一类体积填充效应的趋化模型斑图的形成  

Pattern Formation of a Volume-filling Chemotaxis Model

作　　者：麻作军[1,2] 伏升茂 张丽丽[1] MA Zuojun;FU Shengmao;ZHANG Lili(School of Mathematics and Information Engineering,Longdong University,Qingyang,Gansu,745000,P.R.China;Institute of Applied Mathematics,Longdong University,Qingyang,Gansu,745000,P.R.China;College of Mathematics and Statistics,Northwest Normal University,Lanzhou,Gansu,730070,P.R.China)

机构地区：[1]陇东学院数学与信息工程学院,甘肃庆阳745000 [2]陇东学院应用数学研究所,甘肃庆阳745000 [3]西北师范大学数学与统计学院,甘肃兰州730007

出　　处：《数学进展》2024年第5期1039-1058,共20页Advances in Mathematics（China）

基　　金：甘肃省教育厅创新基金(No.2021B-262)。

摘　　要：本文讨论了一类具有体积填充效应的趋化模型斑图的形成机制,通过对具有细胞动力项、自扩散及趋化性模型的稳定性分析,发现趋化是诱导Turing斑图的重要因素,并得到了趋化模型非平凡平衡态失稳的充分条件.然后利用弱非线性分析理论和多尺度摄动方法讨论了趋化模型在平衡态失稳之后的情形,并建立了相应的振幅方程.当χ>χ_(a)且有一个不稳定模时,给出了模型的定态斑图近似表达式、超临界和次临界边界的分岔条件、双稳态条件和分岔图;当χ>χ_(a)且有两个不稳定模时,利用振幅方程组分析了双模竞争情形,给出模型的定态斑图近似表达式,并用数值模拟方法验证了前面的结论.This paper is devoted to investigating the pattern formation of a volume-filling chemotaxis model.By analyzing the stability of the model with the cell kinetics term,self-diffusion and chemotaxis,it is found that chemotaxis is an important factor for inducing Turing pattern,and sufficient conditions for the nontrivial steady state instability of the chemotaxis model are obtained.Then,the weakly nonlinear analysis theory and multi-scale perturbation method are used to discuss the case of the chemotaxis model after the instability of the steady state,and the corresponding amplitude equations are established.When x>Xa and with an unstable mode,the approximate expression of the steady state pattern,the bifurcation condi-tion of the supercritical and subcritical boundary,the bistable condition,and the bifurcation diagram of the model are obtained.When x>xa and with two unstable modes,the two-mode competition is analyzed by using the amplitude equations,the approximate expressions of the stationary pattern of the model are given,and numerical simulations are used to verify the previous conclusions.

关 键 词：趋化模型 体积填充 弱非线性分析 振幅方程 双稳态 斑图形成 

分 类 号：O175.26[理学—数学]