A New Approach To Analysis Of The Growth Data: Short-term Parameterization
A NEW APPROACH TO ANALYSIS OF THE GROWTH DATA: SHORT-TERM PARAMETERIZATION
OF THE GROWTH EQUATION
V.N. Zolotarev
Odessa Branch Institute of biology of Southern Seas, National
Academy of Sciences of Ukraine
Analysis of the animal growth data is often used in revealing organism response to environmental changes. The size-age relations in animals have been usually expressed by the several growth curves, from which von Bertalanffy curve, Gompertz curve and Brody exponential curve are most frequently used, and the methods calculating for parameters of the growth equations from the data have been described (Walford, 1946; Allen, 1966; Rafail, 1973; Ricker, 1975; Shnute, Fournier, 1980).
The growth equation coefficients established by traditional methods have been usually interpreted as the population characteristics that averaged for the growing period studied. However, in many cases it is necessary to reveal the growth parameters for the short time space, for instance for individual annual or seasonal increments. In this paper, some approaches to estimation of the short-term growth parameters are represented for the von Bertalanffy curve that is well known and most frequently used.
The von Bertalanffy growth equation is usually represented in following form:
(1)
where Lt is the length at time t, L∞ means the asymptotic size, t0 is a constant representing time when Lt = 0, K is a constant defining the rate at which the asymptotic length is approached.
For short-term parameterization the following version of the von Bertalanffy growth model is suitable:
; i = 1, 2, … , M (2)
In this equation M is number of the age classes, i represents the number (from 1 to M), Li is the length at age ti (ti = tx + i – 1, where tx is the age of the first age class), L1 and LM are the first and final lengths at ages tx and tM respectively; k is the fraction at which the distance between two successive lengths shrinks each year (Shnute, Fournier, 1980).
The generalized form of equation (2) is proposed for unequally spaced data (Ratkowsky, 1986). It may be represented as follows:
(3)
where B = (ti – t1)/(tM – t1), M is the number of data points. The generalization (3) collapses to the equation (2) when time intervals represent equally spaced data.
The equation (1) and (2)-(3) are actually two expressions for the same von Bertalanffy growth curve with different sets of parameters. The first uses L∞, K, t0, while the second involves L1, LM, k, and each set of parameters can be transformed to another (Shnute, Fournier, 1980; Ratkowsky, 1986).
The parameters of the equations (2) and (3) are more appropriate to short-term transformation of the growth curve. In many cases the first the first and last means of the body length are known, as well as at least one further Li for the some age ti. The values of L1, LM and Li can be substituted into (2) or (3) and these equations can be solved for k.
The authors of parameterizations (2) and (3) suppose that it is not possible to express analytically a general solution for k in above-mentioned equation, but an initial estimate of k may be obtained by the trial-and-error method (Shnute, Fournier, 1980; Ratkowsky, 1986). However, after simple transformation, equation (2) can be represented as follows:
(4)
where A = (Li – L1)/(LM – L1). This equation has one positive solution for k. If (LM – Li)/(M – i)
New parameterization (4) can be solved for k by the iterative method calculating its successive approaches:
; n = 0, 1, 2, …
The iterative calculations are carried out till the difference between kn and kn+1 will not exceed required accuracy of the estimation for k. To transform parameters of (2) to parameters of (1) the following relations can be applied (Shnute, Fournier, 1980):
(5)
(6)
(7)
Similarly, a new transformation of the equation (3) is
(8)
where B = (ti – t1)/(tM – t1). The equation (8) is solved by the iterative method as in the case of (4). The transformation of the parameters of (8) to parameters of (1) is given by (5) for L∞ and by the following relations for K and t0 (Ratkowsky, 1986):
New parameterization of the von Bertalanffy curve can be applied for the different relations of initial (L1), maximal (LM) and intermediate (Li) lengths. However, when the distance between Li and LM is the annual increment, the coefficient k estimated by the equation (4) or (8) will arrange calculated values of increments for each year in such a fashion that observed and calculated increments will be equal to the last year. In means that estimation of the growth characteristics from data of the last year by the proposed method is the way to short-term parameterization of the growth curve for observed annual increments.
If the data of annual growth rate are available, there is another manner to obtain short-term estimates of the growth parameters. The equation for the annual increments changing with time is derived from (1), as follows:
(9)
where lt is the annual increment at the age t (from initial age tx to maximal age tm), l0 represents the initial length, K is a parameter of the von Bertalanffy growth curve. The logarithmic form of (9) is a linear equation:
where coefficient K represents the slope of the straight line, l0 is y-intercept. For short-term parameterization of the growth equation the straight line must pass over two points, one representing the last year increment ln lm at maximal age tm, another being the midpoint of age interval from tx to tm–1 and of length interval from ln lx to ln lM–1. In this case, the slope of straight line is as follows:
(10)
where M is the number of annual increments (M = tm – tx – 1). The parameters L∞ and t0 are calculated from (5) and (7), substituting k = e–K.
A new approach is especially perspective in studying organisms that show periodical microstructural variations in hard parts (e.g. shells of mollusks, scales and otoliths of fishes). Proper interpretation of skeletal periodicity permits assessment of growth rate during selected time space. Such information may then be used to estimate ontogenetic parameters of the growth equations. The data obtained on the basis of sclerochronological methods and new calculation procedures seem to be useful for monitoring biological effects of contamination at the individual and population levels. In particular, analysis of short-term growth characteristics may be applied in Mussel Population Watch (Zolotarev, Shurova, 1998), since distinct annual shell growth bands were found in many species of Mytilus from many coastal regions (Lutz, Rhoads, 1980; Shurova, Zolotarev, 1988; Zolotarev, 1989).
References
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