January  2015, 20(1): 259-280. doi: 10.3934/dcdsb.2015.20.259

Graph-theoretic approach to stability of multi-group models with dispersal

1. 

Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China, China

Received  February 2014 Revised  August 2014 Published  November 2014

This paper is mainly concerned with the issue of stability for multi-group models with dispersal (MGMD). A system on multi-digraph is used to model the MGMD. The popular single graph-based method has been successfully generalized into multi-digraph-based approach. More precisely, by constructing a Lyapunov function for general MGMD, some simple yet less conservative conditions are derived for the stability of MGMD. Furthermore, the graph-theoretic method on multi-graph is successfully applied on predator-prey model with dispersal and coupled oscillators on two digraphs. Subsequently, numerical results are presented to demonstrate the effectiveness of the proposed new technique.
Citation: Chunmei Zhang, Wenxue Li, Ke Wang. Graph-theoretic approach to stability of multi-group models with dispersal. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 259-280. doi: 10.3934/dcdsb.2015.20.259
References:
[1]

F. Chen and A. Huang, On a nonautonomous predator-prey model with prey dispersal, Appl. Math. Comput., 184 (2007), 809-822. doi: 10.1016/j.amc.2006.06.072.

[2]

H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 4391-4400. doi: 10.1016/j.amc.2011.10.015.

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H. Chen and J. Sun, Stability analysis for coupled systems with time delay on networks, Physica A., 391 (2012), 528-534. doi: 10.1016/j.physa.2011.08.037.

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H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.

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H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discrete Contin. Dyn. Syst.-Ser. B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413.

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N. Hirano and S. Rybicki, Existence of limit cycles for coupled van der Pol equations, J. Differ. Equ., 195 (2003), 194-209. doi: 10.1016/S0022-0396(03)00212-2.

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C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A., 390 (2011), 1747-1762.

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C. Ji, D. Jiang, Q. Yang and N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044.

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Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. Biosci., 120 (1994), 77-98. doi: 10.1016/0025-5564(94)90038-8.

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M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47 doi: 10.1016/j.jmaa.2009.09.017.

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M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canad. Appl. Math. Quart., 17 (2009), 175-187.

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M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.

[13]

W. Li, H. Su and K. Wang, Global stability analysis for stochastic coupled systems on networks, Automatica, 47 (2011), 215-220. doi: 10.1016/j.automatica.2010.10.041.

[14]

W. Li, L. Pang, H. Su and K. Wang, Global stability for discrete Cohen-Grossberg neural networks with finite and infinite delays, Appl. Math. Lett., 25 (2012), 2246-2251. doi: 10.1016/j.aml.2012.06.011.

[15]

W. Li, H. Su, D. Wei and K. Wang, Global stability of coupled nonlinear systems with Markovian switching, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2609-2616. doi: 10.1016/j.cnsns.2011.09.039.

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W. Li, H. Song, Y. Qu and K. Wang, Global exponential stability for stochastic coupled systems on networks with Markovian switching, Syst. Control Lett., 62 (2013), 468-474. doi: 10.1016/j.sysconle.2013.03.001.

[17]

L. Liu, W. Cai and Y. Wu, Global dynamics for an SIR patchy model with susceptibles dispersal, Adv. Differ. Equ., 131 (2012), 1-11. doi: 10.1186/1687-1847-2012-131.

[18]

A. L. Lloyd and V. A. A. Jansen, Spatiotemporal dynamics of epidemics: Synchrony in metapopulation models, Math. Biosci., 188 (2004), 1-16. doi: 10.1016/j.mbs.2003.09.003.

[19]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473.

[20]

Y. Muroya, Y. Enatsu and T. Kuniya, Global stability of extended multi-group sir epidemic models with patches through migration and cross patch infection, Acta Math. Sci., 33 (2013), 341-361. doi: 10.1016/S0252-9602(13)60003-X.

[21]

H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016.

[22]

H. Su, W. Li and K. Wang, Global stability analysis of discrete-time coupled systems on networks and its applications, Chaos, 22 (2012), 033135. doi: 10.1063/1.4748851.

[23]

J. Suo, J. Sun and Y. Zhang, Stability analysis for impulsive coupled systems on networks, Neurocomputing, 99 (2013), 172-177. doi: 10.1016/j.neucom.2012.06.002.

[24]

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.

[25]

J. Wang, J. Zu, X. Liu, G. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate, J. Biol. Syst., 20 (2012), 235-258. doi: 10.1142/S021833901250009X.

[26]

W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001.

[27]

D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.

[28]

C. Xu, X. Tang and M. Liao, Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments, Appl. Math. Comput., 216 (2010), 2920-2936. doi: 10.1016/j.amc.2010.04.004.

[29]

R. Xu and Z. Ma, The effect of dispersal on the permanence of a predator-prey system with time delay, Nonlinear Anal. RWA, 9 (2008), 354-369. doi: 10.1016/j.nonrwa.2006.11.004.

[30]

Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Anal. RWA, 14 (2013), 1434-1456. doi: 10.1016/j.nonrwa.2012.10.007.

[31]

C. Zhang, W. Li and K. Wang, Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling, Appl. Math. Model., 37 (2013), 5394-5402. doi: 10.1016/j.apm.2012.10.032.

[32]

C. Zhang, W. Li, H. Su and K. Wang, A graph-theoretic approach to boundedness of stochastic Cohen-Grossberg neural networks with Markovian switching, Appl. Math. Comput., 219 (2013), 9165-9173. doi: 10.1016/j.amc.2013.03.048.

