Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability

Spatial tumour--immune heterogeneity is a key feature of solid-tumour progression, immune infiltration, and immune exclusion. We develop a computational oncology model in which tumour cells, immune effector cells, and a chemokine signal interact through a reaction--diffusion--chemotaxis system on a bounded tissue domain with no-flux boundaries. Chemokine is produced by tumour cells and tumour--immune contact, recruits immune cells, and guides chemotactic migration. After nondimensionalization, we establish positivity, a tumour-density bound, and immune/chemokine mass estimates. We identify the tumour-free equilibrium, derive the immune-control threshold $\sigma_0>\delta$, and reduce coexistence to a scalar equation. Linear stability analysis about coexistence yields a mode-wise dispersion relation in which chemotaxis appears as a wavenumber amplified coupling, producing finite-wavelength instability above a critical sensitivity. A conservative finite-volume scheme with upwind chemotactic flux verifies the thresholds, dominant unstable modes, sensitivity maps, positivity, convergence, and residual consistency.