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Interspecific allometric scaling of multicellular organisms as an evolutionary process of food chain creation, influenced by mechanical constraints

Metabolic rate of organisms (amount of energy produced per unit time) increases slower than organisms' mass. This phenomenon, when considered across different taxa, is called interspecific allometric scaling. Its fundamental causes remain unknown. We discovered that interspecific allometric scaling occurred as the result of collective adaptation of organisms within the food chain, influenced by mechanical constraints. It appears to be a fundamental principle, affecting life evolution and its organization. On one hand, organisms have to reliably acquire sufficient food for successful reproduction of their populations. This evolutionary need enforces adaptations enhancing metabolic power, when the size of an organism increases. On the other hand, this advantage in metabolic power cannot be too strong - otherwise, the populations of species in the food chain, which the organism feeds on, could be destroyed. The counteraction of these two opposing factors establishes a dynamic balance in the food chain and preserves its continuity. Applying this principle, we introduce biomechanical models and obtain allometric exponents for the maximal and basal metabolic rates considering mammals, reptiles, fish and birds. The results correspond well to previous experimental and other studies, which confirms validity of the discovered principle and explains the fundamental causes of interspecific allometric scaling.


Introduction
Numerous biochemical processes, supporting life existence, its evolution and reproduction, rely on production of energy from acquired nutrients (meaning all kinds of involved substances, including mineral components). In many instances, electromagnetic radiation in a visible or nearby wavelength spectrum is also required, like sunlight for photosynthesis in plants. Most common energy producing biochemical mechanisms use oxygen, although there is a great variety of organisms, which employ anaerobic or both aerobic and anaerobic biochemical reactions for energy production, including humans. Due to their importance for biological, medical, biotechnological and many other applications, energy producing mechanisms and their properties are intensively studied from different perspectives, at all scale levels -from molecular to whole organisms to systems of organisms in different strata.
One of the important directions of such studies is a metabolic allometric scaling (to which allometric studies of other organismal properties often relate, such as scaling of size of limbs, organs, morphological, kinematic features, etc.) This phenomenon is mostly known as a slower increase of metabolic power compared to the increase of organismal mass. The effect was discovered in 19-th century. In 1932, Kleiber [1] published results, which stated two important properties of interspecific (across different taxa) allometric scaling, fascinating many researchers for the next 84 years. The first was that the metabolic rate B (the amount of energy produced per unit time) mathematically is well described by a From that time, this number for a long time became a benchmark, which other studies referred to. One of the important implications of his work was that the idea of such a power scaling was soon applied to other phenomena, which led to discoveries of many other regularities observed with the increase of size of organisms or their constituents [2]. The more detailed excurse on the subject can be found in many works, including [2][3][4][5][6][7].
However, discovering the fundamental causes of metabolic allometric scaling turned out to be a difficult problem, whose solution still did not "surrender" to researchers. The consensus presently can be summarized as follows: (a) allometric scaling is due to cooperative action of multiple causes, but not to a particular physiological mechanism, which was a popular proposition for some time [9,10]; (b) there is no a single universal value of the allometric exponent, common for all organisms, but different taxa may have substantially different allometric exponents [3][4][5].
Two types of allometric scaling are distinguished: when the phenomenon is studied across different taxa, it is called interspecific allometric scaling; when it is considered ontogenetically or for the same species, it is called intraspecific allometric scaling. The causes of the last one were considered in [8]. It turned out that they much relate to cellular properties, modulated by heat dissipation abilities of organisms.
The interspecific allometric scaling, given the results presented in this paper and in works [11][12][13], is rather the consequence of a systemic evolution of an entire food chain, so that the mechanisms, defining intraspecific and interspecific allometric scaling, are different and, in fact, reside in different domains.
The previous study [13] formulated the main principles, which define interspecific allometric scaling, and provided proofs of their validity for the case of geometric similarity.
Using these principles, the values of allometric exponents, corresponding to maximal levels of metabolic activity, were found for mammals and fish, as well as estimates were obtained for the basal metabolic rates. All estimates corresponded to values of allometric exponents, previously obtained in the studies simultaneously considering different taxa. On the other hand, that study made certain "shortcut" approximations, which were justified for the first iteration, whose purpose was rather a proof of concept through obtaining average values. In particular, it was assumed that the limbs' length scales as a 1/3 power of mass. This assumption is based on geometric similarity, although in reality, as we will see, there are substantial variations of this and other values amongst different species, which noticeably affect allometric exponents. Disregard of such finer effects did not allow obtaining values of allometric exponents, precisely matching known values for different classes of animals, in particular the ones presented in studies [3,4]. These studies rely on extensive datasets for different species and use measurable and well founded methods for data processing. We will use the results of these studies for comparison.
Lack of fine structure for allometric exponents in [13], as well as some other "shortcuts", caused fair criticism by commentators, which is addressed in this work (for the author, the non-principle character of such critics was clear from the very beginning, but this fact by no means undervalues the commentators efforts and comments, which very much contributed to the significant advancements reported in this paper, compared to the first version).
An important note related to rather strategic positioning of the subject of interspecific allometric scaling as a whole should be made. The recent developments, described in [11][12][13], unexpectedly exposed the problem from a very different than traditional perspective.
These works presented proofs that the fundamental causes of interspecific allometric scaling originate from the systemic evolution of an entire food chain; namely, that organisms, by virtue of adaptation, with the increase in size acquire a metabolic power advantage such that it is sufficient for their successful reproduction in the given nutritional environment, but which is not too strong to destroy populations of their preys or the food chain's links this and other organisms depend upon. In other words, this phenomenon is the result of adaptation for successful reproduction within the food chain, when organisms have to obtain enough nutrients for successful reproduction, but not too excessively to affect the size of populations of species they feed on, which could jeopardize the dynamic balance of a food chain.
Given the fact that the answer to this problem was previously searched in the conceptually different domains, it might be difficult for the community to accept this although presently well supported, but unconventional idea that such a delicate dynamic balance of the food chain could be the primary cause of this phenomenon. However, once accepted, this evolutional principle will have profound and numerous implications, which will be discussed in this paper.

