Stochastic Modeling of Adaptive Trait Evolution in Phylogenetics: A Polynomial Regression and Approximate Bayesian Computation Approach

In nature, closely related species often exhibit diverse characteristics, challenging simplistic line interpretations of trait evolution. For these species, the evolutionary dynamics of one trait may differ markedly from another, with some traits evolving at a slower pace and others rapidly diversifying. In light of this complexity and concerning the phenomenon of trait relationships that escape line measurement, we introduce a novel general adaptive optimal regression model, grounded on polynomial relationships. This approach seeks to capture intricate patterns in trait evolution by considering them as continuous stochastic variables along a phylogenetic tree. Using polynomial functions, the model offers a holistic and comprehensive description of the traits of the studied species, accounting for both decreasing and increasing trends over evolutionary time. We propose two sets of optimal adaptive evolutionary polynomial regression models of k t h order, named the Ornstein–Uhlenbeck Brownian Motion Polynomial ( OUBMP k ) model and Ornstein–Uhlenbeck Ornstein–Uhlenbeck Polynomial ( OUOUP k ) model, respectively. Assume that the main trait value y t is a random variable of the Ornstein–Uhlenbeck (OU) process and that its optimal adaptive value θ t y has a polynomial relationship with other traits x t for statistical modeling, where x t can be a random variable of Brownian motion (BM) or OU process. As analytical representations for the likelihood of the models are not feasible, we implement an approximate Bayesian computation (ABC) technique to assess the performance through simulation. We also plan to apply models to the empirical study using the two datasets: the longevity vs. fecundity in the Mediterranean nekton group, and the trophic niche breadth vs. body mass in carnivores in a European forest region.