Stability and bifurcation diagram for a one-dimensional shadow reaction-diffusion system of substrate depletion type

Yasuhito Miyamoto; Mizuki Obara; Shingo Yorii

doi:10.3934/dcdsb.2025134

Article Contents

2026, Volume 37: 1-22. Doi: 10.3934/dcdsb.2025134

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Stability and bifurcation diagram for a one-dimensional shadow reaction-diffusion system of substrate depletion type

1.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
2.
Graduate School of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan
3.
Department of Integrated Science, College of Arts and Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

^*Corresponding author: Yasuhito Miyamoto
^*Corresponding author: Yasuhito Miyamoto

Received: March 2025

Revised: June 2025

Early access: August 25, 2025

Published online: August 25, 2025

The first author was supported by JSPS KAKENHI Grant Number 24K00530.

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Abstract

We are concerned with a shadow reaction-diffusion system of substrate depletion type in a finite interval

$\begin{align} \begin{cases} \partial_tA = {\varepsilon}^2A_{xx}+f(\xi)A^3-A & \text{for}\ 0<x<1,\ 0<t<T,\\ \tau\partial_t\xi = \gamma-f(\xi)\int_0^1A^3dx-\xi & \text{for}\ 0<t<T,\\ A_x(0,t) = A_x(1,t) = 0 & \text{for}\ 0<t<T, \end{cases} \end{align}$

where $ A(x,t) $ and $ \xi(t) $ are nonnegative unknown functions, and $ f\in C^1[0,\infty) $ is a nonnegative increasing function satisfying certain conditions. We determine the complete bifurcation diagram of the positive stationary solutions if $ \gamma>0 $ is larger than a certain value. We also study the stability of every positive stationary solution. Specifically, we show that some $ 1 $-mode solutions are stable for small $ \tau>0 $ and large $ \gamma>0 $, and that $ n $-mode solutions, $ n\ge 2 $, are unstable for all $ \tau>0 $. In particular, the case where $ f(\xi) = \xi^2 $ is studied in detail.

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Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35K57, 33E05.

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Figure 1. A schematic picture of the bifurcation diagram $ {\mathcal{S}} $. The horizontal axis is the $ {\varepsilon} $-axis, and the vertical axis represents the space $ X\times{\mathbb{R}} $. Each stationary solution on the solid curve and line is stable for small $ \tau>0 $ and each stationary solution on the dashed curves and lines is unstable for all $ \tau>0 $. $ ({\varepsilon}_1(0),A_0(x,\xi_0(0)),\xi_0(0)) $ is marginally stable for small $ \tau>0 $

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