Allometry - Knowledge (XXG)

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unit body mass, larger birds also have slower wing beat frequencies, allowing them to fly at higher altitudes, longer distances, and faster absolute speeds than smaller birds. Because of the dynamics of lift-based locomotion and the fluid dynamics, birds have a U-shaped curve for metabolic cost and velocity. Because flight, in air as the fluid, is metabolically more costly at the lowest and the highest velocities. On the other end, small organisms such as insects can make gain advantage from the viscosity of the fluid (air) that they are moving in. A wing-beat timed perfectly can effectively uptake energy from the previous stroke (Dickinson 2000). This form of wake capture allows an organism to recycle energy from the fluid or vortices within that fluid created by the organism itself. This same sort of wake capture occurs in aquatic organisms as well, and for organisms of all sizes. This dynamic of fluid locomotion allows smaller organisms to gain advantage because the effect on them from the fluid is much greater because of their relatively smaller size.

1706:. This shows that mammals, regardless of size, have similarly scaled respiratory and cardiovascular systems and the same relative amount of blood: about 5.5% of body mass. This means that for similarly designed marine mammals, a larger individual can travel more efficiently, as it takes the same effort to move one body length. For example, large whales can migrate far distance in the oceans and not stop for rest. It is metabolically less expensive to be larger in body size. This goes for terrestrial and flying animals as well: smaller animals consume more oxygen per unit body mass than larger ones. The metabolic advantage in larger animals makes it possible for larger marine mammals to dive for longer durations of time than their smaller counterparts. That the heart rate is lower means that larger animals can carry more blood, which carries more oxygen. In conjuncture with the fact that mammals reparation costs scales in the order of 591:. An organism which doubles in length isometrically will find that the surface area available to it will increase fourfold, while its volume and mass will increase by a factor of eight. This can present problems for organisms. In the case of above, the animal now has eight times the biologically active tissue to support, but the surface area of its respiratory organs has only increased fourfold, creating a mismatch between scaling and physical demands. Similarly, the organism in the above example now has eight times the mass to support on its legs, but the strength of its bones and muscles is dependent upon their cross-sectional area, which has only increased fourfold. Therefore, this hypothetical organism would experience twice the bone and muscle loads of its smaller version. This mismatch can be avoided either by being "overbuilt" when small or by changing proportions during growth, called allometry. 621:, total length, etc.). A perfectly allometrically scaling organism would see all volume-based properties change proportionally to the body mass, all surface area-based properties change with mass to the power of 2/3, and all length-based properties change with mass to the power of 1/3. If, after statistical analyses, for example, a volume-based property was found to scale to mass to the 0.9th power, then this would be called "negative allometry", as the values are smaller than predicted by isometry. Conversely, if a surface area-based property scales to mass to the 0.8th power, the values are higher than predicted by isometry and the organism is said to show "positive allometry". One example of positive allometry occurs among species of monitor lizards (family 887:

investigated once the role of taxonomy is established. The challenge with this lies in the fact that a shared environment also indicates a common evolutionary history and thus a close taxonomic relationship. There are strides currently in research to overcome these hurdles; for example, an analysis in muroid rodents, the mouse, hamster, and vole type, took into account taxonomy. Results revealed the hamster (warm dry habitat) had lowest BMR and the mouse (warm wet dense habitat) had the highest BMR. Larger organs could explain the high BMR groups, along with their higher daily energy needs. Analyses such as these demonstrate the physiological adaptations to environmental changes that animals undergo.

666:. Sometimes, the two analyses can yield different results, but often they do not. If the expected slope is outside the confidence intervals, allometry is present. If the mass in this imaginary animal scaled with a slope of 5, which was a statistically significant value, then mass would scale very fast in this animal versus the expected value. It would scale with positive allometry. If the expected slope were 3 and in reality, in a certain organism mass scaled with 1 (assuming this slope is statistically significant), it would be negatively allometric. 165: 810:, 1932. This means that larger-bodied species (e.g., elephants) have lower mass-specific metabolic rates and lower heart rates, as compared with smaller-bodied species (e.g., mice). The straight line generated from a double logarithmic scale of metabolic rate in relation to body mass is known as the "mouse-to-elephant curve". These relationships of metabolic rates, times, and internal structure have been explained as, "an elephant is approximately a blown-up gorilla, which is itself a blown-up mouse." 655:

measurement. In biology, this is appropriate because many biological phenomena (e.g., growth, reproduction, metabolism, sensation) are fundamentally multiplicative. Statistically, it is beneficial to transform both axes using logarithms and then perform a linear regression. This will normalize the data set and make it easier to analyze trends using the slope of the line. Before analyzing data, it is important to have a predicted slope of the line to compare the analysis to.

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of gait A and these three coefficients, one could produce gait B, and vice versa. The hypothesis itself is as follows: "animals of different sizes tend to move in dynamically similar fashion whenever the ratio of their speed allows it." While the dynamic similarity hypothesis may not be a truly unifying principle of animal gait patterns, it is a remarkably accurate heuristic.

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the viscosity of the fluid compared to a bacterium in the same medium. The way in which the fluid interacts with the external boundaries of the organism is important with locomotion through the fluid. For streamlined swimmers, the resistance or drag determines the performance of the organism. This drag or resistance can be seen in two distinct flow patterns:

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conditions, evolutionary factors, and the availability of food; a small population of large predators depend on a much greater population of small prey to survive. In an aquatic environment, the largest animals can grow to have a much greater body mass than land animals where gravitational weight constraints are a factor.

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even been combined with evolutionary algorithms to form realistic hypotheses concerning the locomotive patterns of extinct species. These studies have been made possible by the remarkable similarities among disparate species' locomotive kinematics and dynamics, "despite differences in morphology and size".

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Physiological scaling in muscles affects the number of muscle fibers and their intrinsic speed to determine the maximum power and efficiency of movement in a given animal. The speed of muscle recruitment varies roughly in inverse proportion to the cube root of the animal's weight (compare the intrinsic

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slope is 3, meaning in this case, the mass is increasing extremely fast. For example, different sized frogs should be able to jump the same distance according to the geometric similarity model proposed by Hill 1950 and interpreted by Wilson 2000, but in actuality larger frogs do jump longer distances.

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characteristics of animals are similar in a wide range of animal sizes, though muscle sizes and shapes can and often do vary depending on environmental constraints placed on them. The muscle tissue itself maintains its contractile characteristics and does not vary depending on the size of the animal.

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Consequently, the body mass itself can explain the majority of the variation in the BMR. After the body mass effect, the taxonomy of the animal plays the next most significant role in the scaling of the BMR. The further speculation that environmental conditions play a role in BMR can only be properly

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Another example: Force is dependent on the cross-sectional area of muscle (CSA), which is L. If comparing force to a length, then the expected slope is 2. Alternatively, this analysis may be accomplished with a power regression. Plot the relationship between the data onto a graph. Fit this to a power

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Flying organisms such as birds are also considered as moving through a fluid. In scaling birds of similar shape, it has also been seen that larger individuals have less metabolic costs per kg, as expected. Birds also have a variance in wing beat frequency. Beyond the compensation of larger wings per

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Traveling long distances and deep dives are a combination of good stamina and also moving an efficient speed and in an efficient way to create laminar flow, reducing drag and turbulence. In sea water as the fluid, it traveling long distances in large mammals, such as whales, is facilitated by their

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Scaling also has an effect on the performance of organisms in fluid. This is extremely important for marine mammals and other marine organisms that rely on atmospheric oxygen for respiration and survival. This can affect how fast an organism can propel itself efficiently or how long and deep it can

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is very helpful in determining expected slope. This 'expected' slope, as it is known, is essential for detecting allometry because scaling variables are comparisons to other things. Saying that mass scales with a slope of 5 in relation to length doesn't have much meaning unless knowing the isometric

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that absorb energy from the propulsion or momentum of the organism. Scaling also affects locomotion through a fluid because of the energy needed to propel an organism and keep up velocity through momentum. The rate of oxygen consumption per gram body size decreases consistently with increasing body

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The mass and density of an organism have a large effect on the organism's locomotion through a fluid. For example, a tiny organism uses flagella and can effectively move through a fluid it is suspended in, while on the other end of the scale, a blue whale is much more massive and dense relative to

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Alexander incorporates Froude-number analysis into his "dynamic similarity hypothesis" of gait patterns. Dynamically similar gaits are those between which there are constant coefficients that can relate linear dimensions, time intervals, and forces. In other words, given a mathematical description

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Isometric scaling happens when proportional relationships are preserved as size changes during growth or over evolutionary time. An example is found in frogs—aside from a brief period during the few weeks after metamorphosis, frogs grow isometrically. Therefore, a frog whose legs are as long as its

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Arguing that there are a number of analogous concepts and mechanisms between cities and biological entities, Bettencourt et al. showed a number of scaling relationships between observable properties of a city and the city size. GDP, "supercreative" employment, number of inventors, crime, spread of

