Theory in population biology, or biologically inspired mathematics?

TPB welcomes submissions from authors new to the journal, whom we hope to aid in preparing high-quality manuscripts that will succeed in our editorial process. Previous editorials described the topics of emphasis over the course of the journal’s history (Rosenberg, 2013) and commented on three features expected in TPB papers: motivation from a biological perspective, significant mathematical contribution, and interpretation aimed toward advancing biology (Rosenberg, 2014). Long-time readers will recognize a distinction between (1) research contributions to mathematics inspired by progress in theoretical population biology, which are not the focus of the journal, and (2) contributions to theoretical population biology itself, the core interest of TPB. To provide further guidance to authors, this editorial elaborates on the difference between biologically inspired contributions to mathematics and theoretical advances in population biology. Because theory in biology links mathematical and biological topics, theoretical population biology has often led to new advances in mathematics. A differential equation or stochastic process developed for a specific biological problem might open new research areas in themathematical theory of differential equations or stochastic processes, as has happened many times in the history of theoretical biology (Cohen, 2004). More generally, a complex biological problemmight generate a newmathematical entity whose utility in the mathematical sciences is subsequently recognized. The Ewens Sampling Formula first reported in TPB (Ewens, 1972), with its broad impact in probability and statistics (Tavaré and Ewens, 1997), is a celebrated example of this phenomenon. In mature cases, the impact of a significant result in theoretical population biology can have a clear bifurcation into its contributions to biology and its contributions to mathematics. Thus, for instance, the Yule birth process, originally developed for understanding the distribution of the number of species across genera (Yule, 1925), is the starting point for a thriving intellectual tradition in biology that is still motivated in large part by an interest in exploring macroevolutionary phenomena (e.g. Aldous, 2001; Nee, 2006; Stadler, 2013), and that makes regular appearances in TPB (e.g. Wilkinson and Tavaré, 2009; Zhu et al., 2011; Heled, 2012; Lambert and Stadler, 2013). At the same time, however, the Yule birth process has also become a basic object underlying a substantial body of work in stochastic processes, where it often appears as a textbook example stripped entirely of the macroevolutionary context of the problem in which it originated (e.g. Karlin and Taylor, 1975; Grimmett and Stirzaker, 2001). TPBmanuscripts center on contributions to biology—empirical, theoretical, or both. Thus, whereas introducing new terms to equations from theoretical population biology and proving theorems