Methods
The study of interspecific allometric scaling began similarly to explorations in unicellular organisms [11,12], that is in search for particular physiological mechanisms, which collectively could define this phenomenon. However, as the study was proceeding, the logic of the research forced to discard more and more particular mechanisms, until the realization came that there should be some fundamental cause of a general nature that defines this phenomenon. The facts were indicating that this is probably the entire food chain, whose continuity and dynamic balance are conditions of its existence, regulates the "appetites" of creatures composing the food chain, and accordingly their metabolic properties. The metabolic power should be sufficient to acquire enough nutrients for a successful reproduction, on one hand, but not excessively strong to jeopardize the reproduction of neighboring links, which are the species the organism directly feeds on or indirectly interacts with within the food chain.
Of course, environmental conditions change all the time, so that such a balance by its nature is a dynamic one. As it was stated in [11,12], two conditions should be fulfilled for the food chain. First, it cannot be broken. The second postulate states that evolutionary development of organisms equipped them with adaptation capabilities, such, that using combination of different physiological mechanisms and developing new ones, they can adapt to a very wide range of environmental conditions, far exceeding limitations imposed by particular physiological mechanisms. Such mechanisms are the means serving the main purpose of any species -survival of its population. Mechanisms can be enhanced, combined in different ways, new ones can be developed, but these are not mechanisms, which define the limits of evolutionary development, but the need for survival of populations of species in conditions, imposed by environments, which mobilizes all possible resources for this most important evolutionary purpose of any living creature.
Indeed, organisms living in the same habitat, within the reach of each other, eventually create a single food chain. There are no unrelated food chains in the same habitual domain.
Even if species do not feed directly on each other, they share common nutritional environment, like vegetation, seeds, fruits, nuts, even common atmosphere contributes to their linking. This defining role of an entire food chain was the subject of unfounded irony from few (but far from all) commentators. The thing is that the links in the food chain are not straightforward. Such, cats feed not only on mice, but evolutionarily cats developed in such a way that they do not overexploit ability to catch their preys, so that the preys' populations could reproduce in sustainable numbers, for the benefit of their own but also for the cats' populations too.
Examples of high adaptability of living organisms are numerous. Even humankind presents extremely high variability of all characteristics, including metabolic rates. Athletic training and world records is one example, while the great diversity of human populations adapted to different geographical zones and sometimes very specific habitats is another example. Whatever exotic characteristics the environment has, there are almost surely some living organisms finding their "home" there. Each geological period on the Earth, which provided some minimal conditions, had some life forms. There are certain fundamental reasons for such an enormous living capacity of organic matter, some of which can be found in [14][15][16]. However, we will not discuss them here. For our purposes, stating the fact that living organisms could adapt to a very wide range of environmental conditions, which is confirmed by their mere existence in very different geological epochs, and known evolutionary developments, is sufficient evidence that no intrinsic factor could fundamentally limit organismal adaptation, save for some extreme conditions known life forms are principally incompatible with. In [17], a similar idea is formulated as follows: "evolution is constrained by physical laws, but … the effects of these laws can be modified by biological innovation".
Well supported examples of high variability of metabolic properties of unicellular organisms can be found in [11,12], while the same organisms exhibit high regularity of metabolic allometric scaling in the range of masses, which embraces almost eight orders of magnitude. The fact that multicellular organisms exhibit wide range of allometric exponents, depending on the physiological regime [2][3][4][5][6][7][8], also supports the thesis that living organisms can adjust their metabolism to very different environmental conditions. So, the introduction and proof of validity of the principal consideration about the decisive role of an entire food chain in shaping metabolic properties of organisms will be the backbone, the main methodological paradigm of our study, from which other more particular methods will originate.

Bio-mechanical denominators of the food chain
In [2,18], the scaling of bio-mechanical constraints was considered, such as mechanical capacity of limbs to withstand buckling and pressure. Other works studied geometrical, kinematic and dynamical mechanical parameters of organisms and their scaling relationships [19,20]. However, such studies, undoubtedly very useful, did not shed light on the fundamental level mechanisms defining metabolic allometric scaling. In some instances, the discovered scaling patterns do not match the results predicted by models. For instance, in [19] the authors acknowledge that "limb inertial properties do not have the potential to underlie COT scaling" (COT stands for "cost of transport"). Such a relationship, if it existed, could optimize the energy expenditure, but it is not the case.
Thus, in this and other cases, considered in the same work, apparently other than merely optimization factors have more impact on organisms' development. This situation rather speaks in favor of our main idea that the primary evolutionary trend any organism follows to is to adapt, by all possible means, to the environment in order to secure reproduction of its population. In fulfilling this task, the primary role belongs to necessities supporting population reproduction, while these necessities have to be satisfied within the constraints imposed by the need to preserve continuity of the entire food chain and its balanced state.
In such an arrangement, optimization of particular mechanisms and processes becomes of secondary importance.
We argue that the main, the foundational characteristic of the food chain is a complex relationship between the predator and the prey through numerous feedback loops and environmental parameters. (When organisms do not directly feed on each other, they still compete for nutrients through the common nutritional environment.) Neither the predator, nor the prey may have an absolute advantage; otherwise, the dynamic equilibrium in the food chain will be destroyed. This equilibrium equation includes many variables. We will consider those, which most closely relate to metabolism, that is energetic and kinematic ones. In particular, when the speed is of primary importance for the survival of species' populations (which is true for many organisms), this means that a predator and a prey have to have commensurate speeds, with some advantage on the predator's side.