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It has also been shown that living organisms of all shapes and sizes utilize spring mechanisms in their locomotive systems, probably in order to minimize the energy cost of locomotion. The allometric study of these systems has fostered a better understanding of why spring mechanisms are so common,

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Allometry has been used to study patterns in locomotive principles across a broad range of species. Such research has been done in pursuit of a better understanding of animal locomotion, including the factors that different gaits seek to optimize. Allometric trends observed in extant animals have

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To determine whether isometry or allometry is present, an expected relationship between variables needs to be determined to compare data to. This is important in determining if the scaling relationship in a dataset deviates from an expected relationship (such as those that follow isometry). Using

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There are two reasons why logarithmic transformation should be used to study allometry —a biological reason and a statistical reason. Log-log transformation places numbers into a geometric domain so that proportional deviations are represented consistently, independent of the scale and units of

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Data gathered in science do not fall neatly in a straight line, so data transformations are useful. It is also important to remember what is being compared in the data. Comparing a characteristic such as head length to head width might yield different results from comparing head length to body

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is one of the largest determining factors in its size. On land, there is a positive correlation between body mass of the top species in the area and available land area. However, there are a much greater number of "small" species in any given area. This is most likely determined by ecological

629:, the weight of which grows with about the power of 3.325 of its length. A 30-inch (76 cm) muskellunge will weigh about 8 pounds (3.6 kg), while a 40-inch (100 cm) muskellunge will weigh about 18 pounds (8.2 kg), so 33% longer length will more than double the weight. 890:

Energy metabolism is subjected to the scaling of an animal and can be overcome by an individual's body design. The metabolic scope for an animal is the ratio of resting and maximum rate of metabolism for that particular species as determined by oxygen consumption. Oxygen consumption

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of the objects' dimensions. Two objects of different size, but common shape, have their dimensions in the same ratio. Take, for example, a biological object that grows as it matures. Its size changes with age, but the shapes are similar. Studies of ontogenetic allometry often use

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To find the expected slope for the relationship between mass and the characteristic length of an animal (see figure), the units of mass (M) from the y-axis are divided by the units of the x-axis, Length (L). The expected slope on a double-logarithmic plot of L / L is 3

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In addition to studies that focus on growth, allometry also examines shape variation among individuals of a given age (and sex), which is referred to as static allometry. Comparisons of species are used to examine interspecific or evolutionary allometry (see also

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Across a broad range of species, allometric relations are not necessarily linear on a log-log scale. For example, the maximal running speeds of mammals show a complicated relationship with body mass, and the fastest sprinters are of intermediate body size.

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Allometric study of locomotion involves the analysis of the relative sizes, masses, and limb structures of similarly shaped animals and how these features affect their movements at different speeds. Patterns are identified based on dimensionless

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is the skeleton of mammals. The skeletal structure becomes much stronger and more robust relative to the size of the body as the body size increases. Allometry is often expressed in terms of a scaling exponent based on body mass, or body length

1496: = 10), but exhibit it in nature. G. A. Steven observed and documented dolphins moving at 15 knots alongside his ship leaving a single trail of light when phosphorescent activity in the sea was high. The factors that contribute are: 157:), where a small change in overall body size can lead to an enormous and disproportionate increase in the dimensions of appendages such as legs, antennae, or horns The relationship between the two measured quantities is often expressed as a 1297:

The physiological effect of drugs and other substances in many cases scales allometrically. For example, plasma concentration of carotenoids scales to the three-quarter power of mass in nine predatory and scavenger raptor species.

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Many physiological and biochemical processes (such as heart rate, respiration rate or the maximum reproduction rate) show scaling, mostly associated with the ratio between surface area and mass (or volume) of the animal. The

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Max Kleiber contributed the following allometric equation for relating the BMR to the body mass of an animal. Statistical analysis of the intercept did not vary from 70 and the slope was not varied from 0.75, thus:

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neutral buoyancy and have their mass completely supported by the density of the sea water. On land, animals have to expend a portion of their energy during locomotion to fight the effects of gravity.

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R. O. Anderson and R. M. Neumann, Length, Weight, and Associated Structural Indices, in Fisheries Techniques, second edition, B.E. Murphy and D.W. Willis, eds., American Fisheries Society, 1996.

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Many factors go into the determination of body mass and size for a given animal. These factors often affect body size on an evolutionary scale, but conditions such as availability of food and

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the velocity of an organism through fluid, which changes the dynamic of the flow around that organism – the shape of the organism becomes more important for laminar flow as velocity increases.

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distributions indicating that, despite the complexity of their systems, there is a power law dependence of similarity; therefore, universal patterns are observed in diverse animal taxonomy.

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Mechanical design can also determine the maximum allowable size for a species. Animals with tubular endoskeletons tend to be larger than animals with exoskeletons or hydrostatic skeletons.

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Basic physiological design plays a role in the size of a given species. For example, animals with a closed circulatory system are larger than animals with open or no circulatory systems.

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is the number. That "number" is the relationship between the data points. The downside, to this form of analysis, is that it makes it a little more difficult to do statistical analyses.

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is the body mass of the individual. Lung volume is also directly related to body mass in mammals (slope = 1.02). The lung has a volume of 63 ml for every kg of body mass, with the

1733:, this shows having a larger body mass can be advantageous. More simply, a larger whale can hold more oxygen and at the same time demand less metabolically than a smaller whale. 337: 594:

Isometric scaling is often used as a null hypothesis in scaling studies, with 'deviations from isometry' considered evidence of physiological factors forcing allometric growth.

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Duty factors—percentages of a stride during which a foot maintains contact with the ground—remain relatively constant for different animals moving with the same Froude number.

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For inter-species allometric relations related to such ecological variables as maximal reproduction rate, attempts have been made to explain scaling within the context of

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The dynamic similarity hypothesis states that "animals of different sizes tend to move in dynamically similar fashion whenever the ratio of their speed allows it".

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Aquatic mammals, like other mammals, have the same size heart proportional to their bodies. In general, mammals have hearts about 0.6% of their total body mass:

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dependence. Overall metabolic rate in animals is generally accepted to show negative allometry, scaling to mass to a power of ≈ 0.75, known as

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for practical applications to the differential growth rates of the parts of a living organism's body. One application is in the study of various

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Rothman DH, Weitz JS (Nov 2005). "Beyond the '3/4-power law': variation in the intra- and interspecific scaling of metabolic rate in animals".

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length. That is, different characteristics may scale differently. A common way to analyze data such as those collected in scaling is to use

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Blanco, G.; Bautista, L.M.; Hornero-Méndez, D.; Lambertucci, S.A.; Wiemeywer, G.; Sánchez-Zapata, J.A.; Hiraldo, F.; Donazar, J.A. (2014).

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Robinson, Michael; Motta, Philip (2002). "Patterns of growth and the effects of scale on the feeding kinematics of the nurse shark (

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Wilson RS, Franklin CE, James RS (June 2000). "Allometric scaling relationships of jumping performance in the striped marsh frog

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E.L. McCullough, K.J. Ledger, D.M. O'Brien, D.J. Emlen (2015) Variation in the allometry of exaggerated rhinoceros beetle horns.

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at rest being 1/10 the lung volume. In addition, respiration costs with respect to oxygen consumption is scaled in the order of

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West, Brown, and Enquist in 1997 derived a hydrodynamic theory to explain the universal fact that metabolic rate scales as the

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Kerkhoff, A.J.; Enquist, B.J. (2009). "Multiplicative by nature: Why logarithmic transformation is necessary in allometry".

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how limb compliance varies with body size and speed, and how these mechanisms affect general limb kinematics and dynamics.

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dive. Heart mass and lung volume are important in determining how scaling can affect metabolic function and efficiency.

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West, G. B.; Brown, J.H.; Enquist, B. J. (1997). "A general model for the origin of allometric scaling laws in biology".

2214: 899:. Oxygen consumption in species that differ in body size and organ system dimensions show a similarity in their charted V 4916: 4857: 4782: 4777: 4772: 1401: 820: 1434: 395: 4787: 4572: 2350: 1384:

might be used for substances that are eliminated mainly by metabolism, or by metabolism and excretion combined, while

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The resistance to the motion of an approximately stream-lined solid through a fluid can be expressed by the formula:

378:, which does not account for error variance in the independent variable (e.g., log body mass). Other methods include 4408: 4039:

Dickinson MH, Farley CT, Full RJ, Koehl MA, Kram R, Lehman S (April 2000). "How animals move: an integrative view".

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US FDA: Estimating the Safe Starting Dose in Clinical Trials for Therapeutics in Adult Healthy Volunteers, July 2005

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curve (depending on the stats program, this can be done multiple ways), and it will give an equation with the form:

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Human body size and the laws of scaling: physiological, performance, growth, longevity and ecological ramifications

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the length of the organism, as the surface area of just the front 2/3 of the organism has an effect on the drag.

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body will retain that relationship throughout its life, even if the frog itself increases in size tremendously.

