Pattern suppression and recovery under one-way versus two-way chemotactic coupling in hybrid partial differential equation–ordinary differential equation models

We study coupled mass transport in a tumor–microenvironment regime with two motile densities ( S , R ) and non-motile state switching ( P , A ). The populations diffuse and undergo chemotactic drift; ( P , A ) follow pointwise ODE switching. A decoupled inhibitory field D satisfies a damped Neumann heat equation, giving maximum-principle bounds and exponential decay. Together with the pointwise invariant P + A , these identities yield global existence, positivity, and long-time reduction to limiting ( S , R ) kinetics with a unique globally attracting coexistence state. Neumann eigenmode reduction gives closed dispersion relations. The base ( S , R ) reaction–diffusion (RD) block remains stable for all nonconstant modes for any d S , d R > 0, excluding classical Turing destabilization. Chemotaxis is posed via a diffusive cue c , since ∇ A is undefined for non-diffusive A . In one-way damped coupling, the linearized mode matrix is block triangular and leaves the ( S , R ) spectrum unchanged. Two-way coupling adds a feedback rank-one mobility correction, induces effective cross-diffusion, and admits mode growth. We give explicit trace/determinant criteria for unstable Laplacian modes and the resulting instability thresholds.