Analysis of Pattern Selection in Reaction-Diffusion Systems: From Toxic Zooplankton Dynamics to Tumor Growth Modelling

In this thesis, we developed a model to study how patterns form and evolve over time due to the interaction between toxic phytoplankton and zooplankton. Our \mbox{analysis} revealed that nonlinear cross-diffusion plays a crucial role in shaping spatial patterns. We derived amplitude equations to describe the dynamics under nonlinear cross-diffusion, which helped us to understand the transitions and stability of various Turing patterns. Our numerical simulations confirmed the validity of our theoretical results. We found that in the absence of cross-diffusion, the distribution of plankton is homogeneous. However, when cross-diffusivity exceeds a critical value, the spatial distribution of all plankton species becomes inhomogeneous in space, leading to different patterns. This thesis also presents a mathematical model of acid-mediated tumor growth using reaction-diffusion equations in spherical coordinates. Tumor invasion is a complex process involving cell migration and proliferation. Mathematical modelling can aid in understanding the mechanisms by which primary and secondary (metastatic) tumors invade and damage normal cells. A numerical study of acid-mediated tumor growth may provide a better understanding of how to design new experiments or cures for the future. Cancer cells use anaerobic glycolysis, which increases the production of lactic acid. This acidic environment is favourable for tumor growth, and if it persists, normal cells cannot survive and begin to die, thereby facilitating tumor invasion. The results show that the method of lines is a powerful numerical scheme for solving the proposed model, and MATLAB is used to analyze the computed results graphically.