83: 4534: 1985: 659: 625:), in which the limbs are relatively longer in larger-bodied species. The same is true for some fish, e.g. the 1488: > 2.0×106). Also, increase in velocity (V) increases turbulence, which can be proved using the 164: 4084:"Allometric Engineering: An Experimental Test of the Causes of Interpopulational Differences in Performance" 2344: 2080: 224: 4862: 4767: 3173: 3093: 2718:

Bonduriansky, Russell; Day, Troy (2003). "The Evolution of Static Allometry in Sexually Selected Traits".

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The amount by which a leg shortens during a stride (i.e. its peak displacement) is proportional to

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of the law. Methods for estimating this exponent from data can use type-2 regressions, such as

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Dodds PS, Rothman DH, Weitz JS (March 2001). "Re-examination of the "3/4-law" of metabolism".

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Emerson S. B. (September 1978). "Allometry and Jumping in Frogs: Helping the Twain to Meet".

2127: 1484: < 0.5x106), whereas larger, less streamlined organisms produce turbulent flow ( 1045:, which incorporate measures of animals' leg lengths, speed or stride frequency, and weight. 4880: 4815: 4610: 4426: 4216: 4206: 4159: 4108: 4056: 3937: 3929: 3873: 3844: 3836: 3792: 3784: 3742: 3690: 3682: 3636: 3580: 3572: 3528: 3479: 3432: 3422: 3366: 3322: 3267: 3257: 3183: 3138: 3103: 3054: 3013: 2872: 2807: 2727: 2678: 2664:"Seasonal, sexual, and individual variation in endurance and activity metabolism in lizards" 2606: 2581: 2499: 2458: 2448: 2395: 2368: 1773: 807: 1779: 4931: 4638: 4622: 4605: 4412: 1561: 2396: 2180: 1951: 1826: 1500:

the surface area of the organism and its effect on the fluid in which the organism lives.

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The allometric equation can also be acquired as a solution of the differential equation

4890: 4708: 4680: 4665: 3942: 3915: 3797: 3772: 3695: 3670: 3585: 3560: 3483: 3437: 3402: 3272: 3237: 2731: 2488:"Die Abhängigkeit des Hirngewichts von dem Körpergewicht und den geistigen Fähigkeiten" 2374: 2160: 2122: 2054: 2034: 1931: 1911: 1806: 1469: 1468:, where the fluid is relatively uninterrupted after the organism moves through it, and 1443: 1405: 1148: 862: 787: 556: 345: 3773:"Fleximble mechanisms: the diverse roles of biological springs in vertebrate movement" 3532: 68:

The proportionately thicker bones in the elephant are an example of allometric scaling

4910: 4751: 4688: 4593: 4588: 4509: 4467: 4249: 4221: 4186: 3920: 3754: 3187: 2747: 2463: 2428: 1063: 1042: 465: 130: 79: 3965: 3195: 2682: 2511: 2365: – Functional relationship between two quantities (also known as a scaling law) 1111:{\displaystyle k_{\text{leg}}={\frac {\text{peak force}}{\text{peak displacement}}}} 4835: 4746: 4741: 4477: 4171: 4128: 3893: 3511:

Chappell, R. (1989). "Fitting bent lines to data, with applications to allometry".

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size can act much more quickly on a species. Other examples include the following:

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Alexander found that animals of different sizes and masses traveling with the same

648: 4112: 2585: 2377: – Quantitative relations between some key characteristic dimensions of trees 1351:) scale in the same way, leading to blood pressure being constant across species. 27:

Study of the relationship of body size to shape, anatomy, physiology, and behavior

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An online allometric scaler of drug doses based on the above work is available.

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After data are log-transformed and linearly regressed, comparisons can then use

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is a method for manipulating allometric relationships within or among groups.

754:{\displaystyle {\frac {\log _{10}\mathrm {L} ^{3}}{\log _{10}\mathrm {L} }}=3} 112: 3916:"Allometric scaling of xenobiotic clearance: uncertainty versus universality" 2341: – Theory regarding evolutionary trends in the shape of mammalian skulls 472:

or hatching and because they exhibit a large range of body sizes between the

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Hu and Hayton in 2001 discussed whether the basal metabolic rate scale is a

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equation (allometric equation) which expresses a remarkable scale symmetry:

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Alexander, R. McN. (1984). "The gaits of bipedal and quadrupedal animals".

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disease, and even pedestrian walking speeds scale with city population.

4718: 4449: 3849: 3788: 2884: 2819: 2503: 2267: 1005:{\displaystyle \mathrm {frequency} ={\frac {1}{\mathrm {mass} ^{1/3}}}} 896: 880: 607: 457: 146: 108: 3840: 2858:"Scaling of limb proportions in monitor lizards (Squamata: Varanidae)" 2347: – Study of evolutionary changes in physiological characteristics 4163: 3933: 3156:

O'Hara, R.B.; Kotze, D.J. (2010). "Do not log-transform count data".

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In general, smaller, more streamlined organisms create laminar flow (

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might apply for drugs that are eliminated mainly by renal excretion.

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power with body weight. They also showed why lifespan scales as the +

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Bettencourt LM, Lobo J, Helbing D, Kühnert C, West GB (April 2007).

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Bornstein MH, Bornstein HG (19 February 1976). "The Pace of Life".

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Online allometric scaling calculator, with explanation and source.

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Body mass has even more of an effect than speed on limb dynamics.

530: 103:"measurement") is the study of the relationship of body size to 4422: 2329: – Body scale based on waist/hip circumference and height 1415: 1168:

Peak force experienced throughout a stride is proportional to

3238:"Growth, innovation, scaling, and the pace of life in cities" 1472:, where the fluid moves roughly around an organism, creating 480:

stage. Lizards often exhibit allometric changes during their

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The angle swept by a leg during a stride is proportional to

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Principles of legged locomotion identified through allometry

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Allometric relations show as straight lines when plotted on

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of flying bodies (insects, birds, airplanes) and body mass

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Burness, G. P.; Diamond, Jared; Flannery, Timothy (2001).

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The mass-specific work rate of a limb is proportional to

3822:"Allometric deviations of plasma carotenoids in raptors" 2767:

Life's Devices: The Physical World of Animals and Plants

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Sellers, William Irving; Manning, Phillip Lars (2007).

2994:"The dimensions of animals and their muscular dynamics" 1439: 499: 451:

Allometry often studies shape differences in terms of

2217: 2183: 2163: 2130: 2083: 2057: 2037: 1988: 1954: 1934: 1914: 1863: 1829: 1809: 1782: 1712: 1685: 1637: 1267: 1234: 1204: 1174: 1151: 1124: 1084: 933: 865: 823: 693: 398: 348: 289: 227: 178: 4007: 4005: 4003: 4001: 3999: 2323: – Study of the mechanics of biological systems 1664:{\displaystyle {\text{Heart weight}}=0.006{M}^{1.0}} 924:

of the sparrow's flight muscle to that of a stork).

790:

of an individual animal is also subject to scaling.

602:

