Malthusian Relativityι**=7/3ψ
Life history evolution - traits & transitions

Reproductive rate

The competitive interaction fix-points balance reproductive rates against mortality

The population dynamic feed-back of interactive competition is selecting a reproductive rate that is exactly so high that it produces an equilibrium abundance that generates the level of interference of the competitive interaction fix-point.

To deduce this we have e.g. the competitive interaction fix-point for a body mass in evolutionary equilibrium

ι** = 1 / ψ

This level of interference is also a function of the population density (N), approximated here as ι = γι ln N where γι is the density dependence parameter. The fix-point is thus defining the evolution of an equilibrium abundance

N** = exp(ι**ι)

This equilibrium is defined also by density regulation, described for example by the discrete growth rate (λ) that is regulated from its maximum value of λm = pRm down to one λ = 1 = pRmN* -γ at the population dynamic equilibrium (*) [Rm is maximal lifetime reproduction, p the probability to survive to reproduce, and γ the parameter of density regulation]. For the chosen formulation, density regulation defines N* = (pRm)1/γ, and combined with the abundance of the evolutionary equilibrium we find that the competitive interaction fix-point constrains lifetime reproduction to

Rm** = exp(ι**γ/γι)/p

Lifetime reproduction is thus expected to be inversely related to the probability to survive to reproductive. And as this probability was predicted to be body mass invariant in the sections on inter-specific allometries, we expect a body mass invariant lifetime reproduction.