth parametersFord – Walford Method This is a simplest method of estimating the growth parameters for the length data representing equal time intervals. This method is widely applied as the plot could be used to obtain a quick estimate of L and K without calculation also. From the von Bertalanffy’s growth equation it follows a series of algebraic manipulations.Where, a = L∞ * (1-b) and b = exp (-K * Δt) K = - lnb; L∞ = (a /(1- b))L (t + Δt) = a + b*L(t)X = L(t) Y = L (t + Δt)

6. Draw graph using Lt and Lt+1 as ‘X’ and ‘Y’ values respectively and draw a line connecting maximum points (at least three points must be on the straight line).Draw another straight line from the origin at 45° till it meets the first straight line.Mark the point where two lines cross each other and draw a perpendicular to X – axis.Mark L on X – axis.Draw a right triangle on the first straight line and measure tan which gives ‘b’ value.Substitute the ‘b’ value in the equation K = -ln b and estimate ‘K’.

7. Chapman’s method: Chapman (1961) and later Gulland (1969) is again based on constant pairs of intervals. From von Bertalanffy’s growth equation is implies that L (t + Δt) – Lt = C * L∞ - C*L(t)Since K and L∞ are constants, if Δt remain constant and it becomes a linear equation i.e. y = a + bx Where, a = C* L∞ and b = - C C = 1 – exp (-K *Δt) X = L(t) Y = L (t +Δt) – L(t) K = - (1/Δt)*ln(1+b) L∞ = - a/b or a/c

8. Draw graph using L(t) as ‘X’ and L (t +Δt) – L(t) as ‘Y’ values.The downward slope that touches on X – axis indicates L.Draw a right triangle on the slope and mark  angle. Tan  will gives ‘b’ value.Substitute ‘b’ value in the formula K = - (1/Δt)* In ( 1 + b) and estimate ‘K’

9. Gulland and Holt methodThis method is also used for estimating L and K growth parameters. This method is mainly used while dealing with data of variable time intervals between samples. X=Lt = L(t+1) + Lt / 2 Y = ΔL/Δt = [L(t+Δt) – Lt] /[(t+Δt) –t]L = - a/b K = -ba = K*L and b = -K

10. The intersection point between the regression line and the X-axis gives L ∞ .Draw graph using above ‘X’ and ‘Y’ values.Downward slope touching X – axis indicates L.Draw right angle triangle on the slope and mark  angle.Substitute tan  in the formula K = -b and estimate ‘K’.

11. The von Bertalanffy’s plotIt can be used to estimate K and to from age/length data which requires the equation to be rewritten as -ln (1 – Lt/L∞) = - K * to + K*tWhere ‘t’ is the independent variable and the left hand side of the equation as dependent variable, i.e. X=t and Y= -ln (1 – Lt/L∞) slope b = K and intercept a = - K * to K = b t0 = -a/b

12. Draw graph using above ‘X’ and ‘Y’ values.Upward slope, which crosses on X – axis indicates ‘t0’ value.Draw a right angle triangle on the slope and mark  angle.Substitute tan in formula K = b.

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