交叉扩散驱动的SI模型空间斑图  

Spatial pattern of a class of SI models driven by cross diffusion

作　　者：陆源杉 肖敏 万佑红[1] 丁洁[1] 蒋海军[2] Lu Yuan-Shan;Xiao Min;Wan You-Hong;Ding Jie;Jiang Hai-Jun(College of Automation,College of Artificial Intelligence,Nanjing University of Posts and Telecommunications,Nanjing 210023,China;College of Mathematics and Systems Science,Xinjiang University,Urumchi 830047,China)

机构地区：[1]南京邮电大学自动化学院,人工智能学院,南京210023 [2]新疆大学数学与系统科学学院,乌鲁木齐830047

出　　处：《物理学报》2024年第8期62-71,共10页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:62073172,U1703262);江苏省自然科学基金(批准号:BK20221329);江苏省研究生科研与实践创新计划(批准号:KYCX_220968)资助的课题.

摘　　要：目前国内外关于SI模型空间格局的研究大多数局限在自扩散以及系统参数对斑图模式的影响,而关于交叉扩散对空间格局的演化机理研究成果较少.本文建立了一个具有自扩散和交叉扩散的空间流行病模型,研究了在有无自扩散驱动系统不稳定的情况下,交叉扩散对SI模型的稳定性、稳定速度以及斑图结构的影响.研究发现,在无自扩散驱动系统不稳定的情况下,引入交叉扩散能够激发Turing斑图的产生;在自扩散驱动系统不稳定的情况下,交叉扩散可以实现斑图结构的改变;对于SI模型的稳定速度,不论有无自扩散驱动系统不稳定,交叉扩散都影响了其到达稳定所需时间,且在不同的交叉扩散系数下,所需时间也不同.因此,交叉扩散对于SI模型的稳定性、稳定速度、斑图结构都有重要的影响.Currently,most of researches on the spatial patterns of the SI model focus on the influences of self-diffusion and system parameters on pattern formation,but only a few studies involve how cross-diffusion influences the evolution of spatial patterns.In this paper,we establish a spatial epidemic model that considers both selfdiffusion and cross-diffusion and investigate the effects of cross-diffusion on the stability,the rate of stability,and the pattern structure of the SI model with or without self-diffusion-driven system instability.The stability of the non-diffusive system is analyzed,and the conditions for Turing instability in the presence of diffusion terms are elucidated.It is found that when the system is stable under self-diffusion-driven conditions,the introduction of cross-diffusion can change the system's local stability,and produce Turing patterns as well.Furthermore,different cross-diffusion coefficients can generate patterns with different structures.When the system is unstable under self-diffusion-driven conditions,the introduction of cross-diffusion can change the pattern structure.Specifically,when the cross-diffusion coefficient D_(1) for the susceptible individuals is negative,the pattern structure is transformed from spot-stripe patterns into spot patterns,and when it is positive,the pattern structureturns from spot-stripe patterns into labyrinthine patterns,and eventually into a uniform solid color distribution.When the cross-diffusion coefficient D_(2) for the infected individuals is positive,the pattern transformation is similar to when the cross-diffusion coefficient D_(1) for susceptible individuals is negative,the pattern gradually changes into spot patterns.When D_(2) is negative,the pattern structure exhibits a porous structure,eventually it is transformed into a uniform solid color distribution.Regarding the rate of stability of the SI model,in the case of a stable self-diffusion system,the introduction of cross-diffusion may change the rate of system stability,and the larger the cro

关 键 词：Turing不稳定 非线性发病率 交叉扩散 TURING 斑图 

分 类 号：R181[医药卫生—流行病学] O175[医药卫生—公共卫生与预防医学]