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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
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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.
165:
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.
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."
53:
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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.
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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
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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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853:
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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.
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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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1492:. In nature however, organisms such as a 6-foot-6-inch (1.98 m) dolphin moving at 15 knots does not have the appropriate Reynolds numbers for laminar flow (
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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:
1404:(FDA) published guidance in 2005 giving a flow chart that presents the decisions and calculations used to generate the maximum recommended starting dose in drug
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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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2638:. Ostrava, Czech Republic: VSB – Technical University of Ostrava, Faculty of Mechanical Engineering, Department of Applied Mechanics. p. 461.
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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.
3561:"Two explanations for the compliant running paradox: reduced work of bouncing viscera and increased stability in uneven terrain"
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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".
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899:. Oxygen consumption in species that differ in body size and organ system dimensions show a similarity in their charted V
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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
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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:
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1985:
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
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4084:"Allometric Engineering: An Experimental Test of the Causes of Interpopulational Differences in Performance"
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Bonduriansky, Russell; Day, Troy (2003). "The
Evolution of Static Allometry in Sexually Selected Traits".
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2359: – Use of information on the historical relationships of lineages to test evolutionary hypotheses
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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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855:(although the universality of this relation has been disputed both empirically and theoretically)
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802:(BMR) against the animal's own body mass, a logarithmic straight line is obtained, indicating a
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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".
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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.
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2664:"Seasonal, sexual, and individual variation in endurance and activity metabolism in lizards"
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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
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2488:"Die Abhängigkeit des Hirngewichts von dem Körpergewicht und den geistigen Fähigkeiten"
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3773:"Fleximble mechanisms: the diverse roles of biological springs in vertebrate movement"
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The proportionately thicker bones in the elephant are an example of allometric scaling
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2365: – Functional relationship between two quantities (also known as a scaling law)
1111:{\displaystyle k_{\text{leg}}={\frac {\text{peak force}}{\text{peak displacement}}}}
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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
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2377: – Quantitative relations between some key characteristic dimensions of trees
1351:) scale in the same way, leading to blood pressure being constant across species.
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Study of the relationship of body size to shape, anatomy, physiology, and behavior
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3465:"The relation between maximal running speed and body mass in terrestrial mammals"
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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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2605:. Lecture Notes in Morphogenesis. Springer Berlin Heidelberg. pp. 23–73.
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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}
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3916:"Allometric scaling of xenobiotic clearance: uncertainty versus universality"
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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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2472:
2453:
2320:
1676:
1465:
648:
4405:
3885:
3650:
3540:
2690:
2335: – Study of the diversity of functional characteristics of organisms.
1421:
4852:
4703:
4698:
4600:
4418:
3641:
626:
59:
3671:"Estimating dinosaur maximum running speeds using evolutionary robotics"
3309:
1761:
disease, and even pedestrian walking speeds scale with city population.
3849:
3819:
3788:
2884:
2819:
2503:
1005:{\displaystyle \mathrm {frequency} ={\frac {1}{\mathrm {mass} ^{1/3}}}}
112:
3840:
2858:"Scaling of limb proportions in monitor lizards (Squamata: Varanidae)"
2347: – Study of evolutionary changes in physiological characteristics
4830:
4820:
4497:
4163:
3933:
3156:
O'Hara, R.B.; Kotze, D.J. (2010). "Do not log-transform count data".
3018:
2993:
2362:
2353: – Theory concerning metabolism and observed patterns in ecology
2300:
1480:
In general, smaller, more streamlined organisms create laminar flow (
1394:
might apply for drugs that are eliminated mainly by renal excretion.
1311:
power with body weight. They also showed why lifespan scales as the +
921:
803:
622:
215:
158:
98:
87:
4187:"Dinosaurs, dragons, and dwarfs: The evolution of maximal body size"
3236:
Bettencourt LM, Lobo J, Helbing D, Kühnert C, West GB (April 2007).
2876:
2811:
2487:
632:
374:, as these account for the variation in both variables, contrary to
35:
4142:
Bornstein MH, Bornstein HG (19 February 1976). "The Pace of Life".
