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Use of the original von Bertalanffy growth model to describe the growth of barramundi, Lates calcarifer (Bloch).



Abstract--In the original von Bertalanffy growth equation, the rate of change in body mass of an individual is assumed to result from two opposing biological processes: anabolism anabolism: see metabolism.  and catabolism catabolism (kətăb`əlĭz'əm), subdivision of metabolism involving all degradative chemical reactions in the living cell. . Because this differential equation differential equation

Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions.
 cannot be solved analytically, some of its analytically solvable special cases are commonly used, despite their restrictive assumptions. In this study, I used a generalization of the original von Bertalanffy growth equation and some of its commonly used special cases to estimate parameters from a set of tagging data on times at liberty, lengths at release, and lengths at recapture of a centropomid perch (Lates calcarifer Lates calcarifer

farmed finfish in the family Centropomatidae. Called also barramundi, giant sea perch. See Table 23.
) and provide a method for determining the anabolic anabolic

pertaining to or arising from anabolism.


anabolic steroid
steroids with a tissue-building effect. Testosterone is an example of a natural anabolic steroid with the, sometimes undesirable, effect of causing masculinization.
 and catabolic Catabolic
A metabolic process in which energy is released through the conversion of complex molecules into simpler ones.

Mentioned in: Anabolic Steroid Use


catabolic

see catabolism.
 rates of animals in their natural environment. Fitting the original von Bertalanffy growth equation to the tagging data suggests that a 1% increase in body mass of the fish corresponds to a 0.8721% increase in anabolic rate and a 1.0357% increase in catabolic rate. Alternatively, L. calcarifer may be interpreted as exhibiting a strong seasonality in growth: it grows fastest in length at the start of autumn, grows less until a full stop in the middle of winter, shrinks until the middle of spring, and then resumes a positive growth for another cycle. Consequently, it is unnecessary to use the analytically solvable special cases of the original von Bertalanffy growth equation in data analysis, unless their assumptions are validated. I also explain why Pauly's index of growth performance is adequate and propose an index of catabolic performance.

Information on the growth of animals is important for studying their population dynamics Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. , physiology, and biochemistry (Peters, 1983; Calder, 1984; Schmidt-Nielsen, 1984; Reiss, 1989; Xiao, 1998). Many empirical models have been developed to describe the growth of animals macroscopically mac·ro·scop·ic   also mac·ro·scop·i·cal
adj.
1. Large enough to be perceived or examined by the unaided eye.

2. Relating to observations made by the unaided eye.
, including the Gompertz (1825) and logistic growth models (Verhulst, 1838). By contrast, von Bertalanffy (1938) proposed a somewhat mechanistic mech·a·nis·tic
adj.
1. Mechanically determined.

2. Of or relating to the philosophy of mechanism, especially one that tends to explain phenomena only by reference to physical or biological causes.
 growth model for body mass W(a)[is greater than or equal to] 0 of an individual of age a, of the form

dW(a)/da = AW[(a).sup.B] - CW[(a).sup.D],

where A, B, C and D = positive biological constants;

AW[(a).sup.B] = the rate of anabolism (building up of body mass) at age a; and

CW[(a).sup.D] = the rate of catabolism (breaking down of body mass) at age a.

Thus, in this model, the rate of change in body mass of an individual dW(a)/da at age a is assumed to result from two opposing biological processes (anabolism and catabolism). Although the underlying mechanisms may be too complicated for dW(a)/da to be approximated or even interpreted as such, this differential equation has opened up a line of thought for integrating the macroscopic macroscopic /mac·ro·scop·ic/ (mak?ro-skop´ik) gross (2).

mac·ro·scop·ic or mac·ro·scop·i·cal
adj.
1. Large enough to be perceived or examined by the unaided eye.

2.
 growth of animals with certain physiological and biochemical processes (Pauly, 1981). Also, it is fairly general, includes almost all previous deterministic 1. (probability) deterministic - Describes a system whose time evolution can be predicted exactly.

Contrast probabilistic.
2. (algorithm) deterministic - Describes an algorithm in which the correct next step depends only on the current state.
 growth models as its special cases, and forms a basis for identifying the "right" growth models from amongst all its special cases. Consequently, some work has been done to estimate parameters A, B, C, and D to determine the anabolic and catabolic rates of fish (Ursin, 1967; Pauly, 1981).

