Mode-Wise Spectral Criteria for Coupled Mass Transport in Hybrid PDE--ODE Tumor Microenvironments

TL;DR

This paper develops spectral criteria to analyze the stability of coupled mass transport in tumor microenvironments modeled by PDEs and ODEs, revealing conditions for stability and instability in the system.

Contribution

It introduces mode-wise spectral criteria for coupled PDE-ODE models in tumor microenvironments, including explicit instability thresholds and analysis of feedback effects.

Findings

Neumann eigenmode reduction yields closed dispersion relations.

The base reaction-diffusion system remains stable against classical Turing patterns.

Two-way coupling can induce mode growth and instability.

Abstract

We study coupled mass transport in a tumor--microenvironment setting 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 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 $\nabla A$ 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.