^{**}=7/3ψ

# The exponential increase in mass

Unconstrained feed-back selection for an exponential increase in net energy and mass

The feed-back selection of the interactive competition in a population of a multicellular animal will stabilise at the competitive interaction fix-point for a body mass in evolutionary equilibrium when the net energy of the average individual is stable. But unconstrained natural selection is causing an exponential increase in net energy, and this is selecting for an evolutionary steady state where the body mass is increasing exponentially on the per generation time-scale of natural selection (Witting, 1997, 2003, 2017b).

To formulate this, from the discrete growth rate [ λ ∝ p ε t_{r}/w ] we have a selection gradient of unity for net energy on logarithmic scale [ ∂ r / ∂ ln ε = 1 ]. This selection is generating an exponential increase

r_{ε} = d ln ε / d τ = σ^{2} ∂ r / ∂ ln ε = σ^{2}

on the pre-generation time-scale (τ) of natural selection, whenever the additive heritable variance (σ^{2}) is stable (Robertson, 1968; Taylor, 1996). The increased net energy is allocated into reproduction and an increase in the population dynamic equilibrium and its associated interference (ι^{*}). And this interference is selecting for an increase in mass

r_{w} = d ln w / d τ = σ^{2} ∂ r / ∂ ln w = σ^{2} [ ψ ι^{*} – 1 ]

This rate of increase in mass is a consequence of the increase in net energy, and it may thus be rewritten as

d ln w / d τ = [ ∂ ln w / ∂ ln ε ] [ d ln ε / d τ ]

with an invariant selection relation

∂ ln w / ∂ ln ε = 1 / e

that defines mass

w = ∫ [∂ ln w / ∂ ln ε] d ln ε = (ε/ε_{0})^{1/e}

by the inverse of the mass allometry for net energy [ ε = ε_{0} w^{e} ], where e = (2d-1)/2d from the allometric deduction (Witting, 1995, 2017a).

We may thus write the rate of change in mass as

r_{w} = r_{ε} / e

and with r_{ε}=σ^{2} and r_{w}=σ^{2}[ψι^{*}-1], we find the resource bias

ψι^{**} = (4

and interference [ ι^{**} = (4_{e} ε ] and mass [ d w / d τ = r_{w} w ]. The attractor of the evolutionary steady state is illustrated for different evolutionary trajectories in the left plot of Fig. 1, and the right plot shows the steady state on the selection profile relative to the evolutionary equilibrium with a stable body mass.

The increase applies to unconstrained selection in stable environments, and as such it is expected for the largest and dominant species in a competitive guild. Other species may increase at a slower rate, or have a stable or even a declining mass if the access to resources is declining because of inter-specific interactions or environmental variation.

### References

- Robertson, A. 1968. The spectrum of genetic variation. pp. 5--16, In: R. C. Lewontin (ed.) Population Biology and Evolution. Syracuse University Press, New York.
- Taylor, P.D. 1996. The selection differential in quantitative genetics and ESS models. Evolution 50:2106--2110.
- 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. 2003. Major life-history transitions by deterministic directional natural selection. Journal of Theoretical Biology 225:389--406.
- 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.