However, because the differential equation cannot be solved analytically, its analytically solvable special cases are so commonly used that one simple special case has become known in the fisheries fisheries. From earliest times and in practically all countries, fisheries have been of industrial and commercial importance. In the large N Atlantic fishing grounds off Newfoundland and Labrador, for example, European and North American fishing fleets have long  literature as the von Bertalanffy growth equation (Xiao, 1996). Nonetheless, assumptions for its various analytically solvable special cases can be very restrictive. Indeed, although assumed to take a value of 2/3 in that simple special case (Pauly, 1981), constant B can take any value from 2/5 to 5/7, because B is often assumed to satisfy [Beta](B-1)+1=0 and because B in Equation 2 below is known to take any value from 2 1/2 to 3 1/2. It is also possible that catabolic rate is not proportional to body mass (i.e. D [is not equal to] 1). In any case, it is best not to make any assumptions about the values of A, B, C, and D.

Like most growth models, the von Bertalanffy (1938) growth equation is age-dependent. Although it can be modified to consider, implicitly, the seasonal growth of animals and the effects of tagging, a general framework was not available for explicitly incorporating time and time-dependent factors (i.e. ambient temperature Outside temperature at any given altitude, preferably expressed in degrees centigrade.  and food availability) in age-dependent growth models. This prompted Xiao (1999) to derive general age- and time-dependent growth models for animals and to give a comprehensive list of their commonly used special cases. Such models explicitly incorporate age, time, and their dependent factors and are useful for modeling growth at age and time (e.g. from length-at-age data), incremental Additional or increased growth, bulk, quantity, number, or value; enlarged.

Incremental cost is additional or increased cost of an item or service apart from its actual cost.
 growth at age and time increment To add a number to another number. Incrementing a counter means adding 1 to its current value.  (e.g. data on length increment at age and time increment from tagging studies), the effects of tagging, and, if coupled with a proper age- and time-dependent population dynamics model, the effects on the growth of animals of many population characteristics, such as population size.

Because of experimental constraints, such as difficulties in taking continuous measurements (if measured at all), anabolic and catabolic rates of animals are necessarily measured either by restraining them in the laboratory or in the field. Such restraints can cause stress to animals and hence result in biased measurements. Experimental methods should be developed to estimate the anabolic and catabolic rates of animals in as natural an environment as possible.

In this study, I use an age- and time-dependent von Bertalanffy (1938) growth equation and some of its commonly used special cases for estimating the parameters from a set of tagging data on times at liberty, lengths at release, and lengths at recapture of a centropomid perch, barramundi barramundi

see lates calcarifer.
 (Lates calcarifer) and provide a method for determining the anabolic and catabolic rates of animals in their natural environment. I also explain why Pauly's (1981) index of growth performance is adequate and propose an index of catabolic performance.

Model

Let 0 [is less than or equal to] W(a,t) [is less than] [infinity], -[infinity][is less than] [a.sub.0][is less than or equal to] a [is less than] [infinity], -[infinity] [is less than] [t.sub.0] [is less than or equal to] t [is less than] [infinity], denote the body mass of an individual of age a at time t, with an arbitrary reference age [a.sub.0] and an arbitrary reference time [t.sub.0]. The von Bertalanffy (1938) growth equation can be generalized as

(1) dW(a,t)/dt = A(a,t)W[(a,t).sup.B] - C(a,t)W[(a,t).sup.D],

where A(a,t) [is greater than or equal to] 0 and C(a,t) [is greater than or equal to] 0 = functions of age a and time t; and

B and D = positive biological constants.

For a particular functional form of A(a,t) and C(a,t), Equation 1 can be used for estimating its parameters from data on body masses of animals of different ages at different times, or on two distinct body masses of the same individual at different times. If collected at all, such data are collected mostly for terrestrial and occasionally for aquatic animals. What is most commonly gathered for both terrestrial and aquatic animals is, however, one or more linear dimensions of an individual's body, such as its total length at age, or two distinct measurements at different times. Measurements of linear dimensions of an animal contain useful information on its body mass. Indeed, it is well known that body mass W(a,t) is scaled allometrically to body length L(a,t), i.e.

