| 引用本文: | 李华,肖敏,Leszek Rutkowski,万佑红.一类含饱和项Gierer-Meinhardt模型的交叉扩散研究[J].控制理论与应用,2026,43(3):561~572.[点击复制] |
| LI Hua,XIAO Min,Leszek Rutkowski,WAN You-hong.Cross-diffusion research of a class of Gierer-Meinhardt models with saturation terms[J].Control Theory & Applications,2026,43(3):561~572.[点击复制] |
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| 一类含饱和项Gierer-Meinhardt模型的交叉扩散研究 |
| Cross-diffusion research of a class of Gierer-Meinhardt models with saturation terms |
| 摘要点击 477 全文点击 70 投稿时间：2023-10-27 修订日期：2025-11-20 |
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| DOI编号 10.7641/CTA.2025.30701 |
| 2026,43(3):561-572 |
| 中文关键词 Gierer-Meinhardt系统 分岔 反应扩散效应 交叉扩散 图灵不稳定 |
| 英文关键词 Gierer-Meinhardt system bifurcation reaction-diffusion effect cross diffusion Turing instability |
| 基金项目 国家自然科学基金项目(62073172),江苏省自然科学基金项目(BK20221329)资助. |
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| 中文摘要 |
| Gierer-Meinhardt系统是描述化学和生物现象的典型数学模型. 目前国内外关于该系统的时空演化研究仅
仅局限于图灵不稳定和斑图模式,而关于交叉扩散对斑图动力学的影响研究还不深入.为此,本文提出了一个具有
交叉扩散和饱和项的Gierer-Meinhardt模型,旨在研究交叉扩散对Gierer-Meinhardt模型的稳定性,斑图演化速度及
斑图模式的影响规律.通过线性稳定性分析和数值仿真,探讨了交叉扩散系数对斑图动力学行为的影响.研究结果
表明,当自扩散驱动系统稳定时,引入交叉扩散可以驱动系统不稳定而产生图灵斑图;不同的交叉扩散系数会导致
不同的斑图模式,并影响最终形成稳定斑图的时间.当自扩散驱动系统不稳定时,交叉扩散可以驱使系统趋于稳定,
并且在不同交叉扩散系数下,系统稳定到平衡态所需的时间不同.因此,交叉扩散对系统的稳定性、斑图演化速度、
斑图模式都起着重要的作用. |
| 英文摘要 |
| The Gierer-Meinhardt system is a classic mathematical model for describing chemical and biological phe
nomena. Current studies on the spatiotemporal evolution of this system are largely confined to Turing instability and
pattern formation, with limited exploration of the effects of cross-diffusion on pattern dynamics. To address this gap, this
study proposes a Gierer-Meinhardt model incorporating cross-diffusion and saturation terms to investigate the influence of
cross-diffusion on system stability, pattern evolution speed, and pattern formation. Through linear stability analysis and
numerical simulations, the effects of cross-diffusion coefficients on the dynamical behavior of patterns were examined. The
results reveal that cross-diffusion can destabilize a system otherwise stabilized by self-diffusion, leading to Turing patterns.
Different cross-diffusion coefficients result in distinct pattern modes and influence the time required to achieve stable pat
terns. Conversely, in systems where self-diffusion drives instability, cross-diffusion can promote stabilization, with the time
to reach equilibrium varying under different cross-diffusion coefficients. These findings demonstrate that cross-diffusion
plays a crucial role in system stability, pattern evolution speed, and pattern formation. |
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