No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
... its body weight [Snell, 1892]: ...
... Thus study of brain weight scaling and metabolic scaling may, likely will, 48 reveal a principle of physics significant for a wide range of scientific disciplines Snell presented an argument in his 1892 paper that "the brain weight . . . is 75 proportional to the surface area of the body" which implies s = 2/3 but remarks 76 incidentally that "probably the somatic exponent is greater than" 2/3. 77 Snell's 1892 article suggests two possible bases for 2/3 scaling of brain weights 78 [Snell, 1892]. ...
The same universal 4/3 law, itself a consequence of the principle of dimensional capacity, underlies 3/4 scaling of brain weights and metabolisms.
... Knowing the contributions of some key scientists is helpful to understand the essence of allometric growth law (Table 1). Julian Huxley may have been inspired by Snell (1892). In 1891, Otto Snell discovered that the growth rate of mammalian brain size is different from the growth rate of body size (Lee, 1981). ...
The law of allometric growth originated from biology has been widely used in urban research for a long time. Some conditional research conclusions based on biological phenomena have been erroneously transmitted in the field of urban geography, leading to some misunderstandings. One of the misunderstandings is that allometric analysis must be based on average measure. The aim of this paper is at explaining how to correctly understand the law of urban allometric growth by means of the methods of literature analysis and mathematical analysis. The results show the average measures cannot be applied to all types of allometric relationships, and the allometric relationships based on average measures cannot be derived from a general principle. Whether it is an empirical model or a theoretical model of allometric growth, its generation and derivation are independent of the average measures. Conclusions can be reached that the essence of allometric growth lies in that the ratio of two related general relative growth rates is a constant, and this constant represents the allometric scaling exponent and fractal dimension ratio. The average measures are helpful to estimate the allometric scaling exponent value which accords with certain theoretical expectations more effectively.
... The study of size and its biological consequences, known as allometry, has fascinated biologists for centuries (e.g., Snell, 1892;Thompson, 1917;Huxley, 1932;Gould, 1966; see also Stevens 2009). Allometry refers to the covariation between body size and the size of other body parts, capturing a crucial aspect of phenotypic variation across biological levels-from ontogeny and within individuals to variation among individuals, populations, and species. ...
Allometry, the relationship between body size and the size of other body parts, explains a significant portion of morphological variation across biological levels, at the individual level, within and between species. We used external morphology measurements of 6 Triturus (sub)species, focusing on the T. marmoratus species group, to explore allometric parameters within and between taxa. We tested for allometry of sexual size dimorphism in body, head, and limb dimensions and examined whether intraspecific allometry directed evolutionary allometry, as described by Rensch’s rule. Our findings indicated that female-biased trunk and head dimensions exhibited positive allometry, whereas male-biased limb dimensions showed isometric relationships or weak correlations with body size. Morphological divergences between sexes occurred along common allometric slopes, most often through changes in the intercepts. Among taxon, comparisons revealed that (sub)species diverged in the direction of the allometric slopes. In line with Rensch’s rule, sexual size dimorphism in female-biased traits significantly decreased as overall body size increased. However, the observed intraspecific allometric parameters deviated from theoretical expectations because the steepest allometric slopes for female-biased traits were recorded in the larger species. Our results contribute to understanding the dynamics of allometric relationships and sexual dimorphism in amphibians and provide a robust baseline for future comparative analyses.
... To my knowledge, the first use of allometric scale was in biology and dates back to 1891 [2]; it got general recognition in the 1920s [3], for later attracting the attention of urban scholars who, during the last few decades, started to consistently investigate eventual allometries in urban settlements. ...
... The scaling of brain size with body size has typically been characterized by a power law (Snell 1892;Dubois 1898;Jerison 1973), where: ...
The timing and nature of evolutionary shifts in the relative brain size of Primates have been extensively studied. Less is known, however, about the scaling of the brain-to-body size in their closest living relatives, i.e., among other members of Euarchontoglires (Dermoptera, Scandentia, Lagomorpha, Rodentia). Ordinary least squares (OLS), reduced major axis (RMA), and phylogenetic generalized least squares (PGLS) regressions were fitted to the largest euarchontogliran data set of brain and body mass, comprising 715 species. Contrary to previous inferences, lagomorph brain sizes (PGLS slope = 0.465; OLS slope = 0.593) scale relative to body mass similarly to rodents (PGLS = 0.526; OLS = 0.638), and differently than primates (PGLS = 0.607; OLS = 0.794). There is a shift in the pattern of the scaling of the brain in Primates, with Strepsirrhini occupying an intermediate stage similar to Scandentia but different from Rodentia and Lagomorpha, while Haplorhini differ from all other groups in the OLS and RMA analyses. The unique brain-body scaling relationship of Primates among Euarchontoglires illustrates the need for clade-specific metrics for relative brain size (i.e., encephalization quotients; EQs) for more restricted taxonomic entities than Mammalia. We created clade-specific regular and phylogenetically adjusted EQ equations at superordinal, ordinal, and subordinal levels. When using fossils as test cases, our results show that generalized mammalian equations underestimate the encephal-ization of the stem lagomorph Megalagus turgidus in the context of lagomorphs, overestimate the encephalization of the stem primate Microsyops annectens and the early euprimate Necrolemur antiquus, but provide similar EQ values as our new strepsirrhine-specific EQ when applied to the early euprimate Adapis parisiensis.
... In response to debates over the interpretation of his fossils, Dubois undertook research on the allometric relationship between brain weight and body weight. Extending earlier theoretical works (Snell, 1892), he established that brain size was not only related to body weight by a decreasing power function but also depended on a "coefficient of cephalization, " supposed to reflect the degree of development and complexification of the brain (Dubois, 1897). Applying this mathematical relationship to his fossils, Dubois calculated that the cephalization coefficient of Pithecantropus erectus was roughly half that of anatomically modern humans and double that of apes, further confirming the intermediate evolutionary position of the Javanese primate (Dubois, 1899). ...
Man's natural inclination to classify and hierarchize the living world has prompted neurophysiologists to explore possible differences in brain organisation between mammals, with the aim of understanding the diversity of their behavioural repertoires. But what really distinguishes the human brain from that of a platypus, an opossum or a rodent? In this review, we compare the structural and electrical properties of neocortical neurons in the main mammalian radiations and examine their impact on the functioning of the networks they form. We discuss variations in overall brain size, number of neurons, length of their dendritic trees and density of spines, acknowledging their increase in humans as in most large-brained species. Our comparative analysis also highlights a remarkable consistency, particularly pronounced in marsupial and placental mammals, in the cell typology, intrinsic and synaptic electrical properties of pyramidal neuron subtypes, and in their organisation into functional circuits. These shared cellular and network characteristics contribute to the emergence of strikingly similar large-scale physiological and pathological brain dynamics across a wide range of species. These findings support the existence of a core set of neural principles and processes conserved throughout mammalian evolution, from which a number of species-specific adaptations appear, likely allowing distinct functional needs to be met in a variety of environmental contexts.