(2) W(a, t) = [Alpha]L[(a, t).sup.[Beta]],

where [Alpha] and [Beta] = (constant) allometric al·lom·e·try  
n.
The study of the change in proportion of various parts of an organism as a consequence of growth.



al
 parameters (Peters, 1983; Calder, 1984; Schmidt-Nielsen, 1984; Reiss, 1989).

Substitution of Equation 2 into Equation 1 yields

(3) dL(a,t)/dt = 1/[Beta] [[Alpha].sup.B-1] A(a,t)L[(a,t).sup.[Beta](B-1)+1]

- 1/[Beta] [[Alpha].sup.D-1]C(a,t)L[(a,t).sup.[Beta](D-1)+1].

Thus, if [Alpha] and [Beta] are known, as is usually assumed, parameters B and D, and those in A(a,t) and C(a,t) can be estimated from data on length-at-age data, or on two distinct lengths of the same individual at different times.

Although too general to be solved even numerically, Equations i and 3 are useful in formulating ideas. Now, I consider a special case of Equations 1 and 3 for seasonally varying A(a,t) and C(a,t), such that

A(a,t) = [Gamma]/K [K + b cos(2[Pi]/T(t - [t.sub.[Phi]]))] an c(a,t) = K + b cos (2[Pi]/T (t - [t.sub.[Phi]])),

where [Gamma], [Gamma]b/K, T, and [t.sub.[Phi]] are, respectively, the mean, amplitude, period, and time shift for the anabolic process; K, b, T, and [t.sub.[Phi]] are, respectively, the mean, amplitude, period, and time shift for the catabolic process. For this special case, Equations 1 and 3 then become, respectively

(4) dW(a,t)/dt = [K + b cos(2[Pi]/T (t - [t.sub.[Phi]]))] x[[Gamma]/K W[(a,t).sup.B] - W[(a,t).sup.D]]

and

(5) dL(a,t)/dt = 1/[Beta] [K+b cos(2[Pi]/T(t - [t.sub.[Phi]]))]

x[[Gamma]/K[[Alpha].sup.B-1]L[(a,t).sup.[Beta](B-1)+1 - [[Alpha].sup.D-1]L[(a,t).sup.[Beta](D-1)+1]].

Equations 4 and 5 can be solved numerically but not analytically. For comparison and illustration, I now consider five special cases (four of which are reparameterizations of commonly used growth equations) of Equation 5:

If b=0, or if A(a,t) and C(a,t) are constants, Equation 5 becomes

(6) dL(a,t)/dt = 1/[Beta] ([Gamma][[Alpha].sup.B-1]L[(a,t).sup.[Beta](B-1)+1] - K[[Alpha].sup.D-1]L[(a,t).sup.[Beta](D-1)+1]).

If [Beta](B-1)+1=0 (i.e. B-1=-1/[Beta]) and [Beta](D-1)+1=1 (i.e. D=1), Equation 5 becomes

(7) dL(a,t)/dt = 1/[Beta] [K+b cos(2[Pi]/T(t - [t.sub.[Phi]]))]([Gamma]/K [[Alpha].sup.-1/[Beta]]-L(a,t)),

the solution of which as an initial value problem, with [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ], yields

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similar age-dependent models have been given, inter alia [Latin, Among other things.] A phrase used in Pleading to designate that a particular statute set out therein is only a part of the statute that is relevant to the facts of the lawsuit and not the entire statute. , by Appeldoorn (1987), Pauly et al. (1992), Fontoura and Agostinho (1996), and Xiao (1999). Also, notice that bT (in Appeldoorn (1987) and Pauly et al. (1992), b=C and T=1) is a dimensionless quantity In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number.  and is useful for interspecific in·ter·spe·cif·ic  
adj.
Arising or occurring between species.



interspecific also interspecies  

Arising or occurring between species.

Adj. 1.
 comparison of the strength of seasonal growth oscillations oscillations See Cortical oscillations.  (Pauly, 1084, 1985, 1990).

If [Beta](B-1)+1=0 (i.e., B-1=-1/[Beta]), [Beta](D-1)+1=1 (i.e., D=1), and b=0, Equation 5 becomes

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

a reparameterization of what is commonly called in the fisheries literature the von Bertalanffy (1938) growth equation.

