The Fractional MacroEvolution Model: a simple quantitative scaling macroevolution model

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.