Population abundance

The population dynamic equilibrium of selection-regulated dynamics is the competitive interaction fix-point of population dynamic feed-back selection. This implies that the densities of populations are no longer determined by famine from intense competition for depleted resources, as proposed by Malthus in 1798 and elaborated by density-regulation theory (Verhulst, 1838; Pearl and Reed, 1920; Lotka, 1925). The densities of natural populations are instead selected by the density-frequency-dependent gradient (ψ ι) in the access to resources across the individuals in populations.

At the strict evolutionary equilibrium [ ψι^** = 1 ] all the individuals in a population reproduce and survive at the same optimal rates with no population dynamic growth. The competitively superior individuals obtain more resources than the competitively inferior, but they do also invest more resources in each offspring with the end result being that both variants have about similar rates of survival and reproduction (Witting, 1997, 2021).

For species that are selected at the evolutionary steady state with an exponentially increasing body mass (Witting 1997, 2020), the resource gradient is somewhat larger [ ψι^** = (4d - 1) / (2d - 1) ], and so is the population abundance and frequency of competitive encounters. This generates an intra-specific reproductive rate that increases in approximate proportion with body mass, as observed in many species (e.g., Peters 1983; Reiss 1989; Kingsolver and Pfennig 2004).

The evolutionary fixation of the intra-specific resource gradients at ψι^** = 1 and ψι^** = (4d - 1) / (2d - 1) by the attractors of the competitive interaction fix-points are the top-down constraints of natural selection. Yet these gradients are mechanistically determined by the bottom-up processes of the density-frequency-dependent competitive interactions among the individuals. This implies that a large part of the variation in population densities can be explained by the density dependent variation in the behavioural interactions that determines the resource gradient. The intuitive prediction, that population densities increase with an increased availability of resources, follow from a redistribution where more individuals are needed to generate the necessary steepness [ ψι^** = 1 or ψι^** = (4d - 1) / (2d - 1) ] in the distribution of resources across the individuals.

Allometric abundance

A more complex prediction follows from the density dependent connection between the home range and the optimal foraging solution that satisfies the resource distribution constraints [ ψι^** = 1 or ψι^** = (4d - 1) / (2d - 1) ] at the evolutionarily determined population dynamic equilibrium. This selection optimises the home range to minimise the overall regulation [ f_ι f_s ] of the counteracting forces of interference competition (f_ι) and local resource exploitation (f_s). On top of this there is population dynamic feed-back selection for the level of interference competition that matches the level of the competitive interaction fix-points (Fig. 1).

Fig. 1 The optimal home range is selected to minimise the joint regulation by interference competition and local resource exploitation (green arrows), and the population dynamic feed-back selects the population density where the level of interference that is generated by individuals in optimal home ranges matches the level of the competitive interaction fix-point, as determined by the selection attractor on body mass (red arrows). From Witting (2017b).

This selection-based population dynamic equilibrium is not only part of the allometric solution for life history and ecological traits like mass, metabolism, abundance, and home range. It is actually the key that determines the allometric correlations, bacause it is the selected invariant density regulation

f = f_e f_ι f_s = f_e(Nwβ) f_ι(NVH^(d-1)/d) f_s(βH^1/d/V) ∝ w⁰

that determines the exponents of the body mass allometries (Witting, 1995, 2017a). Damuth's (1981, 1987) -3/4 power decline in abundance with mass follows as an essential prediction for species with predominantly two-dimensional interactive behaviour. The corresponding exponent for three-dimensional interaction is -5/6 (see e.g. the Metabolism and Mass section on mrLife.org).

The allometry for abundance tends to vary with the scale of observation (Nee et al., 1991) following expected variation in the distribution of resources across species (Witting, 2021). The exponents are usually around -0.75 across terrestrial species (Damuth, 1981, 1987; Peters, 1983; Nee et al., 1991) on scales where effects from inter-specific interactions are negligible (as assumed in the allometric deduction). The exponent has also been found to decline from -0.79 to -0.86 across species with two and three dimensional behaviour (Pawar et al.,2012); in close agreement with the predicted change from -0.75 to -0.83 (Witting, 1995, 2017).

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References

• Damuth, J. 1981. Population density and body size in mammals. Nature 290:699--700.
• Damuth, J. 1987. Interspecific allometry of population density in mammals and other animals: the independence of body mass and population energy-use. εm Biological Journal of the Linnean Society 31:193--246.
• Kingsolver, J.G., and D.W. Pfennig 2004. Individual-level selection as a cause of Cope's rule of phylogenetic size increase. Evolution 56:1608--1612.
• Lotka, A.J. 1925. Elements of physical biology. Williams and Wilkins, Baltimore.
• Malthus, T.R. 1798. An essay on the principle of population. Johnson, London.
• Nee, S., A.F. Read, J.J.D. Greenwood and P.H. Harvey 1991. The relationship between abundance and body size in British birds. Nature 351:312--313.
• Pawar, S., A.I. Dell and V.M. Savage 2012. Dimensionality of consumer search space drives trophic interaction strengths. Nature 486:485--489.
• Pearl, R., and L.J. Reed 1920. On the rate of growth of the population of the United States since 1790 and its mathematical representation. εm Proceedings of the National Academy of Sciences 6:275--288.
• Peters, R.H. 1983. The ecological implication of body size. Cambridge University Press, Cambridge.
• Reiss, M.J. 1989. The allometry of growth and reproduction. Cambridge University Press, Cambridge.
• Verhulst, P.F. 1838. Notice sur la loi que la population suit dans son accroissement. Corresp. Math. Phys. 10:113--121.
• Witting, L. 1995. The body mass allometries as evolutionarily determined by the foraging of mobile organisms. Journal of Theoretical Biology 177:129--137, https://doi.org/10.1006/jtbi.1995.0231.
• Witting, L. 1997. A general theory of evolution. By means of selection by density dependent competitive interactions. Peregrine Publisher, Århus, 330 pp, URL https://mrLife.org.
• Witting, L. 2017a. The natural selection of metabolism and mass selects allometric transitions from prokaryotes to mammals. Theoretical Population Biology 117:23--42, https://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, https://dx.doi.org/10.1002/ece3.3432.
• Witting, L. 2021. Selection-regulated population dynamic in birds and mammals. Preprint at bioRxiv https://dx.doi.org/10.1101/2021.11.27.470201.