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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 self-diffusion 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 graduallychanges 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 cross-diffusion coefficient$D_1$ for the susceptible individuals, the faster the system stabilizes. When the self-diffusion-driven system is unstable, the cross-diffusion causes the system to change from an unstable state into a locally stable state, and the smaller the susceptible individuals' cross-diffusion coefficient, the slower the rate of system stabilization is. Therefore, cross-diffusion has a significantinfluence on the stability, the rate of stability, and the pattern structure of the SI model.-
Keywords:
- Turing instability /
- nonlinear incidence rate /
- cross diffusion /
- Turing pattern
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图 3 不同交叉扩散取值的空间格局 (a)$ D_2=6 $; (b)$ D_2=8 $; (c)$ D_1=-6, D_2=6 $; (d)$ D_1=-8, D_2=8 $
Fig. 3. The spatial pattern of of different cross diffusion values: (a)$ D_2=6 $; (b)$ D_2=8 $; (c)$ D_1=-6, D_2=6 $; (d)$ D_1=-8, D_2=8 $
图 4 不同取值的交叉扩散系数$D_1$的色散关系曲线及空间格局 (a)色散关系曲线; (b)$D_1$= 0; (c)$D_1=-0.2$; (d) $D_1=-1$; (e)$D_1=1$; (f)$D_1=3.5$
Fig. 4. Dispersion relationship curves and spatial patterns of cross diffusion coefficients $D_1$ with different values: (a) Dispersion relationship curve; (b)$D_1=0$; (c)$D_1=-0.2$; (d)$D_1=-1$; (e)$D_1=1$; (f)$D_1=3.5$
图 5 不同取值的交叉扩散系数$D_2$的色散关系曲线及空间格局 (a)色散关系曲线; (b)$D_2=0$; (c)$D_2=0.5$; (d)$D_2=1$;(e)$D_2=-1$; (f)$D_2=-2$
Fig. 5. Dispersion relationship curves and spatial patterns of cross diffusion coefficients $D_2$ with different values: (a) Dispersion relationship curve; (b)$D_2=0$; (c)$D_2=0.5$; (d)$D_2=1$; (e)$D_1=-1$; (f)$D_1=-2$
图 6 不同交叉扩散系数对应的系统演化图 (a)$D_1=-9$; (b)$D_1=-9.8$; (c)$D_1=-10.2$
Fig. 6. System evolution diagrams corresponding to different cross diffusion coefficients: (a)$D_1=-9$; (b)$D_1=-9.8$; (c)$D_1=-10.2$
图 7 当$ D_1=-9,\; -9.8,\; -10.2 $时, 不同迭代步数稳定到平衡点的网格个数
Fig. 7. The number of grids that stabilize to the equilibrium point with different iteration steps when $ D_1=-9, $$ -9.8, -10.2 $
图 8 不同交叉扩散系数对应的系统演化图 (a)$D_1=4$; (b)$D_1=3.5$; (c)$D_1=3.2$
Fig. 8. System evolution diagrams corresponding to different cross diffusion coefficients: (a)$D_1=4$; (b)$D_1=3.5$; (c)$D_1=3.2$
图 9 当$ D_1=4,\; 3.5,\; 3.2 $时, 不同迭代步数稳定到平衡点的网格个数
Fig. 9. The number of grids that stabilize to the equilibrium point with different iteration steps when $ D_1=4, $$ 3.5, \;3.2 $
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[1] Turing A M 1952 Philos. Trans. R. Soc. London, Ser. B 2 37
Google Scholar
[2] Capone F, Carfora M F, De Luca R, Torcicollo I 2019 Math. Comput. Simul. 165 172