If [Beta](B-1)+1=0 (i.e., B=1) and [Beta](D-1)+1=2 (i.e. D-1=1/[Beta]), Equation 5 becomes

(10) dL(a,t)/dt = [Gamma]/[Beta]K [K + b cos(2[Pi]/T(t - [t.sub.[Phi]]))] x L(a,t)(1-L(a,t)/[Gamma]/K [[Alpha].sup.-1/[Beta]),

the solution of which as an initial value problem, with L(a, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], yields

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a reparameterization of the seasonal logistic growth equation (Xiao, 1999).

If [Beta](B-1)+1=1 (i.e., B=1), [Beta](D-1)+1=2 (i.e., D-1=1/[Beta]), and b=0, Equation 5 becomes

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a reparameterization of the logistic growth equation (Xiao, 1999).

Data and analysis

Equations 8, 9, 11, and 12 are segmented functions (Xiao, 1999); they provide flexibility in analysis of growth data. Thus, by appropriately choosing the value of time t (which is a relative quantity), one can use either segment (a - [a.sub.0] [is less than] t-[t.sub.0] or a - [a.sub.0] [is greater than or equal to] t-[t.sub.0]) for an individual animal or for a group of individuals, or use both segments (a - [a.sub.0] [is less than] t-[t.sub.0] and a - [a.sub.0] [is less than or equal to] t-[t.sub.0]) for a group of individuals. It is, however, more convenient to use only one segment in a single analysis. Indeed, although growth parameters can be estimated by use of either segment of any of Equations 8, 9, 11, and 12, it is easier to use the segment for a - [a.sub.0] [is less than] t-[t.sub.0], by letting time t start before the animals whose growth is to be modeled are born, unless time is allowed to take negative values. Use of the other segment, i.e. that for a - [a.sub.0] [is less than or equal to] t-[t.sub.0], gives identical results, but it requires first calculating L([t.sub.0] + a - t, [t.sub.0]).

The amount of data required to estimate parameters in a growth model is a function of the generality of that model: the more general a model is, the more data it usually requires. Age- and time-dependent growth models generally require knowledge of two ages [a.sub.0] and a, time t, and two sizes L([a.sub.0], t - a + [a.sub.0]) and L(a,t) if a - [a.sub.0] [is less than] t-[t.sub.0]; or knowledge of two times [t.sub.0] and t, age a, and two sizes L([t.sub.0] + a - t, [t.sub.0]) and L(a,t) if a - [a.sub.0] [is less than or equal to] t-[t.sub.0]. However, use of Equations 4, 5, 8, and 11 only requires knowledge of the difference between two ages a - [a.sub.0], time t, and two sizes L([a.sub.0], t - a + [a.sub.0]) and L(a,t); or of the difference between two times t - [t.sub.0], time t, and two sizes L([t.sub.0] + a - t, [t.sub.0]) and L(a,t). By contrast, use of Equations 9 and 12 only requires knowledge of the difference between two ages, a - [a.sub.0], and two sizes, L([a.sub.0], t - a + [a.sub.0]) and L(a,t), or of the difference between two times t - [t.sub.0], and two sizes L([t.sub.0] + a - t, [t.sub.0]) and L(a,t).

Interestingly, a reparameterization of Equation 9 has been widely used to model tagging data (Xiao, 1999), where [a.sub.0] or [t.sub.0] is interpreted as time at release, a or t as time at recapture, a - [a.sub.0] or t - [t.sub.0] as time at liberty, L([a.sub.0], t - a + [a.sub.0]) or L([t.sub.0] + a - t, [t.sub.0]) as size at release, and L(a,t) as size at recapture. It has also been used extensively to model size-at-age data (obtained, say, by aging animals by reading marks in scales and otoliths) (e.g. Moulon et al., 1992), where [a.sub.0] or [t.sub.0] is interpreted as age at birth, a or t as age, L([a.sub.0], t - a + [a.sub.0]) or L([t.sub.0] + a - t, [t.sub.0]) as size at birth, and L(a,t) as size at age. However, it is rare to know two ages and the corresponding sizes of an animal; what are commonly measured are one age and its corresponding size. Consequently, it is common practice to fit Equation 9 to such size-at-age data to estimate age at birth [a.sub.0] or [t.sub.0], as well as the growth parameters, thereby implicitly assuming, for all animals concerned, that the size at birth L([a.sub.0], t - a + [a.sub.0]) or L([t.sub.0] + a - t, [t.sub.0]) is zero and that the age at birth [a.sub.0] or [t.sub.0] is the same. Exactly the same argument applies to Equation 12.