Allometric scaling is any change that deviates from

4798: 4760: 4727: 4679: 4631: 4581: 4565: 4527: 4460: 4415:(For "first in human" clinical trials of new drugs) 3623:Farley CT, Glasheen J, McMahon TA (December 1993). 2249:{\displaystyle V_{\text{opt}}\sim M^{\frac {1}{6}}} 2248: 2197: 2169: 2149: 2107: 2063: 2043: 2017: 1968: 1940: 1920: 1894: 1843: 1815: 1795: 1725: 1698: 1663: 1280: 1250: 1217: 1187: 1157: 1137: 1110: 1004: 871: 848:{\displaystyle {\text{Metabolic rate}}=70M^{0.75}} 847: 753: 440: 354: 331: 269: 203: 4406:FDA Guidance for Estimating Human Equivalent Dose 1023:. However, such ideas have been less successful. 633:Determining if a system is scaling with allometry 441:{\displaystyle {\frac {dy}{y}}=a{\frac {dx}{x}}.} 141:Allometry is a well-known study, particularly in 4293:Plant allometry: The scaling of form and process 3728: 3726: 3724: 3722: 3720: 3718: 3716: 3714: 879:is body mass, and metabolic rate is measured in 145:for its theoretical developments, as well as in 3401:Labra FA, Marquet PA, Bozinovic F (June 2007). 3213: 3211: 3209: 3207: 3205: 2945: 2840: 3766: 3764: 3735:The International Journal of Robotics Research 3664: 3662: 3660: 3559:Daley, Monica A.; Usherwood, James R. (2010). 3231: 3229: 2601:Longo, Giuseppe; Montévil, Maël (2014-01-01). 2531:(Canto ed.). Cambridge University Press. 4434: 4015:Animal Physiology: Adaptation and Environment 3618: 3616: 3614: 3612: 3610: 3608: 3606: 3604: 2596: 2594: 1928:are both inversely proportional to body mass 8: 4018:(5th ed.). Cambridge University Press. 3554: 3552: 3550: 2987: 2985: 2983: 2981: 3771:Roberts, Thomas J.; Azizi, Emanuel (2011). 1066:consistently exhibit similar gait patterns. 4441: 4427: 4419: 2770:. Princeton University Press. p. 39. 1895:{\displaystyle q_{0}\sim M^{\frac {3}{4}}} 496:Isometric scaling and geometric similarity 464:as model organisms both because they lack 4362:. Cambridge: Cambridge University Press. 4359:Scaling: why is animal size so important? 4220: 4210: 3941: 3848: 3796: 3694: 3640: 3584: 3436: 3426: 3308: 3271: 3261: 3177: 3097: 3017: 2462: 2452: 2262:Determinants of size in different species 2235: 2222: 2216: 2187: 2182: 2162: 2135: 2129: 2094: 2082: 2056: 2036: 2018:{\displaystyle t\sim M^{-{\frac {1}{4}}}} 2003: 1999: 1987: 1958: 1953: 1933: 1913: 1881: 1868: 1862: 1833: 1828: 1808: 1787: 1781: 1717: 1711: 1690: 1684: 1655: 1650: 1638: 1636: 1272: 1266: 1239: 1233: 1209: 1203: 1179: 1173: 1150: 1129: 1123: 1098: 1089: 1083: 990: 986: 972: 966: 934: 932: 864: 839: 824: 822: 761:). This is the slope of a straight line. 737: 728: 716: 711: 701: 694: 692: 420: 399: 397: 347: 288: 226: 192: 177: 4335:The allometry of growth and reproduction 4314:The ecological implications of body size 3625:"Running springs: speed and animal size" 2925:. Oxford University Press. p. 111. 2121:the proportionality between the optimal 4012:Schmidt-Nielsen, Knut (10 April 1997). 2386: 1506:the density and viscosity of the fluid. 168:Allometric equation: way of expressions 4269:McMahon, T. A.; Bonner, J. T. (1983). 2108:{\displaystyle A\sim M^{\frac {7}{8}}} 1623:> 2.0×10 = turbulent flow threshold 1412:Allometric scaling in fluid locomotion 501:Scaling range for different organisms 270:{\displaystyle \log y=a\log x+\log k,} 3403:"Scaling metabolic rate fluctuations" 613:Dialogues Concerning Two New Sciences 587:Isometric scaling is governed by the 7: 1617:< 0.5×10 = laminar flow threshold 1374:power of body mass. The exponent of 3777:The Journal of Experimental Biology 3220:Environmental Physiology of Animals 2662:, T. Jr.; P. L. Else (March 1987). 2299:An animal's habitat throughout its 332:{\displaystyle \ln y=a\ln x+\ln k,} 3675:Proceedings of the Royal Society B 3484:10.1111/j.1469-7998.1983.tb02087.x 2732:10.1111/j.0014-3820.2003.tb01490.x 1769:Some examples of allometric laws: 982: 979: 976: 973: 959: 956: 953: 950: 947: 944: 941: 938: 935: 738: 712: 25: 2556:(2nd ed.). New York: Dover. 1433:to comply with Knowledge (XXG)'s 911:Allometric muscle characteristics 662:with 95% confidence intervals or 606:. A classic example discussed by 115:and behaviour, first outlined by 4885: 4876: 4875: 4518: 3188:10.1111/j.2041-210X.2010.00021.x 3158:Methods in Ecology and Evolution 2992:Hill, A.V. (November 12, 1949). 2357:Phylogenetic comparative methods 1420: 490:Phylogenetic comparative methods 382:models and a particular kind of 51: 34: 4296:. University of Chicago Press. 4275:. Scientific American Library. 4251:Size, function and life history 3966:"Allometric Scaling Calculator" 2683:10.1152/ajpregu.1987.252.3.R439 2398:The Statistical Theory of Shape 895:and maximum oxygen consumption 4338:. Cambridge University Press. 4317:. Cambridge University Press. 4082:Sinervo, B.; Huey, R. (1990). 3513:Journal of Theoretical Biology 3297:Journal of Theoretical Biology 3123:Journal of Theoretical Biology 2394:Small, Christopher G. (1996). 2339:Cranial evolutionary allometry 1027:Allometry of legged locomotion 1: 4113:10.1126/science.248.4959.1106 3533:10.1016/S0022-5193(89)80141-9 2921:Pennycuick, Colin J. (1992). 2586:10.1016/j.anbehav.2015.08.013 1803:is proportional to body mass 1321:power and heart rate as the - 372:reduced major axis regression 4858:Standard anatomical position 4783:Glossary of plant morphology 4778:Glossary of dinosaur anatomy 4773:Anatomical terms of location 4254:. Harvard University Press. 4191:Proc. Natl. Acad. Sci. U.S.A 4061:10.1126/science.288.5463.100 3878:10.1126/science.276.5309.122 3407:Proc. Natl. Acad. Sci. U.S.A 3242:Proc. Natl. Acad. Sci. U.S.A 2433:Proc. Natl. Acad. Sci. U.S.A 1756:In characteristics of a city 1402:Food and Drug Administration 794:Metabolic rate and body mass 384:principal component analysis 4377:Samaras, Thomas T. (2007). 2611:10.1007/978-3-642-35938-5_2 2554:Problems of Relative Growth 2525:Thompson, D'Arcy W (1992). 2351:Metabolic theory of ecology 2031:mass transfer contact area 1021:metabolic theory of ecology 664:reduced major axis analysis 4948: 3747:10.1177/027836498400300205 3472:Journal of Zoology, London 3143:10.1016/j.jtbi.2008.12.026 3086:Journal of Zoology, London 2552:Huxley, Julian S. (1972). 2427:Damuth J (February 2001). 1610:Notable Reynolds numbers: 1596:= axial length of organism 1251:{\displaystyle M^{-0.034}} 143:statistical shape analysis 93: 82: 4896:Index of anatomy articles 4871: 4516: 3371:10.1017/S1464793105006834 3108:10.1017/S0952836902000493 2634:Frydrýšek, Karel (2019). 