ESTIMATION OF GROWTH RATE IN ANIMALS BY MARKING 



EXPERIMENTS 



By Milton J. Lindner, Fishery Research Biologist 



Effective action to conserve an animal resource 

 must be based on such vital statistics as age com- 

 position of the stock and rates of mortality, re- 

 placement, and growth. In some species of ani- 

 mals, but unfortunately not in all, age and growth 

 are indicated by the occurrence and spacing of 

 rings on such hard parts as scales, bones, and shells. 

 In studying shrimp (Penaeus setiferus) along the 

 southern coast of the United States I found the 

 problem of age determination particularly diffi- 

 cult, since no skeletal structures are carried over 

 from one molt to another. Researchers studying 

 other Crustacea report the same difficulty, with the 

 result that growth of such important species as 

 lobsters and crabs is imperfectly known. 



The method for estimating growth described 

 here can be used wherever growth follows geo- 

 metric progression. It can be used in laboratory 

 feeding experiments and with animals of unknown 

 ages, and it is particularly useful when the age of 

 animals cannot be ascertained by any other means. 

 Although I stress the growth aspect in this paper, 

 it is obvious that the method can be used also for 

 determining age (only within the limits for which 

 the technique is valid for growth) simply by plot- 

 ting the results in the usual manner with size 

 against time. 



For equal time intervals Walford (194G) has 

 demonstrated for many animals that growth, 

 above the point of inflection on the curve of abso- 

 lute growth, may be plotted as a straight line. 

 This is accomplished by plotting the length at 

 ages 1, 2, 3, 4 ... on the A' axis against length 

 at ages 2, 3, 4, 5 . . . n + 1 on the Y axis. The 

 slope of this line is k. On the line, the length rela- 

 tions are such that: 



t - l ~ kn T 



■'-'n — i h'-^n + l- 



-U 1 _ k k >->—L> 



and Z„ 



L x 



l-k 



L a represents the ultimate length, which is also 

 the point where this transformed growth line in- 

 tersects the 45° line, or where X= Y. 



Walford's transformation, which is based on the 

 sizes of animals of known ages, can be modified to 

 determine the growth rate of animals of unknown 

 ages. This can be done because the time intervals 

 are uniform or constant. As a consequence, if we 

 take the lengths of a group of animals of unknown 

 ages and of varying sizes on a certain day and 

 measure these same animals again 1 year later, we 

 shall have their respective lengths at age n and at 

 age n+1 year. Plotting lengths n against n+1 

 will result in Walford's transformation for one 

 time interval, in this instance, 1 year. If we 

 should measure these same animals 2 years later 

 and plot them in the same fashion (n against 

 ri +2), we should have a growth line representing 

 the increment for 2-year intervals. From the 

 relations between these lines we can arrive at the 

 growth rate of an animal for any time interval we 

 choose. The time interval, of course, must be 

 constant. 



A graphical representation of the relation be- 

 tween transformation lines obtained by plotting 

 n against n+1, n+2, n+3, and so on, is shown hi 

 figure 1. Line A represents the transformed 

 growth for one time interval, or Walford's trans- 

 formation (n against n+1), line B for two time 

 intervals (n against n+2), line C for three inter- 

 vals in against ?i + 3) and so on. 



In each instance, the length of the horizontal 

 lines extended from Z 2 to intersect line A, from L 3 

 to line B, from L K to line C, and so on, are equal. 

 The length of each line is also equal to In. Con- 

 sequently, if we know the sizes of a number of 

 animals of different ages on a certain date and can 

 measure these same animals (or representative 

 samples of them) at successively equal later in- 

 tervals of time, we can determine their growth line 

 (Walford line) for the time interval chosen. An 



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