一类带负交叉扩散项的SIR传染病模型的空间Turing斑图
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摘要
本文研究了带有负交叉扩散项的SIR传染病模型的空间斑图动力学问题.利用稳定性理论和Hopf分支理论,获得了Turing失稳的条件以及Turing斑图的存在区域,并且利用Matlab软件模拟获得了不同类型的Turing斑图,比如点状、条状以及二者共存等Turing斑图.通过负交叉扩散诱导出规则斑图,推广了负扩散效应对空间斑图的形成具有巨大影响的结果.
In this paper, spatial pattern of SIR epidemic model with negative cross diffusion is considered. By performing a linear approach around the positive steady states of the model and Hopf bifurcation theorem, sufficient conditions are obtained for the Turing instability. And Turing region in which there are plenty of complicate spatial patterns is derived. Finally, some numerical simulations are given to certify that Turing patterns, such as spot, stripe and mixture of spot-stripe patterns. The regular pattern is induced by negative cross diffusion, which generalizes the results that negative cross diffusion has great influence on the spatial pattern formation.
In this paper, spatial pattern of SIR epidemic model with negative cross diffusion is considered. By performing a linear approach around the positive steady states of the model and Hopf bifurcation theorem, sufficient conditions are obtained for the Turing instability. And Turing region in which there are plenty of complicate spatial patterns is derived. Finally, some numerical simulations are given to certify that Turing patterns, such as spot, stripe and mixture of spot-stripe patterns. The regular pattern is induced by negative cross diffusion, which generalizes the results that negative cross diffusion has great influence on the spatial pattern formation.
引文
[1] Li Jing, Sun Guiquan, Jin Zhen. Pattern formation of an epidemic model with time delay[J]. Phys.Stat. Mech. Appl., 2014, 403(6):100-109.
[2] Wang Weiming, Liu Houye, Cai Yongli, Li Zhenqing. Turing pattern selection in a reaction-diffusion epidemic model[J]. Chinese Phys. B, 2011, 20(7):286-297.
[3]宋礼,靳祯,孙桂全.一具有非线性发生率传染病模型的稳定性和霍普夫分支[J].生物数学学报,2009,24(2):207-212.
[4]宋美,刘广臣,郭洪霞.一类具有非线性发生率的带时滞的SIR传染病模型的局部渐进稳定性[J].生物数学学报,2011, 26(2):255-262.
[5]马知恩,周义仓.常微分方程定性与稳定性方法[M].北京:科学出版社,2001.
[6]杜艳可,徐瑞.两类具有空间扩散的SIR传染病模型的斑图形成[J].工程数学学报,2010, 26(5):1168-1179.
[7] Sun Guiquan, Li Li, Jin Zhen. Effect of noise on the pattern formation in an epidemic model[J].Numer. Meth. Part. Diff. Equ., 2007, 34(10):2401-2408.
[8] Wang Yi, Wang Jianzhong, Zhang Li. Cross diffusion-induced pattern in an SI model[J]. Appl.Math. Comp., 2010, 217(5):1965-1970.
[9] Zheng Qianqian, Shen Jianwei. Pattern formation in the FitzHugh-Nagumo model[J]. Comp. Math.Appl., 2015, 70(5):1082-1097.
[10]欧阳颀.非线性科学和斑图动力学导论[M].北京:北京大学出版社,2010.
[11]曹玉林,马建萍,赵炎鑫.基于脉冲微分方程的移动自组网病毒传播免疫模型[J].四川大学学报(自然科学版),2016, 53(2):295-304.
[12] Zhao Hongyong, Zhang Xuebing, Huang Xuanxuan. Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion[J]. Appl. Math. Comp., 2015, 266(C):462-480.
[13] Su Ying, Wei Junjie, Shi Junping. Hopf bifurcations in a reaction-diffusion population model with delay effect[J]. J. Diff. Equ., 2009, 247(4):1156-1184.
[2] Wang Weiming, Liu Houye, Cai Yongli, Li Zhenqing. Turing pattern selection in a reaction-diffusion epidemic model[J]. Chinese Phys. B, 2011, 20(7):286-297.
[3]宋礼,靳祯,孙桂全.一具有非线性发生率传染病模型的稳定性和霍普夫分支[J].生物数学学报,2009,24(2):207-212.
[4]宋美,刘广臣,郭洪霞.一类具有非线性发生率的带时滞的SIR传染病模型的局部渐进稳定性[J].生物数学学报,2011, 26(2):255-262.
[5]马知恩,周义仓.常微分方程定性与稳定性方法[M].北京:科学出版社,2001.
[6]杜艳可,徐瑞.两类具有空间扩散的SIR传染病模型的斑图形成[J].工程数学学报,2010, 26(5):1168-1179.
[7] Sun Guiquan, Li Li, Jin Zhen. Effect of noise on the pattern formation in an epidemic model[J].Numer. Meth. Part. Diff. Equ., 2007, 34(10):2401-2408.
[8] Wang Yi, Wang Jianzhong, Zhang Li. Cross diffusion-induced pattern in an SI model[J]. Appl.Math. Comp., 2010, 217(5):1965-1970.
[9] Zheng Qianqian, Shen Jianwei. Pattern formation in the FitzHugh-Nagumo model[J]. Comp. Math.Appl., 2015, 70(5):1082-1097.
[10]欧阳颀.非线性科学和斑图动力学导论[M].北京:北京大学出版社,2010.
[11]曹玉林,马建萍,赵炎鑫.基于脉冲微分方程的移动自组网病毒传播免疫模型[J].四川大学学报(自然科学版),2016, 53(2):295-304.
[12] Zhao Hongyong, Zhang Xuebing, Huang Xuanxuan. Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion[J]. Appl. Math. Comp., 2015, 266(C):462-480.
[13] Su Ying, Wei Junjie, Shi Junping. Hopf bifurcations in a reaction-diffusion population model with delay effect[J]. J. Diff. Equ., 2009, 247(4):1156-1184.