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Journal ArticleDOI

A new graphic method of describing the growth of animals.

01 Apr 1946-The Biological Bulletin (Marine Biological Laboratory)-Vol. 90, Iss: 2, pp 141-147
TL;DR: Growth curves, when conventionally plotted as length on age, are difficult to compare and classify; the usual mathematical methods of fitting them, such as the logistic and the Gompertz are rather laborious and incon venient for application to large numbers of individuals.
Abstract: Growth curves, when conventionally plotted as length on age, are difficult to compare and classify. Moreover, the usual mathematical methods of fitting them, such as, the logistic and the Gompertz are rather laborious and incon venient for application to large numbers of individuals. Fortunately, for many purposes, it is unnecessary to describe the whole growth curve; for the part below the inflection point is completed early and the part above the inflection point—the “? self-inhibiting― phase, covers the period of life when differences in growth are likely to be most striking. That phase of the growth curve can be appropriately represented by a straight line, the charac teristics of which can be treated statistically, by the following graphic method: Using arithmetic graph paper, with body length represented along both the x axis and along the y axis, plot length at ages 1, 2, 3, 4, 5@ n on the x axis against length at ages 2, 3, 4, 5, 6. n + 1, respectively, on the y axis. For several species on which I have found published length data, these points fall along a straight line. This line can be regarded as a sort of transformation of the usual growth curve, and in the following discussion I will call it that. The nine examples given in Figures 1—3are based on average lengths of large samples. When lengths of individual specimens are plotted by this method, a straight-line relationship is still obvious, though the points deviate more widely from the linethan when averages are used. These deviationsdoubtless result from several causes, among which random variations in environmental experience and errors of observation readily suggest themselves. For a few species the published growth data failed to produce a straight line. In these cases, the course of growth may differ from that in other animals; or the observed anomalies may reflect some artifactual effect in the data. Among those species for which this “? transformation― results in a straight line, the growth increments corresponding to equal time intervals, for example, be tween years of age (12 —? 11,13 —? 12,14 —? i@, @l,—? i@), have the following inter relations; where 1,, refers to the length at any given age, i.e., at the end of any given time interval :2
Citations
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Book ChapterDOI
TL;DR: Length varies with the condition of the fish—for example, whether it is alive, recently killed, after rigor mortis has set in, or at different intervals of time after preservation in formalin or alcohol.
Abstract: Fish lengths have been measured in many different ways The differences arise from choosing different reference points near the anterior end and near the posterior end of the fish, and from using different methods of making the measurement Methods of making the measurement include using calipers; using a tape held along the curve of the body; laying the fish on a measuring board with the front end pressed against an upright piece; laying the fish on a board with a movable cross hair above it, attached to an indicator running along a scale In theory, any combination of reference points and methods might be used, but practice is considerably more restricted Some of the commoner combinations have special names; these are provided in the chapter, together with the reference points used In addition to these methodological differences, length varies with the condition of the fish—for example, whether it is alive, recently killed, after rigor mortis has set in, or at different intervals of time after preservation in formalin or alcohol

1,150 citations

Book
01 Jan 1984
TL;DR: This manual discusses length-Weight Relationships, Parameter Estimation, and total mortality from the oldest animal in the catch and further inferences from length-converted catch curves.
Abstract: ..................................................... xv 1. How to Use this Manual ...................................... 1 2. Length-Weight Relationships ................................... 5 Introduction ............................................. 5 Parameter Estimation ...................................... 5 3. Meph Selection ............................................. 10 Introduction ............................................. 10 Trawl Mesh Selection ...................................... 10 Gillnet Selection .......................................... 12 Using a Selection Curve to Adjust Catch Samples ................. 17 4. Fish Growth ............................................... 23 Introduction ............................................. 23 Data Needed for Parameter Estimation ........................ 26 Methods for Parameter Estimation ............................ 28 The von Betalanffy plot ................................. 29 The Ford-Walford plot ................................... 31 The Gulland and Holt plot ................................ 33 The Munro plc%......................................... 35 Fitting seasonally oscillating length-growth data ................ 37 Extended Gulland and Holt plot ............................ 39 Growth: A Concluding Program .............................. 41 5. Total, Natural and Fishing Mortalities ............................ 52 Introduction ............................................. 52 Estimating Total Mortality .................................. 53 Total mortality from the oldest animal in the catch ............. 53 Total mortality from the mean size in the catch ................ 55 Estimation of Z from cumulative plots ....................... 56 Catch curves and length-converted catch curves ................ 58 Further inferences from length-converted catch curves ........... 64 Estimating Z from a pseudo-catch curve ...................... 66 Simultaneous Estimation of Z and K ........................... 67

1,023 citations


Cites methods from "A new graphic method of describing ..."

  • ...Of all methods used for estimating the parameters of the VBGF, the Ford-Walford plot (Ford 1933; Walford 1946) is the most commonly used....

