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We consider a generalized version of the birth-death (BD) and death-birth(DB) processes introduced by Kaveh et al (2015), in which two constantfitnesses, one for birth and the other for death, describe the selectionmechanism of the population. Rather than constant fitnesses, in this paper weconsider more general frequency-dependent fitness functions (allowing anysmooth functions) under the weak-selection regime. For a large populationstructured as a star graph, we provide approximations for the fixationprobability which are solutions of certain ODEs (or systems of ODEs). For theDB case, we prove that our approximation has an error of order 1/N, where N isthe size of the population. These approximations are obtained in the samespirit of Chalub and Souza (2016) albeit with quite different techniques. Thegeneral BD and DB processes contain, as special cases, the BD-* and DB-* (where* can be either B or D) processes described in Hadjichrysanthou et al (2011) --this class includes many examples of update rules used in the literature.

Continuous approximations for the fixation probability of the Moran processes on star graphs