... A beam's volume V has 3/2 the dimensional 27 capacity of A. 28 Generalizing and extending this dimensional capacity ratio, 1D radiation in 29 3D space has 4/3 the dimensional capacity of 3D space. (Kleiber, 1932), Peto's paradox (Peto,42 1977; Nunney, 2020), brain weight scaling (Snell, 1892) and expanding cosmo- Cancer cells travel 1D paths in a 3D animal body, thus scaling by 4/3 relative 52 to mass M. But 3/4 metabolic scaling results in cancer prevalence being invariant 53 for differently sized animals. ...
Since 4/3 scaling, which applies to bounded, finite systems, as in metabolic scaling, Peto's paradox and brain weight scaling, also applies to unbounded expanding cosmological space, 4/3 scaling is probably valid and the mathematics of WBE 1997 is probably wrong.
... The logarithmic transformation was first used in allometric research late in the 19 th century when several investigators independently examined variation in relative size of the mammalian brain (Snell, 1892;Dubois, 1897;Lapicque,1898). The basic idea then, as now, was to linearize a curvilinear distribution in the original bivariate observations by transforming them to logarithms, fit a straight line to the transformations and then back-transform (exponentiate) the resulting equation to form a two-parameter power equation on the arithmetic (linear) scale. ...
The field of biological allometry has been dominated since early in the last century by the logarithmic transformation, which is widely perceived to be necessary for the proper analysis of bivariate data relating the size of a structure or the intensity of a process to some measure of body size. Some investigators argue that transformation is needed to align the analysis with underlying theory; others assert that transformation is required to describe multiplicative growth in living substance; and still other workers believe that transformation is necessary to accommodate multiplicative variation in the response variable (heteroscedasticity) and/or a lognormal distribution for residuals from the fitted equation. None of these beliefs is true. Moreover, constraints imposed by logarithmic transformation typically result in data being ‘fitted’ to a predetermined statistical model instead of a model being fitted to the data, thereby leading in many instances to erroneous perceptions of pattern in the data, misinterpretation of the findings and misdirection for future research. Robust statistical models with different functional form and different assumptions about random error can be fitted directly to the original data by non-linear regression, thereby obviating transformation altogether. The utility of the regression protocol is illustrated in a re-analysis of published data.
... Allen's scenarios have implications for allometry 26 -i.e. the relationship between the absolute size of appendage and body size [27][28][29] . If Allen was right, and body size determines how the relative size of appendages increases with temperature (a stronger Allen's rule relationship in larger animals), then the allometry of appendage size should vary across temperature gradients, with the increase in the absolute size of appendages (against body size) expected to be steeper in warm climates and milder in cold climates. ...
Animals tend to decrease in body size (Bergmann’s rule) and elongate appendages (Allen’s rule) in warm climates. However, it is unknown whether these patterns depend on each other or constitute independent responses to the thermal environment. Here, based on a global phylogenetic comparative analysis across 99.7% of the world’s bird species, we show that the way in which the relative length of unfeathered appendages co-varies with temperature depends on body size and vice versa. First, the larger the body, the greater the increase in beak length with temperature. Second, the temperature-based increase in tarsus length is apparent only in larger birds, whereas in smaller birds, tarsus length decreases with temperature. Third, body size and the length of beak and tarsus interact with each other to predict the species’ environmental temperature. These findings suggest that the animals’ body size and shape are products of an evolutionary compromise that reflects distinct alternative thermoregulatory adaptations.
... Allometry-the study of biological scaling-examines how the phenotype of organisms changes in proportion with changes in body mass on log scale (Snell 1892;Thompson 1917;Huxley 1932;Kleiber 1932). Mass (w) is seen as the independent variable with the remaining phenotypic traits (x) following as an allometric function ...
I use data-based life history models for 9,488 species of birds and 4,865 species of mammals to illustrate natural selection causes for the evolution of inter-specific body mass allometries. Each model integrates the growth and demography of individuals with the life history energetics and population ecology of the species. I show i) how the primary selection of resource handling and mass-specific metabolism generates the net energy of individuals, ii) how the selected net energy generates a population dynamic feedback selection where intra-specific interactive competition selects body masses that scale in proportion with net energy on the timescale of natural selection, iii) how the primary selection of metabolism selects an allometric curvature where the residual mass-specific metabolism---relative to the expectation of the mass-rescaling allometry---is an initially declining function of mass in terrestrial placentals and birds, but not in marsupials and bats, iv) how the selection of body mass buffers ecological variation in survival, and v) how the joint selection of mass and optimal foraging selects the exponents of body mass allometries from the dominant spatial dimensionality of the foraging ecology.
... In terms of energy, the rate of ATP synthesis at the cell surface must meet the rate of ATP consumption by the whole cell volume. However, surface area decreases relative to volume as cells grow larger-surface area scales with the square of the linear dimension, whereas volume scales with the cube of the linear dimension [29][30][31] . A developmental or evolutionary increase in cell volume thus poses a challenge to cells because, if internal volumes remain active, processes that are carried out at the cell surface (for example, respiration or nutrient transport) will, at some cell volume, be unable to support processes that occur in the cytoplasm (for example, protein synthesis). ...
The origin of eukaryotic cell size and complexity is often thought to have required an energy excess supplied by mitochondria. Recent observations show energy demands to scale continuously with cell volume, suggesting that eukaryotes do not have higher energetic capacity. However, respiratory membrane area scales superlinearly with the cell surface area. Furthermore, the consequences of the contrasting genomic architectures between prokaryotes and eukaryotes have not been precisely quantified. Here, we investigated (1) the factors that affect the volumes at which prokaryotes become surface area-constrained, (2) the amount of energy divested to DNA due to contrasting genomic architectures and (3) the costs and benefits of respiring symbionts. Our analyses suggest that prokaryotes are not surface area-constrained at volumes of 100‒103 µm3, the genomic architecture of extant eukaryotes is only slightly advantageous at genomes sizes of 106‒107 base pairs and a larger host cell may have derived a greater advantage (lower cost) from harbouring ATP-producing symbionts. This suggests that eukaryotes first evolved without the need for mitochondria since these ranges hypothetically encompass the last eukaryotic common ancestor and its relatives. Our analyses also show that larger and faster-dividing prokaryotes would have a shortage of respiratory membrane area and divest more energy into DNA. Thus, we argue that although mitochondria may not have been required by the first eukaryotes, eukaryote diversification was ultimately dependent on mitochondria. Analysing the energetic constraints on prokaryotic cell size, the energetic implications of eukaryotic genome architecture, and the presence of endosymbionts, the authors suggest that mitochondria were not required for the initial origin of eukaryotes, but did facilitate their subsequent diversification and expansion.
