Is it environment or life that drives macroevolution? A recent analysis of the massive paleobiology database argues that the answer depends on the timescale. At short timescales, less than 40 million years, it is the environment, at longer timescales, life can effectively adapt. Both the environment and life are scaling—they fluctuate over the full range of scales from millions to hundreds of millions of years (the megaclimate regime). In this paper, we present a simple model of this scaling “crossover” phenomenon. The model has some unusual features: it is fully random and is based on fractional (rather than classical integer-ordered) differential equations.
The model is driven by temperature (a proxy for the environment) and the turnover rate (a proxy for life); it has two exponents, a cross-over time and two correlations, yet it is able to reproduce not only the statistics of the temperature, diversity, extinction, origination, and turnover rates, but it also effectively reproduces the pairwise correlations between them, and this over the whole range of timescales. If forced deterministically, it gives the response to bolide impact or other sharp forcing events.
Scaling fluctuation analyses of marine animal diversity and extinction and origination rates based on the Paleobiology Database occurrence data have opened new perspectives on macroevolution, supporting the hypothesis that the environment (climate proxies) and life (extinction and origination rates) are scaling over the “megaclimate” biogeological regime (from ≈1 Myr to at least 400 Myr). In the emerging picture, biodiversity is a scaling “crossover” phenomenon being dominated by the environment at short timescales and by life at long timescales with a crossover at ≈40 Myr. These findings provide the empirical basis for constructing the Fractional MacroEvolution Model (FMEM), a simple stochastic model combining destabilizing and stabilizing tendencies in macroevolutionary dynamics, driven by two scaling processes: temperature and turnover rates.
Macroevolution models are typically deterministic (albeit sometimes perturbed by random noises) and are based on integer-ordered differential equations. In contrast, the FMEM is stochastic and based on fractional-ordered equations. Stochastic models are natural for systems with large numbers of degrees of freedom, and fractional equations naturally give rise to scaling processes.
The basic FMEM drivers are fractional Brownian motions (temperature, T) and fractional Gaussian noises (turnover rates, E+) and the responses (solutions), are fractionally integrated fractional relaxation noises (diversity [D], extinction [E], origination [O], and E– = O– E). We discuss the impulse response (itself representing the model response to a bolide impact) and derive the model's full statistical properties. By numerically solving the model, we verified the mathematical analysis and compared both uniformly and irregularly sampled model outputs with paleobiology series.
