^{**}=7/3ψ

# The selection of optimal density regulation

The natural selection of density regulation selects for allometric correlations

The cost of interactive competition (f_{ι}) is selecting for small home ranges with no overlap and no interactions between individuals (Fig. 1; left plot; red curve). This is counterbalanced by foraging self-inhibition (f_{s}), where local resources are overexploited if home ranges are too small (blue curve). The joint regulation is therefore selecting for an intermediate home range, where foraging is optimal and regulation minimal, i.e., where the f_{ι}f_{s} product is at a maximum (green curve).

The optimal home range

H^{**} ∝ ( V / β )^{d}

is found (Witting, 1995, 2017a) to depend on the foraging speed (V), the mass specific metabolism (β), and the spatial dimensionality of the foraging behaviour (d). And with mass specific metabolism being affected by mass-rescaling, we find that the selection of the density regulation optimum is part of the overall mass-rescaling, where the life history is selected in response to the evolutionary changes in mass.

The optimal home range is also found to be independent of the population density and the level of interactive competition in the population. So where the selected home range is affected by the mass that is evolving by the population dynamic feed-back of interactive competition, it is unaffected by the level of interference competition that is selected by the same feed-back selection. The result is the evolution of a joint selection attractor, where feed-back selection is adjusting the population density to the level where the interference competition between individuals in optimal home ranges is matching the level of the competitive interaction fix-point [ ι^{**} = 1/ψ ] that defines the selection attractor on mass (Fig. 1; right plot).

This level of interference is invariant of the selected mass, and so is the local resource exploitation at the selection attractor. And with the interference level in high-energy species reflecting the overall exploitation of the resource, we expect density regulation as a whole to be body mass invariant

f(N) = f_{e}(Nwβ) f_{ι}(NVH^{(d-1)/d}) f_{s}(βH^{1/d}/V) ∝ w^{0}

The joint attractor is therefore selecting for life history correlations that satisfy

Nwβ ∝ NVH^{(d-1)/d} ∝ βH^{1/d}/V ∝ w^{0}

### 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. 2017a. 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.
- Witting, L. 2017b. The natural selection of metabolism and mass selects lifeforms from viruses to multicellular animals. Ecology and Evolution 7:9098--9118, http://dx.doi.org/10.1002/ece3.3432.