3.1 Growth by length and weight
3.2 Conversion of length to age
3.3 Exercises
For population analysis it is desirable to express the growth of fish in a mathematical expression. The basic requirement is an expression which will give the size (in terms of length or weight) at any given age which agrees with the observed data of size at age, and which is in a mathematical form which can be incorporated reasonably easily in expressions for yield. Strictly, most population analysis is concerned more directly with growth rate, i.e. increase in weight or length per unit time, rather than with the size at various ages, because many problems in fishery assessment are essentially a matter of comparing weight gained by growth against that lost by natural mortality. Sometimes, for instance when considering the effect of an increase in size of first capture, it is particularly important to know the rate of growth over a comparatively short part of the total life span  that is, to know how long it will take a fish to grow from the original size of first capture to the new. There are therefore good reasons for preferring, other things being equal, a method of fitting equations to growth rate data, rather than simply size at age.
Other desirable features of a growth equation are that the computational work involved in fitting it to observed data should be small, that the number of constants used should be few, that as far as possible these constants should have some biological meaning, and that if extrapolated to ages beyond those used in fitting the equation, the equation should not lead to unreasonable results.
There exists a considerable and growing literature on growth equations, which covers a wide range of possible equations, none of which seems to be entirely satisfactory in all situations. In fact it is most unlikely that a simple formula would always be able to describe the growth of even a single fish through most of its life, in which there could be greatly different conditions of food supply, reproductive strain, etc. These notes do not attempt to give a complete survey of growth equations, but will be mainly concerned with one particular equation, that ascribed to von Bertalanffy (1938), which satisfies the two most important criteria  it fits most of the observed data of fish growth, and can be incorporated readily into stock assessment models.
If the length of a fish, crustacean, or lamellibranch is plotted against age, the result is usually a curve of which the slope continuously de creases with increasing age, and which approaches an upper asymptote parallel to the Xaxis (see Figure 3.1.1). Curves of weight at age also approach an upper asymptote, but usually form an asymmetrical sigmoid, the inflexion occurring at a weight of about one third of the asymptotic weight (see Figure 3.1.2).
FIGURE 3.1.1.  Curve of growth by length.
FIGURE 3.1.2.  Curve of growth by weight.
If the rate of growth in length is plotted against length, the result is often well fitted by a straight line, cutting the Xaxis at a point L_{¥ } beyond which the fish will not grow. This is of course the asymptote of the plot of length on age. If the rate of growth in length is linearly related to length, then in mathematical terms
_{} ................... (3.1)
where L_{¥ } is the value of l for which the rate of growth is zero. This equation is in the general form of a straight line relation, y = ax + b, with a =  K, b = K L_{¥ }. To integrate this differential equation, we write it in the form
_{} (see section 2)
and therefore  log (L_{¥ }  l) = Kt + constant
or L_{¥ }  l = Kt + constantor l = L_{¥ }  constant × e^{Kt}
Putting t_{0} as the theoretical age at which l = 0 we can obtain the value of the constant from the relation
_{}
then
_{} ................. (3.2)
Equation (3.1) is identical with that deduced by von Bertalanffy (1938) on physiological grounds. He considered that growth in weight was the resultant of the difference between anabolic and katabolic factors, taken as proportional to surface area and weight respectively
_{}
or, putting w µ l^{3}, s µ l^{2}
_{}
In fitting the growth curve the instantaneous rate of growth will not be known, but only lengths at certain times  often one year apart when they are derived from age determination, but at irregular intervals if derived from tagging data. If the times t_{1}, t_{2} are close together, a close approximation to the instantaneous rate of growth is given by _{}, where l_{1}, l_{2} are the lengths at times t_{1}, t_{2} respectively. If this growth rate is plotted against the mean length_{}, then a plot corresponding to equation (3.1) is obtained; the intercept on the Xaxis gives an estimate of L_{¥ }, and the slope an estimate of  K.
For data at equal time intervals of length T, from equation (3.2)
_{}_{}
_{}
l_{t+T}  l_{T} = (L_{¥ }  l_{t})(1  e^{KT}) ............. (3.3)
The plot of increment l_{t+T}  l_{T}, against initial length, l_{t}, therefore gives a line, slope  (1  e^{KT}), and an intercept, on the Xaxis, of L_{¥ } (see Figure 3.2.1). The important special curve is T = 1 year, when the slope =  (1  e^{K}). An alternative form of equation (3.3) is
l_{t+T}  l_{T} = L_{¥ } (1  e^{KT}) + l_{t}e^{KT} ..............(3.4)
which, for T = 1, is the wellknown FordWalford plot, of l_{t+i} against l_{t}, which gives a straight line, slope e^{K}, and an intercept on the 45° line, where l_{t} = l_{t+}_{1}, of L_{¥ } (see Figure 3.2.2). This plot is essentially the same as the plot of increment (l_{t+1}  l_{t}) against initial length. The points will seem to fit the FordWalford line better, but because the intersection of the regression line and the 45° line is very oblique, L_{¥ } is in fact estimated with equal precision by the two plots, and if the fitting is done graphically rather than by calculating regression lines, e.g. by least squares, greater errors in drawing are likely to be introduced in the plot of l_{t+1} against l_{t}.
FIGURE 3.2.1.  Estimation of growth constants. Plot of length increment against initial length.
FIGURE 3.2.2.  Plot of length against length one year earlier (FordWalford plot).
L_{¥ } and K can be determined directly from the lines fitted to these plots either by regression analysis, or by eye; then t_{0} can be estimated from equation (3.2) for any particular observation of length at age. For this, equation (3.2) is best rewritten as
_{}
or
_{}............... (3.5)
While an estimate of t_{0} can be thus obtained for each age for which the mean length is known, these estimates will not be equally good. Those from old fish will be highly variable, because a small difference in l_{t} makes a big difference to the estimate of t_{0} when l_{t} is nearly equal to L_{¥ }, while the mean length of the youngest fish may be biased because only the bigger fish of these ages appear in the catches. The best estimate of t_{0} would appear to be the mean of the estimates of t_{0} obtained from the younger, but fully recruited, age groups.
