Allometry

 ( Branches Of Biology ) 

In general, smaller, more streamlined organisms create laminar flow ( R < 0.5x106), whereas larger, less streamlined organisms produce turbulent flow ( R > 2.0×106). Also, increase in velocity (V) increases turbulence, which can be proved using the Reynolds equation. In nature however, organisms such as a dolphin moving at 15 knots does not have the appropriate Reynolds numbers for laminar flow ( R = 10⁷), 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:

• the surface area of the organism and its effect on the fluid in which the organism lives.
• 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.
• the density and viscosity of the fluid.
• the length of the organism, as the surface area of just the front 2/3 of the organism has an effect on the drag.

The resistance to the motion of an approximately stream-lined solid through a fluid can be expressed by the formula: C _fρ(total surface) V²/2, where:

V = velocity

ρ = density of fluid

C _f = 1.33 R − 1 (laminar flow)

R = Reynolds number

The Reynolds number R is given by R = VL/ ν, where:

V = velocity

L = axial length of organism

ν = kinematic viscosity (viscosity/density)

R < 0.5 million = laminar flow threshold

R > 2.0 million = turbulent flow threshold

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

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: $\backslash text\{Heart\; weight\}\; =\; 0.006\; \{M\}^\{1.0\}$, where M 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 tidal volume at rest being 1/10 the lung volume. In addition, respiration costs with respect to oxygen consumption is scaled in the order of $M^\{0.75\}$. 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 $M^\{0.75\}$, 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.

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 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.

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 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.

• Kleiber's law, metabolic rate $q\_0$ is proportional to body mass $M$ raised to the $3/4$ power:

: $q\_\{0\}\; \backslash sim\; M^\{\backslash frac\; 3\; 4\}$

• breathing and heart rate $t$ are both inversely proportional to body mass $M$ raised to the $1/4$ power:

: $t\; \backslash sim\; M^\{-\backslash frac\{1\}\{4\}\}$

• mass transfer contact area $A$ and body mass $M$:

: $A\; \backslash sim\; M^\{\backslash frac\; 7\; 8\}$

• the proportionality between the optimal cruising speed $V\_\{opt\}$ of flying bodies (insects, birds, airplanes) and body mass $M$ raised to the power $1/6$:

: $V\_\backslash text\{opt\}\; \backslash sim\; M^\{\backslash frac\; 1\; 6\}$

• Physiological design

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.

• Mechanical design

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.

• Habitat

An animal's habitat throughout its evolution 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 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.

• (also known as a scaling law)
• Urban scaling – Quantitative relations between urban characteristics and city population size

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• Thomas T. Samaras (2025). 9781600214080, Nova Publishers. . ISBN 9781600214080

• FDA Guidance for Estimating Human Equivalent Dose (For "first in human" clinical trials of new drugs)