Abstract: Local environmental interactions are a major factor in determining the success of a new mutant in structured populations. Spatial variations of concentration of a resource change the fitness of competing strategies locally and thus can drastically change the outcome of evolutionary process in unintuitive ways. Environmental interactions can be asymmetric, i.e. the same local resource value affects the fitness of strategies differently. The question is how such local environmental variations in network population structures change the condition for selection and fixation probability of an advantageous (or deleterious) mutant. We consider linear graph structure and focus on the case where resources have a spatial periodic pattern. Our model covers several biologically relevant cases. We numerically calculate fixation probability and fixation time for a Moran birth-death process as fitness heterogeneity and period vary. The fixation probability is affected by not only the level of fitness heterogeneity, but also spatial scale of resource variations set by period of distribution T . For most (weak) asymmetric environmental interactions the chance of success of a mutant increases with heterogeneity. We identify conditions for which a previously deleterious mutant (in a uniform environment) becomes beneficial as fitness heterogeneity is increased. We observe cases where the fixation probability of both mutant and resident types are less than their neutral value, 1/N , simultaneously. This corresponds to potential coexistence of resident and mutant types. Finally, we discuss the effect of ‘fitness shift’ where the fitness function of two types has a phase difference. This happens when there are more than one type of resources in the environment. We observe significant increase (or decrease) in the fixation probability of the mutant as a result of such phase shift.