## Reconciling data and theory

Bird & Mammal Populations (BMP) reconcile data and natural selection theory to estimate the inter-specific variation in life history and population ecological traits.

The observed covariance of the traits (X) is often described as body mass (w) allometries X ∝ w^{x} with linear relations (ln* X ∝ x *ln* w) on double logarithmic scale. The allometric exponents (x) are explained (Witting, 1995, 2017) by the ecological foraging that generates net energy for the population dynamic feed-back selection of mass, with Fig. 1 showing the parallel alignment of data and theory. *

*This selection balances the cost (f*_{s}) of the local exploitation of individuals in home ranges against the cost (f_{ι}) of interference competition in overlapping home ranges. This selects the home range (H ∝ w^{h}) that minimises the joint cost of foraging (f_{s} f_{ι}), selecting the following invariant arguments of the two cost functions:

* f*_{s}(β^{2}H^{1/d}) ∝ f_{s}(w^{2b+h/d}) ∝ f_{s}(w^{0}) and f_{ι}(H^{(d-1)/d}/wβ^{2}) ∝ f_{ι}(w^{(d-1)h/d-1-2b}) ∝ f_{ι}(w^{0})

*with β ∝ w*^{b} being mass-specific metabolism, and d the dimensionality of the ecological stacking of home ranges (see Witting 2023). These arguments produce the lines that align with BMP data and estimates in the plots to the right in Fig. 1.

*The left plot shows the corresponding theoretical exponents, with the h=-2db and h=d(1+2b)/(d-1) lines obtained from the equation above satisfying the invariance of the arguments. The intercepts between the invariance lines are the selected exponents where the allometric constraints on local resource exploitation are reconciled with those on interference competition.*

*Fig. 1 Right: The selected invariant arguments of the cost of local exploitation (top) and cost of interference competition (bottom) expressed as allometric relations on double logarithmic scale. Red lines are the theoretical prediction, black lines the outline of BMP data, and blue, green, and yellow dots BMP estimates (placentals only). Left: The associated theoretical deduction of the allometric exponents for home range (h=1) and mass-specific metabolism (b=-1/2, -1/4, -1/6 for d=1, 2, 3) given by the intercepts between the invariance lines of the exponents for local exploitation [h=-2db; declining lines] and interference competition [h=d(1+2b)/(d-1); increasing lines].*

*Axes of natural selection*

*The reconciliation of theory with BMP data and estimates in Fig. 1 is one component of population dynamic feed-back selection. More generally, the selection explains the life history and ecological variation among species from the selected and naturally occurring variation in a few basic traits; referred to as the axes of natural selection.*

*For traits like net energy, annual reproduction, body mass and population abundance, the four plots in Fig. 2 show how the BMP medians of major taxa are explained by variation in the axes of selection. Being define as the product ε=αβ between resource handling (α) and mass-specific metabolism (β, also metabolic pace), median net energy (ε) in the top left plot aligns almost as a product of the medians on α and β.*

*Fig. 2 The median net energy (ε), annual reproduction (m), body mass (w), and population abundance (n) of bird and mammal taxa as explained by resource handling (α), mass-specific metabolism (β), adult mortality (q*_{ad}), juvenile survival (l_{m}), level of intra-specific interference (ι), home range (h), and encounter rate (v).

*The red dots in the lower left plot illustrate that body mass is selected in an approximate proportion to net energy. This follows an allometric relation w ∝ ε*^{2d/(2d-1)} behind the deduction in Fig. 1, where mass is selected by the interactive competition that is generated by the population dynamic growth that follows from the reproductive potential of the average net energy in a population.

*The w ∝ ε*^{2d/(2d-1)} dependence of mass on net energy, however, assumes no additional variation in adult and juvenile survival. With increased mortality, more energy is needed to produce the abundance and interactive competition that selects mass, implying a smaller mass for the same amount of energy. The green and blue dots show how body mass is adjusted to an almost perfect proportional dependence, when net energy is corrected for adult and juvenile survival. The top right plot illustrates a sub-component of this adjustment, where annual reproduction is selected in proportion to overall mortality so that it generates the abundance and interactive competition that is necessary for the selection of mass.

*The main attractor of population dynamic feedback selection is this necessary level of interactive competition, which is selected exactly so high that the interactive selection of mass is balanced against the quality-quantity trade-off that selects for a decline in mass. And with interactive competition occurring mainly in the overlap of home ranges (which is defined as the product between home range area and population density) we find a population density that is selected in proportion to the level of interference, and in inverse proportion to the home range area and the encounter rate between individuals in the overlap of home ranges (lower right plot).*