Stability analysis of a fractional order differential equation model of a brain tumor growth depending on the density

In this paper, fractional-order is introduced into a brain tumor model that is known as GBM. This model shows a population of a brain tumor that has two-strains of tumors: sensitive tumor cells and resistant tumor cells. This system will explain a brain tumor model, that has a monoclonal origin (sensitive tumor cells), and produces, after reaching a specific density, another tumor with different growth-rate and treatment susceptibility. In this work, we have shown that the model possesses non-negative solutions. Furthermore, we have studied the stability, the existence and the uniqueness of the constructed model. To investigate the conditions for an extinction of the tumor cells, we have modified this system and have considered the Allee threshold. Numerical simulations have given a detailed description of the behavior of the model at the end of the paper.