1473:
603:
481:
363:
154:
42:
3982:
Online allometric scaling calculator, with explanation and source.
4449:
2267:
1075:
Body mass has even more of an effect than speed on limb dynamics.
896:
880:
607:
457:
146:
108:
674:
103:"measurement") is the study of the relationship of body size to
916:
543:
517:
461:
150:
2329: – Body scale based on waist/hip circumference and height
1168:
Peak force experienced throughout a stride is proportional to
3238:"Growth, innovation, scaling, and the pace of life in cities"
3235:
1472:, where the fluid moves roughly around an organism, creating
569:
495:
480:
stage. Lizards often exhibit allometric changes during their
477:
469:
452:
104:
2851:
2429:"Scaling of growth: plants and animals are not so different"
2261:
1228:
The angle swept by a leg during a stride is proportional to
1057:
Principles of legged locomotion identified through allometry
678:
Allometric relations show as straight lines when plotted on
3155:
3120:
2371: – A biological rule concerning sexual size dimorphism
2317: – A concept in plant biology concerning plant biomass
2157:
of flying bodies (insects, birds, airplanes) and body mass
530:
4185:
Burness, G. P.; Diamond, Jared; Flannery, Timothy (2001).
2845:
1411:
1261:
The mass-specific work rate of a limb is proportional to
4038:
3822:"Allometric deviations of plasma carotenoids in raptors"
2767:
Life's
Devices: The Physical World of Animals and Plants
3669:
Sellers, William Irving; Manning, Phillip Lars (2007).
2994:"The dimensions of animals and their muscular dynamics"
4184:
451:
Allometry often studies shape differences in terms of
3622:
3400:
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:
3906:
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).
910:
790:
of an individual animal is also subject to scaling.
602:
Allometric scaling is any change that deviates from
4415:(For "first in human" clinical trials of new drugs)
4081:
3623:Farley CT, Glasheen J, McMahon TA (December 1993).
3040:
2249:{\displaystyle V_{\text{opt}}\sim M^{\frac {1}{6}}}
4141:
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
3996:
3294:
2658:
1026:
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
4908:
4293:Plant allometry: The scaling of form and process
3863:
3728:
3726:
3724:
3722:
3720:
3718:
3716:
3714:
3453:
879:is body mass, and metabolic rate is measured in
145:for its theoretical developments, as well as in
4352:
4011:
3401:Labra FA, Marquet PA, Bozinovic F (June 2007).
3213:
3211:
3209:
3207:
3205:
2945:
2840:
2834:
2717:
1755:
793:
3766:
3764:
3735:The International Journal of Robotics Research
3668:
3664:
3662:
3660:
3559:Daley, Monica A.; Usherwood, James R. (2010).
3558:
3231:
3229:
2601:Longo, Giuseppe; Montévil, Maël (2014-01-01).
2531:(Canto ed.). Cambridge University Press.
4434:
4268:
4015:Animal Physiology: Adaptation and Environment
3711:
3618:
3616:
3614:
3612:
3610:
3608:
3606:
3604:
3504:
3079:
2797:
2600:
2596:
2594:
1928:are both inversely proportional to body mass
4178:
4018:(5th ed.). Cambridge University Press.
3857:
3770:
3554:
3552:
3550:
3356:
3202:
2987:
2985:
2983:
2981:
3771:Roberts, Thomas J.; Azizi, Emanuel (2011).
3761:
3657:
3459:
3226:
1066:consistently exhibit similar gait patterns.
4448:
4441:
4427:
3958:
3813:
3601:
2920:
2770:. Princeton University Press. p. 39.
2591:
2485:
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:
3732:
3694:
3640:
3584:
3547:
3436:
3426:
3308:
3271:
3261:
3177:
3097:
3017:
2978:
2633:
2462:
2452:
2262:Determinants of size in different species
2018:{\displaystyle t\sim M^{-{\frac {1}{4}}}}
1744:
761:). This is the slope of a straight line.
4335:The allometry of growth and reproduction
4314:The ecological implications of body size
3625:"Running springs: speed and animal size"
3510:
2957:
2925:. Oxford University Press. p. 111.
