Carl Boettiger

Modeling stabilizing selection: expanding the Ornstein-Uhlenbeck model of adaptive evolution.

Comparative methods used to study patterns of evolutionary change in a continuous trait on a phylogeny range from Brownian motion processes to models where the trait is assumed to evolve according to an Ornstein–Uhlenbeck (OU) process. Although these models have proved useful in a variety of contexts, they still do not cover all the scenarios biologists want to examine. For models based on the OU process, model complexity is restricted in current implementations by assuming that the rate of stochastic motion and the strength of selection do not vary among selective regimes. Here, we expand the OU model of adaptive evolution to include models that variously relax the assumption of a constant rate and strength of selection. In its most general form, the methods described here can assign each selective regime a separate trait optimum, a rate of stochastic motion parameter, and a parameter for the strength of selection. We use simulations to show that our models can detect meaningful differences in the evolutionary process, especially with larger sample sizes. We also illustrate our method using an empirical example of genome size evolution within a large flowering plant clade.

IS YOUR PHYLOGENY INFORMATIVE? MEASURING THE POWER OF COMPARATIVE METHODS

Phylogenetic comparative methods may fail to produce meaningful results when either the underlying model is inappropriate or the data contain insufficient information to inform the inference. The ability to measure the statistical power of these methods has become crucial to ensure that data quantity keeps pace with growing model complexity. Through simulations, we show that commonly applied model choice methods based on information criteria can have remarkably high error rates; this can be a problem because methods to estimate the uncertainty or power are not widely known or applied. Furthermore, the power of comparative methods can depend significantly on the structure of the data. We describe a Monte Carlo‐based method which addresses both of these challenges, and show how this approach both quantifies and substantially reduces errors relative to information criteria. The method also produces meaningful confidence intervals for model parameters. We illustrate how the power to distinguish different models, such as varying levels of selection, varies both with number of taxa and structure of the phylogeny. We provide an open‐source implementation in the pmc (“Phylogenetic Monte Carlo”) package for the R programming language. We hope such power analysis becomes a routine part of model comparison in comparative methods.

Quantifying limits to detection of early warning for critical transitions.

Catastrophic regime shifts in complex natural systems may be averted through advanced detection. Recent work has provided a proof-of-principle that many systems approaching a catastrophic transition may be identified through the lens of early warning indicators such as rising variance or increased return times. Despite widespread appreciation of the difficulties and uncertainty involved in such forecasts, proposed methods hardly ever characterize their expected error rates. Without the benefits of replicates, controls or hindsight, applications of these approaches must quantify how reliable different indicators are in avoiding false alarms, and how sensitive they are to missing subtle warning signs. We propose a model-based approach to quantify this trade-off between reliability and sensitivity and allow comparisons between different indicators. We show these error rates can be quite severe for common indicators even under favourable assumptions, and also illustrate how a model-based indicator can improve this performance. We demonstrate how the performance of an early warning indicator varies in different datasets, and suggest that uncertainty quantification become a more central part of early warning predictions.

Tipping points: From patterns to predictions

Truly generic signals warning of tipping points are unlikely to exist, warn Carl Boettiger and Alan Hastings, so researchers should study transitions specific to real systems.

Early warning signals and the prosecutor's fallacy

Early warning signals have been proposed to forecast the possibility of a critical transition, such as the eutrophication of a lake, the collapse of a coral reef or the end of a glacial period. Because such transitions often unfold on temporal and spatial scales that can be difficult to approach by experimental manipulation, research has often relied on historical observations as a source of natural experiments. Here, we examine a critical difference between selecting systems for study based on the fact that we have observed a critical transition and those systems for which we wish to forecast the approach of a transition. This difference arises by conditionally selecting systems known to experience a transition of some sort and failing to account for the bias this introduces—a statistical error often known as the prosecutors fallacy. By analysing simulated systems that have experienced transitions purely by chance, we reveal an elevated rate of false-positives in common warning signal statistics. We further demonstrate a model-based approach that is less subject to this bias than those more commonly used summary statistics. We note that experimental studies with replicates avoid this pitfall entirely.

Fluctuation domains in adaptive evolution

We derive an expression for the variation between parallel trajectories in phenotypic evolution, extending the well known result that predicts the mean evolutionary path in adaptive dynamics or quantitative genetics. We show how this expression gives rise to the notion of fluctuation domains-parts of the fitness landscape where the rate of evolution is very predictable (due to fluctuation dissipation) and parts where it is highly variable (due to fluctuation enhancement). These fluctuation domains are determined by the curvature of the fitness landscape. Regions of the fitness landscape with positive curvature, such as adaptive valleys or branching points, experience enhancement. Regions with negative curvature, such as adaptive peaks, experience dissipation. We explore these dynamics in the ecological scenarios of implicit and explicit competition for a limiting resource.