  Evolutionary models on graphs, as an extension of the Moran process, have two
major implementations: birth-death (BD) models (or the invasion process) and
death- birth (DB) models (or voter models). The isothermal theorem states that
the fixation probability of mutants in a large group of graph structures (known
as isothermal graphs, which include regular graphs) coincides with that for the
mixed population. This result has been proven by Lieberman et al (Nature 433:
312-316, 2005) in the case of BD processes, where mutants differ from the wild
types by their birth rate (and not by their death rate). In this paper we
discuss to what extent the isothermal theorem can be formulated for DB
processes, proving that it only holds for mutants that differ from the wild
type by their death rate (and not by their birth rate). For more general BD and
DB processes with arbitrary birth and death rates of mutants, we show that the
fixation probabilities of mutants are different from those obtained in the
mass-action populations. We focus on spatial lattices and show that the
difference between BD and DB processes on 1D and 2D lattices are non-small even
for large population sizes. We support these results with a generating function
approach that can be generalized to arbitrary graph structures. Finally, we
discuss several biological applications of the results.