... scientists have attempted to explain animal growth using mathematical functions. Snell (1892) was the first to create a function to describe animal brain weight in relation to body weight. Huxley (1924;1932) later used Snell's function to describe allometric growth in animals as y = bx k , where y expresses the size or weight of a body fraction, x represents the size or live weight of the animal, and b and k are constants. ...
This doctoral thesis generates basic data to determine the nutrient and energy requirements of growing Fleckvieh (German Simmental) beef bulls under different feeding regimes. Hence, analyses of body composition and nutrient and energy accretion rates were performed in growing Fleckvieh bulls fed rations with different energy concentrations. The results illustrate allometric growth patterns in cattle and demonstrate how more sustainable feeding concepts can be realized.
Seventy-two Fleckvieh bull calves, representing the current genetic level, were customarily reared. During the fattening period, the calves were allocated to normal energy and high-energy treatment groups fed 11.6 and 12.4 megajoule metabolizable energy per kilogram dry matter, respectively. Differences in dietary energy concentrations were achieved by varying the amounts of concentrates and maize silage in the feeding rations. Bulls from both feeding groups were slaughtered in a serial slaughter trial at final live weights of 120, 200, 400, 600, and 780 kilograms. During slaughter and subsequent beef cutting, the bulls’ bodies were dissected into individual tissue fractions, which were homogenized and analyzed for their chemical composition. Regression modeling was applied to determine the body composition of the growing bulls. The first derivative of the individual equations was used to calculate the gain composition and describe changes in body proportions throughout the growth process.
The study results are presented in two publications. The first publication specifies fattening and slaughter performance in current Fleckvieh bulls at defined final weights and under feeding regimes with varying energy concentrations. The results demonstrate that Fleckvieh bulls at the current genetic level feature increased growth potential and final weights, compared to Fleckvieh bulls in previous decades. Current Fleckvieh bulls efficiently exploited energy and nutrients in the offered feed and exhibited increased daily weight gains, leading to high final weights. High-energy fed bulls showed increased growth performance compared to bulls fed a regular-energy diet. Differences in carcass composition and meat quality traits in growing bulls from both treatment groups were not recorded. Consequently, Fleckvieh bulls fed rations with lower energy concentrations needed more time to reach the highest target weight. However, slaughter performance and meat quality traits were comparable to those of bulls fed high-energy diets.
The second publication assesses body tissue composition, body chemical composition, and the composition of body weight gain in growing Fleckvieh bulls fed rations with varying energy concentrations. The results indicate that feed containing varying amounts of energy did not alter body composition or energy and nutrient accretion rates in growing Fleckvieh beef bulls. During growth, unequal changes in body tissue gain and nutrient accretion, attributable to allometric cattle growth, became apparent. Comparison with earlier research reveals that current Fleckvieh bulls with high live weights feature lower rates of crude protein accretion but higher crude fat and energy accumulation rates than bulls in previous decades. Hence, feeding recommendations for growing Fleckvieh bulls must be regularly adjusted to suit energy and nutrient requirements and increase daily weight gain and target weights.
In summary, growing Fleckvieh bulls in the normal and high-energy feed intake groups demonstrated similar body composition, carcass composition, composition of gain, and meat quality traits. It can be concluded that feeding high-energy rations is not necessary from a metabolic standpoint. Fleckvieh bulls can be fattened using lower energy, roughage-rich rations according to their physiological advantages as ruminants, which also contributes to animal welfare. Future feeding concepts should aim to increase roughage feeding in cattle nutrition to reduce competition for resources used in livestock feed and human food production. Furthermore, phase feeding should be used to feed growing cattle according to their energy and nutrient requirements and reduce nitrogen excretion and the resulting environmental impacts. Hence, more sustainable cattle feeding concepts can contribute to resource conservation, environmental protection, and increased animal welfare.
... In biology, the allometric dependence has become widespread [22][23][24][25][26][27][28][29][30]. Such models for biomass components of the genus Quercus spp. ...
Current climate changes caused by greenhouse gas emissions alter landscape and environmental conditions, increase instability in many ecosystems, and increase the global role of forest cover. We attempted to model Quercus spp. single-tree biomass using the data from 500 sample trees distributed along the trans-Eurasian hydrothermal gradients. Today, several equations for the tree biomass involve both morphological-structural characteristics of trees and stands, and climate indicators as independent variables. The models make it possible to predict changes in biomass due to shifts in climate trends, but do not show the contribution of climate variables to the explanation of biomass variability. This variability depends on both the species of the tree and stand, and the structure of a model. The models designed show to what extent the deviation from the classical allometric model caused by the inclusion of additional independent variables, increases the contribution of climate variables to the explanation of biomass variability. The model shows the greatest contribution when it includes age, stem diameter, tree height, and their combined effect. The 3D-interpretation of the "best" model showed a propeller-shaped dependence of the components of oak tree biomass on temperatures and precipitation. The shape of this dependence is a mirror image of a similar dependence for the biomass of trees of two-needled pines and larches. This may be due to the functioning traits of leaved and coniferous species. Opposed Reviewers: Additional Information: Question Response Abstract: Current climate changes caused by greenhouse gas emissions alter landscape and environmental conditions, increase instability in many ecosystems, and increase the global role of forest cover. We attempted to model Quercus spp. single-tree biomass using the data from 500 sample trees distributed along the trans-Eurasian hydrothermal gradients. Today, several equations for the tree biomass involve both morphological-structural characteristics of trees and stands, and climate indicators as independent variables. The models make it possible to predict changes in biomass due to shifts in climate trends, but do not show the contribution of climate variables to the explanation of biomass variability. This variability depends on both the species of the tree and stand, and the structure of a model. The models designed show to what extent the deviation from the classical allometric model caused by the inclusion of additional independent variables, increases the contribution of climate variables to the explanation of biomass variability. The model shows the greatest contribution when it includes age, stem diameter, tree height, and their combined effect. The 3D-interpretation of the "best" model showed a propeller-shaped dependence of the components of oak tree biomass on temperatures and precipitation. The shape of this dependence is a mirror image of a similar dependence for the biomass of trees of two-needled pines and larches. This may be due to the functioning traits of leaved and coniferous species.
... Allometric equation.-Changes between parts of an organism and their proportions are generally described by the allometric equation (Snell 1892;Huxley 1924), given by a power law formula: y = bx a (1) where y and x are variables that express the dimension of some parts or components, b is a constant, and a is the law's exponent, or in this case, the allometric coefficient. This equation implies that change in one quantity (x) results in a proportional relative change in another (y). ...