Horizontal motion of limbs
First we consider the minimal mechanical energy requirements for motion, assuming that the predator and the prey move with the same speed. This will give us the base allometric exponent, to which the components due to other factors, such as a certain speed advantage, will be added later. (Decomposing the total allometric exponent into components corresponding to different affecting factors, is a general approach, which was successfully applied in studies [11][12][13].) Speed is achieved through the motion of limbs. Fig. 1A presents their rotational motion. However, when displacement S is small compared to the limbs' length, mechanically, the translational motion of a center of mass is an accurate approximation. In [19], it was estimated to be of about 1/3.) This location of COM for mammalians, as was found in [19], scales according to geometric similarity, that is remains unchanged. The authors acknowledge: "For all subgroups, fore-and hindlimb COM position scales according to geometric similarity, … indicating that limb mass distribution remains unchanged with respect to increasing body mass." Similarly, the average backward velocities of centers of limbs cb V are equal too.
We want to know the ratio of energies, accordingly E and e, required to move limbs by the distance S for the same time. For the forward movement, we should take into account the increase of velocity of centers of masses from the backward to forward velocities, which, as we found, are the same for both animals and equal to ) ( Then, the ratio of kinetic energies (in the bodies' systems of coordinates) is as follows. If we assume geometric similarity, that is , then Eqn 1 transform to the following.
In other words, in order to support the same speed, the bigger animal needs more energy proportionally to the ratio of limbs' masses in the 2/3 power, which is the value of the "mechanical" part of an allometric exponent in case of geometric similarity.
In fact, the ratio ) / ( L l scales differently for different groups of animals, which, as we will find out, is one of the reasons why different groups of living organisms have different allometric exponents in the same physiological state.
The term (l/L) in Eqn 1 takes into account that in order to cover the same distance S for the same time, the smaller animal needs to make each step faster and to make ) / ( l L times more steps.
The notion of geometric similarity includes both proportional increase of limbs' lengths, and also increase of limbs' mass proportionally to mass of whole organisms. The last assumption is fulfilled with high accuracy for mammals [19]. The authors conclude, "Across quadrupedal mammals, limb mass scales isometrically with body mass". (Note that our approach allows taking into account insignificant positive allometry of 1.01 and 1.03 for fore-and hindlimb mass increase discovered in [19], if necessary.) The assumption about 3-D increase of limbs is also supported with reasonable accuracy by studies of body proportions to weight in different animals, reviewed in [18].
Similarly, we can find the ratio of energies during the backward movement of limbs.
In this case, the limbs push off the supporting surface with the force proportional to mass, and cover the same distance S. Let k be a coefficient of proportionality for the force, which is the same for both animals due to the same geometry of motion. Then, the ratio of energies is If we assume geometric similarity, then Eqn 4 transforms into In Eqn 4, the term (l/L) accounts for the fact that the smaller animal has to make more steps -same as in Eqn 1. Note that we deal with the whole body masses M and m. The bones of a bigger animal might be disproportionally heavier, but we will account for this factor later.
Now, let us consider the forward movement of limbs as rotation. In recent years, this approach received attention, since it was discovered that energetic costs of limbs' swinging comprise 8 to 33% of the total locomotor costs [19,21]. In our case, we are interested in the ratio of kinetic energies, which is as follows: Here, I denotes moments of inertia of rotating limbs, Ω and ω are angular speeds, accordingly of the bigger and smaller animals. Since the apex angles are the same for both animals, the smaller animal has to exercise a greater angular speed by L/l times, in order to provide the same horizontal speed of animal's body.
In case of geometric similarity Eqn 6 transforms to Similar to Eqns 6 and 7, we can consider the backward motion of limbs (accounting for the fact that a smaller animal has to make L/l times more steps). In this case, the moment of inertia will include both the limbs', and a part of body's mass (since in quadrupedal animals two limbs can be simultaneously involved). This consideration will not change the final ratio, since, given the aforementioned result from [19] about isometrical scaling of limbs' mass, we can substitute instead the mass of whole animals or equally apportioned parts of whole masses (when two limbs are involved in a backward movement).
So, the ratios of kinetic energies in both modes of motion are described by the same , if we account for isometric scaling of limb mass). In case of geometric similarity, these expressions produce the allometric exponent of 2/3.

Vertical displacements
Let us consider the situation when the apex angles are different, while both animals make equal steps (Fig. 1B). The striding angles are accordingly Here, we used the first two terms of the Taylor's series representation of cosine. Using Eqns 8 and 9, the ratio of potential energies, required to overcome the force of gravity, is In case of geometric similarity, the same allometric exponent is equal to 2/3.
So, the ratio of potential energies is the same as for kinetic energies.
However, when the apex angles are equal and steps have different length, Here, we took into account that a smaller animal makes (L/l) more steps.
The comparison of potential energy versus the kinetic energy for particular motion scenarios shows that the energy of vertical displacements is relatively small compared to kinetic energy and energy expenditures for other purposes. Besides, real animals significantly reduce vertical displacements, compared to our model, by bending limbs in joints. Videos showing chasing and escaping quadrupedal animals convincingly demonstrate that vertical oscillations of animals' bodies in such situations are very small.
From the evolutionary perspective, since the vertical oscillations require additional energy, it makes sense to minimize them, and this is the evolutionary path the development of animals followed. Nonetheless, more rigorous studies could consider this factor.
Note that other more complex forms of motion can be always decomposed into combination of rotational and translation movements, which we considered. So, the obtained formulas, in fact, have more general appeal than only for description of particular motion scenarios. The presented models can be used as a basis for more sophisticated mechanical modeling, which, in principle, can introduce adjustments of the second order.
However, as we will see, the introduced models provide an adequate mathematical description of the phenomenon for our purposes.

Proportionality of energy expenditures for moving to velocity
This is an important subject for our studies. Sometimes, it is not understood well, because kinetic energy is proportional to square of velocity, which misled some people to think that the energy expenditures for moving should be also proportional to square of velocity. In fact, this is not so. It was found experimentally in [22] that the velocity of animals is proportional to used energy, although the authors could not find an explanation.
We found that the theoretical rationale for this fact is as follows. Let us compare energies of two animals moving with different speeds due to different lengths of strides, while making strides for the same time T. The animals have accordingly masses m and M, steps' width 1 s and 2 s , and speeds 1 V and ) / ( The first animal spends energy to cover the distance 2 s . The second animal, for the same distance, Eqn 13 states that the kinetic energies, when the speed increase is due to longer steps, indeed, are proportional to animals' masses and speeds (but not to squares of speeds!).
Bigger animals make longer steps, so that our assumption is justified. The authors of [19] also mention that the stride angles in bigger and smaller animals are close, which means that bigger animals, indeed, make bigger steps.
Let us prove that the used energy is proportional to velocities for potential energy too, using a vertical ascent. The scenario can be generalized for other forms of motion and for adding more affecting factors.
The potential energy E required to ascend the height H is equal to is the acceleration of a free fall. If a climber is able to develop power W, then the ascending time T will be W mgH Thus, we proved the result obtained experimentally in [22] that the energy costs for motion are proportional to velocity. We did not take into account the air resistance and some other possible secondary factors; however, in case of more detailed studies they can be accounted for too.

Scaling of velocity and other characteristics with mass increment
We will need a mathematical method for finding allometric exponents for speed increase with relation to mass increase. This suggestion implicitly assumes that the speed, similarly to metabolic rate, also changes as a power function. Results in work [19] confirm this as follows: "physiologically equivalent speeds are positively allometric with body mass, scaling approximately as 21 . 0 M ". So, we can use a power function for the velocity increase too.
Let us consider evolutionary change in velocity when each next evolutionary stage produces a bigger animal that has a greater speed. Such an increase, manifested generally as an increase of metabolic rate in bigger organisms, is not a hypothetical proposition. For instance, it is observed in unicellular organisms as the increase of amount of nutrients acquired per unit surface in bigger organisms [12]. The rationale is that the bigger organisms could originate because of the predispositionally higher metabolism compared to competitors. Once organisms become bigger, they have to maintain higher metabolic capacities in order to reliably acquire nutrients for the reproduction of their populations.
For instance, in mammals, such an advantage in many instances is transformed into an increase in maximal speed for each successive developmental phase, when we ascend the food chain considering bigger animals.
Let us consider x hypothetical evolutionary development stages, each producing a bigger organism with mass x M . The relative mass increase is by g times at each stage.
is the allometric exponent for velocity). Then, the velocity x v at x phase is as follows.
where 0 v is the velocity at the initial stage.
Substituting the value of x g from Eqn 14 into Eqn 15, we obtain In other words, the allometric exponent for velocity v b does not depend on the number of developmental stages, nor on the mass increment g.
The solution of Eqn 16 is as follows.
Below, we assume that the density is constant, so that mass is proportional to volume, V M ∝ , and consequently 0 0 The obtained Eqns 16 and 17 are valid for any other value, whose change is associated with mass and can be described by a power function.

