A fitted operator method for tumor cells dynamics in their micro-environment

Abstract

In this paper, we consider a quasi non-linear reaction-diffusion model designed to mimic tumor cells’ proliferation and migration under the influence of their micro-environment in vitro. Since the model can be used to generate hypotheses regarding the development of drugs which confine tumor growth, then considering the composition of the model, we modify the model by incorporating realistic effects which we believe can shed more light into the original model. We do this by extending the quasi non-linear reaction-diffusion model to a system of discrete delay quasi non-linear reaction-diffusion model. Thus, we determine the steady states, provide the conditions for global stability of the steady states by using the method of upper and lower solutions and analyze the extended model for the existence of Hopf bifurcation and present the conditions for Hopf bifurcation to occur. Since it is not possible to solve the models analytically, we derive, analyze, implement a fitted operator method and present our results for the extended model. Our numerical method is analyzed for convergence and we find that is of second order accuracy. We present our numerical results for both of the models for comparison purposes.