American Society of Naturalists

A membership society whose goal is to advance and to diffuse knowledge of organic evolution and other broad biological principles so as to enhance the conceptual unification of the biological sciences.

“Beyond Brownian motion and the Ornstein-Uhlenbeck process: Stochastic diffusion models for the evolution of quantitative characters”

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Simone P. Blomberg, Suren I. Rathnayake, and Cheyenne M. Moreau (Feb 2020)

New evolutionary models for continuous traits, with an R package! Diffusion models get a closer evolutionary examination

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New stochastic diffusion models for the evolution of quantitative traits

Consider the way population geneticists model the evolution of gene frequencies over time, or the way quantitative genetic models are used for the understanding of trait evolution in response to natural selection. The evolutionary history of organisms does not enter into the mathematical equations describing the evolution of genes or traits. The models are “ahistorical” and hence are termed “microevolutionary” models: models that describe the evolution of traits (or genes) over short time scales. In contrast, a key principle of “macroevolutionary” models is that history matters. We cannot truly comprehend the evolution of traits without an understanding of the context of trait evolution gained by comparing multiple species over “deep time” using the so-called “comparative method.”

Biologists who are interested in the evolution of quantitative traits (such as body mass, limb length, etc.) often wish to understand the tempo and mode of trait evolution over “deep time.” They ask questions such as, “Do some organisms evolve faster than others?”, “Has there been a shift in the mean value of a trait at some point in an organism’s evolutionary history?”, “How strong is natural selection in determining trait values?”, and “How are multiple traits related to each other and to the environment?” One way that biologists attempt to answer these questions is to fit mathematical models to trait data which describe the way evolution unfolds over time and along an evolutionary tree. Such models are called “diffusions,” as they model the path of a trait through evolutionary time as if it were a particle diffusing through a medium, being affected by both deterministic and stochastic forces.

Most current diffusion models of evolution are designed for traits that follow the Normal distribution: the bell-shaped curve. In a new paper in The American Naturalist, a team from the University of Queensland in Australia, led by Dr. Simone Blomberg, describe two new models of trait evolution that do not require the traits to be described by a bell curve. These new models allow the study of quantitative traits that are not easily examined using current methods, such as lifespan or sex ratio. Using methods first developed for use in physics and quantitative finance, the team also demonstrate how to derive new, different models of evolution, how to understand their properties, and how to fit them to trait data on an evolutionary tree. They provide software tools to do this, using modern computer-intensive statistical techniques.

The UQ team hope their new paper in The American Naturalist will inspire biologists to become more adventurous in modelling trait data over “deep time” and throw off the shackles of the Normal distribution!


Gaussian processes such as Brownian motion and the Ornstein-Uhlenbeck process have been popular models for the evolution of quantitative traits and are widely used in phylogenetic comparative methods. However, they have drawbacks which limit their utility. Here we describe new, non-Gaussian stochastic differential equation (diffusion) models of quantitative trait evolution. We present general methods for deriving new diffusion models, and develop new software for fitting non-Gaussian evolutionary models to trait data. The theory of stochastic processes provides a mathematical framework for understanding the properties of current and future phylogenetic comparative methods. Attention to the mathematical details of models of trait evolution and diversification may help avoid some pitfalls when using stochastic processes to model macroevolution.