Mesosaurs were small amphibious tetrapods that lived in western Gondwana during the early Permian or even earlier,
when temperate Carboniferous–Permian conditions initiated after the glaciations that affected the southern region of
Pangea. In this contribution, we applied traditional linear regression morphometrics to analyse proportions of both the
skull and limb bones in more than 100 mesosaur specimens. The analyses revealed that all mesosaur bones scale remarkably
close to a model of geometrical similarity (isometry), and that this pattern is particularly strong in long bones
and also in the skull. These results indicate that juvenile and adult mesosaurs do not display appreciable change in bone
proportions, meaning that there are few or no noticeable differences between them during growth. The well-defined
isometry, and particularly, the high interrelation between metatarsals and phalanges permit us to suggest that the mesosaur
hind limb is subject to notable modularity. This evidence strongly argues that the differences previously described to
support three mesosaur species in Western Gondwana, might instead reflect natural intraspecific variability, taphonomic
features or even possible sexual dimorphism, as recently suggested. Our study also reinforces the general plesiomorphic
structure of the mesosaur skeleton, which along with some cranial specializations for ecological fitness and the evidence
of strong isometric growth as we demonstrate herein, may suggest new hypotheses of relationships for mesosaurs which
thus would position them as more basal amniotes than previously thought.
... This phenomenon is based on allometric approaches that take body weight into consideration. The idea is to account for the influence of body weight on brain weight [Snell, 1892;Dubois, 1897;Frick and Nord, 1963;Stephan et al., 1988]. In wild species, however, some data obtained through the allometric method could be better understood as somatization rather than encephalization, e.g., Towe and Mann [1995] and Mann et al. [2018]. ...
The avian class is characterized by particularly strong variability in their domesticated species. With more than 250 breeds and highly efficient commercial lines, domestic chickens represent the outcome of a really long period of artificial selection. One characteristic of domestication is the alterations in brain size and brain composition. The influence of domestication on brain morphology has been reviewed in the past, but mostly with a focus on mammals. Studies on avian species have seldom been taken into account. In this review, we would like to give an overview about the changes and variations in (brain) morphology and behavior in the domestic chicken, taking into consideration the constraints of evolutionary theory and the sense or nonsense of excessive artificial selection.
... In it, Hirt and colleagues have provided an extensive data compilation of 458 measured values of maximum running speeds, gathered from 199 animal species, which they plotted doublelogarithmically against an animal's body size. Such plots have long been used as the visualising counterpart of mathematically capturing, by means of a power law equation (Snell, 1892;Huxley, 1924), the effect of body size on any biological property-that is, the property's allometric scaling: ''. . . the structural and functional consequences of a change in size or in scale . . ." (Schmidt-Nielsen and Pedley, 1977). ...
The maximum running speed of legged animals is one evident factor for evolutionary selection-for predators and prey. Therefore, it has been studied across the entire size range of animals, from the smallest mites to the largest elephants, and even beyond to extinct dinosaurs. A recent analysis of the relation between animal mass (size) and maximum running speed showed that there seems to be an optimal range of body masses in which the highest terrestrial running speeds occur. However, the conclusion drawn from that analysis-namely, that maximum speed is limited by the fatigue of white muscle fibres in the acceleration of the body mass to some theoretically possible maximum speed-was based on coarse reasoning on metabolic grounds, which neglected important biomechanical factors and basic muscle-metabolic parameters. Here, we propose a generic biomechanical model to investigate the allometry of the maximum speed of legged running. The model incorporates biomechanically important concepts: the ground reaction force being counteracted by air drag, the leg with its gearing of both a muscle into a leg length change and the muscle into the ground reaction force, as well as the maximum muscle contraction velocity, which includes muscle-tendon dynamics, and the muscle inertia-with all of them scaling with body mass. Put together, these concepts' characteristics and their interactions provide a mechanistic explanation for the allometry of maximum legged running speed. This accompanies the offering of an explanation for the empirically found, overall maximum in speed: In animals bigger than a cheetah or pronghorn, the time that any leg-extending muscle needs to settle, starting from being isometric at about midstance, at the concentric contraction speed required for running at highest speeds becomes too long to be attainable within the time period of a leg moving from midstance to lift-off. Based on our biomechanical model, we, thus, suggest considering the overall speed maximum to indicate muscle inertia being functionally significant in animal locomotion. Furthermore, the model renders possible insights into biological design principles such as differences in the leg concept between cats and spiders, and the relevance of multi-leg (mammals: four, insects: six, spiders: eight) body designs and emerging gaits. Moreover, we expose a completely new consideration regarding the muscles' metabolic energy consumption, both during acceleration to maximum speed and in steady-state locomotion.
... Initially, allometry meant linkage of trait and whole body sizes for an organism. Rooted in interpretations by Snell (1892), and Thompson (1992), the concept advanced into a study subject with the seminal work of (Huxley 1932). He embedded it in the theory of constant relative growth by two body parts and formulated though the scaling equation (Huxley 1932), ...
Examining sigmoidal allometries in geometrical space can be carried away by direct nonlinear regression or generalized additive modeling approaches. Nevertheless, producing consistent estimates of breakpoints characterizing phases composing sigmoidal heterogeneity could be problematic. Here, we explain how the paradigm of weighted multiple–phase allometries embraced by the mixture structure of the total output of a first-order Takagi–Sugeno–Kang fuzzy model can carry on this task in a direct, intuitive and efficient way. Present calibration tasks relied on log-transformed amniote testes mass allometry data. The considered TSK fuzzy model approach not only offers a way to back the assumption that analyzed testes mass allometry is sigmoidal in geometrical space but beyond this, it provided meaningful estimates for transition among involved phases. Results confirm previously raised views on the superior capabilities of the addressed fuzzy approach to validating prior subjective knowledge in allometry.
... Initially, allometry meant linkage of trait and whole body sizes for an organism. Rooted in interpretations by Snell (1892), and Thompson (1992), the concept advanced into a study subject with the seminal work of (Huxley 1932). He embedded it in the theory of constant relative growth by two body parts and formulated though the scaling equation (Huxley 1932), ...
In this paper, we approach the Multi-Objective Portfolio Optimization Problem with trapezoidal fuzzy parameters. As to the best of our knowledge, there are not reports of this version of the problem. In this work, a formulation of the problem and a solution algorithm is presented for the first time. Traditionally, this kind of algorithm uses the crowding distance density estimator, therefore, we propose substituting this estimator for the Spatial Spread Deviation to improve the distribution of the solutions in the Pareto fronts. We apply a defuzzification process that permits to measure the algorithm performance using the commonly used real metrics. The computational experiments use a set of problem instances and the metrics of hypervolume and generalized spread. The results obtained are encouraging as they confirm the feasibility of the proposed approach.