[33]

C. Zhang, W. Li and K. Wang, A graph-theoretic approach to stability of neutral stochastic coupled oscillators network with time-varying delayed coupling, Math. Meth. Appl. Sci., 37 (2014), 1179-1190. doi: 10.1002/mma.2879.

[34]

L. Zu, D. Jiang and F. Jiang, Existence, stationary distribution, and extinction of predator-prey system of prey dispersal with stochastic perturbation, Abstract Appl. Anal., 2012 (2012), 1-24.

show all references

References:
[1]

F. Chen and A. Huang, On a nonautonomous predator-prey model with prey dispersal, Appl. Math. Comput., 184 (2007), 809-822. doi: 10.1016/j.amc.2006.06.072.

[2]

H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 4391-4400. doi: 10.1016/j.amc.2011.10.015.

[3]

H. Chen and J. Sun, Stability analysis for coupled systems with time delay on networks, Physica A., 391 (2012), 528-534. doi: 10.1016/j.physa.2011.08.037.

[4]

H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.

[5]

H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discrete Contin. Dyn. Syst.-Ser. B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413.

[6]

N. Hirano and S. Rybicki, Existence of limit cycles for coupled van der Pol equations, J. Differ. Equ., 195 (2003), 194-209. doi: 10.1016/S0022-0396(03)00212-2.

[7]

C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A., 390 (2011), 1747-1762.

[8]

C. Ji, D. Jiang, Q. Yang and N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044.

[9]

Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. Biosci., 120 (1994), 77-98. doi: 10.1016/0025-5564(94)90038-8.

[10]

M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47 doi: 10.1016/j.jmaa.2009.09.017.

[11]

M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canad. Appl. Math. Quart., 17 (2009), 175-187.

[12]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.

[13]

W. Li, H. Su and K. Wang, Global stability analysis for stochastic coupled systems on networks, Automatica, 47 (2011), 215-220. doi: 10.1016/j.automatica.2010.10.041.

[14]

W. Li, L. Pang, H. Su and K. Wang, Global stability for discrete Cohen-Grossberg neural networks with finite and infinite delays, Appl. Math. Lett., 25 (2012), 2246-2251. doi: 10.1016/j.aml.2012.06.011.

[15]

W. Li, H. Su, D. Wei and K. Wang, Global stability of coupled nonlinear systems with Markovian switching, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2609-2616. doi: 10.1016/j.cnsns.2011.09.039.

[16]

W. Li, H. Song, Y. Qu and K. Wang, Global exponential stability for stochastic coupled systems on networks with Markovian switching, Syst. Control Lett., 62 (2013), 468-474. doi: 10.1016/j.sysconle.2013.03.001.

[17]

L. Liu, W. Cai and Y. Wu, Global dynamics for an SIR patchy model with susceptibles dispersal, Adv. Differ. Equ., 131 (2012), 1-11. doi: 10.1186/1687-1847-2012-131.

[18]

A. L. Lloyd and V. A. A. Jansen, Spatiotemporal dynamics of epidemics: Synchrony in metapopulation models, Math. Biosci., 188 (2004), 1-16. doi: 10.1016/j.mbs.2003.09.003.

[19]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473.

[20]

Y. Muroya, Y. Enatsu and T. Kuniya, Global stability of extended multi-group sir epidemic models with patches through migration and cross patch infection, Acta Math. Sci., 33 (2013), 341-361. doi: 10.1016/S0252-9602(13)60003-X.

[21]

H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016.

[22]

H. Su, W. Li and K. Wang, Global stability analysis of discrete-time coupled systems on networks and its applications, Chaos, 22 (2012), 033135. doi: 10.1063/1.4748851.

[23]

J. Suo, J. Sun and Y. Zhang, Stability analysis for impulsive coupled systems on networks, Neurocomputing, 99 (2013), 172-177. doi: 10.1016/j.neucom.2012.06.002.

[24]

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.

[25]

J. Wang, J. Zu, X. Liu, G. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate, J. Biol. Syst., 20 (2012), 235-258. doi: 10.1142/S021833901250009X.

[26]

W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001.

[27]

D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.

[28]

C. Xu, X. Tang and M. Liao, Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments, Appl. Math. Comput., 216 (2010), 2920-2936. doi: 10.1016/j.amc.2010.04.004.

[29]

R. Xu and Z. Ma, The effect of dispersal on the permanence of a predator-prey system with time delay, Nonlinear Anal. RWA, 9 (2008), 354-369. doi: 10.1016/j.nonrwa.2006.11.004.

[30]

Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Anal. RWA, 14 (2013), 1434-1456. doi: 10.1016/j.nonrwa.2012.10.007.

[31]

C. Zhang, W. Li and K. Wang, Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling, Appl. Math. Model., 37 (2013), 5394-5402. doi: 10.1016/j.apm.2012.10.032.

[32]

C. Zhang, W. Li, H. Su and K. Wang, A graph-theoretic approach to boundedness of stochastic Cohen-Grossberg neural networks with Markovian switching, Appl. Math. Comput., 219 (2013), 9165-9173. doi: 10.1016/j.amc.2013.03.048.

[33]

C. Zhang, W. Li and K. Wang, A graph-theoretic approach to stability of neutral stochastic coupled oscillators network with time-varying delayed coupling, Math. Meth. Appl. Sci., 37 (2014), 1179-1190. doi: 10.1002/mma.2879.

[34]

L. Zu, D. Jiang and F. Jiang, Existence, stationary distribution, and extinction of predator-prey system of prey dispersal with stochastic perturbation, Abstract Appl. Anal., 2012 (2012), 1-24.

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