2524:
2426:
2121:the proportionality between the optimal
780:
673:
163:
4376:
4075:
4012:Schmidt-Nielsen, Knut (10 April 1997).
3217:
1506:the density and viscosity of the fluid.
168:Allometric equation: way of expressions
14:
4909:
4310:
4289:
4269:McMahon, T. A.; Bonner, J. T. (1983).
4247:
2551:
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,}
4422:
4331:
3403:"Scaling metabolic rate fluctuations"
2760:
2393:
613:Dialogues Concerning Two New Sciences
597:
587:Isometric scaling is governed by the
2991:
1617:< 0.5×10 = laminar flow threshold
1415:
1374:power of body mass. The exponent of
1292:
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
1031:
332:{\displaystyle \ln y=a\ln x+\ln k,}
24:
4240:
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:
4943:
4399:
2556:(2nd ed.). New York: Dover.
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
4135:
4032:
3985:
3966:"Allometric Scaling Calculator"
3900:
3394:
3350:
3288:
3149:
3114:
3073:
3034:
2951:
2939:
2914:
2905:
2791:
2754:
2711:
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
2652:
2627:
2570:
2545:
2518:
2479:
2420:
2394:Small, Christopher G. (1996).
2387:
2339:Cranial evolutionary allometry
1027:Allometry of legged locomotion
499:
13:
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
2381:
1803:is proportional to body mass
1764:
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
7:
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
2308:
2031:mass transfer contact area
1433:to comply with Knowledge's
1021:metabolic theory of ecology
669:
664:reduced major axis analysis
136:
10:
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:
4798:
4760:
4727:
4679:
4631:
4581:
4565:
4527:
4516:
4460:
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:
4419:
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:
4370:
4346:
4325:
4304:
4283:
4262:
4243:
4241:Further reading
4238:
4183:
4179:
4150:(5544): 557–9.
4140:
4136:
4086:
4080:
4076:
4047:(5463): 100–6.
4037:
4033:
4026:
4010:
3997:
3990:
3986:
3975:
3973:
3964:
3963:
3959:
3905:
3901:
3862:
3858:
3824:
3818:
3814:
3769:
3762:
3731:
3712:
3667:
3658:
3621:
3602:
3565:Biology Letters
3557:
3548:
3509:
3505:
3496:
3494:
3490:
3467:
3458:
3454:
3413:(26): 10900–3.
3399:
3395:
3355:
3351:
3310:physics/0007096
3293:
3289:
3234:
3227:
3216:
3203:
3179:10.1.1.466.9313
3154:
3150:
3119:
3115:
3099:10.1.1.524.9341
3078:
3074:
3039:
3035:
2990:
2979:
2972:
2956:
2952:
2944:
2940:
2933:
2919:
2915:
2910:
2906:
2897:
2895:
2891:
2877:10.2307/1565513
2860:
2854:Garland T., Jr.