    [...]

Journal ArticleDOI
TL;DR: Thirty age groups are described and illustrated, which are related to the progress of eruption and wear of the six teeth in each side of the lower jaw, which indicate an upper age limit of about 60 years, which is compatible with fuller information on Indian elephants.
Abstract: Summary 1. The paper is based on a collection of 385 lower jaws of the African elephant (Loxodonta a. africana Blumenbach) from western Uganda. 2. Thirty age groups are described and illustrated, which are related to the progress of eruption and wear of the six teeth in each side of the lower jaw. 3. Correct identification of individual teeth in the series is essential and several checks are described. Thus, when length is plotted against width for the six teeth the points fall into six well - defined groups. Frequency distributions of laminary indices and lamella numbers support the belief that teeth have been correctly identified, but overlapping, distributions preclude identification on these characters alone. 4. The presence of abnormal seventh molars in four jaws is described. 5. Chronological ages have been assigned to the relative age groups. Information on age and growth of captive animals has been considered and the general form of the growth curve established. Arbitrary estimates of the intervals between successive age groups were made and subsequently checked against growth and seasonal ridges on the roots. These indicate an upper age limit of about 60 years, which is compatible with fuller information on Indian elephants. 6. Growth curves support the validity of the ages assigned. Asymptotic heights at shoulder (∞) are respectively 317 cm and 298 cm for males from Murchison Falls and Queen Elizabeth National Parks. For females no distinction is made (because the sample from the latter area is small) and ∞ is 272 cm. 7. Growth in weight has also been established but shows no significant differences between the two populations. The mean maximum weight of females is 2766 kg (6100 lb), and of males 5450 kg (12000 lb). Growth apparently continues throughout life. 8. Tusk growth is analysed. There is a linear increase in weight of female tusks from eruption at 1–3 years up to the oldest group, with an indicated mean combined weight of 17.7 kg (39 lb) at 60 years. Male tusks show an increasing rate of growth throughout life to a mean combined weight of about 109 kg (240 lb) at 60 years. Big tusks are generally the result of prolonged growth; extremely big tusks probably result from prolonged and above average rate of growth. 9. Seasonal and annual incremental layers on the root 3 of the teeth are briefly described; these give an objective estimate of the chronological intervals between the relative age groups. 10. Field age criteria are presented which are derived from these growth curves. 11. The age at puberty in the female elephant is discussed and compared with earlier conclusions. There is evidence of a retardation in recent years in the Murchison Falls National Park (South bank) population and of a lengthening of the mean calving interval. This considerable depression of the reproductive rate, which is almost certainly density dependent, agrees with the observed lower recruitment in this population. 12. Survivorship curves constructed from material representing 325 natural deaths are presented for the two populations. If estimates of the age at puberty and the reproductive rate are taken into account, the expected differences in recruitment are found. 13. The calculated mean expectation of life is less than 15 years. 14. Mean individual weight is estimated at about 3800 lb.

443 citations

Book ChapterDOI
01 Jan 1957

286 citations

References
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Book
01 Jan 1917
TL;DR: This book is an application of some of the concepts of physical science and sundry mathematical methods to the study of organic form and is like one of Darwin's books, well-considered, patiently wrought-out, learned, and cautious.
Abstract: Introduction John Tyler Bonner VII 1. Introductory 2. On magnitude 3. The forms of cells 4. The forms of tissues, of cell-aggregates 5. On spicules and spicular skeletons 6. The equiangular spiral 7. The shapes of horns and of teeth or tusks 8. On form and mechanical efficiency 9. On the theory of transformations, or the comparison of related forms 10. Epilogue Index.

4,470 citations

Journal ArticleDOI

1,158 citations

Book
01 Jan 1908

139 citations

Journal ArticleDOI
TL;DR: Current mathematical expressions for the course of growth are discussed and a formula used which emphasises Minot's conception of a growth-rate constantly declining with age is found to graduate the extensive data in clam growth with significant accuracy.
Abstract: 1. The present paper is a study of the growth of a clam ( Siliqua patula ) under natural conditions and over a wide range of latitude. 2. Various constants derived from the growth data are compared for the different localities. For this species, over the range considered, growth in the southern localities as compared with the northern is initially more rapid but less sustained, leads to a smaller total length and is associated with a shorter life span. 3. Reasons are presented for considering the relative growth-rate as a particularly significant constant leading to more sound biological conclusions than the use of the absolute growth-rate. 4. On the basis of the relative growth-rate, current mathematical expressions for the course of growth are discussed and a formula used which emphasises Minot's conception of a growth-rate constantly declining with age. This expression L = Be - ce - ce - kt , in which L = length at time t , e = base of natural logarithms, and B , c and k are constants, is found to graduate the extensive data in clam growth with significant accuracy. Published with the permission of the United States Commissioner of Fisheries.

66 citations