Universality of evolutionary trajectories under arbitrary competition dynamics

The assumption of constant population size is central in population genetics. It led to a large body of results, that are robust to modeling choices and that have proven successful to understand evolutionary dynamics. In reality, allele frequencies and population size are both determined by the interaction between a population and the environment. Relaxing the constant-population assumption have two big drawbacks. It increases the technical difficulty of the analysis, and it requires specifying a mechanism for the saturation of the population size, possibly making the results contingent on model details. Here, we develop a framework that encompasses a great variety of systems with an arbitrary mechanism for population growth limitation. By using techniques based on scale separation for stochastic processes, we are able to calculate analytically properties of evolutionary trajectories, such as the fixation probability. Remarkably, these properties assume a universal form with respect to our framework, which depends on only three parameters related to the inter-generation timescale, the invasion fitness, and the carrying capacity of the strains. In other words, different systems, such as Lotka-Volterra or a chemostat model (contained in our framework), share the same evolutionary outcomes after a proper re-mapping of their parameters. An important and surprising consequence of our results is that the direction of selection can be inverted, with a population evolving to reach lower values of invasion fitness.