2603:Perspectives on Organisms 1908:breathing and heart rate 204:{\displaystyle y=kx^{a},} 4545:morphological plasticity 4535:Bacterial cell structure 1726:{\displaystyle M^{0.75}} 1699:{\displaystyle M^{0.75}} 1446:may contain suggestions. 1431:may need to be rewritten 1281:{\displaystyle M^{0.11}} 1218:{\displaystyle M^{0.30}} 1188:{\displaystyle M^{0.97}} 1138:{\displaystyle M^{0.67}} 798:In plotting an animal's 660:least squares regression 376:least-squares regression 3428:10.1073/pnas.0704108104 3263:10.1073/pnas.0610172104 3059:10.1242/jeb.203.12.1937 2958:Gibbings, J.C. (2011). 2345:Evolutionary physiology 2150:{\displaystyle V_{opt}} 4927:Ecological experiments 4863:Transcendental anatomy 4768:Anatomical terminology 4311:Peters, R. H. (1983). 4290:Niklas, K. J. (1994). 4248:Calder, W. A. (1984). 4212:10.1073/pnas.251548698 3687:10.1098/rspb.2007.0846 3577:10.1098/rsbl.2010.0175 3327:10.1006/jtbi.2000.2238 3082:Ginglymostoma cirratum 2865:Journal of Herpetology 2454:10.1073/pnas.051011198 2333:Comparative physiology 2250: 2199: 2171: 2151: 2109: 2065: 2045: 2019: 1970: 1942: 1922: 1896: 1845: 1817: 1797: 1750:Allometric engineering 1745:Allometric engineering 1727: 1700: 1665: 1282: 1252: 1219: 1189: 1159: 1139: 1112: 1006: 873: 849: 755: 683: 442: 356: 333: 271: 205: 169: 4540:cellular morphologies 4332:Reiss, M. J. (1989). 3970:Clymer.altervista.org 3218:Willmer, Pat (2009). 3043:Limnodynastes peronii 2251: 2200: 2172: 2152: 2110: 2066: 2046: 2020: 1971: 1943: 1923: 1897: 1846: 1818: 1798: 1796:{\displaystyle q_{0}} 1728: 1701: 1666: 1331:power. Blood flow (+ 1283: 1253: 1220: 1190: 1160: 1140: 1118:, is proportional to 1113: 1017:dynamic energy budget 1007: 874: 850: 781:Physiological scaling 756: 677: 443: 368:major axis regression 357: 334: 272: 206: 167: 4811:Anatomical variation 4494:Microscopic anatomy 3642:10.1242/jeb.185.1.71 2961:Dimensional Analysis 2946:Schmidt-Nielsen 1984 2923:Newton Rules Biology 2841:Schmidt-Nielsen 1984 2402:. Springer. p. 2327:Body roundness index 2274:Physiological design 2215: 2181: 2177:raised to the power 2161: 2128: 2081: 2055: 2035: 1986: 1952: 1932: 1912: 1861: 1827: 1807: 1780: 1710: 1683: 1635: 1567:The Reynolds number 1341:) and resistance (- 1265: 1232: 1202: 1172: 1149: 1122: 1082: 931: 863: 821: 800:basal metabolic rate 691: 640:dimensional analysis 396: 346: 287: 225: 176: 63:(Dasyurus maculatus) 4917:Branches of biology 4826:Form classification 4694:Neanderthal anatomy 4552:Colonial morphology 4490:Comparative anatomy 4473:Superficial anatomy 4383:. Nova Publishers. 4354:Schmidt-Nielsen, K. 4203:2001PNAS...9814518B 4156:1976Natur.259..557B 4105:1990Sci...248.1106S 4053:2000Sci...288..100D 3525:1989JThBi.138..235C 3419:2007PNAS..10410900L 3319:2001JThBi.209....9D 3254:2007PNAS..104.7301B 3170:2010MEcEv...1..118O 3135:2009JThBi.257..519K 3010:1949Natur.164R.820. 2677:(3 Pt 2): R439–49. 2486:Otto Snell (1892). 2445:2001PNAS...98.2113D 2198:{\displaystyle 1/6} 1969:{\displaystyle 1/4} 1844:{\displaystyle 3/4} 1606:(viscosity/density) 1604:kinematic viscosity 502: 4841:History of anatomy 4528:Bacteria and fungi 4411:2009-05-11 at the 3789:10.1242/jeb.038588 3359:Biological Reviews 3222:. Wiley-Blackwell. 3053:(Pt 12): 1937–46. 2528:On Growth and Form 2504:10.1007/BF01843462 2315:Biomass allocation 2246: 2195: 2167: 2147: 2105: 2061: 2041: 2015: 1966: 1938: 1918: 1892: 1841: 1813: 1793: 1723: 1696: 1661: 1554:− 1 (laminar flow) 1539:= density of fluid 1408:from animal data. 1278: 1248: 1215: 1185: 1155: 1135: 1108: 1002: 869: 845: 751: 684: 680:double-logarithmic 649:log-transformation 598:Allometric scaling 500: 438: 352: 329: 267: 201: 170: 126:On Growth and Form 18:Allometric scaling 4904: 4903: 4737:Amphibian anatomy 4729:Other vertebrates 4671:Arthropod cuticle 4649:Insect morphology 4644:Gastropod anatomy 4557:Lichen morphology 4483:brain morphometry 4390:978-1-60021-408-0 4369:978-0-521-31987-4 4345:978-0-521-42358-8 4324:978-0-521-28886-6 4303:978-0-226-58081-4 4282:978-0-7167-5000-0 4261:978-0-674-81070-9 4025:978-0-521-57098-5 3872:(5309): 122–126. 3841:10.1111/ibi.12155 2971:978-1-84996-317-6 2932:978-0-19-854021-2 2777:978-0-691-02418-9 2726:(11): 2450–2458. 2645:978-80-248-4263-9 2563:978-0-486-61114-3 2538:978-0-521-43776-9 2413:978-0-387-94729-7 2284:Mechanical design 2243: 2225: 2170:{\displaystyle M} 2102: 2064:{\displaystyle M} 2044:{\displaystyle A} 2011: 1941:{\displaystyle M} 1921:{\displaystyle t} 1889: 1816:{\displaystyle M} 1776:, metabolic rate 1641: 1490:Reynolds equation 1461: 1460: 1435:quality standards 1293:Drug dose scaling 1158:{\displaystyle M} 1106: 1105: 1104:peak displacement 1102: 1092: 1000: 872:{\displaystyle M} 827: 743: 619:snout–vent length 581: 580: 433: 412: 380:measurement-error 355:{\displaystyle a} 16:(Redirected from 4939: 4889: 4879: 4878: 4816:Anatomical plane 4714:Elephant anatomy 4611:Plant morphology 4522: 4443: 4436: 4429: 4420: 4394: 4373: 4349: 4328: 4307: 4286: 4272:On Size and Life 4265: 4235: 4234: 4224: 4214: 4197:(25): 14518–23. 4182: 4176: 4175: 4164:10.1038/259557a0 4139: 4133: 4132: 4099:(4959): 1106–9. 4088: 4079: 4073: 4072: 4036: 4030: 4029: 4009: 3994: 3989: 3983: 3981: 3979: 3977: 3962: 3956: 3955: 3945: 3934:10.1208/ps030429 3904: 3898: 3897: 3861: 3855: 3854: 3852: 3826: 3817: 3811: 3810: 3800: 3768: 3759: 3758: 3730: 3709: 3708: 3698: 3681:(1626): 2711–6. 3666: 3655: 3654: 3644: 3620: 3599: 3598: 3588: 3556: 3545: 3544: 3508: 3502: 3501: 3499: 3498: 3492: 3486:. Archived from 3469: 3463:Jr., T. (1983). 