Google Scholar
[3] Ali I, Saleem M T 2023 Mathematics 11 1459
Google Scholar
[4] Hu J, Zhu L, Peng M 2022 Inf. Sci. 596 501
Google Scholar
[5] Ruiz-Baier R, Tian C 2013 Nonlinear Anal. Real World Appl. 14 601
Google Scholar
[6] Sun G Q, Jin Z, Liu Q X, Li L 2007 J. Stat. Mech. Theory Exp. 2007 P11011
Google Scholar
[7] 刘若琪, 贾萌萌, 范伟丽, 贺亚峰, 刘富成 2022 71 248201
Google Scholar
Liu R Q, Jia M M, Fan W L, He Y F, Liu F C 2022 Acta Phys. Sin. 71 248201
Google Scholar
[8] 张荣培, 王震, 王语, 韩子健 2018 67 050503
Google Scholar
Zhang R P, Wang Z, Wang Y, Han Z J 2018 Acta Phys. Sin. 67 050503
Google Scholar
[9] Giri A, Pramod Jain S, Kar S 2020 Chemphyschem 21 1608
Google Scholar
[10] 王楠, 肖敏, 蒋海军, 黄霞 2022 71 180201
Google Scholar
Wang N, Xiao M, Jiang H J, Huang X 2022 Acta Phys. Sin. 71 180201
Google Scholar
[11] 王凌志, 周先春, 陈铭 2019 信息与控制 48 559
Google Scholar
Wang L Z, Zhou X C, Chen M 2019 Inf. Control. 48 559
Google Scholar
[12] Pastor-Satorras R, Castellano C, Van Mieghem P, Vespignani A 2015 Rev. Mod. Phys. 87 925
Google Scholar
[13] Wang W, Cai Y, Wu M 2012 Nonlinear Anal. Real. World Appl. 13 2240
Google Scholar
[14] 阮中远 2020 中国科学: 物理学 力学 天文学 50 010507
Google Scholar
Ruan Z Y 2020 Sci. Sin-Phys. Mech. Astron. 50 010507
Google Scholar
[15] Wang L, Li X 2014 Chin. Sci. Bull. 59 3511
Google Scholar
[16] Sun G Q, Jusup M, Jin Z, Wang Y, Wang Z 2016 Phys. Life Rev. 19 43
Google Scholar
[17] Guin L N, Acharya S 2017 Nonlinear Dyn. 88 1501
Google Scholar
[18] Zhao L, Wang Z C, Ruan S 2020 Nonlinear Anal. Real World Appl. 51 102966
Google Scholar
[19] Zheng Q, Pandey V, Shen J, Xu Y, Guan L 2022 EPL 137 42002
Google Scholar
[20] Kuniya T, Wang J 2018 Nonlinear Anal. Real World Appl. 43 262
Google Scholar
[21] Ahmed N, Fatima M, Baleanu D, Nisar K S, Khan I, Rafiq M, Rehman M A U, Ahmad M O 2020 Front. Phys. 7 220
Google Scholar
[22] Wang W, Gao X, Cai Y, Shi H, Fu S 2018 J. Franklin Inst. 355 7226
Google Scholar
[23] Sun G Q 2012 Nonlinear Dyn. 69 1097
Google Scholar
[24] Kerner E H 1957 Bull. Math. Biol. 19 121
Google Scholar
[25] Fan Y 2014 Appl. Math. Comput. 228 311
Google Scholar
[26] Ghorai S, Poria S 2016 Chaos Solitons Fractals 91 421
Google Scholar
[27] Aly S, Khenous H B, Hussien F 2015 Int. J. Biomath. 8 1550006
Google Scholar
[28] Triska A, Gunawan A Y, Nuraini N 2022 J. Math. Computer Sci. 27 1
Google Scholar
[29] Brauer F, Driessche P V D 2001 Math. Biosci. 171 143
Google Scholar
[30] Chinviriyasit S, Chinviriyasit W 2010 Appl. Math. Comput. 216 395
Google Scholar
[31] Simon C P, Jacquez J A 1992 SIAM J. Appl. Math. 52 541
Google Scholar
[32] Hethcote H W, van den Driessche P 1991 J. Math. Biol. 29 271
Google Scholar
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