... Following this introduction is an attempted translation from German to English of an 1892 article by Dr. Otto Snell about allometry as it pertains to mammalian brain size: Die Abhängigkeit des Hirngewichtes von dem Körpergewicht und den geistigen Fähigkeiten [Snell, 1892]. Following the translation is a copy of the original 1892 article from the journal in which it is found. ...
English translation from the original German of Otto Snell's 1892 article.
... In fact, the first studies on the subject appeared long before the book (e.g. 2 ), but it was Thomson's work that laid the foundations for this discipline, which, following the studies of Julian Huxley 3,4 , became a major fundamental and applied area of science [5][6][7][8] . Allometry and scaling of living systems are being studies within that area to this day. ...
Revealing scaling rules is necessary for understanding the morphology, physiology and evolution of living systems. Studies of animal brains have revealed both general patterns, such as Haller's rule, and patterns specific for certain animal taxa. However, large-scale studies aimed at studying the ratio of the entire neuropil and the cell body rind in the insect brain have never been performed. Here we performed morphometric study of the adult brain in 37 insect species of 26 families and ten orders, ranging in volume from the smallest to the largest by a factor of more than 4,000,000, and show that all studied insects display a similar ratio of the volume of the neuropil to the cell body rind, 3:2. Allometric analysis for all insects shows that the ratio of the volume of the neuropil to the volume of the brain changes strictly isometrically. Analyses within particular taxa, size groups, and metamorphosis types also reveal no significant differences in the relative volume of the neuropil; isometry is observed in all cases. Thus, we establish a new scaling rule, according to which the relative volume of the entire neuropil in insect brain averages 60% and remains constant.
... Therefore, the removal of uncertainties related to the assessment of forest cover biological productivity and biodiversity is of paramount importance. Since the 19th century, researchers have noted that the relationships between the mass of individual parts and the whole organism in different species are well described by the so-called self-similarity function, or allometric one (Snell 1892, Dubois 1897, Huxley 1932, Gould 1966, Zar 1968, Ishchenko 1969, Mina and Клевезаль 1976, Kofman 1986, Gelashvili et al. 2013. Recently, a comparative analysis of the accuracy of different methods for determining the biological productivity of some tree species was fulfilled, and it was shown that allometric models designed at a tree scale give a smaller prediction error compared to models performed at the forest stand scale (Zeng et al. 2018). ...
The first attempt of modeling changes in the aboveground additive component composition of larch (genus Larix spp.) tree biomass, according to the Trans-Eurasian hydrothermal gradients of Eurasia on the database compiled for the structure of harvest biomass in a number of 510 sample trees is fulfilled. The adequacy of the obtained regularities is determined by the level of variability 87-99 % explained by the proposed regression models. For the central territory of European Russia, characterized by the mean annual temperature of January-10 °C and the mean annual precipitation of 400 mm, the increase in temperature by 1°C at the constant level of precipitation causes on Larix spp. trees of the equal age and sizes, the decrease in the aboveground, stem, needle and branches by 0.4, 0.3, 1.4 и 1.3 %, respectively. For the same region, in equal-sized trees, the increase in precipitation by 100 mm at a constant annual temperature in January causes the decrease of the aboveground and stem biomass by 1.2 and 1.7%, respectively, and the increase of needle and branches biomass by 4.0 and 6.0%, respectively. The development of such models for the main forest-forming species of Eurasia will make it possible to predict changes in the productivity of the forest cover of Eurasia in connection with climate change.
... This empirical observation was subsequently made between many other biological observables and was first expressed mathematically as an allometric relation by Snell [316]: ...
Complex systems often exhibit amazingly regular behavior through allometry relations (ARs) between their functional attributes and their size. An empirical allometry relation (EAR) between two properties of a complex system relates the average functionality [Formula: see text] and the average size [Formula: see text][Formula: see text] = a[Formula: see text] where the allometry coefficient a and allometry exponent b are empirical constants fit to data. This EAR is a static relation and is found in every sub-discipline of Natural History. Herein we establish, both empirically and theoretically, that for some classes of evolving technology systems, the empirical allometry coefficient a is not constant, but is strongly dependent on the historical time at which the technology system originated. Specifically, we construct an EAR with a time-dependent coefficient, using the evolution of a broad class of military systems, over the last ten centuries, ranging from a medieval bowman to a modern rifleman, from a horse-drawn cannon to a tank. This time-dependent EAR is derived from fundamental considerations involving complexity, scaling, and renormalization group theory. The theory entails an information as well as complexity generalization of the traditional allometry coefficient that combines system size with the technology knowledge of the new system’s developers (time-dependence). It relates technology ARs to technology evolution relations, such as Moore’s Law, with implications for technology drawn from biological evolution analysis.
... Variation in relative size of the vertebrate brain has been a topic of major interest to students of allometry since the founding of the discipline late in the 19 th century (Snell, 1892;Dubois, 1897;Lapicque, 1898). Many investigations have addressed variation across adults of different species (evolutionary allometry), and many others have focused on variation across adults of the same species (static allometry). ...
The concept of biphasic, loglinear growth of the vertebrate brain is based on graphical displays of logarithmic transformations of the original measurements. Such displays commonly give the appearance of two distinct mathematical distributions-one set of observations following a steep trajectory at the low end of the size range and another set following a shallow trajectory at the high end. However, the appearance of two distributions is an artefact resulting from the logarithmic transformations. Observations of brain mass vs. body mass in each of the eight vertebrate species examined in the current investigation conform to a single mathematical distribution that is well described by a single equation fitted to the original, untransformed data by non-linear regression. Data for carp, chickens, kangaroos and rabbits are described by three-parameter power equations whereas those for dolphins and primates are described by exponential functions that rise rapidly to a maximum. The brain continues to grow throughout life in carp, chickens, kangaroos and rabbits but not in dolphins and primates. Future investigations of relative growth of the brain should be based on graphical and analytical study of observations expressed on the native mathematical scale. ADDITIONAL KEYWORDS: loglinear allometry-ontogenetic allometry.
... where y represents the brain weight, b is the intercept of the allometric regression with the 183 abscissa, x is the body weight and a is the slope of the regression (Snell, 1892). To obtain 184 reliable slopes, the data should originate from a sample that covers a reasonable body weight 185 range and whose individual members are part of a biologically significant group, for example, a 186 taxonomic unit. ...
During domestication, many different chicken breeds have been developed that show many alterations compared to their wild ancestors and large variability in parameters such as body size, colouring, behaviour and even brain morphology. Among the breeds, one can differentiate between commercial and non-commercial strains, and commercial strains do not usually show variability as high as that in non-commercial breeds but exhibit a high production rate of eggs (or meat). The breeding of high-performing laying hens, including the housing conditions of hens, is often a focus of concern for animal welfare, and to date, little is known about the correlation between housing conditions and artificial selection on brain structure.
