   Malthusian Relativityι**=7/3ψ
The bend of evolutionary time

The time-scale of natural selection may contract, or dilate, with the evolution of mass

To predict body mass evolution over millions of years, as captured by the fossil record, we need to transfer the predicted increase [ d w / d τ = rw w ] on the per generation time-scale of natural selection into physical time [ d w / d t = rw w / τ ]. To do this we need first of all to predict the correlated evolution [ ∂ ln τ / ∂ ln w = τ ] between generation time (τ) and mass (w).

For this we will use the inverse relationship between biotic time and mass specific metabolism [ τ ∝ 1 / β ], and the importance of the pre-mass selected exponential increase in mass specific metabolism

rββ = d ln ββ / d τ = σ2

for the exponential increase in net energy

rε = rββ + rα

and mass (Witting, 2017b), rw = rε / ε = (rββ + rα) / ε.

Let us define

ββ = rββ / rw = rββ ε / (rββ + rα)

and express the per generation exponential increase in mass and the pre-mass component of metabolism as a function of the rate of increase in mass (rw):

wτ = w0 erw τ , and ββ,τ = ββ,0 eββ rw τ

We may then solve the mass equation for time τ=ln(wτ/w0)/rw and recall the allometric deduction (Witting, 1995, 2017a) where ε=(2d-1)/2d. Insert these two equations and ββ=rββ ε / (rββ + rα) into the exponential expressions for pre-mass metabolism and obtain the following exponent

ββ = (2d-1) / 2d (1+rα/rββ)

for the metabolic-rescaling [ ββ,τ ∝ wτββ ] of the allometry for mass specific metabolism [ wβ ∝ wββ wβw ], where β = ββ + βw. Then, as the exponent for mass-rescaling is βw = -1/2d, we obtain the allometric exponent for mass specific metabolism

β = [ 2(d-1)/(1+rα/rββ) ñ 1/(1+rββ/rα) ] / 2d

as a function of the rate of increase in the pre-mass component of mass specific metabolism over the rate of increase in resource handling (rββ/rα; Fig. 1). Fig. 1 The mass-rescaling exponent (green), metabolic-rescaling exponent (blue), and final allometric exponent (red) for mass specific metabolism as a function of the selected rate of increase in metabolic pace (rββ) over the selected rate of increase in resource handling (rα). From Witting (2017b)

Now that we have the metabolic-rescaling exponent for mass specific metabolism (ββ) as a function of the rββ/rα-ratio, we can extend our deduction and obtain all the allometric exponents for the body mass evolution of a phylogenetic lineage in time. This complete set of allometric exponents is given in Table 1 for selected rββ/rα-ratios and interactive competition in one, two and three spatial dimensions.

The main purpose of our study was a prediction of the evolution of natural selection time, as defined by the selected relationship between generation time and mass. With this relation being given by the inverse of mass specific metabolism, we find that the time-scale of natural selection is evolving as

∂ ln τ / ∂ ln w = τ = [ 1/(1+rββ/rα) ñ 2(d-1)/(1+rα/rββ) ] / 2d

This time-scale is dilating to the 1/4, or 1/6, power of mass when the rββ/rα-ratio is zero and mass is evolving exclusively from an increase in the handling of resources. And it is contracting to the -1/2, or -2/3, power of mass when the rββ/rα-ratio is infinite and mass is evolving exclusively from metabolic acceleration. Table. 1 The theoretical exponents for exponential body mass evolution in time, as they evolve from allometric rescaling with selection on mass and metabolism. The exponents depend on the spatial dimensionality of the interactive behaviour of individuals, and on the rββ/rα-ratio that describes the relative importance of metabolism for the evolution of mass. Traits include resource handling (α), mass specific metabolism (β), time-periods (τ), survival (p), lifetime reproduction (R), rate of population increase (r), home-range (h), population density (n), and the rate of change in mass in physical time (dt). From Witting (2017b).

### References

• 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. 2016. The natural selection of metabolism bends body mass evolution in time. Preprint at bioRxiv http://dx.doi.org/10.1101/088997.
• Witting, L. 2017. The natural selection of metabolism and mass selects allometric transitions from prokaryotes to mammals. Theoretical Population Biology 117:23--42, http://dx.doi.org/10.1016/j.tpb.2017.08.005.