  We consider a generalized version of the birth-death (BD) and death-birth
(DB) processes introduced by Kaveh et al (2015), in which two constant
fitnesses, one for birth and the other for death, describe the selection
mechanism of the population. Rather than constant fitnesses, in this paper we
consider more general frequency-dependent fitness functions (allowing any
smooth functions) under the weak-selection regime. For a large population
structured as a star graph, we provide approximations for the fixation
probability which are solutions of certain ODEs (or systems of ODEs). For the
DB case, we prove that our approximation has an error of order 1/N, where N is
the size of the population. These approximations are obtained in the same
spirit of Chalub and Souza (2016) albeit with quite different techniques. The
general 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.