2852:Christian, A.;
2850:
2846:
2839:
2835:
2812:10.2307/2407721
2796:
2792:
2782:
2780:
2778:
2759:
2755:
2716:
2712:
2703:
2701:
2697:
2666:
2657:
2653:
2646:
2632:
2628:
2621:
2599:
2592:
2575:
2571:
2564:
2550:
2546:
2539:
2523:
2519:
2492:Arch. Psychiatr
2484:
2480:
2425:
2421:
2414:
2392:
2388:
2384:
2311:
2264:
2235:
2231:
2222:
2218:
2216:
2213:
2212:
2187:
2182:
2179:
2178:
2162:
2159:
2158:
2135:
2131:
2129:
2126:
2125:
2094:
2090:
2082:
2079:
2078:
2056:
2053:
2052:
2036:
2033:
2032:
2003:
1999:
1995:
1987:
1984:
1983:
1958:
1953:
1950:
1949:
1933:
1930:
1929:
1913:
1910:
1909:
1881:
1877:
1868:
1864:
1862:
1859:
1858:
1833:
1828:
1825:
1824:
1808:
1805:
1804:
1787:
1783:
1781:
1778:
1777:
1767:
1758:
1747:
1717:
1713:
1711:
1708:
1707:
1690:
1686:
1684:
1681:
1680:
1655:
1650:
1649:
1638:
1636:
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:
1272:
1268:
1266:
1263:
1262:
1239:
1235:
1233:
1230:
1229:
1209:
1205:
1203:
1200:
1199:
1179:
1175:
1173:
1170:
1169:
1150:
1147:
1146:
1129:
1125:
1123:
1120:
1119:
1098:
1089:
1085:
1083:
1080:
1079:
1078:Leg stiffness,
1059:
1034:
1029:
1019:theory and the
990:
986:
972:
971:
966:
934:
932:
929:
928:
913:
902:
894:
864:
861:
860:
839:
835:
824:
822:
819:
818:
796:
783:
737:
728:
724:
723:
716:
711:
710:
701:
697:
696:
694:
692:
689:
688:
672:
635:
600:
589:square–cube law
557:Vascular plants
498:
422:
420:
401:
399:
397:
394:
393:
362:is the scaling
347:
344:
343:
288:
285:
284:
226:
223:
222:
192:
188:
177:
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:
4935:
4934:
4929:
4924:
4919:
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:
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:
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:
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:
4232:
4228:
4223:
4218:
4213:
4208:
4204:
4200:
4196:
4192:
4188:
4181:
4173:
4169:
4165:
4161:
4157:
4153:
4149:
4145:
4138:
4130:
4126:
4122:
4118:
4114:
4110:
4106:
4102:
4098:
4094:
4093:
4085:
4078:
4070:
4066:
4062:
4058:
4054:
4050:
4046:
4042:
4035:
4027:
4021:
4017:
4016:
4008:
4006:
4004:
4002:
4000:
3993:
3988:
3972:. 13 May 2012
3971:
3967:
3961:
3953:
3949:
3944:
3939:
3935:
3931:
3927:
3923:
3922:
3921:AAPS PharmSci
3917:
3913:
3909:
3903:
3895:
3891:
3887:
3883:
3879:
3875:
3871:
3867:
3860:
3851:
3846:
3842:
3838:
3834:
3830:
3823:
3816:
3808:
3804:
3799:
3794:
3790:
3786:
3782:
3778:
3774:
3767:
3765:
3756:
3752:
3748:
3744:
3740:
3736:
3729:
3727:
3725:
3723:
3721:
3719:
3717:
3715:
3706:
3702:
3697:
3692:
3688:
3684:
3680:
3676:
3672:
3665:
3663:
3661:
3652:
3648:
3643:
3638:
3634:
3630:
3626:
3619:
3617:
3615:
3613:
3611:
3609:
3607:
3605:
3596:
3592:
3587:
3582:
3578:
3574:
3570:
3566:
3562:
3555:
3553:
3551:
3542:
3538:
3534:
3530:
3526:
3522:
3518:
3514:
3507:
3493:on 2018-08-31
3489:
3485:
3481:
3477:
3473:
3466:
3462:
3456:
3448:
3444:
3439:
3434:
3429:
3424:
3420:
3416:
3412:
3408:
3404:
3397:
3388:
3384:
3380:
3376:
3372:
3368:
3364:
3360:
3353:
3344:
3340:
3336:
3332:
3328:
3324:
3320:
3316:
3311:
3306:
3302:
3298:
3291:
3283:
3279:
3274:
3269:
3264:
3259:
3255:
3251:
3247:
3243:
3239:
3232:
3230:
3221:
3214:
3212:
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2948:, p. 237
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2894:on 2016-11-30
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2762:Vogel, Steven
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2700:on 2020-10-25
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2439:(5): 2113–4.
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2369:Rensch's rule
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1165:is body mass.
1152:
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1064:Froude number
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512:Length range
511:
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466:parental care
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131:Julian Huxley
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80:Ancient Greek
77:
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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:
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4180:
4147:
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3974:. Retrieved
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3912:W. L. Hayton
3902:
3869:
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3780:
3776:
3741:(2): 49–59.
3738:
3734:
3678:
3674:
3635:(1): 71–86.
