Stability analysis and simulations of tumor growth model based on system of reaction-diffusion equation in two-dimensions

In this study, we introduce a novel framework for exploring the dynamics of tumor growth and an evolution model for two-stage carcinogenic mutations in two-dimensions based on a system of reaction-diffusion equations. It is shown theoretically that the system is globally stable in the absence of both delay and diffusion. The inclusion of diffusion does not destabilize the system, while including delay does capture the key elements of how normal cells convert into cancer cells. To further validate these results, several numerical experiments are performed for different parameter values involved in the model equation. These parameter values are chosen in the sense that they have some biological meanings using the steady states of the equilibrium points. For the purpose of simulation, a stable Euler scheme is used for temporal discretization, while a Fourier spectral method is used for space variables, which is a natural choice due to the periodic boundary conditions in the model equation. The numerical simulation results further confirm our theoretical justification.