Alternatively, only L_{¥ } is estimated from the plots of increment against initial length. From this _{} can be calculated, and plotted against t. From equation (3.5) this should give a straight line, slope  K intercept on the taxis equal to t_{0}.
Data on length at annual intervals can be obtained and tabulated in various ways. Probably the most straightforward is the analysis of the growth of a single fish  the length at the end of each year of life being obtained from measurements along an axis of a scale, otolith or other hard structure. Alternatively, the growth of a single yearclass can be followed over a period of years  i.e. for the 1956 yearclass the increment in the fifth year of life is the difference between the mean length of fiveyearold fish at the end of 1961 and fouryearold fish at the end of 1960. Both these methods are concerned with the growth of a group of fish during their life, the growth during each year of life providing one point in the plot. Thus the growth increments refer to different calendar years; for example for the 1956 yearclass the increment in the third year of life is put on in 1959, possibly under conditions very different to those in 1961, the fifth year of life. Growth in a particular year can be studied by taking the growth of different yearclasses  e.g. data of growth in 1960 are given by the growth of the 1956 yearclass in its fifth year of life, of the 1957 yearclass in its fourth year of life  each yearclass provides one point in the plot. This is a particularly useful method for studying the effect of the environment  food, temperature, density, etc. on growth. Finally, growth can be estimated from a single year's data, e.g. from data in 1960 the increment in the fifth year of life can be estimated from the difference in lengths of the 1955 and 1956 yearclasses, the fourth year's increment from the difference between the 1956 and 1957 yearclasses, etc. These differences do not, in fact, correspond to the growth of any particular fish or group of fish, and the method should be used only when other methods cannot. An improvement is to take the average size at each age over a number of years, but in this case it is usually possible, and more satisfactory, to analyse the growth of the individual yearclasses or in each calendar year.
The weight of a fish is usually closely proportional to the cube of its length, so that from equation (3.2)
_{} ................ (3.6)
where W_{¥ } is the asymptotic weight corresponding to the asymptotic length L_{¥ }.
Similarly, if weight is proportional to the nth power of the length, _{}.
When growth data are given in terms of weight, fitting of growth curves is most easily done by using the cuberoot of the weight as an index of length, fitting this to the equations of growth in length as above, and finally cubing to return to weight.
A more general growth equation has been given by Richards (1959); with a slight change from his notation this is
_{}.............. (3.7)
For various values of m this equation becomes one or other of the common growth equations. Thus if _{}, equation (3.7) becomes the same as equation (3.2) for the von Bertalanffy curve. If m = 2 the equation, with some rearrangement, becomes the autocatalytic equation
_{}
and for m = 0, the monomolecular equation
w_{t} = W_{¥ } (1  ae^{kt})
and it can be shown that in the limit as m ® 1, the equation becomes the Gompertz log w_{t} = log W_{¥ } (1  ae^{Kt}). This curve satisfies a relation similar to that of equation (3.1), only in terms of log length rather than length, i.e.
_{}
The expressions above give the weight or length of a fish in terms of its age. Sometimes, the inverse procedure is required, i.e. it is desired to know the age of fish of a given length, e.g. selection data are normally in terms of length, but for incorporation in yield equations need to be expressed in terms of age.
If the seasonal growth pattern is very marked, and all fish of a particular age are nearly the same length, the actual mean age of fish of a given length may be quite different from the mean age determined from the average annual growth curve, particularly at the beginning and end of the season of rapid growth. If this is the case, the conversion of length to age is best made empirically from the observed agelength curve. Also, particularly if there is much individual variation in growth, the curve of mean length at a given age will be different from the curve of mean age at a given length (just as generally the regression line of y on x is different from the regression line of x on y). Usually, however, it is sufficient to convert length to age from a growth equation fitted to all the observed data of mean length at age, e.g. the von Bertalanffy equation
_{}
To obtain t in terms of l we divide both sides by L_{¥ }, and subtract from unity, giving
_{}
Taking natural logs of both sides gives
_{}
and therefore
_{}................. (3.8)
The span of age t_{2}  t_{1} between two lengths l_{1}, l_{2} is therefore
_{} ................. (3.9)
1. The von Bertalanffy growth equation contains three parameters (L_{¥ }, K and t_{0}), which can therefore be estimated from three observations of length at age. From equation (3.2) write down the lengths at times t  1, t, t + 1, and the increments in length between t  1 and t and between t and t + 1. Hence determine an estimate for K from the ratio of these increments; by substituting this estimate for K in equation (3.2), or otherwise, find an estimate for L_{¥ }, in terms of l_{t} and the increments.
2. Kitahama (1955) gives the following sizeatage data for the Pacific herring, Clupea pallasii C. and V., caught off the western coast of Hokkaido from 191054. The lengths are averages over a fortyfive year period; ages determined from scale rings.
Age 
Total length 
years 
cm 
3 
25.70 
4 
28.40 
S 
30.15 
6 
31.65 
7 
32.85 
8 
33.65 
9 
34.44 
10 
34.97 
11 
35.56 
12 
36.03 
13 
35.93 
14 
37.04 
15 
37.70 
W = 7.8 × 10^{33} L^{3} g(a) Construct a worksheet and determine L_{¥ }, W_{¥ }, K and t_{0}.
(b) Calculate the theoretical curves of growth in length and weight over the range 015 years.
3. Yokota (1951) gives the following data for growth of the shark Mustelus kanekonis (age determined from seasonal changes in length of claspers).
Age 
Weight 
groups 
g 
I ............... 
375 
II ............... 
1 519 
III ............... 
2 430 
IV ............... 
3 247 
W = 7.4 × 10^{3} L^{3} g(a) Determine L_{¥ }, W_{¥ }, K and a working value for t_{0}.
(b) Calculate curves of growth in length and weight up to age group VIII.
4. Posgay (1953) tagged sea scallops, and obtained the following data on growth during the ten months (approximately) between tagging and recapture.