Based on an allometric approach, we compared the relative brain sizes of two inbred strains of laying hens (WLA and R11) with those of seven other non-commercial chicken breeds.Additionally, we examined the brain composition of laying hens and analysed the relative sizes of the telencephalon, hippocampus, tectum opticum and cerebellum. Half of WLA and R11 lines were kept in floor-housing systems, and the other half were kept in a single cage-housing system.
Both strains of laying hens showed significantly smaller brains than the other chicken breeds.Additionally, there was a significant difference between WLA and R11 hens, with R11 hens having larger brains. There was no difference in the relative brain sizes of floor-housed and cage-housed hens. WLA and R11 hens did not differ in their brain composition, but floor-housed hens showed a significantly larger cerebellum than cage-housed hens.
Apparently, pure breeding over a long time and strong artificial selection for a high production of eggs is accompanied by (unintentional) selection for smaller brains. Further studies may also reveal differences in brain composition and the influence of housing conditions on brain composition.
We measured the BASAL breathing frequency following an overnight fast in adult, non‐pregnant/non‐lactating, inactive mammals ranging in body mass from 15 to 5520 kg. The data included results from 338 individual animals from 34 species that were divided into terrestrial, semi‐aquatic (Otariidae and Phocidae) and aquatic mammals. Following attempts to limit the collection of breathing frequency using a basal definition and to correct the analysis phylogenetically, our results suggest that there are differences in the allometric mass‐exponent between terrestrial and aquatic/semi‐aquatic mammals. An allometric regression model, whereby both body mass and breathing frequency were transformed using log10, suggested that the allometric mass exponent for terrestrial mammals (−0.303) was different from both aquatic mammals (−0.124) and semi‐aquatic mammals (−0.091). For semi‐aquatic mammals, the breathing frequency was lower in water, but we detected no association between the breathing frequency and the temperature of the medium (water or air). We propose that allometric studies of cardiorespiratory function should, if possible, adhere to the basal definition during data collection, similar to that used for metabolic rate. Such data will provide valuable information for comparative medicine of large species that are difficult to study, for which controlled baseline data might be difficult to obtain.
Describing Scaling Relationships
Scaling relationships are a central feature of global ecology, quantifying general biological patterns across broad spatial and temporal scales. Traditionally characterised as scale‐invariant power laws, the scope of biological scaling has expanded in recent decades to include log–log curvilinearity and exponential functions. In macroecology and biogeography, a major focus is on quantifying these general relationships using empirical data, comparing observations across datasets and testing their consistency with theoretical predictions. This is typically accomplished by fitting linear models to log‐transformed data, estimating slopes (representing scaling exponents or exponential rate constants) and 95% confidence intervals (CIs), and evaluating whether these CIs align with empirical observations or theoretical predictions.
Challenges of Existing Methods
The accuracy of general slope estimates depends critically on the distribution of data across the range of the abscissa. When observations are unevenly distributed, with clustering in some portions of the range, slope and CI estimates become biased toward regions of higher data density. This imbalance increases the risk of type I or II errors, potentially leading to erroneous conclusions in comparisons of data with observations or predictions.
Bootstrapping Enables Accurate Estimates of Scaling Relationships
We introduce a novel bootstrapping approach to address data imbalance in biological scaling analyses that improves the accuracy of general slope and CI estimates. This method enables more precise comparisons with empirical observations and theoretical predictions. We validate the approach by accurately reproducing a known slope from plant height‐diameter data. Additionally, we demonstrate that fitting linear models to imbalanced and balanced metabolic rate‐body mass data yields different slope estimates, leading to different conclusions regarding agreement between data and theory. Finally, we evaluate three common data processing methods and show that model fits to balanced data are superior for reliable quantification of general scaling relationships.
The conceptual and historical context, the character and the implications, of an equation that characterizes metabolic scaling are discussed and explored. Then the equation is used to derive Kleiber's empirically found 3/4 metabolic scaling.
Differential scaling of a part or process of an organism relative to a reference, often organism weight, is an anisometry, a non-proportional scaling. An allometric scaling is an anisometric scaling that reverses the effect of differential scaling of parts on a change in size. Thus an allometry maintains the homeostasis of a physiological ratio. Metabolic scaling maintains the cellular ratio of energy supply to energy used homeostatic. Cells only get the energy they require, not more, and so do not overheat. An allometric scaling achieves homeostasis.
To explain why cognition evolved requires, first and foremost, an analysis of what qualifies as an explanation. In terms of physics, causes are forces and consequences are changes in states of substance. Accordingly, any sequence of events, from photon absorption to focused awareness, chemical reactions to collective behavior, or from neuronal avalanches to niche adaptation, is understood as an evolution from one state to another toward thermodynamic balance where all forces finally tally each other. From this scale-free physics perspective, energy flows through those means and mechanisms, as if naturally selecting them, that bring about balance in the least time. Then, cognitive machinery is also understood to have emerged from the universal drive toward a free energy minimum, equivalent to an entropy maximum. The least-time nature of thermodynamic processes results in the ubiquitous patterns in data, also characteristic of cognitive processes, i.e., skewed distributions that accumulate sigmoidally and, therefore, follow mostly power laws. In this vein, thermodynamics derived from the statistical physics of open systems explains how evolution led to cognition and provides insight, for instance, into cognitive ease, biases, dissonance, development, plasticity, and subjectivity.
Allometry refers to the relationship between the size of a trait and that of the whole body of an organism. Pioneering observations by Otto Snell and further elucidation by D’Arcy Thompson set the stage for its integration into Huxley’s explanation of constant relative growth that epitomizes through the formula of simple allometry. The traditional method to identify such a model conforms to a regression protocol fitted in the direct scales of data. It involves Huxley’s formula-systematic part and a lognormally distributed multiplicative error term. In many instances of allometric examination, the predictive strength of this paradigm is unsuitable. Established approaches to improve fit enhance the complexity of the systematic relationship while keeping the go-along normality-borne error. These extensions followed Huxley’s idea that considering a biphasic allometric pattern could be necessary. However, for present data composing 10 410 pairs of measurements of individual eelgrass leaf dry weight and area, a fit relying on a biphasic systematic term and multiplicative lognormal errors barely improved correspondence measure values while maintaining a heavy tails problem. Moreover, the biphasic form and multiplicative-lognormal-mixture errors did not provide complete fit dependability either. However, updating the outline of such an error term to allow heteroscedasticity to occur in a piecewise-like mode finally produced overall fit consistency. Our results demonstrate that when attempting to achieve fit quality improvement in a Huxley’s model-based multiplicative error scheme, allowing for a complex allometry form for the systematic part, a non-normal distribution-driven error term and a composite of uneven patterns to describe the heteroscedastic outline could be essential.