3457: 3451: 3450: 3440: 3430: 3398: 3392: 3390: 3354: 3348: 3346: 3312: 3292: 3286: 3285: 3275: 3265: 3233: 3224: 3223: 3215: 3200: 3199: 3181: 3153: 3147: 3146: 3118: 3112: 3111: 3101: 3077: 3071: 3070: 3038: 3032: 3031: 3021: 3019:10.1038/164820b0 2989: 2976: 2975: 2955: 2949: 2943: 2937: 2936: 2918: 2912: 2909: 2903: 2902: 2900: 2899: 2893: 2887:. Archived from 2862: 2849: 2843: 2838: 2832: 2831: 2795: 2789: 2788: 2786: 2784: 2758: 2752: 2751: 2715: 2709: 2708: 2706: 2705: 2699: 2693:. Archived from 2668: 2656: 2650: 2649: 2631: 2625: 2624: 2598: 2589: 2578:Animal Behaviour 2574: 2568: 2567: 2549: 2543: 2542: 2522: 2516: 2515: 2483: 2477: 2476: 2466: 2456: 2424: 2418: 2417: 2401: 2391: 2255: 2253: 2252: 2247: 2245: 2244: 2236: 2227: 2226: 2223: 2204: 2202: 2201: 2196: 2191: 2176: 2174: 2173: 2168: 2156: 2154: 2153: 2148: 2146: 2145: 2114: 2112: 2111: 2106: 2104: 2103: 2095: 2070: 2068: 2067: 2062: 2050: 2048: 2047: 2042: 2024: 2022: 2021: 2016: 2014: 2013: 2012: 2004: 1975: 1973: 1972: 1967: 1962: 1947: 1945: 1944: 1939: 1927: 1925: 1924: 1919: 1901: 1899: 1898: 1893: 1891: 1890: 1882: 1873: 1872: 1850: 1848: 1847: 1842: 1837: 1822: 1820: 1819: 1814: 1802: 1800: 1799: 1794: 1792: 1791: 1732: 1730: 1729: 1724: 1722: 1721: 1705: 1703: 1702: 1697: 1695: 1694: 1670: 1668: 1667: 1662: 1660: 1659: 1654: 1642: 1639: 1456: 1453: 1447: 1424: 1416: 1393: 1392: 1388: 1383: 1382: 1378: 1373: 1372: 1368: 1363: 1362: 1358: 1350: 1349: 1345: 1340: 1339: 1335: 1330: 1329: 1325: 1320: 1319: 1315: 1310: 1309: 1305: 1287: 1285: 1284: 1279: 1277: 1276: 1257: 1255: 1254: 1249: 1247: 1246: 1224: 1222: 1221: 1216: 1214: 1213: 1194: 1192: 1191: 1186: 1184: 1183: 1164: 1162: 1161: 1156: 1144: 1142: 1141: 1136: 1134: 1133: 1117: 1115: 1114: 1109: 1107: 1103: 1100: 1099: 1094: 1093: 1090: 1032:Methods of study 1011: 1009: 1008: 1003: 1001: 999: 998: 994: 985: 967: 962: 878: 876: 875: 870: 854: 852: 851: 846: 844: 843: 828: 825: 760: 758: 757: 752: 744: 742: 741: 733: 732: 722: 721: 720: 715: 706: 705: 695: 503: 447: 445: 444: 439: 434: 429: 421: 413: 408: 400: 361: 359: 358: 353: 338: 336: 335: 330: 276: 274: 273: 268: 210: 208: 207: 202: 197: 196: 155:Hercules beetles 102: 97: 91: 86: 55: 38: 21: 4947: 4946: 4942: 4941: 4940: 4938: 4937: 4936: 4907: 4906: 4905: 4900: 4867: 4794: 4788:leaf morphology 4756: 4723: 4719:Giraffe anatomy 4675: 4639:Decapod anatomy 4627: 4623:Soil morphology 4606:Plant life-form 4577: 4561: 4523: 4514: 4456: 4447: 4413:Wayback Machine 4402: 4397: 4391: 4376: 4370: 4352: 4346: 4331: 4325: 4310: 4304: 4289: 4283: 4268: 4262: 4247: 4243: 4241:Further reading 4238: 4184: 4183: 4179: 4150:(5544): 557–9. 4141: 4140: 4136: 4086: 4081: 4080: 4076: 4047:(5463): 100–6. 4038: 4037: 4033: 4026: 4011: 4010: 3997: 3990: 3986: 3975: 3973: 3964: 3963: 3959: 3906: 3905: 3901: 3863: 3862: 3858: 3824: 3819: 3818: 3814: 3770: 3769: 3762: 3732: 3731: 3712: 3668: 3667: 3658: 3622: 3621: 3602: 3565:Biology Letters 3558: 3557: 3548: 3510: 3509: 3505: 3496: 3494: 3490: 3467: 3459: 3458: 3454: 3413:(26): 10900–3. 3400: 3399: 3395: 3356: 3355: 3351: 3310:physics/0007096 3294: 3293: 3289: 3235: 3234: 3227: 3217: 3216: 3203: 3179:10.1.1.466.9313 3155: 3154: 3150: 3120: 3119: 3115: 3099:10.1.1.524.9341 3079: 3078: 3074: 3040: 3039: 3035: 2991: 2990: 2979: 2972: 2957: 2956: 2952: 2944: 2940: 2933: 2920: 2919: 2915: 2910: 2906: 2897: 2895: 2891: 2877:10.2307/1565513 2860: 2854:Garland T., Jr. 2852:Christian, A.; 2851: 2850: 2846: 2839: 2835: 2812:10.2307/2407721 2797: 2796: 2792: 2782: 2780: 2778: 2760: 2759: 2755: 2717: 2716: 2712: 2703: 2701: 2697: 2666: 2658: 2657: 2653: 2646: 2633: 2632: 2628: 2621: 2600: 2599: 2592: 2575: 2571: 2564: 2551: 2550: 2546: 2539: 2524: 2523: 2519: 2492:Arch. Psychiatr 2485: 2484: 2480: 2426: 2425: 2421: 2414: 2393: 2392: 2388: 2384: 2311: 2264: 2231: 2218: 2213: 2212: 2179: 2178: 2159: 2158: 2131: 2126: 2125: 2090: 2079: 2078: 2053: 2052: 2033: 2032: 1995: 1984: 1983: 1950: 1949: 1930: 1929: 1910: 1909: 1877: 1864: 1859: 1858: 1825: 1824: 1805: 1804: 1783: 1778: 1777: 1767: 1758: 1747: 1713: 1708: 1707: 1686: 1681: 1680: 1649: 1633: 1632: 1562:Reynolds number 1549: 1522:(total surface) 1521: 1457: 1451: 1448: 1438: 1425: 1414: 1406:clinical trials 1390: 1386: 1385: 1380: 1376: 1375: 1370: 1366: 1365: 1360: 1356: 1355: 1347: 1343: 1342: 1337: 1333: 1332: 1327: 1323: 1322: 1317: 1313: 1312: 1307: 1303: 1302: 1295: 1268: 1263: 1262: 1235: 1230: 1229: 1205: 1200: 1199: 1175: 1170: 1169: 1147: 1146: 1125: 1120: 1119: 1085: 1080: 1079: 1078:Leg stiffness, 1059: 1034: 1029: 1019:theory and the 971: 929: 928: 913: 902: 894: 861: 860: 835: 819: 818: 796: 783: 724: 723: 710: 697: 696: 689: 688: 672: 635: 600: 589:square–cube law 557:Vascular plants 498: 422: 401: 394: 393: 362:is the scaling 344: 343: 285: 284: 223: 222: 188: 174: 173: 153:species (e.g., 139: 121:D'Arcy Thompson 73: 72: 71: 70: 69: 67: 66: 56: 47: 46: 45: 41:Skeleton of an 39: 28: 23: 22: 15: 12: 11: 5: 4945: 4943: 4935: 4934: 4929: 4924: 4919: 4909: 4908: 4902: 4901: 4899: 4898: 4893: 4883: 4872: 4869: 4868: 4866: 4865: 4860: 4855: 4850: 4849: 4848: 4838: 4833: 4828: 4823: 4818: 4813: 4808: 4802: 4800: 4799:Related topics 4796: 4795: 4793: 4792: 4791: 4790: 4780: 4775: 4770: 4764: 4762: 4758: 4757: 4755: 4754: 4749: 4744: 4739: 4733: 4731: 4725: 4724: 4722: 4721: 4716: 4711: 4706: 4701: 4696: 4691: 4685: 4683: 4677: 4676: 4674: 4673: 4668: 4666:Spider anatomy 4663: 4662: 4661: 4656: 4646: 4641: 4635: 4633: 4629: 4628: 4626: 4625: 4620: 4619: 4618: 4608: 4603: 4598: 4597: 4596: 4585: 4583: 4579: 4578: 4576: 4575: 4569: 4567: 4563: 4562: 4560: 4559: 4554: 4549: 4548: 4547: 4542: 4531: 4529: 4525: 4524: 4517: 4515: 4513: 4512: 4507: 4506: 4505: 4500: 4492: 4487: 4486: 4485: 4475: 4470: 4464: 4462: 4458: 4457: 4448: 4446: 4445: 4438: 4431: 4423: 4417: 4416: 4401: 4400:External links 4398: 4396: 4395: 4389: 4374: 4368: 4350: 4344: 4329: 4323: 4308: 4302: 4287: 4281: 4266: 4260: 4244: 4242: 4239: 4237: 4236: 4177: 4134: 4074: 4031: 4024: 3995: 3984: 3957: 3899: 3856: 3835:(3): 668–675. 3812: 3783:(3): 353–361. 3760: 3710: 3656: 3600: 3571:(3): 418–421. 3546: 3519:(2): 235–256. 3503: 3478:(2): 157–170. 3452: 3393: 3365:(4): 611–662. 3349: 3287: 3248:(17): 7301–6. 3225: 3201: 3164:(2): 118–122. 3148: 3129:(3): 519–521. 3113: 3092:(4): 449–462. 3072: 3033: 2977: 2970: 2950: 2938: 2931: 2913: 2904: 2871:(2): 219–230. 2844: 2833: 2806:(3): 551–564. 2790: 2776: 2753: 2710: 2651: 2644: 2636:Biomechanika 1 2626: 2619: 2590: 2580:109: 133–140. 2569: 2562: 2544: 2537: 2517: 2498:(2): 436–446. 