Body mass evolution is bend by the dilation and contraction of natural selection time
From the selected exponent (Witting 2016) between biotic time and mass
τ• = ∂ ln τ / ∂ ln w = [ 1/(1+rββ/rα) – 2(d-1)/(1+rα/rββ) ] / 2d
we can transfer the per generation exponential increase in mass [ d w / d τ = rw w ] into an allometry for the predicted rate of change in mass in physical time (t)
d w / d t = rw w / τ = rw wx
with the following exponent x = 1 – τ• = [ 1/(1+rββ/rα) + 2/(1+rα/rββ) ] [ 2d-1 ] / 2d
The log-linear trajectory of the exponential increase is thus found to be bend in physical time whenever the time-scale of natural selection is evolving with the evolutionary increase in mass, i.e. whenever τ• ≠ 0 and x ≠ 1.
This bend is illustrated in Fig 1 together with fossil data for four scale dependent bending exponents. When the rββ/rα-ratio is zero (blue curve), net energy and mass is increasing exclusively from an exponential increase in resource handling, and the generation time is evolving only by mass-rescaling with an allometric exponent of 1/4 (2D) or 1/6 (3D). This increase in the generation time with mass is dilating the time-scale of natural selection, and slowing down the increase in mass in physical time, with a downward bend trajectory that is identified by a dw/dt exponent (x) of 3/4 in 2D and 5/6 in 3D.
At the other limit where the rββ/rα-ratio is approaching infinity (red curve), net energy and mass is increasing exclusively because of metabolic acceleration. The time-scale of natural selection is then contracting with a τ• exponent of -1/2 in 2D and -2/3 in 3D. This is accelerating the increase in physical time, with an upward bend trajectory that is identified by a dw/dt exponent (x) of 3/2 in 2D and 5/3 in 3D.
In between the two limits, there is a slightly upward bend trajectory (green) when the rββ/rα-ratio is unity and resource handling and metabolic pace are equally important for the exponential increase in mass. And a log-linear trajectory (yellow) when there are no evolutionary changes in time, which is the case when the rββ/rα-ratio is 1/(2d-2).
These four bending cases represent expectations for different modes of selection, and this is discussed and compared with fossil data in the next four sub-sections.
- Witting, L. 2016. The natural selection of metabolism bends body mass evolution in time. Preprint at bioRxiv http://dx.doi.org/10.1101/088997.