Malthusian Relativityι**=7/3ψ
The bend of evolutionary time

Evolution of metabolism and mass in time

The selected increase in metabolism, resource handling, and net energy generates an evolutionary steady state with interactive selection for a body mass that increases exponentially on the per generation time-scale of natural selection. The resulting trajectory of mass over time is log-linear in biotic time, but the linearity is bend in physical time whenever the generation time is evolving with the evolution of mass.

This bend is dependent on the relative importance of metabolism for the evolution of mass, and on the spatial dimensionality of the interactive behaviour; a prediction that holds across four scale dependent bending exponents in the fossil record. These cover the complete range of possibilities, with masses that evolve from increased resource handing to masses that increase exclusively because of an evolutionary increase in mass specific metabolism.

The exponential increase in mass

Given primary selection for an exponential increase in the net energy of a multicellular animal, the population dynamic feed-back selection is attracted to the competitive interaction fix-point of the evolutionary steady state. This attractor has an invariant level of interactive competition that selects the increase in energy into a body mass that increases exponentially on the per generation time-scale of natural selection.

The exponential increase applies to unconstrained selection in stable environments. 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 changes.

The evolution of natural selection time

To predict the body mass evolution that is captured by the fossil record over millions of years, we need to transfer the predicted increase [ d w / d τ = rw w ] on the per generation time-scale of natural selection into physical time [ d w / d t = rw w / τ ]. For this we need to predict the correlated evolution [ ∂ ln τ / ∂ ln w = τ ] between generation time (τ) and mass (w).

This correlation is given by the exponent (τ) of the body mass allometry [ τ ∝ wτ ] for generation time, as it is selected in an evolutionary lineage over time. The exponent depends on the mass-rescaling of the life history, and on the rββ/rα-ratio that describes the relative importance of the selected increase in the pre-mass component of mass specific metabolism (rββ) for the exponential increase in net energy [ rε = rββ + rα ] and mass [ rw = rε / ε ] [ rα is the selected increase in resource handling, and ε the exponent between net energy and mass; ε ∝ wε ]. The selection relation of the allometric exponent for generation time is given by the allometric solution of the selection equations, and it is predicted to be

∂ ln τ / ∂ ln w = τ = [ 1/(1+rββ/rα) 2(d-1)/(1+rα/rββ) ] / 2d

where d is the spatial dimensionality of the interactive behaviour in the population.

This equation describes how the time-scale of natural selection is dilating or contracting dependent upon the underlying selection of the mass specific metabolism and resource handling that generate the net energy for the evolutionary increase in mass.

Evolution bend by time dilation and contraction

The exponentially increasing mass on the per generation time-scale of natural selection [ d w / d τ = rw w ] is then described in physical time (t) by an allometry

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 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.

Compared with the log-linear trajectory of exponential increase, this dilation (τ > 0) or contraction (τ < 0) of natural selection time is bending the mass trajectories of the fossil record from concave (x < 1), over linear (x = 1) to strongly convex (x > 1). Just as General Relativity predicts the bending of light around the sun, so does Malthusian Relativity predict a bending of evolutionary trajectories that holds across four scale dependent bending powers (see below).

Fast evolution the decelerating mass

Let us start with the most familiar case of evolutionary bending that occurs when the rββ/rα-ratio is zero. As we can expect a rather constant pre-mass selection on metabolism (stable rββ), the rββ/rα-ratio is about zero when body mass evolution is fast and the increase in resources handling is outrunning evolution in the pre-mass component of mass specific metabolism [ rα >> rββ ].

With a rββ/rα-ratio around zero, this evolution has almost no time contraction, and it is dominated by the time dilation that is caused by the mass-rescaling of the life history. This implies a generation time that increases to the 1/4 power of mass in 2D, and 1/6 power in 3D; as typically observed for inter-specific allometries. For evolution in physical time, the time dilation is generating a downward bend mass trajectory, with a dw/dt-exponent of 3/4 in 2D and 5/6 in 3D.

With this trajectory being predicted for the fastest body mass evolution, it can be expected for the evolution of maximum species size during pulses of evolutionary radiation. This is apparently not the case in whales, but the dw/dt-exponent is only slightly larger than 3/4 for the evolution of maximum mass across trunked, and all terrestrial, mammals.

