# Cooperative families - the unconstrained group

Cooperatively reproducing families are selected by unconstrained selection for an exponential increase in mass

Unconstrained selection with exponential mass: The multicellular animal with pair-wise sexual reproduction can evolve into a cooperatively reproducing family by unconstrained feed-back selection for an exponentially increasing mass (Witting, 1997, 2002).

The sexually reproducing pair is predicted for a high-energy organism with a stable net energy. Yet, unconstrained selection generates an exponential increase in net energy, and this generates an evolutionary steady state where the exponential increase in energy is selected into an exponential increase in mass (Witting, 1997, 2003, 2016). The invariant interference of the selection attractor [ ι^{**} = (4d - 1) / (2d - 1) ψ ] is then, as illustrated in Fig. 1 right, given by a resource bias exponent that is selected to

ψι^{**} = (4d - 1) / (2d - 1)

with ψι^{**} being 3 for 1D, 7/3 ≈ 2.3 for 2D, and 11/5 ≈ 2.2 for 3D interactions.

Fig. 1 Left: The exponential increase in the mass of fossil horses over 57 million years of evolution [ From Witting (1997) with data from MacFadden (1986) ]. Right: An illustration of the unconstrained feed-back selection that predicts an exponential increase in body mass (2D). The selection attractor is a steady state where an exponential increase in net energy is selected into an exponential increase in mass (Witting, 1997, 2003, 2016). The invariant interference of the attractor [ ι^{**} = (4d - 1) / (2d - 1) ψ ] is given by a resource bias exponent that is selected to ψι^{**} = (4d - 1) / (2d - 1). The extra dots on each circle are the offspring workers of the reproducing unit.

Allometries: The body mass allometries for animals with exponentially increasing masses and stable masses are predicted to be the same.

Life history: The life history is also predicted to be about the same, except for a small increase in the interactive quality of the reproducing unit. With a level of interference that is about twice as large as the level in species with a non-evolving mass, the reproducing unit is selected to contain a single or a few extra interacting individuals. And with the interactive quality of the male being transferred by sexual reproduction to potential offspring workers, there is a diminishing return in the interactive quality that can be gained by increasing the number of males that participate in sexual reproduction. The result is selection for co-operative families, where the extra interactors are sexually produced offspring workers that evolve at the cost of unknown forms of sexual reproduction with several males per female (Witting, 1997, 2002).

### References

- MacFadden, B.J. 1986. Fossil horses from ``Eohippus" (Hyracotherium) to Equus: scaling, Cope's Law, and the evolution of body size. Paleobiology 12:355--369.
- 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. 2002. From asexual to eusocial reproduction by multilevel selection by density dependent competitive interactions. Theoretical Population Biology 61:171--195.
- 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.