Mixed updating in structured populations

Evolutionary graph theory (EGT) studies the effect of population structure on evolutionary dynamics. The vertices of the graph represent the N individuals. The edges denote interactions for competitive replacement. Two standard update rules are death-Birth (dB) and Birth-death (Bd). Under dB, an individual is chosen uniformly at random to die, and its neighbors—the individuals on adjacent vertices of the graph— compete to fill the vacancy proportional to their fitness. Under Bd, an individual is chosen for reproduction proportional to fitness, and its offspring replaces a randomly chosen neighbor on an adjacent vertex. Here we study mixed updating between those two scenarios. In each time step, with probability δ the update is dB and with the remaining probability it is Bd. We study fixation probabilities and times as functions of δ under neutral evolution and constant selection. Despite the fact that fixation probabilities and times can be increasing, decreasing, or non-monotonic in δ, we prove that nearly all unweighted undirected graphs have short fixation times and provide an efficient algorithm to estimate their fixation probabilities. Finally, we prove exact formulas for fixation probabilities on cycles, stars, and more complex structures and classify their sensitivities to δ.Author summary: Evolutionary graph theory models how a population’s interaction structure shapes the fate of a newly arising mutant. Outcomes depend not only on that structure but also on the timing of the birth and death events in each update. Two standard choices, death-Birth (dB) and Birth-death (Bd), can give strikingly different predictions on the very same structure. Most previous work studies one rule alone. We mix them, taking each step to be dB with probability δ and Bd otherwise. We ask how the fixation probability (the chance a mutant takes over) and the fixation time depend on δ. Both can behave in surprisingly complicated ways, rising, falling, or reversing direction as δ changes, yet striking regularities emerge. For a neutral mutant, the fixation probability averaged over all starting locations is exactly 1/N on every structure and for every δ, and a structure is insensitive to δ when its individuals all have roughly the same number of neighbors. When dB and Bd are equally likely (δ=1/2), fixation is fast, for neutral mutants and under constant selection alike. We also derive exact formulas for cycles and stars, and give an efficient algorithm to estimate fixation probabilities on almost any structure.