^{**}=7/3ψ

# The interactive selection of mass

Multicellular animals are selected by the interactive competition of a fully developed population dynamic feed-back

When the maximum resource bias evolves to unity [ ψι^{*} = 1 ], the mass of the self-replicating cell is selected to the limit where the metabolic pathways are fully developed and the dependence of mass specific metabolism on mass is vanishing [ β_{β}^{•} → 0 ]. Any further increase in mass is then in the hands of selection by density dependent interactive competition.

The selection gradient on mass is then

∂ r / ∂ ln w_{i|wi=w} = ψ ι^{*} – 1

with the level of interference at the population dynamic equilibrium

ι^{*} = f [ n^{*}( r ∝ – ln w ) ]

generating a fully developed population dynamic feed-back, where the interactive selection on mass is dependent upon the average mass in the population. The feed-back selection occurs because the selection of extra mass is dependent on a resource bias above unity [ ψ ι^{*} > 1 ], and this occurs only when the level of interference competition is larger than ι^{*} > 1/ψ; a level that is dependent on a sufficiently large equilibrium abundance [ ι^{*} = f ( n^{*} ) ], that is dependent on a sufficiently large population dynamic growth rate [ n^{*}(r) ], that is dependent on a sufficiently small body mass [ r ∝ – ln w ].

As illustrated in Fig. 1., for organisms with stable net energy, the result is a body mass that is selected as an energetic buffer that is adjusting the resource bias in the population [ (w_{i}/w)^{ψι} ] to an equilibrium attractor with an exponent of unity

ψ ι^{**} = 1

The associated body masses are selected beyond the minimum that is required for the metabolism of the cell, and there is therefore no longer selection for a single-celled individual. Multicellularity may thus evolve from the increased functionality that can be obtained from the division of a single large cell into a multitude of smaller cooperating cells.

_{ψι*} = ψ ι^{*} log (w_{i}/w) ]. The red line is fitness from the quality-quantity trade-off [ r_{w} = - log (w_{i}/w) ]. And green lines are overall fitness [ r=r_{ψι*}+r_{w} ], with w_{i} being the mass of the ith variant in the population, and w the average mass. The selection attractor has a resource bias of unity ψ ι^{**} = 1 and body mass invariant fitness.

It is possible to visualize the mass attractor of population dynamic feed-back selection by including the population dynamic feed-back of the density dependent interference competition into the equations. This is done in Fig. 2, that compares the fitness profiles (landscapes), selection gradients and selection integrals across three populations that differ in the average net energy per individual.

_{i}; i subscript: within population variants; _{i}/∂w_{i}; _{i}/∂w_{i})dw; no subscript: population average;

### References

- Witting, L. 2016. The natural selection of metabolism and mass selects lifeforms from viruses to multicellular animals. Preprint at bioRxiv http://dx.doi.org/10.1101/087650.