Chemotactic Feedback Controls Patterning in Hybrid Tumor--Stroma Model

Motivated by an ongoing collaboration with clinical oncologists and pathologists, we develop a hybrid partial differential equation--ordinary differential equation (PDE--ODE) framework that captures (i) competition between susceptible and resistant phenotypes, (ii) stromal state switching, and (iii) a clinically realistic open-loop, single-dose therapeutic agent $I$ with diffusion and clearance. Clinical management of solid tumors is increasingly limited by spatial heterogeneity and therapy-induced resistance niches that are difficult to predict from well-mixed models. We establish a rigorous mathematical backbone with forward invariance of the nonnegative cone and global-in-time well-posedness. Exploiting the decoupled drug equation $\partial_t I=d_I\Delta I-\gamma_I I$, we prove a long-time reduction during washout and show that the damped base dynamics admit no diffusion-driven (Turing-type) instability. We then formulate a directionality--damping principle: unidirectional (open-loop) sensing yields at most transient focusing, whereas bidirectional (closed-loop) feedback reshapes the effective mobility and produces explicit thresholds separating stable homogeneity, finite-band patterning (resistance niche formation), and aggregation when strong parabolicity is violated. Reproducible simulations corroborate this classification and highlight when flux regularization is required for physical realism.