Bifurcation Structure and Control Interpretation in pH-Responsive Tumor– Immune Dynamics: A Reaction–Diffusion Framework

Abstract Living systems often regulate their microenvironment through feedback mechanisms that couple biochemical activity, transport processes, and energetic constraints. In tumors, extracellular acidity represents a critical environmental variable that suppresses immune activity and shapes therapeutic response. Here, we analyze the dynamical and spatial structure of pH-responsive tumor–immune interactions using a biologically motivated reaction–diffusion framework with nonlinear feedback. We model tumor growth, immune effector dynamics, and extracellular acidity as a coupled control system in which alkalinization acts as an environmental modulation mechanism subject to transport and metabolic constraints. Using bifurcation analysis and numerical phase-space exploration, we identify distinct dynamical regimes corresponding to immune suppression, partial control, and tumor clearance. These regimes are organized by well-defined bifurcation boundaries that depend on kinetic suppression strength, diffusive transport scales, and alkalinization efficiency. Our analysis reveals that tumor control is not determined by biological potency alone, but instead emerges from the interplay between nonlinear feedback, spatial diffusion, and energetic limitations. In particular, we show how diffusion-limited transport and metabolic cost impose hard constraints on the effectiveness and scalability of pH-modulating strategies, leading to sharp transitions between controllable and uncontrollable regimes. Sensitivity analyses demonstrate that the qualitative phase structure is robust across broad parameter ranges. This work provides a physical interpretation of tumor–immune regulation as a constrained reaction–diffusion control problem and offers a unifying dynamical framework for understanding environmentally mediated control in biological systems. The results have implications for the design of pH-modulating immunotherapies and more broadly for biological systems in which organisms actively modify their environment under transport and energetic constraints.