3632:
3629:J. Exp. Biol
3628:
3568:
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3512:
3506:
3495:. Retrieved
3488:the original
3475:
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3085:
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3075:
3050:
3047:J. Exp. Biol
3046:
3042:
3036:
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2964:. Springer.
2960:
2953:
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2916:
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2896:. Retrieved
2889:the original
2868:
2864:
2847:
2836:
2803:
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2793:
2781:. Retrieved
2766:
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2719:
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2702:. Retrieved
2695:the original
2674:
2671:Am J Physiol
2670:
2654:
2635:
2629:
2602:
2577:
2572:
2553:
2547:
2527:
2520:
2495:
2491:
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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:
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1533:V = velocity
1527:
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1512:
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1485:
1481:
1479:
1466:laminar flow
1462:
1449:
1440:You can help
1430:
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450:
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341:
279:
213:
140:
124:
119:in 1892, by
99:
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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. M. 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:
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4144:Nature
4127:
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1976:power:
1851:power:
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1477:size.
1442:. The
917:muscle
859:where
561:10,000
506:Group
468:after
462:snakes
453:ratios
342:where
218:form,
151:insect
100:métron
95:μέτρον
4594:fruit
4222:64714
4168:S2CID
4125:S2CID
4087:(PDF)
3890:S2CID
3825:(PDF)
3751:S2CID
3491:(PDF)
3468:(PDF)
3383:S2CID
3339:S2CID
3305:arXiv
3192:S2CID
3024:S2CID
2892:(PDF)
2881:JSTOR
2861:(PDF)
2816:JSTOR
2744:S2CID
2698:(PDF)
2667:(PDF)
2508:S2CID
2464:33381
1647:0.006
1244:0.034
570:Algae
478:adult
470:birth
105:shape
89:állos
84:ἄλλος
4452:and
4385:ISBN
4364:ISBN
4340:ISBN
4319:ISBN
4298:ISBN
4277:ISBN
4256:ISBN
4227:PMID
4117:PMID
4065:PMID
4020:ISBN
3978:2015
3948:PMID
3882:PMID
3829:Ibis
3803:PMID
3701:PMID
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3278:PMID
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3063:PMID
2966:ISBN
2927:ISBN
2824:PMID
2785:2014
2772:ISBN
2736:PMID
2687:PMID
2640:ISBN
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2558:ISBN
2533:ISBN
2469:PMID
2408:ISBN
1719:0.75
1692:0.75
1274:0.11
1211:0.30
1181:0.97
1131:0.67
915:The
881:kcal
841:0.75
682:axes
548:1000
535:1000
531:Fish
522:1000
476:and
4217:PMC
4207:doi
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1811:M
1789:0
1785:q
1715:M
1688:M
1673:M
1652:M
1644:=
1621:R
1615:R
1600:ν
1594:L
1588:V
1581:ν
1579:/
1573:R
1569:R
1558:R
1552:R
1547:f
1543:C
1537:ρ
1524:V
1515:C
1494:R
1486:R
1482:R
1454:)
1450:(
1437:.
1391:3
1387:2
1381:4
1377:3
1371:4
1367:3
1361:3
1357:2
1348:4
1344:3
1338:4
1334:3
1328:4
1324:1
1318:4
1314:1
1308:4
1304:3
1288:.
1270:M
1258:.
1237:M
1225:.
1207:M
1195:.
1177:M
1153:M
1127:M
1096:=
1087:k
996:3
992:/
988:1
983:s
980:s
977:a
974:m
969:1
964:=
960:y
957:c
954:n
951:e
948:u
945:q
942:e
939:r
936:f
891:V
867:M
837:M
830:=
775:n
769:=
767:y
749:3
746:=
739:L
718:3
713:L
687:(
617:(
436:.
431:x
427:x
424:d
418:a
415:=
410:y
406:y
403:d
350:a
327:,
324:k
315:+
312:x
303:a
300:=
297:y
265:,
262:k
253:+
250:x
241:a
238:=
235:y
199:,
194:a
190:x
186:k
183:=
180:y
78:(
65:.
20:)
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