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.
Highlights
Before highlighting the system of non linear reaction-diffusion models modeling an in-vitro situation of tumor cells and their micro-environment with regard to its growth and metastasis derived and experimented in [23] and simulated in [12], we would like to mention that Friedman and Kim in [12] mentioned that tumor cells proliferate at different rates and migrate in different patterns depending on the micro-environment in which they are embedded
In an effort to understand the interaction between tumor cells, fibroblasts and/or myofibroblasts at an early stage of cancer, Friedman and Kim in [12] simulated the model derived in [23] an in-vitro model as
Denoting the required time by τ, this implies that we extend the quasi non-linear reactiondiffusion model simulated in [12] to mimic tumor cells’ proliferation and migration under the influence the micro-environment in vitro in equation (1), to a discrete delay quasi non-linear reaction-diffusion model
Summary
Department of Mathematics and Applied Mathematics, University of the Western Cape, Cape Town 7535, South Africa Copyright c 2019 the authors. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
