中北大学学报（自然科学版）2025，Vol.46Issue(3)：396-404,410,10.DOI:10.62756/jnuc.issn.1673-3193.2024.03.0015
具有交叉扩散的Gierer-Meinhardt系统斑图演化
Pattern Formation Evolution of the Gierer-Meinhardt System with Cross-Diffusion
摘要
Abstract
The Gierer-Meinhardt system is a typical class of reaction-diffusion systems that have been extensively studied in the field of biological pattern formation due to its excellent dynamical characteris-tics.Currently,the majority of domestical and international research,on the spatiotemporal evolution of this system is limited to the instability and Turing patterns driven by self-diffusion,while there are very little research on the instability driven by cross-diffusion.This paper proposed a class of Gierer-Meinhardt reaction-diffusion system with cross-diffusion terms.By utilizing characteristic polynomial eigenvalue analysis and stability theorems,a linear stability analysis of the unique positive equilibrium point of the Gierer-Meinhardt system was carried out.The necessary conditions for the occurrence of Turing instabil-ity of the Gierer-Meinhardt system were identified.The cross-diffusion coefficient was selected as the bifurcation parameter,which further revealed the influence mechanism of cross-diffusion on Turing instabil-ity of the Gierer-Meinhardt system.In addition,through the numerical simulations,this paper explored the response mechanism of cross-diffusion for the evolution of Turing patterns of the Gierer-Meinhardt sys-tem.It is found that when the system driven by self-diffusion is stable,cross-diffusion can induce Turing instability in the Gierer-Meinhardt system and generate unevenly structured Turing patterns.When the system driven by self-diffusion is unstable,cross-diffusion can not only achieve the transformation of Gierer-Meinhardt system pattern structures but also change the evolution speed of patterns.Specifically,the further the cross-diffusion coefficient is from the bifurcation threshold necessary to induce Turing insta-bility,the larger the proportion of point-like patterns in the spatial structure of the system's Turing pat-terns,and the faster the rate of evolution.Therefore,cross-diffusion plays a crucial role in the genera-tion,transformation,and evolution speed of patterns in the Gierer-Meinhardt system.关键词
交叉扩散/Gierer-Meinhardt系统/图灵斑图/反应扩散方程Key words
cross-diffusion/Gierer-Meinhardt system/Turing patterns/reaction-diffusion equations分类
数学引用本文复制引用
鲁言财,肖敏,曹进德,王正新..具有交叉扩散的Gierer-Meinhardt系统斑图演化[J].中北大学学报（自然科学版）,2025,46(3):396-404,410,10.基金项目
国家自然科学基金(62073172) （62073172）
江苏省自然科学基金(BK20221329) （BK20221329）