The barramundi L. calcarifer is a protandrous pro·tan·drous  
adj.
Of or relating to a flower in which the anthers release their pollen before the stigma of the same flower is receptive.



pro·tan
 fish found in estuaries and other coastal areas of the Indo-West Pacific The Indo-West Pacific, or IWP, is a zoogeographical region spanning the entire Indian Ocean including the Red Sea and the Pacific Ocean as far as the Caroline Islands but short of the Marshall Islands.  (Griffin, 1987). Between August 1977 and June 1980, 4933 barramundi with a body total length range of about 10-100 cm were captured by a combination of lure fishing, tidal trap, seine Seine (sān, Fr. sĕn), Lat. Sequana, river, c.480 mi (770 km) long, rising in the Langres Plateau and flowing generally NW through N France.  and gill net. Fish were measured to the nearest cm, tagged with Floy FT-2 dart tags for fish [is greater than] 35 cm and FD-67 anchor tags The HTML code for creating a link to another page or to a particular section within a page. It is also commonly called an "h-ref." See HREF.  for fish [is less than] 35 cm, and released in rivers flowing into the Van Diemen Gulf Van Diemen Gulf () is a gulf between Arnhem Land and Melville Island in northern Australia. It is connected to the Timor Sea by the Clarence Strait (near the city of Darwin) and Dundas Strait.  and the Gulf of Carpentaria Noun 1. Gulf of Carpentaria - a wide shallow inlet of the Arafura Sea in northern Australia
Carpentaria

Australia, Commonwealth of Australia - a nation occupying the whole of the Australian continent; Aboriginal tribes are thought to have migrated from
 of northern Australia The term northern Australia is generally considered to include the States and territories of Australia of Queensland and the Northern Territory. The part of Western Australia (WA) north of latitude 26° south — a definition widely used in law and State government policy  (Davis and Reid, 1982). Of those tagged, 312 of a total length of 23-92 cm with a mean of 60 [+ or -] 13 (mean [+ or -] SE) cm were recaptured, but only 308 were used in the analysis below due to incomplete recapture information. The time at liberty ranged from zero to 932 d, with a mean of 219 [+ or -] 211 d, and the length increment from -21 to 35 cm, with a mean of 6 [+ or -] 8 cm. Negative increments in length are often observed in a tagging experiment, because tagged animals can shrink in size immediately after tagging, or because of recording errors at both release and recapture. The estimates of allometric parameters for barramundi, used in the present paper, were those obtained by Reynolds (1978): [Alpha]=1.06 x [10.sup.-5] kg x cm-[Beta] and [Beta] = 3.02.

Let [a.sub.0] or [t.sub.0] denote time at release, a or t time at recapture, a - [a.sub.0] or t - [t.sub.0] time at liberty, L([a.sub.0], t - a + [a.sub.0]) or L([t.sub.0] + a - t, [t.sub.0]) the length of a fish at release, and L(a,t) its length at recapture. Equation 6 and the segments of Equations 8, 9, 11, and 12 for a - [a.sub.0] [is less than] t-[t.sub.0], were fitted [t.sub.0] the tagging data, by using the nonlinear A system in which the output is not a uniform relationship to the input.

nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input.
 least squares method least squares method