2478: 2419: 2412: 2385: 2383: 2380: 2379: 2378: 2375:Tree allometry 2372: 2366: 2360: 2354: 2348: 2342: 2336: 2330: 2324: 2318: 2310: 2307: 2306: 2305: 2296: 2295: 2291: 2290: 2286: 2285: 2281: 2280: 2276: 2275: 2263: 2260: 2259: 2258: 2257: 2256: 2242: 2239: 2234: 2230: 2221: 2207: 2206: 2194: 2190: 2186: 2166: 2144: 2141: 2138: 2134: 2123:cruising speed 2118: 2117: 2116: 2115: 2101: 2098: 2093: 2089: 2086: 2073: 2072: 2060: 2051:and body mass 2040: 2028: 2027: 2026: 2025: 2010: 2007: 2002: 1998: 1994: 1991: 1978: 1977: 1965: 1961: 1957: 1948:raised to the 1937: 1917: 1905: 1904: 1903: 1902: 1888: 1885: 1880: 1876: 1871: 1867: 1853: 1852: 1840: 1836: 1832: 1823:raised to the 1812: 1790: 1786: 1766: 1763: 1757: 1754: 1746: 1743: 1720: 1716: 1693: 1689: 1658: 1653: 1648: 1645: 1625: 1624: 1618: 1608: 1607: 1597: 1591: 1565: 1564: 1555: 1545: 1540: 1534: 1517: 1511: 1510: 1507: 1504: 1501: 1470:turbulent flow 1459: 1458: 1428: 1426: 1419: 1413: 1410: 1294: 1291: 1290: 1289: 1275: 1271: 1259: 1245: 1242: 1238: 1226: 1212: 1208: 1196: 1182: 1178: 1166: 1154: 1132: 1128: 1097: 1088: 1076: 1073: 1070: 1067: 1058: 1055: 1043:Froude numbers 1033: 1030: 1028: 1025: 1013: 1012: 997: 993: 989: 984: 981: 978: 975: 970: 965: 961: 958: 955: 952: 949: 946: 943: 940: 937: 912: 909: 900: 892: 868: 857: 856: 842: 838: 834: 831: 826:Metabolic rate 795: 792: 788:metabolic rate 782: 779: 750: 747: 740: 736: 731: 727: 719: 714: 709: 704: 700: 671: 668: 638:tools such as 634: 631: 599: 596: 579: 578: 575: 572: 566: 565: 562: 559: 553: 552: 549: 546: 540: 539: 536: 533: 527: 526: 523: 520: 514: 513: 510: 507: 497: 494: 449: 448: 437: 432: 428: 425: 419: 416: 411: 407: 404: 351: 340: 339: 328: 325: 322: 319: 316: 313: 310: 307: 304: 301: 298: 295: 292: 280:or similarly, 278: 277: 266: 263: 260: 257: 254: 251: 248: 245: 242: 239: 236: 233: 230: 212: 211: 200: 195: 191: 187: 184: 181: 138: 135: 58:Skeleton of a 57: 50: 49: 48: 40: 33: 32: 31: 30: 29: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 4944: 4933: 4930: 4928: 4925: 4923: 4920: 4918: 4915: 4914: 4912: 4897: 4894: 4892: 4888: 4884: 4882: 4874: 4873: 4870: 4864: 4861: 4859: 4856: 4854: 4851: 4847: 4844: 4843: 4842: 4839: 4837: 4834: 4832: 4829: 4827: 4824: 4822: 4819: 4817: 4814: 4812: 4809: 4807: 4804: 4803: 4801: 4797: 4789: 4786: 4785: 4784: 4781: 4779: 4776: 4774: 4771: 4769: 4766: 4765: 4763: 4759: 4753: 4752:Shark anatomy 4750: 4748: 4745: 4743: 4740: 4738: 4735: 4734: 4732: 4730: 4726: 4720: 4717: 4715: 4712: 4710: 4709:Horse anatomy 4707: 4705: 4702: 4700: 4697: 4695: 4692: 4690: 4689:Human anatomy 4687: 4686: 4684: 4682: 4678: 4672: 4669: 4667: 4664: 4660: 4657: 4655: 4652: 4651: 4650: 4647: 4645: 4642: 4640: 4637: 4636: 4634: 4632:Invertebrates 4630: 4624: 4621: 4617: 4614: 4613: 4612: 4609: 4607: 4604: 4602: 4599: 4595: 4592: 4591: 4590: 4589:Plant anatomy 4587: 4586: 4584: 4580: 4574: 4571: 4570: 4568: 4564: 4558: 4555: 4553: 4550: 4546: 4543: 4541: 4538: 4537: 4536: 4533: 4532: 4530: 4526: 4521: 4511: 4510:Morphometrics 4508: 4504: 4501: 4499: 4496: 4495: 4493: 4491: 4488: 4484: 4481: 4480: 4479: 4476: 4474: 4471: 4469: 4468:Gross anatomy 4466: 4465: 4463: 4459: 4455: 4451: 4444: 4439: 4437: 4432: 4430: 4425: 4424: 4421: 4414: 4410: 4407: 4404: 4403: 4399: 4392: 4386: 4382: 4381: 4375: 4371: 4365: 4361: 4360: 4355: 4351: 4347: 4341: 4337: 4336: 4330: 4326: 4320: 4316: 4315: 4309: 4305: 4299: 4295: 4294: 4288: 4284: 4278: 4274: 4273: 4267: 4263: 4257: 4253: 4252: 4246: 4245: 4240: 4232: 4228: 4223: 4218: 4213: 4208: 4204: 4200: 4196: 4192: 4188: 4181: 4178: 4173: 4169: 4165: 4161: 4157: 4153: 4149: 4145: 4138: 4135: 4130: 4126: 4122: 4118: 4114: 4110: 4106: 4102: 4098: 4094: 4093: 4085: 4078: 4075: 4070: 4066: 4062: 4058: 4054: 4050: 4046: 4042: 4035: 4032: 4027: 4021: 4017: 4016: 4008: 4006: 4004: 4002: 4000: 3996: 3993: 3988: 3985: 3972:. 13 May 2012 3971: 3967: 3961: 3958: 3953: 3949: 3944: 3939: 3935: 3931: 3927: 3923: 3922: 3921:AAPS PharmSci 3917: 3913: 3909: 3903: 3900: 3895: 3891: 3887: 3883: 3879: 3875: 3871: 3867: 3860: 3857: 3851: 3846: 3842: 3838: 3834: 3830: 3823: 3816: 3813: 3808: 3804: 3799: 3794: 3790: 3786: 3782: 3778: 3774: 3767: 3765: 3761: 3756: 3752: 3748: 3744: 3740: 3736: 3729: 3727: 3725: 3723: 3721: 3719: 3717: 3715: 3711: 3706: 3702: 3697: 3692: 3688: 3684: 3680: 3676: 3672: 3665: 3663: 3661: 3657: 3652: 3648: 3643: 3638: 3634: 3630: 3626: 3619: 3617: 3615: 3613: 3611: 3609: 3607: 3605: 3601: 3596: 3592: 3587: 3582: 3578: 3574: 3570: 3566: 3562: 3555: 3553: 3551: 3547: 3542: 3538: 3534: 3530: 3526: 3522: 3518: 3514: 3507: 3504: 3493:on 2018-08-31 3489: 3485: 3481: 3477: 3473: 3466: 3462: 3456: 3453: 3448: 3444: 3439: 3434: 3429: 3424: 3420: 3416: 3412: 3408: 3404: 3397: 3394: 3388: 3384: 3380: 3376: 3372: 3368: 3364: 3360: 3353: 3350: 3344: 3340: 3336: 3332: 3328: 3324: 3320: 3316: 3311: 3306: 3302: 3298: 3291: 3288: 3283: 3279: 3274: 3269: 3264: 3259: 3255: 3251: 3247: 3243: 3239: 3232: 3230: 3226: 3221: 3214: 3212: 3210: 3208: 3206: 3202: 3197: 3193: 3189: 3185: 3180: 3175: 3171: 3167: 3163: 3159: 3152: 3149: 3144: 3140: 3136: 3132: 3128: 3124: 3117: 3114: 3109: 3105: 3100: 3095: 3091: 3087: 3083: 3076: 3073: 3068: 3064: 3060: 3056: 3052: 3048: 3044: 3037: 3034: 3029: 3025: 3020: 3015: 3011: 3007: 3004:(4176): 820. 3003: 2999: 2995: 2988: 2986: 2984: 2982: 2978: 2973: 2967: 2963: 2962: 2954: 2951: 2948:, p. 237 2947: 2942: 2939: 2934: 2928: 2924: 2917: 2914: 2908: 2905: 2894:on 2016-11-30 2890: 2886: 2882: 2878: 2874: 2870: 2866: 2859: 2855: 2848: 2845: 2842: 2837: 2834: 2829: 2825: 2821: 2817: 2813: 2809: 2805: 2801: 2794: 2791: 2779: 2773: 2769: 2768: 2763: 2762:Vogel, Steven 2757: 2754: 2749: 2745: 2741: 2737: 2733: 2729: 2725: 2721: 2714: 2711: 2700:on 2020-10-25 2696: 2692: 2688: 2684: 2680: 2676: 2672: 2665: 2661: 2655: 2652: 2647: 2641: 2637: 2630: 2627: 2622: 2620:9783642359378 2616: 2612: 2608: 2604: 2597: 2595: 2591: 2587: 2583: 2579: 2573: 2570: 2565: 2559: 2555: 2548: 2545: 2540: 2534: 2530: 2529: 2521: 2518: 2513: 2509: 2505: 2501: 2497: 2493: 2489: 2482: 2479: 2474: 2470: 2465: 2460: 2455: 2450: 2446: 2442: 2439:(5): 2113–4. 