Locomotion as a factor shaping metabolism of whole organisms
During the motion of limbs, other than mechanical energy expenditures occur (average muscle efficiency is in the range of 20 -35%). How does this fit into our approach? This is where we come to an important consideration. Locomotion is the primary function supporting organisms' existence, regardless of the motion mode, like by virtue of limbs, flagella, fins, tail in fish, body movements, etc. Even if the animals use ambush tactics, at the end they have to overcome a prey in direct contact.
Locomotion is the main function, the reference base, to which metabolism of organisms is adjusted evolutionarily and ontogenetically, so that the metabolic allometric scaling defined by locomotion function should be propagated through the entire organism.
This is a very natural arrangement, which stems from the most important and literally vital need to secure population reproduction by mobilizing all resources, while preserving a dynamic balance and continuity of the food chain (otherwise, there will be no source of nutrients). This is a principle consideration, whose consequence is that although the found allometric exponent is based on mechanical energy required for motion, biochemical metabolic activities supporting this energy production have to adjust to the same scaling.
(When other factors play a major role, then evolutionary paths become different.)

Accounting for the increase of skeleton mass
It was shown in [24] and discussed in [2] that the mass of mammals' skeletons scales as the total mass at a power of 04 . 0 08 . 1 ± , so that the allometric exponent for the skeleton weight . This fact has important implications for the metabolic rate. Recall that endurance athletes increase their physical capacities not so much through the weight gain, but mostly through the boosting the metabolic capacities of existing muscles and supporting physiological mechanisms and systems.
The same situation happens with animals when their skeletal mass increases. In order to better understand this, let us imagine an extreme scenario. Suppose that we found three people with successively increasing weight of 60, 70 and 80 kg, but with the same relative skeleton mass of 30% (accordingly 18, 21 and 24 kg). All of them have proportional to weight maximal oxygen consumption. In order to simulate an extreme 40% increase of the relative skeleton mass per each weight increase, the second participant should put a backpack with 14.5 kg, and the third one 55.9 kg. Then let all of them do the same amount of physical work (for instance, carrying bricks of 10-30 kg upstairs in a high-rise construction, in addition to backpacks) for several months, after which the overall maximal oxygen consumption is tested again. Even though this parameter is considered as a conservative one, we will find a greater difference than before the test, even if we account for some possible weight gain (this sort of endurance physical training with outcomes regarding the oxygen consumption was described in [25,26]).
Thus, adding the passive weight gives muscles and other constituents an additional metabolic training, which supersedes some possible weight gain, if any. Given the fact that the increase of metabolic power has to be proportional to the weight load, it means that the metabolic gain should be proportional to the skeleton weight increase, and consequently scales in the same way. For instance, this consideration will change Eqn 6 as follows.
Here, we accounted for the earlier discussed fact that the mass of limbs scales isometrically with the body mass [19]. The case of geometric similarity was presented in [13].
Previously, we discussed that metabolism of organisms is adjusted to the most important organismal function for the survival -to motion, which means that the energy produced for mechanical motion is proportional to the total energy produced by the organism. Thus, Eqn 18 is valid for the metabolic output of whole organisms too.

Calculating scaling of metabolic rate required to support increase of animal speed
The allometric exponent which we considered so far represented the minimum value, which corresponds to equal velocities of a predator and a prey, while in order to catch the prey, the predator needs greater velocity and accordingly higher energy. (One of the commentators noted that high speed might serve as a safeguard for younger animals from predators during reproduction period, while the speed decrease in ageing animals dooms them for destruction, but without jeopardizing the population's reproduction. This suggestion actually makes an argument in favor of our idea about the role of the food chain balance in the life evolution and organization.) How much greater the velocity of a predator should be? One of the important variables is the relative speed of a predator compared to the prey. We will estimate how the value of allometric exponent b above some threshold value b t (in case of geometric similarity b t =0.722 [13]) translates into the speed advantage, assuming that the energy increase is proportional to velocity (this property was considered in the section "Proportionality of energy expenditures for moving to velocity" above). Index 't' denotes threshold values.
, since velocities are proportional to energies. Then, Eqn 19 transforms to the following.
On the other hand, the left part of Eqn 16 can be presented as the ratio of velocities. Since the threshold velocities of large and small animals are equal, that is Increase of speed from mouse weighing 44 g to animals like caribou weighing 54. 5 Of course, we can find the speed increase for any value of mass increment; for instance, per certain mass increase corresponding to a well expressed developmental stage.

Scaling of limb length
Eqn 18 describes ratio of energies required to support motion from the purely mechanical perspective. It includes the ratio of limb lengths. Let us find it. For quadrupedal mammals, it was done in [19]. Depending on the studied group, and fore-or hindlimbs, the allometric exponent noticeably varies. The authors acknowledge that "forelimb slopes range from 0.30 (Rodentia) to 0.42 (Carnivora), while hindlimb slopes range from 0.27 (Rodentia) to 0.42 (Carnivora)". For hindlimbs (which generally consume more energy than forelimbs [19]), the slope is 0.37, for forelimbs, it is 0.4. The authors of [19], apparently based on weighing considerations, use this value for characterization of the whole tested group. They say, "Limb length increases disproportionately with body mass via positive allometry (length / body mass 0.40 )". Our own studies based on data from [23,28], and presented in Table 1