Statistical method for finding a line or curve—the line of best fit—that best represents a correspondence between two measured quantities (e.g., height and weight of a group of college students).
, under the assumptions that T=365.25 d, time started (i.e. time t=0) on 1 January 1960 (see Xiao [1999] for its significance), and errors in L(a,t) follow independent normal distributions, with a mean of L (a,t) and a constant variance of [[Sigma].sup.2] (Table 1). In these calculations, Equation 6 was numerically solved as an initial value problem with L(a, t)|[sub.t = [t.sub.0]] = L([t.sub.0] + a - t, [t.sub.0]) for a - [a.sub.0] [is greater than or equal to] t-[t.sub.0] using the fourth order Runge-Kutta method (Beyer, 1978). A likelihood ratio test suggests that Equation 8 is significantly different from Equation 9 ([F.sub.2,304]=48.6892, P [is less than] 0.0001); and Equation 11 is significantly different from Equation 12 ([F.sub.2,304]=45.3460, P [is less than] 0.0001). Thus, Equations 8 and 11, and their associated estimates of parameters seem adequate for describing the tagging data. Selection among Equations 6, 8, and 11 is difficult because little is known of the underlying mechanisms of the growth process.

[TABULAR DATA 1 NOT REPRODUCIBLE IN ASCII]

Discussion

Fitting of the original von Bertalanffy growth model (Eq. 6) to the tagging data for barramundi suggests that its anabolic rate changes proportionally with the B=0.8721 power of its body mass and that its catabolic rate changes proportionally (D=1.03567 [approximately equals] 1) with its body mass (i.e. at a 1:1 ratio). Such an estimate of B is 9.0125% higher than that (B [approximately equals] 4/5) obtained for many fish under laboratory conditions (Pauly, 1981). More data are needed to examine the generality of this finding. By contrast, little information is available on the value of D. Nonetheless, it is interesting that the catabolic rate of barramundi increases proportionally with its body mass; a 1% increase in body mass corresponds to about a 1% increase in catabolic rate. Consequently, it is unnecessary to use the analytically solvable special cases of the original von Bertalanffy growth equation in data analysis, unless their assumptions are validated.

Alternatively, like many tropical and subtropical sub·trop·i·cal  
adj.
Of, relating to, or being the geographic areas adjacent to the Tropics.


subtropical
Adjective

of the region lying between the tropics and temperate lands

 species offish off·ish  
adj.
Inclined to be distant and reserved; aloof.



offish·ly adv.

off
 (Appeldoorn, 1987; Pauly et al., 1992), barramundi may be interpreted as exhibiting a strong seasonal growth. For both models (Eqs. 8 and 11), its growth rate reaches its maximum on 3 or 4 March (i.e. at the start of autumn), slows down to zero on 17 July (i.e. in the middle of winter), reaches its minimum on 2 or 3 September (i.e. at the start of spring), returns to zero on 19 or 20 October (i.e. in the middle of spring), and comes back to its maximum rate on 3 or 4 March (i.e. at the start of autumn) (Fig. 1). Thus, its length grows fastest on 3 or 4 March (i.e. at the start of autumn), grows less until a full stop on 17 July (i.e. in the middle of winter), shrinks until 19 or 20 October (i.e. in the middle of spring), and resumes a positive growth for another cycle. Thus, barramundi does not grow in length for three months in a year, from 17 July (i.e. in the middle of winter) to 19 or 20 October (i.e. in the middle of spring). However, Equations 8 and 11 have different assumptions and predict different amplitudes of seasonally varying growth rate. Such a strong seasonality in growth rate might be related to seasonal changes in the availability of food and in water temperature.

[Figure 1 ILLUSTRATION OMITTED]

Similarly, tagging may adversely affect the growth of barramundi perch and bias estimates of parameters in a growth model, where its effects are not taken into proper account. In fact, Xiao (1994) has already interpreted the same set of data in terms of the effects of tagging. However, it is impossible to identify the right model from all possible models, because of the inductive inductive

1. eliciting a reaction within an organism.

2.


inductive heating
a form of radiofrequency hyperthermia that selectively heats muscle, blood and proteinaceous tissue, sparing fat and air-containing tissues.
 nature of modeling and because of our poor understanding of the underlying mechanisms of growth and how tagging affects growth. In a preliminary analysis, I have constructed a model, and have attempted (but failed) to estimate, simultaneously, both the effects of tagging and seasonally varying growth rates Growth Rates

The compounded annualized rate of growth of a company's revenues, earnings, dividends, or other figures.

Notes:
Remember, historically high growth rates don't always mean a high rate of growth looking into the future.
. Such a failure is not surprising because the amount of information in a set of tagging data is limited. Further progress can be made only by better understanding the underlying mechanisms of growth.

