Malthusian Relativityι**=7/3ψ
The unfolding of population dynamic feed-back selection

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 [ ββ → 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 wi|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 [ (wi/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.

Fig. 1 The evolution of the intra-specific resource bias by selection by density dependent interactive competition. Blue lines are fitness from the density dependent resource bias [ rψι* = ψ ι* log (wi/w) ]. The red line is fitness from the quality-quantity trade-off [ rw = - log (wi/w) ]. And green lines are overall fitness [ r=rψι*+rw ], with wi 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.

Fig. 2 Fitness landscapes/profiles (ri; i subscript: within population variants; left), selection gradients (∂ri/∂wi; middle) and integrals (∫ (∂ri/∂wi)dw; no subscript: population average; right) for body mass (w). The body mass selection by density dependent interactive competition has no visualisation between fitness landscapes and selection integrals. Populations with stable energetic states have equilibrium attractors with stable masses (open circles). Unconstrained selection has steady state attractors (solid circles) with exponential increase in energetic state and mass, with the arrow showing selection integral evolution. The illustrated steady state is for two dimensional interactions with similar heritable variation in energetic state and mass. Colours represent populations with different attractors, and the multiple fitness profiles per population have different average variants with zero absolute fitness at population dynamic equilibrium, with clear coloured curves having average variants at evolutionary equilibrium or steady state. From Witting (2016).

References

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