Principles in the article Galileo I: Principles of differential scaling and of allometry are here applied to Galileo's animal bone scaling, Kleiber's Law, brain weight scaling and Peto's Paradox.
This article intends to provide a summary of current principles of differential scaling and of allometry, based on Galileo's scaling in his Two New Sciences.
Writing the recently posted article on why WBE is not correct illuminated the question of why allometric scaling does not account for the hierarchical scaling of the circulatory system. I have wondered about that for about 15 years. But on completing the WBE article, the solution presented itself in a minute. Leading to this article.
Galileo showed that magnitudes of a beam's volume and cross-sectional area scale differently due to their different dimensions. But physical dimension is more fundamental than the mathematical scaling it induces. Some twentieth century problems unsolved using scaling --- metabolic scaling and dark energy are examples--- can be solved using dimensions. Revisiting Galileo's 1638 scaling approach helps reveal principles based on dimension missing from physics.
A flowchart display of the logical steps.
A reasoning flowchart shows the pivotal extension from 3D to 4D in the argument for universal 4/3 scaling.
Thermodynamics is regarded as a universal but not foundational theory because its laws for macroscopic quantities have not been derived from microscopic entities. Thus, to root thermodynamics into the fundamental substance, atomism is revived, thinking that the light quantum is the indivisible and permanent element. Assuming the same basic building blocks constitute everything, the state of any system can be quantified by entropy, the logarithmic probability measure multiplied by Boltzmann’s constant. Then, the change in entropy expresses the system’s evolution toward thermodynamic balance with its surroundings. These natural processes consuming free energy in the least time accumulate sigmoidally, resulting in skewed distributions found throughout nature. In this way, thermodynamics makes sense of phenomena across disciplines and provides a holistic worldview to address questions such as what the world is, how we know about it, what is the meaning of life and how we should live.
This article is part of the theme issue ‘’Thermodynamics 2.0: Bridging the natural and social sciences (Part 1)’.
Generalizing Galileo's strength of materials scaling implies 4/3 scaling is a universal physical law, arising in brain weight / body weight scaling, metabolic scaling, Peto's paradox, black body radiation, the fractal envelope of Brownian motion, the expansion of space in cosmology and so on.
Numerous instances of allometric quarter scaling occur in plants and animals. This article makes the case for quarter scaling and related 4/3 scaling as universal physical laws.
Allometry—the study of proportional growth of body parts, and the relationship of body size to an organism’s morphology, physiology and behaviour—is a fundamental influencer of ecological and evolutionary diversity. Allometric studies can focus on scaling across an individual's development (ontogenetic allometry), among individuals at the same developmental stage (static allometry), and among species (evolutionary allometry). The key assumption in allometry is that an organism’s body size is a critical factor in shaping its biology, so biological scaling underpins biological diversity. This commentary accompanies a special issue that collates original research papers on the wide-ranging ecological and evolutionary implications of biological scaling. We discuss the common themes uniting each contribution, such as how ontogenetic allometry facilitates evolutionary allometry, how size influences feeding performance and trophic niche, methodology in allometry and size estimation, and allometry in sexual selection. In doing so we highlight areas of particular need for future studies to better understand the role of allometry in evolutionary ecology.
Comparing scale factors may help analyze metabolic scaling. This approach implies circulatory systems distributing blood-energy-scale by a 4/3 power of body mass. Invariance of the rate of energy use per cell, regardless of mammal size, requires 3/4 scaling of 4/3 scaled energy distribution. Summary
The origin of eukaryotic cell size and complexity is thought by some to have required an energy excess provided by mitochondria, whereas others claim that mitochondria provide no energetic boost to eukaryotes. Recent observations show that energy demand scales continuously and linearly with cell volume across both prokaryotes and eukaryotes, and thus suggest that eukaryotes do not have an increased energetic capacity over prokaryotes. However, amounts of respiratory membranes and ATP synthases scale super-linearly with cell surface area. Furthermore, the energetic consequences of the contrasting genomic designs between prokaryotes and eukaryotes have yet to be precisely quantified. Here, we investigated (1) potential factors that affect the cell volumes at which prokaryotes become surface area-constrained, and (2) the amount of energy that is divested to increasing amounts of DNA due to the contrasting genomic designs of prokaryotes and eukaryotes. Our analyses suggest that prokaryotes are not necessarily constrained by their cell surfaces at cell volumes of 10 ⁰ ‒10 ³ μm ³ , and that the genomic design of eukaryotes is only slightly advantageous at genomes sizes of 10 ⁶ ‒10 ⁷ bp. This suggests that eukaryotes may have first evolved without the need for mitochondria as these ranges hypothetically encompass the Last Eukaryote Common Ancestor and its proto-eukaryotic ancestors. However, our analyses also show that increasingly larger and fast-dividing prokaryotes would have a shortage of surface area devoted to respiration and would disproportionally divest more energy to DNA synthesis at larger genome sizes. We thus argue that, even though mitochondria may not have been required by the first eukaryotes, the successful diversification of eukaryotes into larger and more active cells was ultimately contingent upon the origin of mitochondria.
Significance
There has been a lot of theorizing about the evolution of eukaryotes from prokaryotes, but no consensus seems to be on the horizon. Our quantitative analyses on the required amount of respiratory membrane, and the amount of energy diverted to DNA synthesis, by both prokaryotes and eukaryotes, suggest that mitochondria provided rather small advantages to the first eukaryotes, but were advantageous for the macro-evolutionary diversification of eukaryotes. This conclusion provides a middle road in the debate between those that claim that the origin of eukaryotes required a massive energy boost provided by mitochondria, and those that argue that the origin of mitochondria did not represent a quantum leap in energetic advantages to eukaryotes.
4/3 laws are often based on a ratio of the degrees of freedom of two systems. Applicable phenomena include dark energy, 3/4 scaling of metabolism (Kleiber's Law) and of brain weights, 4/3 scaling of wind eddies, isotropically radiated energy, Peto's cancer paradox, information distribution in social networks. How should 4/3 laws be introduced? By general principles of scaling? By invariance? From a particular instance followed by generalization? By generalizing Galileo's analysis of animal bone supporting body weight? By a common pattern among diverse phenomena?
Understanding the diversity of brain morphology is important to understand the evolution
of cognitive ability and how ecology and phylogeny have influenced the variation in brain
complexity. I examined the morphological variation of the brain in the shark order Lamniformes
based on museum specimens and literature. Where I illustrate a wide range of morphological
diversity in lamniform brains, my study shows that there is a strong positive correlation between
brain size and body size, and that sharks with a larger brain tend to have a more foliated
cerebellum, whereas the body weight over brain weight did not correlate with cerebellar
complexity. In addition, the brain size is found to be affected by ontogeny where younger
individuals tend to have larger brains than older conspecific individuals. I also demonstrate that
different sizes of different parts of the brain with different functions reflect different lifestyles.