This work also puts some of Pauly's (1981) work into perspective. For example, [[Alpha].sup.-1/[Beta]][Gamma]/K and K/[Beta] in Equation 9 can be interpreted respectively as the average maximum size and growth rate of a species. As Pauly (1981) proposed, the product ([[Alpha].sup.-1/[Beta]][Gamma]/K)(K/[Beta]) = [[Alpha].sup.-1/[Beta]][Gamma]/[Beta] is indeed an index of growth performance because it is in direct proportion to the mean anabolic rate. Similarly, [[Alpha].sup.-1/[Beta]][Gamma]/K and [Gamma]/[Beta] in Equation 12 can be interpreted respectively as the average maximum size and growth rate of a species. The quotient quotient - The number obtained by dividing one number (the "numerator") by another (the "denominator"). If both numbers are rational then the result will also be rational.  ([Gamma]/[Beta])/([[Alpha].sup.-1/[Beta]][Gamma]/K) = [[Alpha].sup.1/[Beta]]K/[Beta] is an index of catabolic performance, because it is in direct proportion to the mean catabolic rate.

Finally, anabolic and catabolic rates of animals can be estimated from data from a mark-recapture experiment on two distinct lengths of the same individual measured at different times. Thus, the present work has demonstrated a way to estimate anabolic and catabolic rates of animals. Such field-based estimates can be compared with those obtained under laboratory conditions.

Acknowledgments

I wish to thank three anonymous referees for their very constructive and valuable comments on the manuscript, Kate Watt (SARDI SARDI South Australian Research and Development Institute (Australia)  Aquatic Sciences Centre) for producing Table 1, and Roland K. Griffin (Northern Territory Department of Primary Industry and fisheries) and Tim L. O. Davis (CSIRO CSIRO Commonwealth Scientific & Industrial Research Organization (Australia)  Division of Fisheries) for supplying L. calcarifer data. Roland K. Griffin also provided some references on L. calcarifer.

Literature cited

Appeldoorn, R. S. 1987. Modification of a seasonally oscillating os·cil·late  
intr.v. os·cil·lat·ed, os·cil·lat·ing, os·cil·lates
1. To swing back and forth with a steady, uninterrupted rhythm.

2.
 growth function for use with mark-recapture data. J. Cons. Int. l'Explor. Mer 43:194-198.

Beyer, W. H. 1978. CRC (Cyclical Redundancy Checking) An error checking technique used to ensure the accuracy of transmitting digital data. The transmitted messages are divided into predetermined lengths which, used as dividends, are divided by a fixed divisor.  handbook of mathematical sciences, 6th ed. CRC Press, Inc., Boca Raton Boca Raton (bō`kə rətōn`), city (1990 pop. 61,492), Palm Beach co., SE Fla., on the Atlantic; inc. 1925. Boca Raton is a popular resort and retirement community that experienced significant industrial development in the 1970s and 80s. , FL, 860 p.

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Fontoura, N. F., and A. A. Agostinho. 1996. Growth with seasonally varying temperatures: an expansion of the von Bertalanffy growth model. J. fish Biol. 48:569-584.

Gompertz, B. 1825. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115(1):513-585.

Griffin, R. K. 1987. Life history, distribution, and seasonal migration of barramundi in the Daly River, Northern Territory Daly River is a river and a town in the Northern Territory of Australia. At the 2001 census, Daly River had a population of 623.[1] History
The traditional owners of the area are the Malak Malak people who live both in Nauiyu and at Wooliana downstream from
, Australia. Am. Fish. Soc. Symp. 1:358-363.

Moulton, P. M., T. I. Walker, and S. R. Saddlier. 1992. Age and growth studies of gummy shark The gummy shark, Mustelus antarcticus, is a shark in the family Triakidae, It is a slender grey shark with white spots along the body and flat plate-like teeth for crushing its prey with a maximum length between 157 cm (male) and 175 cm (female). , Mustelus antarcticus Gunther, and school shark The school shark, tope shark, soupfin shark or snapper shark, Galeorhinus galeus, is a hound shark of the family Triakidae, the only member of the genus Galeorhinus, found worldwide in subtropical seas at depths of up to 550 m. , Galeorhinus galeus (Linnaeus), from southern-Australian waters. Aust. J. Mar. Freshwater Res. 43:1241-1267.

