Reproductive rate

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 = γ_ι ln N where γ_ι is the density dependence parameter. The fix-point is thus defining the evolution of an equilibrium abundance

N^** = exp(ι^**/γ_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 λ_m = pR_m down to one λ = 1 = pR_mN^{* -γ} at the population dynamic equilibrium (*) [R_m 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^* = (pR_m)^1/γ, and combined with the abundance of the evolutionary equilibrium we find that the competitive interaction fix-point constrains lifetime reproduction to

R_m^** = exp(ι^**γ/γ_ι)/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.

Evidence

The probability to survive to reproduce is independent of body mass across 114 bird species (Fig. 1, left), and lifetime reproduction is body mass invariant across 221 species (middle).

And maximal yearly reproduction is inversely related to time-scaled survival (R_y ∝ 1/Tp) across 132 species of birds (Fig. 1, right), indicating a maximal lifetime reproduction that is inversely related to the probability to survive and reproduce [as R_yT ∝ R_m ∝ 1/p].

There is also plenty of evidence that shows that populations with increased levels of anthropogenic or natural mortality evolve increased reproductive rates (e.g., Reznick and Bryga, 1987; Reznick et al., 1996; Broman, 2000; Sinclair et al., 2002; Coltman et al., 2003; Olsen et al., 2004; Edeline and Carlson, 2007).

Fig. 1 The body mass (w) dependence for (left) the probability to survive to reproduce (p) and (middle) maximal lifetime reproduction (R_m) across 114 and 221 species of bird. For 132 bird species, the right plot shows maximal yearly reproduction as a function of time-scaled survival (Tp, where T is the age of maturity). p is estimated as yearly survival raised to the power of the age of maturity. From Witting (1997).

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References

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