Some ecological specializations are reflected in the brain anatomy of certain lamniforms, such as
adaptations to deep-water (Mitsukurina), filter feeding (Megachasma), tail-based prey hunting
(Alopias), and thunniform swimming (Lamnidae). My study also shows that more derived
lamniform taxa (e.g., Alopiidae, Cetorhinidae, and Lamnidae) have highly foliated cerebellum,
where limited foliation is regarded as plesiomorphic. Lamnids have relatively small brain,
whereas alopiids have a large brain, where a mid-sized brain can be interpreted as plesiomorphic.
My study represents the first investigation into the morphological variation and diversity of the
brain focusing on lamniforms and demonstrates how ecological factors such as habitat, diet, and
behavior drive brain evolution.
The degrees of freedom of a network that distributes energy measures network output capacity. The degrees of freedom of a network using that energy measures use capacity. The ratio of degrees of freedom --- 4 --- of a system distributing energy and the corresponding system using energy --- with 3 degrees of freedom --- gives a 4/3 ratio of capacities. A 4/3 ratio of capacities accounts for the 4/3 fractal envelope of Brownian motion, Richardson wind eddy scaling, Stefan's Law, and expansion of cosmological space, among other phenomena, and implies existence of dual contemporaneous reference frames. Degrees of freedom supplies useful heuristics that give, perhaps, glimpses of physical laws.
This article is another attempt to generalize dimensional capacity.
This is the English version of Chapter 4. Method, of the book: From energy budget of small hydrobionts to the flow of energy through hydroecosystem, from 2015. I'm preparing step by step the English edition of this Monograph.
Лесные экосистемы, являясь поглотителями атмосферного углерода, играют важную роль в сокращении выбросов CO2 и предотвращении потепления климата. Авторы попытались смоделировать биомассу модельных деревьев Quercus L.,используя данные 500 деревьев, распределенных вдоль транс-евразийских гидротермических градиентов. На сегодняшний день предложено несколько моделей биомассы деревьев и древостоев, включающих в качестве независимых переменных как их морфолого-структурные характеристики, так и климатические показатели. Существующие модели позволяют прогнозировать изменения биомассы вследствие сдвигов климатических трендов, но не показывают вклада климатических переменных в объяснение изменчивости биомассы, которая зависит как от вида дерева и древостоя, так и от структуры модели. Разработанные модели показывают, в какой степени отклонение от классической аллометрической модели, вызванное включением дополнительных независимых переменных, увеличивает вклад климатических переменных в объяснение изменчивости биомассы. Этот вклад наибольший, когда модель включает возраст, диаметр ствола, высоту дерева и их совокупный эффект. 3D-интерпретация«лучшей» модели показала пропеллеро-образную зависимость компонентов биомассы дуба от температуры и осадков, форма которой является зеркальным отражением аналогичной зависимости для биомассы деревьев двухвойных сосен и лиственниц. Это может быть связано с особенностями функционирования лиственных и хвойных видов.
Forest ecosystems, as sinks of atmospheric carbon, play an important role in reducing CO2 emissions and preventing annual temperatures from rising. We attempted to model Quercus spp. single-tree biomass using the data from 500 sample trees distributed along the trans-Eurasian hydrothermal gradients. Today, several models of the biomass of trees and stands have been proposed, including both morphological-structural characteristics of trees and stands, and climate indicators as independent variables. The models make it possible to predict changes in biomass due to shifts in climate trends, but do not show the contribution of climate variables to the explanation of biomass variability, which depends on both the species of the tree and stand, and the structure of a model. The models designed show to what extent the deviation from the classical allometric model caused by the inclusion of additional independent variables, increases the contribution of climate variables to the explanation of biomass variability. The model shows the greatest contribution when itincludes age, stem diameter, tree height, and their combined effect. The 3D- interpretation of the «best» model showed a propeller-shaped dependence of the components of oak tree biomass on temperatures and precipitation, the shape of which is a mirror image of a similar dependence for the biomass of trees of two-needled pines and larches. This may be due to the functioning traits of leaved and coniferous species.
The maximum running speed of legged animals is one evident factor for evolutionary selection—for predators and prey. Therefore, it has been studied across the entire size range of animals, from the smallest mites to the largest elephants, and even beyond to extinct dinosaurs. A recent analysis of the relation between animal mass (size) and maximum running speed showed that there seems to be an optimal range of body masses in which the highest terrestrial running speeds occur. However, the conclusion drawn from that analysis—namely, that maximum speed is limited by the fatigue of white muscle fibres in the acceleration of the body mass to some theoretically possible maximum speed—was based on coarse reasoning on metabolic grounds, which neglected important biomechanical factors and basic muscle-metabolic parameters. Here, we propose a generic biomechanical model to investigate the allometry of the maximum speed of legged running. The model incorporates biomechanically important concepts: the ground reaction force being counteracted by air drag, the leg with its gearing of both a muscle into a leg length change and the muscle into the ground reaction force, as well as the maximum muscle contraction velocity, which includes muscle-tendon dynamics, and the muscle inertia—with all of them scaling with body mass. Put together, these concepts’ characteristics and their interactions provide a mechanistic explanation for the allometry of maximum legged running speed. This accompanies the offering of an explanation for the empirically found, overall maximum in speed: In animals bigger than a cheetah or pronghorn, the time that any leg-extending muscle needs to settle, starting from being isometric at about midstance, at the concentric contraction speed required for running at highest speeds becomes too long to be attainable within the time period of a leg moving from midstance to lift-off. Based on our biomechanical model we, thus, suggest considering the overall speed maximum to indicate muscle inertia being functionally significant in animal locomotion. Furthermore, the model renders possible insights into biological design principles such as differences in the leg concept between cats and spiders, and the relevance of multi-leg (mammals: four, insects: six, spiders: eight) body designs and emerging gaits. Moreover, we expose a completely new consideration regarding the muscles’ metabolic energy consumption, both during acceleration to maximum speed and in steady-state locomotion.
During its 40 years of development, the assumptions of the Dynamic Energy Budget (deb) theory for metabolic organisation turned out hard to replace. To understand this, a reasoning is here presented for why its standard model has no alternatives with a comparable level of simplicity and will never have them. Energy and mass conservation rules are essential to quantify the eco-physiological development of an individual organism thermodynamically. These rules strongly constrain the mathematical modelling of this development. In combination with consistency with a small set of stylised empirical facts, the freedom of modelling the skeleton of the model is reduced to a single one: the standard deb model. This skeleton can, however, be extended in many different ways to capture particular ‘details’. The key-message of this paper is that the more simple metabolic models become, the more constraining are consistency conditions.
ResearchGate has not been able to resolve any references for this publication.