2438: 2434: 2430: 2423: 2420: 2415: 2409: 2405: 2400: 2399: 2390: 2387: 2381: 2376: 2373: 2370: 2369:Rensch's rule 2367: 2364: 2361: 2358: 2355: 2352: 2349: 2346: 2343: 2340: 2337: 2334: 2331: 2328: 2325: 2322: 2319: 2316: 2313: 2312: 2308: 2302: 2298: 2297: 2293: 2292: 2288: 2287: 2283: 2282: 2278: 2277: 2273: 2272: 2271: 2269: 2261: 2240: 2237: 2232: 2228: 2219: 2211: 2210: 2209: 2208: 2192: 2188: 2184: 2164: 2142: 2139: 2136: 2132: 2124: 2120: 2119: 2099: 2096: 2091: 2087: 2084: 2077: 2076: 2075: 2074: 2058: 2038: 2030: 2029: 2008: 2005: 2000: 1996: 1992: 1989: 1982: 1981: 1980: 1979: 1963: 1959: 1955: 1935: 1915: 1907: 1906: 1886: 1883: 1878: 1874: 1869: 1865: 1857: 1856: 1855: 1854: 1838: 1834: 1830: 1810: 1788: 1784: 1775: 1774:Kleiber's law 1772: 1771: 1770: 1764: 1762: 1755: 1753: 1751: 1744: 1742: 1738: 1734: 1718: 1714: 1691: 1687: 1678: 1674: 1656: 1651: 1646: 1643: 1629: 1622: 1619: 1616: 1613: 1612: 1611: 1605: 1601: 1598: 1595: 1592: 1589: 1586: 1585: 1584: 1582: 1578: 1574: 1570: 1563: 1559: 1556: 1553: 1548: 1544: 1541: 1538: 1535: 1532: 1531: 1530: 1528: 1525: 1520: 1516: 1508: 1505: 1502: 1499: 1498: 1497: 1495: 1491: 1487: 1483: 1478: 1475: 1471: 1467: 1455: 1445: 1441: 1436: 1432: 1429:This article 1427: 1423: 1418: 1417: 1411: 1409: 1407: 1403: 1398: 1395: 1352: 1299: 1292: 1273: 1269: 1260: 1243: 1240: 1236: 1227: 1210: 1206: 1197: 1180: 1176: 1167: 1165:is body mass. 1152: 1130: 1126: 1095: 1086: 1077: 1074: 1071: 1068: 1065: 1064:Froude number 1061: 1060: 1056: 1054: 1050: 1046: 1044: 1038: 1031: 1026: 1024: 1022: 1018: 995: 991: 987: 968: 963: 927: 926: 925: 923: 918: 910: 908: 904: 898: 888: 884: 882: 866: 840: 836: 832: 829: 817: 816: 815: 811: 809: 808:Kleiber's law 805: 801: 793: 791: 789: 780: 778: 776: 772: 768: 762: 748: 745: 734: 729: 725: 717: 707: 702: 698: 681: 676: 669: 667: 665: 661: 656: 652: 650: 644: 641: 632: 630: 628: 624: 620: 615: 614: 609: 605: 597: 595: 592: 590: 585: 576: 573: 571: 568: 567: 563: 560: 558: 555: 554: 550: 547: 545: 542: 541: 537: 534: 532: 529: 528: 524: 521: 519: 516: 515: 512:Length range 511: 508: 505: 504: 495: 493: 491: 485: 483: 479: 475: 471: 467: 466:parental care 463: 459: 454: 435: 430: 426: 423: 417: 414: 409: 405: 402: 392: 391: 390: 387: 385: 381: 377: 373: 369: 365: 349: 326: 323: 320: 317: 314: 311: 308: 305: 302: 299: 296: 293: 290: 283: 282: 281: 264: 261: 258: 255: 252: 249: 246: 243: 240: 237: 234: 231: 228: 221: 220: 219: 217: 198: 193: 189: 185: 182: 179: 172: 171: 166: 162: 160: 156: 152: 148: 144: 136: 134: 132: 131:Julian Huxley 128: 127: 122: 118: 114: 110: 106: 101: 96: 90: 85: 81: 80:Ancient Greek 77: 64: 61: 54: 44: 37: 19: 4846:19th century 4836:Hertwig rule 4805: 4747:Fish anatomy 4742:Bird anatomy 4616:reproductive 4478:Neuroanatomy 4379: 4358: 4334: 4313: 4292: 4271: 4250: 4194: 4190: 4180: 4147: 4143: 4137: 4096: 4090: 4077: 4044: 4040: 4034: 4014: 3987: 3974:. Retrieved 3969: 3960: 3925: 3919: 3912:W. L. Hayton 3902: 3869: 3865: 3859: 3832: 3828: 3815: 3780: 3776: 3741:(2): 49–59. 3738: 3734: 3678: 3674: 3635:(1): 71–86. 3632: 3629:J. Exp. Biol 3628: 3568: 3564: 3516: 3512: 3506: 3495:. Retrieved 3488:the original 3475: 3471: 3455: 3410: 3406: 3396: 3362: 3358: 3352: 3300: 3296: 3290: 3245: 3241: 3219: 3161: 3157: 3151: 3126: 3122: 3116: 3089: 3085: 3081: 3075: 3050: 3047:J. Exp. Biol 3046: 3042: 3036: 3001: 2997: 2964:. Springer. 2960: 2953: 2941: 2922: 2916: 2907: 2896:. Retrieved 2889:the original 2868: 2864: 2847: 2836: 2803: 2799: 2793: 2781:. Retrieved 2766: 2756: 2723: 2719: 2713: 2702:. Retrieved 2695:the original 2674: 2671:Am J Physiol 2670: 2654: 2635: 2629: 2602: 2577: 2572: 2553: 2547: 2527: 2520: 2495: 2491: 2481: 2436: 2432: 2422: 2397: 2389: 2321:Biomechanics 2265: 1768: 1759: 1748: 1739: 1735: 1677:tidal volume 1672: 1640:Heart weight 1630: 1626: 1620: 1614: 1609: 1599: 1593: 1587: 1580: 1576: 1572: 1571:is given by 1568: 1566: 1557: 1551: 1546: 1542: 1536: 1533:V = velocity 1527: 1523: 1518: 1514: 1512: 1493: 1485: 1481: 1479: 1466:laminar flow 1462: 1449: 1440:You can help 1430: 1399: 1396: 1353: 1300: 1296: 1051: 1047: 1039: 1035: 1014: 914: 905: 889: 885: 858: 812: 797: 784: 774: 770: 766: 763: 685: 657: 653: 645: 636: 611: 601: 593: 586: 582: 486: 450: 388: 341: 279: 213: 140: 124: 119:in 1892, by 99: 88: 75: 74: 62: 4853:Physiognomy 4704:Dog anatomy 4699:Cat anatomy 4601:Plant habit 3976:15 December 3850:10261/98308 3303:(1): 9–27. 1452:August 2022 627:muskellunge 577:10 to 10 m 564:10 to 10 m 551:10 to 10 m 538:10 to 10 m 525:10 to 10 m 216:logarithmic 123:in 1917 in 60:tiger quoll 4922:Physiology 4911:Categories 4761:Glossaries 4573:Structures 4454:morphology 3928:(4): E29. 3497:2010-03-16 2898:2010-03-15 2704:2009-01-23 2382:References 1590:= velocity 1101:peak force 133:in 1932. 117:Otto Snell 113:physiology 4831:Gracility 4821:Body plan 4806:Allometry 4503:molecular 4498:histology 3755:120138903 3174:CiteSeerX 3094:CiteSeerX 2800:Evolution 2748:221262390 2720:Evolution 2363:Power law 2301:evolution 2229:∼ 2088:∼ 2001:− 1993:∼ 1875:∼ 1583:, where: 1444:talk page 1241:− 922:frequency 883:per day. 804:power-law 735:⁡ 708:⁡ 623:Varanidae 321:⁡ 309:⁡ 294:⁡ 259:⁡ 247:⁡ 232:⁡ 159:power law 92:"other", 76:Allometry 4881:Category 4566:Protists 4409:Archived 4356:(1984). 4231:11724953 4121:17733374 4069:10753108 3952:12049492 3914:(2001). 3908:T. 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Hu 3807:21228194 3705:17711833 3595:20335198 3447:17578913 3379:16221332 3335:11237567 3282:17438298 3196:92046364 3067:10821750 2856:(1996). 2828:28567959 2783:29 March 2764:(1988). 2740:14686522 2512:30692188 2473:11226197 2309:See also 1765:Examples 1671:, where 1474:vortices 1145:, where 773:, where 670:Examples 604:isometry 482:ontogeny 474:juvenile 364:exponent 214:or in a 137:Overview 43:elephant 4681:Mammals 4659:Odonata 4654:Diptera 4450:Anatomy 4199:Bibcode 4172:4176349 4152:Bibcode 4129:3068221 4101:Bibcode 4092:Science 4049:Bibcode 4041:Science 3943:2751218 3894:3140271 3886:9082983 3866:Science 3798:3020146 3696:2279215 3651:8294853 3586:2880072 3541:2607772 3521:Bibcode 3461:Garland 3438:1904129 3415:Bibcode 3387:8546506 3343:9168199 3315:Bibcode 3273:1852329 3250:Bibcode 3166:Bibcode 3131:Bibcode 3028:4082708 3006:Bibcode 2885:1565513 2820:2407721 2691:3826408 2660:Garland 2441:Bibcode 2294:Habitat 2268:habitat 1529:where: 1400:The US 1389:⁄ 1379:⁄ 1369:⁄ 1359:⁄ 1346:⁄ 1336:⁄ 1326:⁄ 1316:⁄ 1306:⁄ 897:VO2 max 610:in his 608:Galileo 574:100,000 544:Mammals 518:Insects 509:Factor 458:lizards 147:biology 129:and by 109:anatomy 4932:Scales 4891:Portal 4582:Plants 4461:Fields 4387: 4366: 4342: 4321: 4300: 4279: 4258: 4229: 4219: 4170: 4144:Nature 4127: 4119: 4067: 4022: 3950: 3940: 3892: 3884: 3805: 3795: 3753: 3703: 3693: 3649: 3593: 3583: 3539: 3445: 3435: 3385: 3377: 3341: 3333: 3280: 3270: 3194: 3176: 3096: 3065: 3026: 2998:Nature 2968: 2929: 2883: 2826: 2818: 2774: 2746: 2738: 2689: 2642: 2617: 2560: 2535: 2510: 2471: 2461: 2410: 1976:power: 1851:power: 1550:= 1.33 1477:size. 1442:. 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