Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations

TL;DR

This paper investigates the stability of nonconstant stationary solutions in a coupled reaction-diffusion and ODE system modeling precancerous cell growth, extending previous results to more general conditions and analyzing stability via spectral methods.

Contribution

It extends prior work by considering a general Hill-type production function and full parameter set, providing new stability conditions for spatial patterns.

Findings

Derived conditions for linearized stability of spatial patterns.

Extended existence results to more general production functions.

Applied spectral analysis and perturbation theory to stability analysis.

Abstract

In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous cells, whose proliferation is controled by growth factors diffusing in the extracellular medium and binding to the cell surface. We extend the results on the existence of nonhomogenous stationary solutions obtained in the previous work of one of the authors to a general Hill-type production function and full parameter set. Using spectral analysis and perturbation theory we derive conditions for the linearized stability of such spatial patterns.