Exact results for fixation probability of bithermal evolutionary graphs

One of the most fundamental concepts of evolutionary dynamics is the “fixation” probability, i.e. the probability that a mutant spreads through the whole population. Most natural communities are geographically structured into habitats exchanging individuals among each other and can be modeled by an evolutionary graph (EG), where directed links weight the probability for the offspring of one individual to replace another individual in the community. EGs have recently spurred huge interest, as it has been shown that some topology can amplify or suppress the effect of beneficial mutations. Very few exact analytical results however are known for EGs. In this article we show that the use of a new technique, the fixed point of probability generating function, allows us to compute the exact fixation probability for a large subset of bithermal graphs. We also show by numerical simulations that the computed solution holds for all bithermal graphs. Moreover, the analytical solution allows us to clarify the opposing consequences of birth–death versus death–birth processes as amplifier or suppressor of beneficial mutations for the same bithermal topology.