Evolutionary speed increase and its conversion to increase of maximal metabolic rate
If we ascend along the food chain, as mass of organisms increases, we observe increasing animals' velocity, when they exercise about the maximal metabolic activity (chase or flee).
Many herbivorous and carnivorous animals coexist in "predator -prey" pairs, so that herbivorous species evolutionarily had to adjust their speeds to predators and vice versa, in order for both populations to not become extinct. Such speed increase in predators, in fact, is a manifestation of the increase of metabolic power of animals when their mass increases.
In this regard, it is very similar to the increase of nutrient consumption per unit surface in unicellular organisms discovered in [12]. In both instances, the foundational cause is the principle of preservation the food chain and its dynamic balance, while the increases of animals' speed or of the nutrient consumption per unit surface in cells are particular manifestations of this principle. A bigger predator, in order to get a prey, objectively has to have higher speed and power to overcome the prey, and so they were developed evolutionarily in such a way. Otherwise, they won't be able to reproduce their population.
On the other hand, the predator's speed and power advantage cannot be excessive; otherwise, the prey's population will be quickly destroyed. Keeping this delicate dynamic balance of the entire food chain is the only way for all organisms to survive together. Of course, such a balance is not a permanent thing; some species can reduce in quantities, or disappear or excessively reproduce. This will cause the disturbance of the balance, which will be compensated in some way in order to restore the dynamic balance of the food chain.
(Recall the disturbance caused by introduction of rabbits in Australia, for which were no natural predators.) Of course, the speed is only one possible manifestation of metabolic power. The more critical for the animal's survival the speed is, the more objective measure for the maximal metabolic rate it is. Most species in the range from several tens grams to the order of one hundred kilos rely on speed for their survival. In smaller species, like small lizards, the speed may not be an accurate manifestation of their metabolic power. Such, in [31], the author discovered that the actual maximum metabolic rate for small lizards is 3.9 times greater than their metabolic capacity required for the maximal running speed on a level surface. Indeed, Fig. 4 shows a noticeable dispersion of velocities for different animals.
This is the reflection of the fact that the maximal metabolic capacity, although strongly correlated with the speed for many animals, is not the only factor, which animals rely upon for their successful reproduction, but they exercise all possible means to secure the population survival.
Big animals like bulls, elephants rely not only on speed, but also on their power and other means, like horns, so that their weaponry is more diverse in this regard, and the speed is not necessarily an accurate indicator of their maximal metabolic power. Nonetheless, in situations, when such animals do not need to develop high speed of the whole body, they still have to move their body parts fast to match or exceed the speed of enemies or their preys. It is very similar to the principle of circuit training for athletes. In this case, different groups of muscles are trained at a maximal capacity separately, and nonetheless, this accordingly improves the overall metabolic capacity, when the entire body exercises [26,32]. (There are certain physiological reasons for that, which we won't discuss here.) Using data for speed and mass from . This value is significantly less than the value of 0.21 found in [19]. The difference is explained by inclusion of heavy animals in our dataset, which, as we previously discussed, rely not so much on speed but other means for survival. For another dataset of 13 animals, presented in Table 1 in Supplementary material (SM) placed at the end of the article, the allometric exponent is 032 . 0 209 . 0 ± , which is very close to the value of 0.21 from [19].
For the dataset 1 of 6 athletic animals from where l b is the allometric exponent for scaling limbs' length, sk b is the allometric exponent for the skeleton mass, v b is the allometric exponent for scaling the animal speed.
So, we found that the allometric exponent for the maximal metabolic rate is Substituting the earlier obtained numerical values of allometric exponents into Eqn 24, we find the allometric exponent for the general mammalian for a maximal metabolic rate. In [33], experiments yielded the value of 0.942 for a different set of animals. So, there is a correspondence between our approximate evaluation and experimental results for athletic animals too. A more accurate study should take into account also scaling of limb lengths for athletic animals, which could be different, and, of course, the dataset of animals for theoretical evaluation should be the same as in experiments.
There are other experimental studies of allometric scaling for the maximal metabolic rate supporting our results. Reviews [6,7] quote values of 0.86, 0.87 and 0.88 for the maximal metabolic rate in mammals, which are practically the same as the value of 871 . 0 , which we obtained.

Allometric scaling for the basal metabolic rate
Now, knowing the maximal metabolic rates, we can estimate the basal metabolic rate. For that, we can use the fact that the basal metabolic rate is usually a stable fraction of the maximal metabolic rate [2,33], usually about 1/10, although the highly trained athletes may have 1/20 and even 1/30. So, we have to find, how the known fraction of a metabolic power translates into the value of allometric exponent.
Using Taylor series' representation, we can show that for small values of increments of allometric exponent (which is our case), the following formula can be used for approximate estimation.
where B denotes metabolic rate; M is mass, indexes 'max' and 'mec' denote the maximal metabolic rate and the 'mechanical' part of the allometric exponent, defined by scaling of limbs and the skeleton mass, equal to ), corresponding to the basal metabolic rate.
Eqn 25 means that the fraction of metabolic rate k for small increments of allometric exponent can be assumed as equal to a fraction of increment p of the allometric exponent.
It can be proved as follows.
Here, we used the Taylor series' expansions for the terms  These results very well correspond to findings from the authoritative works [3,4], that is to the value of 0.686 for the basal metabolic rate [4], and it is also very close to the standard metabolic rate of 0.678 [3].
Note that the earlier obtained values for the maximal metabolic activity will not change. What happened, we just assigned 1/10 of the increase of the allometric exponent due to maximal physical exercising to the basal metabolic rate.
In case of geometric similarity, we obtain the value of 029 .
, which is very close to Kleiber's original estimation of 0.74.
So, the found values of allometric exponents correspond to experimentally obtained values very well. Thus, we may conclude that our theory indeed explains the origin of interspecific allometric scaling in mammals.

