The adaptation front equation explains diversification hotspots and living-fossilization

Taxonomic turnovers are common in the evolutionary histories of biological communities. Such turnovers are often associated with the emergence and diversification of groups that have achieved fundamental innovations beneficial in various ecological niches. In the present study, we show that such innovation-driven turnovers could be analyzed using an equation that describes the dynamics of zero-fitness isoclines in a two-dimensional trait space comprising a “fundamental trait” (describing fundamental innovation) and a “niche trait” (describing niche position) or with its higher-dimensional extensions. Our equation allows analytical prediction of evolutionary source–sink dynamics along the niche axis for an arbitrary unimodal (or multimodal with weak separation) carrying capacity distribution. The prediction was confirmed by numerical simulation under different assumptions for resource competition, reproduction, and mutation. In the simulated evolution, biodiversity sources are the central niches having higher carrying capacities than the outer niches, allowing species there the faster evolutionary advancement in fundamental traits and their repeated diversification into outer niches, which outcompete the indigenous less advanced species. The outcompeted species go extinct or evolve directionally toward the far outer niches of the far slower advancement because of the far lower carrying capacities. In consequence of this globally acting process over niches, species occupying peripheral (i.e., the outermost) niches can have significantly primitive fundamental traits and deep divergence times from their closest relatives, and thus, they correspond to living fossils. The extension of this analysis for multiple geographic regions showed that living fossils are also expected in geographically peripheral regions for the focal species group. Significance Statement We developed a new equation for analyzing the long-term coevolution of many species through their directional evolution, evolutionary branching, and extinction in an arbitrary trait space comprising traits describing niche positions and traits describing fundamental innovations. This equation describes the many coevolving species as a fluid, as in the study of galaxy dynamics in astronomical physics. This equation may be used to explain how innovation-driven taxonomic turnovers generate diversification hotspots and coldspots among niches or geographical regions, accompanied by continuous production of “living fossil” species in peripheries, in a logic equivalent to that deduced by Darwin and Darlington from empirical observations. ### Competing Interest Statement The authors have declared no competing interest.