Unconstrained feed-back selection for an exponential increase in net energy and mass
The feed-back selection of the interactive competition in a population of a multicellular animal will stabilise at the competitive interaction fix-point for a body mass in evolutionary equilibrium when the net energy of the average individual is stable. But unconstrained natural selection is causing an exponential increase in net energy, and this is selecting for an evolutionary steady state where the body mass is increasing exponentially on the per generation time-scale of natural selection (Witting, 1997, 2003, 2017b).
To formulate this, from the discrete growth rate [ λ ∝ p ε tr/w ] we have a selection gradient of unity for net energy on logarithmic scale [ ∂ r / ∂ ln ε = 1 ]. This selection is generating an exponential increase
rε = d ln ε / d τ = σ2 ∂ r / ∂ ln ε = σ2
on the pre-generation time-scale (τ) of natural selection, whenever the additive heritable variance (σ2) is stable (Robertson, 1968; Taylor, 1996). The increased net energy is allocated into reproduction and an increase in the population dynamic equilibrium and its associated interference (ι*). And this interference is selecting for an increase in mass
rw = d ln w / d τ = σ2 ∂ r / ∂ ln w = σ2 [ ψ ι* – 1 ]
This rate of increase in mass is a consequence of the increase in net energy, and it may thus be rewritten as
d ln w / d τ = [ ∂ ln w / ∂ ln ε ] [ d ln ε / d τ ]
with an invariant selection relation
∂ ln w / ∂ ln ε = 1 / e
that defines mass
w = ∫ [∂ ln w / ∂ ln ε] d ln ε = (ε/ε0)1/e
by the inverse of the mass allometry for net energy [ ε = ε0 we ], where e = (2d-1)/2d from the allometric deduction (Witting, 1995, 2017a).
We may thus write the rate of change in mass as
rw = rε / e
and with rε=σ2 and rw=σ2[ψι*-1], we find the resource bias
ψι** = (4
and interference [ ι** = (4
The increase applies to unconstrained selection in stable environments, and as such it is expected for the largest and dominant species in a competitive guild. Other species may increase at a slower rate, or have a stable or even a declining mass if the access to resources is declining because of inter-specific interactions or environmental variation.
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