^{**}=7/3ψ

# Mass rescaling allometries

The mass-rescaling selection 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 ∝ w^{b}

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

f_{e}[εN] ∝ f_{ι}[VNH^{(d-1)/d}/f_{e}] ∝ f_{s}[βH^{1/d}/V] ∝ w^{0}

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 w^{x} 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, 2016 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∝w^{t}; abundance: N∝w^{n}; home range: H∝w^{h}; pace and mass specific metabolism: β∝w^{b}; energetic state: ε∝w^{e}; resource handling: α∝w^{a}; survival: P∝w^{p}].

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), and the corresponding exponents for three dimensional interactions are 1/6 and 5/6, and 1/2 for one dimensional interactions.

### 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.
- 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.
- Witting, L. 1997. A general theory of evolution. By means of selection by density dependent competitive interactions. Peregrine Publisher, Århus, 330 pp, URL http://mrLife.org.
- Witting, L. 2016. The natural selection of metabolism and mass selects allometric transitions from prokaryotes to mammals. Preprint at bioRxiv http://dx.doi.org/10.1101/084624.