Spike patterns in a reaction-diffusion-ode model with Turing instability

TL;DR

This paper investigates spike pattern formation in a coupled reaction-diffusion-ODE system relevant to early carcinogenesis, revealing a novel nonstationary pattern phenomenon through numerical simulations and analysis.

Contribution

It introduces a numerical approach for simulating coupled reaction-diffusion-ODE systems and uncovers a new pattern formation mechanism involving nonstationary structures converging to Dirac deltas.

Findings

Emergence of periodic and irregular spike patterns due to diffusion-driven instability

Development of a finite element method with adaptive mesh for accurate simulations

Identification of nonstationary structures tending to Dirac delta sums

Abstract

We explore a mechanism of pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focused on the model of early carcinogenesis proposed by Marciniak-Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non-diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns due to diffusion-driven instability. To control the accuracy of simulations, we develop a numerical code based on the finite element method and adaptive mesh. Simulations, supplemented by numerical analysis, indicate a novel pattern formation phenomenon based on the emergence of nonstationary structures tending asymptotically to the sum of Dirac deltas.