Interspecific metabolic scaling in reptiles
For reptiles, we used a limited set of data for 12 specimen: from work [23] in order to find scaling of limb lengths relative to mass, and from [30,31] to find the scaling of speed relative to mass. The data are presented in tabular and graphical forms in Supplementary Material. The data show well expressed trends with very reasonable diversion of data from trending lines in both instances. Speed for lizards C. variegates and E. skiltoniaus was increased by 3.9 times according to estimations of the real metabolic capacity done in [31].
Scaling of the skeleton mass relative to the total lizard mass was studied in [34].
Skeletal mass varied from 0.15 to 972.75 g. The authors conclude "in lizards, skeletal mass scales … negatively allometrically with body mass". The estimated value for the slope was 0.716. On the other hand, the authors found that "Body mass and SVL ("snout-vent length" -YS) were highly correlated, and the slope of the RMA regression was not different from the slope predicted for isometry. Similarly, skeletal mass and SVL were highly correlated, and again the slope of the RMA regression was not significantly different from the prediction for isometry." However, if SVL scales about isometrically with the body mass, and the skeletal mass scales about isometrically with SVL, then the scaling of the skeletal mass to the body mass should not differ as much from the isometrical scaling, as the authors found. This negative allometry was deduced on a small amount of data (the authors acknowledge that "Skeletal mass scaled with negative allometry against body mass for this small subset"). So, given the above considerations, there is a high probability that the estimation was done with a large error.
Regression analysis of available data produced the following. Allometric scaling for limbs is 016 .
, for speed 017 . 0 . For reptiles, we assume that we do know neither the allometric exponent sk b for skeletal mass (except that it might be slightly negative), nor the fraction k (corresponding to basal metabolic rate) of the increment v b , corresponding to maximal metabolic capacity. However, we can find both values solving the system of two equations, derived from Eqn 24. In [35], the authors obtained the allometric exponent of 0.889 for active reptiles, although there is no guarantee that this was the maximal possible metabolic activity. On the other hand, the authors of [3] obtained the value of 0.768 for the standard metabolic rate in reptiles (which is close to basal metabolic rate). So, we can write the system of equations as follows. (27) Substituting , and solving the system of equations, we find 862 . These numbers are doubtful to be true. In mammals, the value of k is about 1/10 and less, so that it is unlikely that in ectotherms, some of which can live without food for weeks and even months, the value of k is as high.
The value 862 seems as a little bit low too.
Experiments show that a greater value of allometric exponent for the maximal metabolic rate in reptiles is possible. In review [6], the author mentions "squamate reptiles exhibit a greater range of scaling exponents (0.27-1.26)". Since k < 1, then it follows from the system of equations Eqn 27 that for the specimen, which we considered, , then k=0.032, which seem as possible numbers for reptiles. Indeed, for nine specimen of varanid lizards the allometric exponents were found to be 0.89 and 0.97 for 25 and 35 0 C accordingly (review [6]), so that the obtained numbers are realistic ones. Then, in this case we have the following dataset: for the maximal metabolic rate 92 , for the basal metabolic rate 767 , and the allometric exponent for the skeletal mass is 898 . The unintended surprise for this scenario is that the allometric exponent of 767 . 0 very well matches the value of allometric exponent of 0.768 for the standard metabolic rate in reptiles obtained in [3], which is our main reference.
Thus, although for reptiles we did not obtain the allometric exponents for the maximal and basal metabolic rates directly, like for mammals, we showed that our results agree with available experimental data. Besides, even though we have had two unknown parameters, the proposed method still allowed attaining useful results, predict important characteristics, and do cross-verification of obtained values. Maybe even more important is that the whole concept of integrity of a food chain and developed on its basis methods provides clear directions for future studies of metabolism in reptiles, setting priority for (a) finding the scaling of skeletal mass relative to the total mass, and (b) finding the fraction of the maximal metabolic rate corresponding to the basal metabolic rate, which then can be compared to values we obtained.

Interspecific allometric scaling in fish
For fish, the speed is an objective indicator of the metabolic power, since the water resistance in the range of velocities the overwhelming majority of fish swim in can be considered proportional to speed. (At extreme velocities, fish and sea animals like dolphins still can reduce turbulence by forming sort of wrinkles on the skin, thus extending the range of speeds where the water current around fish bodies still resembles laminar flows.) The main contours of bodies of fast swimming fishes were evolutionarily shaped more or less similarly, much due to adaptation to hydraulic resistance during motion. Although the relationships between the body length and volume are different for different fishes, the same species scale in size in a way close to geometric similarity. Such, in [36], the author found that the length relates to mass (regardless of the size) at a power of 2.8 for dace, 3 for trout and 3.2 for goldfish. Our own study based on photos from books and the Internet showed that in fast fishes of different sizes, from centimeters to several tens of centimeters, the proportions between lengths of tails and bodies do not vary significantly, which is apparently the consequence of the uniform body optimization caused by hydraulic resistance. So, for our estimation purposes, we assumed the geometric similarity. However, the scaling of tail length and surface with the increase of size requires further studies.
Besides, fish use body movements for the propulsion too.
For fish, the equivalent of limb length, which we used in our formulas for mammals, is the length of tail, which is the main propeller in fish. The equivalent of the length of strides in mammals for fish is the tail beat frequency. It was studied in [37]. The results showed that the speed is directly proportional to the beat frequency. This may be not so surprising, but nonetheless a remarkable result for our purposes, which means that the "mechanical" part of the allometric exponent in fish can be described by the same formulas, which were derived for mammals (except for the vertical oscillations, which we do not have in fish).
Similar considerations can be applied to fins, whose length we assumed to increase proportionally to the length of fish. (Even if there are some diversions from this assumption, they are not of importance for our purposes, since the tail is the main fish propeller at high speeds, when fish manifests the highest metabolic power.) We used data from [36,37] (total 33 fishes, divided into two datasets of 14 and 19) in order to find the allometric exponent related to speed increase. (Few measurements were discarded when fish obviously swam below maximum speed.) Data are presented in tabular and graphical forms in SM. The graph for the 14 fish dataset shows that the suitability of data in this case is rather questionable, although the trend is clearly expressed. As the author of [37] mentioned, it is difficult to figure out, if a fish swims at a maximum possible speed, and this factor might be the cause of observed dispersion of data.
The value proportional to mass for the 19 fish dataset was calculated as a fish length at a power of 2.8 for dace, 3 for trout and 3.2 for goldfish, according to [36].
Allometric scaling for speed for the first dataset is 038 .
; for the second dataset the value of 076 .
(0.32 for dace, 0.288 for trout and 0.244 for goldfish). In the last case, the average slope was found as the average of slopes for three fishes. This was done because each fish has a different shape, which results in different water resistance. As a consequence, the intercepts of regression lines are different for all three fishes. Thus, finding the average slope as an average of three slopes is more appropriate in this case, than calculating a single slope for all fishes.
The scaling of skeleton mass in fish with increase of mass, according to [38], is equal to 1.03. Now, we can use Eqn 24 and calculate the allometric exponent for the maximal metabolic rate for the second dataset as follows.
For the first dataset, this value is 038 .
. For small fish, with the exclusion of the 1 g marginal "champion", the result is 11 .
(although the "champion" maybe was one of few fishes, which really swam at a maximum possible speed). So, according to our calculations, the possible range of allometric exponents for fish is 03 .
These results comply with experimentally found value of allometric exponent of 0.97 for the maximal metabolic rate, and 0.78 at rest, for a sockeye salmon [39]. In review [6], the value of 0.974 for the rainbow trout is mentioned; in the same review, the reprinted Fig. 4 shows regression lines for different teleostic fishes, whose allometric slopes range from about 2/3 to slightly over 1. The values of 78 . 0 87 . 0 ± , and 0.793 as the mean scaling exponent, are quoted too. In [3], the authors obtained the value of 0.879 for the standard metabolic rate in fish, normalized to 38 0 C and 20 0 C.
All these numbers are within the range, which we obtained. Of course, it would be very useful for verification to find a basal metabolic rate, but, unlike in case of mammals, we have no estimations, which fraction of the maximal metabolic rate it constitutes, and it is likely that this fraction is different in different fishes. If we assume that the standard metabolic rate is equal to the basal metabolic rate, then, using Eqn 24 and the value of 0.879 from work [3], we can estimate the required increment for the basal metabolic rate above the base value as follows: , or about 66% of the maximal metabolic rate increase, which is high. For the value of mean allometric exponent of 0.793 from review [6], this will give 36%, which is probably still high. Is such high basal metabolic rate a specific feature of fish, or some other factors are involved, like scaling of tails' lengths and surfaces?
We have no answer. If the tail increases slower than the whole body, then the "mechanical" constituent of the allometric exponent for fish will be greater, which will accordingly reduce the fraction for the basal metabolic rate. (Photos of tuna, indeed, show that the relative length of its tail is lesser than in smaller fish.) So, further studies are needed, which can go two ways: the best one would be to know how the tail lengths scale with the mass increase. Another approach would be to find the fraction corresponding to the basal metabolic rate, and then, using Eqn 24, calculate the scaling exponent for tails.
In any case, the outcome of the fish study is that the allometric exponent for the maximal metabolic rate, which we found, is in a good agreement with published experimental data, while finding the allometric exponent for the basal metabolic rate requires further studies.

