# Mass rescaling allometries

The natural selection optimum for density regulation explains the 3/4 and 5/6 exponents of body mass allometries

A large amount of the phenotypic variation across natural species is explained by body mass allometries (Kleiber, 1932; Peters, 1983; Calder, 1984), where traits like mass specific metabolism (β) are given as power functions of mass

β ∝ wb

with the exponent (b) being the slope on double logarithmic scale; ln(β) ∝ b ln(w).

The allometric exponents that describe the mass-rescaling response of the life history to the evolutionary changes in mass are given primarily by the invariant density regulation

fe[εN] ∝ fι[VNH(d-1)/d/fe] ∝ fs[βH1/d/V] ∝ w0

that evolves from the population dynamic feed-back selection on mass.

With foraging speed (V) being proportional with biotic time (T) on the body mass axis (Garland, 1983; Calder, 1984), we may exchange V with T in these functions, and insert power relations wx for the relevant traits in the invariant regulation. Combined with i) the ε=αβ relation between net energy, resource handling and pace, ii) the λ = 1 condition of the population dynamic equilibrium, and iii) the T ∝ 1/β scaling from metabolic trade-off selection, we obtain the following equations for the allometric exponents (see Witting, 1995, 2017 for details):

t + n + (d-1)h/d = 0,

b – t + h/d = 0,

n + e = 0,

t = - b,

e = 1 + b,

a = 1,

p + t + e = 1,

[time periods: T∝wt; abundance: N∝wn; home range: H∝wh; pace and mass specific metabolism: β∝wb; energetic state: ε∝we; resource handling: α∝wa; survival: P∝wp].

When these equations are solved, we obtain the results in Table 1. 1/4 and 3/4 exponents follow from two dimensional interactions (2D; d=2). The corresponding exponents for three dimensional interactions are 1/6 and 5/6, and the exponents are 1/2 for interactions in one dimension. Table 1 The theoretically deduced mass-rescaling exponents for net energy (ε), mass specific metabolism (β), survival (p), home-range (H), lifetime reproduction (R), population density (N), rate of population increase (r), and time periods like lifespan (τ). For details see Witting (1995, 1997, 2017).

## Evidence

The theoretically predicted exponents for two dimensional interactions are compared with empirical exponents for mammals, reptiles and birds in Table 2. Table 2 Theoretical 2D exponents compared with empirical estimates for mammals, reptiles and birds. β:mass specific metabolism; τ:time periods like lifespan; N:animal abundance; H:home range; p:survival; R:lifetime reproduction; r:rate of exponential increase. For references see Witting (1997).

I have found no convincing 1D cases, but the 2D-3D-transition is supported by taxa from unicells to mammals when grouped according to empirical estimates of exponent for mass specific metabolism (Table 3). Most terrestrial and benthic taxa are classified as 2D, and pelagic taxa and primates as 3D. The 2D-3D transition is also supported by empirical estimates of the exponents for lifespan and population density (Fig. 1 and Table 4). Table 3 Taxa grouped as 2D or 3D from empirical estimates of the exponent for mass specific metabolism. From Witting (2017). Fig. 1 Left: The body mass (w) allometry for lifespan (T) among 195 species of terrestrial mammals where the exponents is 0.25 (SE:0.04, open circles), and among 40 species of pelagic mammals (taxa Cetacea, Pinnipedia, and Sirenia) where the exponent is 0.16 (SE:0.02, solid circles). Data from Nowak (1991); figure from Witting (1997). Right: The population density (N) allometry for 123 and 131 organisms with two- (solid circles) and three dimensional (open circles) systems, with estimated exponents of -0.79 (SE:0.09) and -0.86 (SE:0.08). Data from Pawar et al. (2012). Table 4 The 2D-3D transition as apparent in empirical estimates of the allometric exponent for mass specific metabolism (average estimates from table 2), lifespan (mammals from Nowak, 1991), and animal abundance (from Pawar et al., 2012). From Witting (2017).

The natural selection of metabolism and mass selects allometric transitions from prokaryotes to mammals

Trends in Ecology and Evolution 13:25 (1998)Download

Body mass allometries caused by physiological or ecological constraints?

A general theory of evolution. By means of selection by density dependent competitive interactions.

The body mass allometries as evolutionarily determined by the foraging of mobile organisms

## References

• Calder, W. A.I. 1984. Size, function, and life history. Harvard University Press, Cambridge.
• Garland, T. 1983. Scaling the ecological cost of transport to body mass in terrestrial mammals. The American Naturalist 121:571--587.
• Kleiber, M. 1932. Body and size and metabolism. Hilgardia 6:315--353.
• Nowak, R.M. 1991. Walker's mammals of the world, volume I--II. 5th ed. The Johns Hopkins University Press, Baltimore.
• Pawar, S., A.I. Dell and V.M. Savage 2012. Dimensionality of consumer search space drives trophic interaction strengths. Nature 486:485--489.
• Peters, R.H. 1983. The ecological implication of body size. Cambridge University Press, Cambridge.
• Witting, L. 1995. The body mass allometries as evolutionarily determined by the foraging of mobile organisms. Journal of Theoretical Biology 177:129--137, https://doi.org/10.1006/jtbi.1995.0231.
• Witting, L. 1997. A general theory of evolution. By means of selection by density dependent competitive interactions. Peregrine Publisher, Århus, 330 pp, URL https://mrLife.org.
• Witting, L. 2017. The natural selection of metabolism and mass selects allometric transitions from prokaryotes to mammals. Theoretical Population Biology 117:23--42, https://dx.doi.org/10.1016/j.tpb.2017.08.005.