Pauly, D. 1981. The relationships between gill surface area and growth performance in fish: a generalization of von Bertalanffy's theory of growth. Meeresforschung/Rep. Mar. Res. 28:251-282.

1984. Fish population dynamics in tropical waters: a manual for use with programmable calculators A limited-function computer capable of working with only numbers and not alphanumeric data. . ICLARM ICLARM International Centre for Living Aquatic Resources Management (Philippines)  Stud. Rev. 8:40-40.

1985. The population dynamics for short-lived species, with emphasis on squids. NAFO NAFO Northwest Atlantic Fisheries Organization
NAFO Network Adapter Failover (Sun Microsystems)
NAFO National Association of Fire Officers (UK)
NAFO National Association of Fire Officials
 Sci. Counc. Stud. 9:143-154.

1990. Length-converted catch curves and the seasonal growth of fishes. Fishbyte 8:33-38.

Pauly, D., M. Soriano-Bartz, M. J. Moreau, and A. Jarre-Teichmann.

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Peters, R. E. 1983. The ecological implications of body size. Cambridge Univ. Press, Cambridge, 329 p.

Reiss, M. J. 1989. The allometry al·lom·e·try
n.
The study of the change in proportion of various parts of an organism as a consequence of growth.



allometry

measurement of the changes in shape of an animal relative to increases in its size.
 of growth and reproduction. Cambridge Univ. Press, Cambridge, 182 p.

Reynolds, L. F. 1978. The population dynamics of barramundi Lates calcarifer (Pisces: Centropomidae) in Papua New Guinea Papua New Guinea (păp`ə, –y . MSc thesis, Univ. Papua New Guinea, Port Moresby Port Moresby (môrz`bē), town (1990 pop. 193,242), capital of Papua New Guinea, on New Guinea island and on the Gulf of Papua. Rubber, gold, and copra are exported. Port Moresby was founded by Capt. John Moresby, who landed there in 1873. , 249 p.

Schmidt-Nielsen, K. 1984. Scaling: why is animal size so important? Cambridge Univ. Press, Cambridge, 241 p.

Ursin, E. 1967. A mathematical model
Note: The term model has a different meaning in model theory, a branch of mathematical logic. An artifact which is used to illustrate a mathematical idea is also called a mathematical model and this usage is the reverse of the sense explained below.
 of some aspects of fish growth, respiration respiration, process by which an organism exchanges gases with its environment. The term now refers to the overall process by which oxygen is abstracted from air and is transported to the cells for the oxidation of organic molecules while carbon dioxide (CO , and mortality. J. Fish. Res. Board Can. 24:2355-2453.

Verhulst, P. F. 1838. Notice sur la loi que la population pursuit dans son accroissement. Corresp. Math. Phy. 10:113-121.

von Bertalanffy, L.

1938. A quantitative theory of organic growth. Human Biol. 10:181-213.

Xiao, Y. 1994. Growth models with corrections for the retardative effects of tagging. Can. J. Fish. Aquat. Sci. 51:263-267.

1996. How does somatic somatic /so·mat·ic/ (so-mat´ik)
1. pertaining to or characteristic of the soma or body.

2. pertaining to the body wall in contrast to the viscera.


so·mat·ic
adj.
 growth rate affect otolith otolith /oto·lith/ (o´to-lith) statolith.

o·to·lith
n.
1. Any of numerous minute calcareous particles found in the inner ear of certain lower vertebrates and in the statocysts of many
 size in fishes? Can. J. Fish. Aquat. Sci. 53:1675-1682.

1998. What are the units of the parameters in the power function for the length-weight relationship? Fish. Res. 35:247-249.

1999. General age- and time-dependent growth models for animals. Fish. Bull. 97:690-701.

Yongshun Xiao SARDI Aquatic Sciences Centre 2 Hamra Avenue West Beach Southern Australia The term southern Australia is generally considered to include the States and territories of Australia of New South Wales, Victoria, South Australia, Tasmania and the Australian Capital Territory.  5024, Australia E-mail address See Internet address.

e-mail address - electronic mail address
: yongshun.iao@bigpond.com
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