Interspecific metabolic scaling in birds
For birds, we should find the scaling of wing span depending on the birds' mass, scaling of wing mass relative to the body mass, which we do not know, and scaling of some parameter, characterizing a maximal metabolic capacity, for which we have not much choice but to consider speed. Neither we know the fraction of a maximal metabolic capacity corresponding to the basal metabolic rate. Speed of birds depends on air resistance and their aerodynamic shapes. The air resistance at the upper range of birds' speed increases faster than linearly, which is a factor also out of our control. The wing movements are well described by the rotational model for limbs that we discussed previously (Eqn 6). Such are the prerequisites we have for the task.
Data for 30 birds in the tabular and graphical forms are presented in SM. Data for the wing span and speed show well expressed trends and reasonable dispersions. The source [30] provides ranges of values for the wing spans and masses, so that we used two separate tables corresponding to maximum and minimum values; thus, effectively, we had 60 entries. Unfortunately, we did not have speeds for smaller birds, so that we used the same speed (given the facts that many birds tend to fly in flocks, and so to maintain the same speed, the difference should not be substantial.) We computed allometric exponents separately for each table (the values differed very little though, 0.396 and 0.402), and then found the average values.
The allometric exponent for the wing span relative to mass is 031 .
. The allometric exponent for the speed relative to mass is 04 .
. The allometric exponent for the skeletal mass in birds, according to [2,24], is in the range of 1.068-1.079.
Work [27] confirmed but slightly corrected results from [24] We can estimate the basal metabolic rate similarly to mammals, assuming that k=1/10, .
The obtained values agree with available experimental data very well. First of all, the reference work [3] presents the value of 0.644 for the standard metabolic rate, which should be slightly lower than the basal metabolic rate. Even so, this number matches our values of 0.651 and 0.657. Review [6] quotes results of numerous studies for birds regarding allometric exponents for the basal metabolic rate (0.69, 0.681, 0.667), field metabolic rate (0.53), maximal metabolic rate (0.84, 0.88).
Review [6] also mentions that a flying hummingbird has an allometric exponent of 0.88-0.95. We did not include it into our data set, since the data point was an outlier located far above the overall trends, both for the wing span and the speed. However, the specific of this point indirectly confirms the high metabolic rate of a hummingbird.
The obtained value of allometric exponent of 05 . 0 771 . 0 ± for the maximal metabolic rate is noticeably less than the quoted values of 0.84 and 0.88. The most likely reason is that we excluded high metabolic performers like falcons, eagles from our dataset as outliers with regard to the main array of data, since we did not have sufficient statistics for such birds. The addition of a statistically representative dataset for such birds will increase the value of v b , without noticeable impact on the scaling of wing span (we verified this assumption by calculations), which will accordingly increase the value of x b .
So, we may conclude that the allometric exponent for the maximal metabolic rate agrees with experimental data too; to the extent our limited dataset allowed to do this.

Discussion
Due to availability of data, we presented a well supported study for mammals, which convincingly confirmed the validity of the main discovery -the foundational principle of this study that interspecific allometric scaling is the consequence of an evolutionary development of an entire food chain in such a way that it preserves its continuity and a dynamically balanced state. This balance assumes that all animals are able to obtain sufficient amount of food for the reproduction of their populations without jeopardizing reproduction of populations of their preys.
For fish, reptiles and birds we had less data. Even though, we still were able to confirm that the same foundational principle, indeed, defines the interspecific allometric scaling of these classes of living organisms too. Studies [11,12] for unicellular organisms also confirmed that their interspecific allometric scaling is subjected to the same discovered evolutionary principle. So, presently, there is a large body of proofs of validity of this discovery.
As we could see, the total interspecific allometric exponent is the result of action of several factors, when the mass of organisms increases: (a) Scaling of limb masses (tails for fish); (b) Distribution of inertial masses between moving limbs; (c) Scaling of limb lengths; (d) Scaling of skeleton masses; (e) Scaling of the maximal metabolic power; (f) Fractions of the maximal metabolic power corresponding to basal and other specific metabolic rates.
However, as it was said above, for many practical purposes one can assume that 1 = m b .
Distribution of inertial masses between moving limbs means not only masses of limbs, but also distribution of the body mass relative to acceleration thrusts. Forelimbs provide more support and cushion for the landing, while the hindlimbs are the main propellers in animals [19]. For our purposes, that fact was not important, since we used an average length of limbs. However, if the difference in fore-and hindlimbs is substantial, or high accuracy is required, then the specific of mass distribution should be taken into account.
Scaling of limb lengths (tail in fish), as we have seen for all considered classes of animals, is an important parameter, significantly affecting the value of allometric exponents. For the typical proportions in animals, this parameter is far more important than scaling of limb mass. Geometry of limb movement (in particular, the striding angles) is also a factor that can noticeably affect the overall energy expenditures. Fortunately, many animals use similar geometrical patterns for movement, which in many instances was evolutionarily optimized by similar environmental factors, like force of gravity, hydraulic resistance, landscape, terrain, etc. The similarity of striding angles for mammals was confirmed by studies in [19].
Scaling of skeleton mass, as our study showed, turned out to be an important factor for all classes of animals. We saw how such an uncertainty in case of reptiles affected the value of allometric exponents. Fortunately, in some instances it can be found through other parameters, which we did using Eqn 24. Scaling of skeleton mass occurs differently in different species, although mechanical constraints seem as one of the main reasons of this scaling.
Scaling of the maximal metabolic power is probably one of the most important and difficult to find parameters. The reason is that the metabolic power is manifested in many different ways. We saw that in small lizards the actual metabolic power was 3.9 times greater than one could deduce from the running speed [31]. As we saw in case of mammals and fish, the maximal speed is an adequate measure of metabolic power only when this is the major factor supporting animal existence. In case of some animals, like elephants, the speed is not necessarily of such high importance, and so in such cases speed as a measure of the maximal metabolic power is not an accurate parameter. However, in general, the maximal speed is a good indication of the maximal metabolic power, as we have seen in our study. The thing is how to get animals to run, or fish to swim, at a maximum speed.
Overall, the area of maximal metabolic output and the ways of its manifestation is a very promising, interesting and practical one, which awaits studies. Overall, we can summarize the main results of general importance discovered in the study as follows.
(a) Interspecific allometric scaling is defined by continuous evolutionary development of the entire food chain from the smallest organisms to the biggest ones, for which a predator and a prey adjust their metabolism for performing movements with the speed, force and duration sufficient to acquire sufficient amount of food for a successful reproduction, but at the same time without destroying populations of their preys, thus supporting a dynamic balanced state of the entire food chain. The direct "predator -prey" relationships in the food chain are most visible ones and, in many instances, are the most definitive for the metabolic characteristics of living organisms. However, in addition, organisms in the food chain relate to each other through numerous feedback loops, through common nutritional environment and other common environmental parameters, so that the dynamic balance and continuity of the food chain is supported by many mechanisms and diverse interrelationships of organisms. Thus, this principle should not be considered as the one confined to direct "predator -prey" relationships only. In fact, this is a backbone fundamental principle guiding how the food chain organizes and evolutionarily develops, at all scale levels, from the smallest unicellular organisms to the biggest animals.
(b) Allometric scaling of energy requirements due to mechanical constraints imposed on animals' propelling extremities, the whole body and associated body parts propagate through the entire organism, because of the primary importance of animals' motion characteristics for successful reproduction.
(c) Mechanical (meaning set of different physical parameters) characteristics of propelling extremities adjust to evolutionarily optimal motion characteristics of animals; in many instances, these characteristics relate to animals' speed as a primary factor supporting their existence and successful reproduction.
(d) Evolutionarily, the metabolic power increases with the increase of mass (in particular, expressed as an increase of animals' velocities in a certain range of sizes), since a bigger animal has to overcome its prey (if it is a predator) or be able to protect itself from predators (like big herbivores). Such an objective and measureable increase of metabolic power is propagated through the entire food chain in all life domains we studied, including unicellular organisms [11,12]. Greater metabolic power can be manifested by different adaptation means, of which the speed is an important one. On the other hand, the requirement of a balanced state of the food chain caps the increase of metabolic power of bigger animals, so that they could not destroy populations of other animals they are linked to through the food chain. Of course, life developed many forms. However, the size increase was and continues to be a backbone of the evolutionary process, from which other developmental branches followed and will follow. (Imagine evolution without the size increase!) Together, the discussed factors, in our opinion, are the principle causes of interspecific allometric scaling.

