The problem of gradual versus punctuated change within phyletic lineages can be understood in terms of the homogeneity of evolutionary dynamics. Hypotheses of punctuated change imply that the rules governing evolutionary change shift over time such that the normal dynamics of stasis are temporarily suspended, permitting a period of net evolutionary change. Such explanations are members of a larger class of models in which evolutionary dynamics are in some way heterogeneous over time. In this paper, I develop a likelihood-based statistical framework to evaluate the support for this kind of evolutionary model. This approach divides evolutionary sequences into nonoverlapping segments, each of which is fit to a separate evolutionary model. Models with heterogeneous dynamics are generally more complex—they require more parameters to specify— than uniform evolutionary models such as random walks and stasis. The Akaike Information Criterion can be used to judge whether the greater complexity of punctuational models is offset by a sufficient gain in log-likelihood for these models to be preferred.

I use this approach to analyze three case studies for which punctuational explanations have been proposed. In the first, a model of punctuated evolution best accounted for changes in pygidial morphology within a lineage of the trilobite Flexicalymene, but the uniform model of an unbiased random walk remains a plausible alternative. Body size evolution in the radiolarian Pseudocubus vema was neither purely gradual nor completely pulsed. Instead, the best-supported explanation posited a single, pulsed increase, followed later by a shift to an unbiased random walk. Finally, for the much-analyzed claim of “punctuated gradualism” in the foraminifera Globorotalia, the best-supported model implied two periods of stasis separated by a period of elevated but not inherently directional evolution. Although the conclusions supported by these analyses generally refined rather than overturned previous views, the present approach differs from those prior in that all competing interpretations were formalized into explicit statistical models, allowing their relative support to be unambiguously compared.