SHELL DIAMETER  

mm  
At tagging ................ 
64 
69 
71 
94 
104 
105 
110 
117 
117 
126 
At recapture .............. 
98 
102 
93 
115 
120 
126 
125 
127 
136 
138 
(a) Calculate L_{¥ }, and K.(b) Taking t_{0} = 0, calculate and plot the curve of growth in length up to six years.
5. Ochiai (1956) gives the following data for the growth of the soles Heteromycteris japonicus and Pardachirus pavoninus.
Age 
Standard length  

H. japonicus 
P. pavoninus 
Years 
cm  
1............... 
6.09 
10.00 
2............... 
8.10 
14.80 
3............... 
9.20 
(a) Determine arithmetically L_{¥ }, K and t_{0} for each species (assume t_{0} = Q for P. pavoninus).(b) Demonstrate the patterns of growth in weight, assuming a cubic relation of weight to length.
6. Moiseev (1946) gives the following data for growth (total length in cm) of the flatfish Limanda aspera in Peter the Great bay, Japan sea.
Age

Sex

SAMPLING YEAR 

1928 
1930 
1931 
1932 
1933 

1


5.15 
7.60 
6.58 
7.72 
8.34 

5.77 
7.22 
7.20 
9.50 
7.34 

2


10.25 
13.60 
11.70 
13.80 
14.10 

11.77 
11.88 
13.20 
15.50 
12.68 

3


16.40 
19.40 
17.42 
18.82 
19.44 

17.41 
19.00 
18.80 
21.00 
18.68 

4


20.45 
24.00 
21.22 
22.92 
24.32 

22.13 
25.00 
23.60 
25.50 
25.32 

5


25.05 
26.80 
25.42 
26.38 
27.66 

26.51 
26.12 
27.60 
29.50 
27.32 

Index of total demersal catch..... 
1 
1 
5.5 
8.0 
7.7 
(a) Analyse sexual differences in growth.(b) Bearing in mind that largescale fishing was started in the bay only in the season 192930, and that L. aspera formed 80 percent of the demersal catch, examine the yeartoyear variation in growth. The mean age composition by number, during 193133, was as follows:
Age 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
Percent 
0.6 
4.7 
16.1 
33.3 
27.9 
11.3 
3.4 
1.7 
0.9 
0.2 
(c) Estimate the total mortality coefficient and examine the growth data for bias in mean length at age, due to selection.(This part of the exercise is to be completed after sections 5 and 7.)