Combining intraspecific and interspecific allometric scaling
In [8], an explanation of intraspecific allometric scaling was presented. According to it, intraspecific allometric exponents can vary in a wide range (from 2/3 to one). The study did not take into account scaling of the skeleton mass. This was justified by limited experimental evidences that this effect is well expressed for intraspecific allometric scaling, if at all. Such, the study [31] found that the skeleton mass of lizards actually has a negative allometric exponent. So, the subject requires further studies.
Given this consideration, we see the allometric scaling phenomenon as follows. For each species, the intraspecific allometric scaling is represented by a line with a slope between 2/3 and one. However, the regression lines, representing the interspecific allometric scaling of the basal, standard, maximal and other gradations of metabolic rates for different animal classes, have their specific slopes.

Conclusion
The summary of obtained allometric exponents versus available data is presented in Table   2. As we can see, results correspond to data found in the literature. The causes of observed differences were discussed earlier. * For the estimated value of allometric exponent for the skeleton mass of 0.898.
The main result of this study is the formulation of the principle of integrity of an entire food chain and its dynamic balance as a condition of existence of the whole food chain and each organism, which managed to occupy a space there. It is not a ruthless and brutal competition when everybody tries to destroy everybody else. This is a cruel world, but this is not a world that knows no restrictions for the common good and fair rules. It is as much destructive as a constructive one. This is a balanced world, in which measure is the norm, the rule of the game, but not the exception. There are many important things in this arrangement, which humans could learn for their benefit.
Validation of general concepts and theories is always a tedious and long process. In such endeavors, the validity is established through analysis of very different scenarios.
These are multiple proofs of different nature, which are much more important for such theories, than one solid proof [12]. We considered different classes of animals: mammals, reptiles, fish, birds. Besides, earlier we presented proofs that the same theory convincingly explains interspecific allometric scaling in unicellular organisms [11,12]. Together, these results provide a solid and convincing case for the validity of the theory.
Of course, since this is a general theory, it is by no means restricted by analyzed organisms, but the presented conceptual approaches and methods can be used for other groups of living organisms, in different strata; not only for classes of animals, but also for smaller or larger stratifications created on the basis of different criteria.
As it was stated in [12], the study, which includes not only this article, but the previous works too, opens new promising areas for scientific and practical research. Effectively, the theory proposes a whole framework for scientific studies and practical applications, which can be used in many areas of biology and medicine.
One of the important consequences of the study was the recognition that the seemingly particular problem of interspecific allometric scaling all of a sudden became tightly tied to foundational principles of life origin, development and organization at the level of an entire food chain. This discovery originated from the application of the general growth mechanism [14][15][16], also called the General growth law, to unicellular organisms, when, as a by-product result, allometric relationships between transportation and non-transportation metabolic costs were discovered. (This general growth mechanism was also the foundation and the major tool in the allometric studies of unicellular organisms presented in [11,12].) However, eventually, the principle of continuous food chain and its proof came from the studies of interspecific allometric scaling, and, namely through them, the theory received convincing proofs of its validity. So, now the subject acquired a much broader perspective, which potentially could bring lots of new developments. However, nothing will happen until this discovery will be recognized by a scientific community.
The other important consequence of this discovery is that it shifts the focus of the present biological paradigm from particular physiological and biochemical mechanisms, as determinants of organic life, to better understanding that all these mechanisms are servants providing a great range of adaptive flexibility, which allows organisms to adapt to a wide range of environments. If we think for a moment, it could not be otherwise, since all these mechanisms were developed during evolution, and belong to organisms, which manage to survive, and consequently their adaptation capabilities match a great variety of previous and present environmental conditions. This continuous adaptation is guided by certain rules, determinants of high level, like the aforementioned General growth law, which together provide systemic evolution, and one of these determinants is the discovered principle of a balanced evolution of an entire food chain, whose continuity and dynamic balance is a condition of survival of every one and all organisms together, composing the food chain. This balance by its nature is a dynamic one; links can disappear, new links can emerge, but the food chain, following the fundamental principle of continuity and balance, will restore its continuity and establish a new balance, because these are inherent properties of a food chain and of the organic world as a whole, the fundamental principles its existence is based upon.    Fig. 3. Fish speed versus mass for data from [37], in logarithmic scale.