Evolution across niches the symmetrical case

For balanced and unconstrained selection on resource handling and metabolic pace we expect a rββ/rα-ratio around unity. This is the expected base case for the evolution of maximum size in a taxonomic group that diversity by unconstrained and unbiased natural selection across ecological niches. Larger species will then emerge when lineages evolve into new niches where increased resource exploitation is possible.

A rββ/rα-ratio around one is also what we find for maximum body mass evolution in four out of five mammalian clades, relating both to 2D (even-toed ungulates and carnivores) and 3D (whales and primates) evolution. This evolution has a time dilation from mass-rescaling, and a moderate time contraction from pre-mass selection on metabolism, with an overall contraction that bend the body mass trajectory in physical time slightly upward with a dw/dt-exponent of 9/8 in 2D and 5/4 in 3D.

For this unconstrained selection in time we have a generation time that is declining with mass with a -1/8 exponent in 2D (-1/4 in 3D), instead of increasing with the familiar 1/4 exponent (1/6 in 3D). This change of sign in the evolutionary direction of time is induced by the selected acceleration of mass specific metabolism that is generating half of the energy for the selection increase in mass.

Evolution within a niche the accelerating mass

A rββ/rα-ratio of infinity is the conceptually more extreme and intriguing case, although it is related to the simple situation of body mass evolution within an ecological niche.

For evolution within niches we expect species that are adapted to the resources they exploit; with resource handling evolving to an evolutionary optimum. Once at the handling optimum, the net energy of the species may continue to increase by selection due to metabolic acceleration. This will generate a rββ/rα-ratio that approaches infinity, with maximum time contraction from a generation time that scales to the -1/2 power of mass in 2D, and the -2/3 power in 3D. The result is a body mass trajectory that is bend strongly upward in physical time with a dw/dt-exponent of 3/2 in 2D and 5/3 in 3D.

The evolution of horses over the past 57 million years is maybe the best example of within lineage evolution; with fossil data being spot on with an estimated dw/dt-exponent of 1.50±0.17. As the mass of the horse was selected from about 25 to 500 kg over 57 million years of evolution, we calculate a lifespan that declined from about 90 to 20 years over the same period of time. Not only is the strong decline in lifespan unexpected by our common (and apparently flawed) conception from inter-specific allometries. But maybe even more conceptually intriguing is the result that the size increase from 25 to 500 kg happened only because the horse was selected by a constant acceleration of its biochemical and behavioural processes.

Maximum animal mass over 3.5 billion years of evolution at the metabolic limit

Where the three bending cases above follow from ecological differences, a fourth is distinguished by a metabolic limit on process speed. With persistent selection for an exponential increase in mass specific metabolism, macro evolution may proceed to an upper biochemical limit on metabolic pace. Organisms at the limit can evolve an increased mass from increased resource handling and/or increased resource availability. Such an increase will cause an allometric downscaling of metabolism, with the allowance for a subsequent selection increase by metabolic acceleration. The result is a mass specific metabolism that evolves around an upper bound on process speed.

This evolution is log-linear in physical time, because the upward bend from time contraction due to accelerated pace is balanced against the downward bend from mass-rescaling. The increase in maximum animal mass on the macro evolutionary scale across 3.5 billion years of evolution on Earth was not uniform, but it was almost log-linear with a dw/dt-exponent of 1.07±0.02.

The evolution of curved allometries

We predict linear inter-specific allometries with 1/4 exponents in 2D, and 1/6 exponents in 3D, when a clade diversifies into evolutionary lineages with variable resource handling across ecological niches. But the evolutionary diversification in resource handling will come to a stop when species become well adapted to their niches, and this creates a transition where the background selection of metabolism becomes more and more important for the evolutionary diversification in mass.

This background selection on metabolism and mass is mass invariant; yet it affects mainly the smaller species as they evolve through a larger number of generations than the larger species. And this creates an upward bend in the lower size range of the allometry over time, a bend that explains the curvature that is observed in the metabolic scaling of mammals.

The evolution of allometric outliers

Species that evolve a small or large body mass at an early stage relative to the other species in a clade, will evolve over a larger or smaller number of generations that the main clade. They will thus have a tendency to evolve a higher or lower metabolic rate, and this may explain outlier species like shrews (Soricidae) with strongly increased metabolism, and bowhead whales (Balaena mysticetus) with smaller metabolism than expected from mass alone.