Myeloproliferative neoplasm (MPN) is a hematologic malignancy characterized by the clonal outgrowth of hematopoietic cells with a somatically acquired mutation most commonly in JAK2 (JAK2V617F). This mutation endows upon myeloid progenitors cytokine independent growth and consequently leads to excessive production of myeloid lineage cells. It has been previously suggested that inflammation may play a role in the clonal evolution of JAK2V617F mutants. In particular, it is possible that one or more cellular kinetic parameters of hematopoietic stem cells (HSCs) are affected by inflammation, such as division or death rates of cells, and the probability of HSC differentiation. This suggests a mechanism that can steer the outcome of the cellular competition in favor of the mutants, initiating the disease. In this paper we create a number of mathematical evolutionary models, from very abstract to more concrete, that describe cellular competition in the context of inflammation. It is possible to build a model axiomatically, where only very general assumptions are imposed on the modeling components and no arbitrary (and generally unknown) functional forms are used, and still generate a set of testable predictions. In particular, we show that, if HSC death is negligible, the evolutionary advantage of mutant cells can only be conferred by an increase in differentiation probability of HSCs in the presence of inflammation, and if death plays a significant role in the dynamics, an additional mechanism may be an increase of HSC's division-to-death ratio in the presence of inflammation. Further, we show that in the presence of inflammation, the wild type cell population is predicted to shrink under inflammation (even in the absence of mutants). Finally, it turns out that if only the differentiation probability is affected by the inflammation, then the resulting steady state population of wild type cells will contain a relatively smaller percentage of HSCs under inflammation. If the division-to-death rate is also affected, then the percentage of HSCs under inflammation can either decrease or increase, depending on other parameters.