  Resource are often not uniformly distributed within a population. Spatial
variations of concentration of a resource, change the fitness of competing
strategies locally. The notion of fitness varying with respect to both genotype
and environment is important in modeling cancer initiation, microbial evolution
and evolution of drug resistance. Environmental interactions can be asymmetric,
that is, they affect the fitness of one type more than the other. The question
is how local environmental variations in network population structures change
the selection dynamics in a finite population setting. We consider
one-dimensional lattice population structures with spatial fitness
distributions with a periodic pattern. Heterogeneity is determined by standard
deviation of fitnesses and period. The model covers biologically relevant
limits of two-habitat subdivided populations and randomly-distributed resources
in high- and low-periods. We numerically calculate fixation probability and
fixation times for a constant population birth-death process as fitness
heterogeneity and period vary. We identify levels of heterogeneity for which a
previously deleterious mutant, in a uniform environment, becomes beneficial. In
other regimes of the problem we observe unexpected behavior where the fixation
probability of both types are larger than their neutral value at the same time.
This coincides with an exponential increase in time to fixation as a function
of population size, which points to significant slow-down in selection process
and the potential for coexistence between types in realistic time scales. We
also discuss `fitness shift' model where the fitness function of one type is
identical to the other up to a constant spatial shift. This leads to
significant increase (or decrease) in the fixation probability of the mutant
depending the value of the shift.
