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 & 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. doi: 10.1016/j.amc.2006.06.072. Google Scholar

[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. doi: 10.1016/j.amc.2011.10.015. Google Scholar

[3]

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

[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. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar

[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. doi: 10.3934/dcdsb.2012.17.2413. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[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. doi: 10.1016/j.jmaa.2009.09.017. Google Scholar

[11]

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

[12]

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

[13]

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

[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. doi: 10.1016/j.aml.2012.06.011. Google Scholar

[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. doi: 10.1016/j.cnsns.2011.09.039. Google Scholar

[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. doi: 10.1016/j.sysconle.2013.03.001. Google Scholar

[17]

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

[18]

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

[19]

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

[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. doi: 10.1016/S0252-9602(13)60003-X. Google Scholar

[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. doi: 10.1016/j.nonrwa.2011.11.016. Google Scholar

[22]

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

[23]

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

[24]

H. R. Thieme, Mathematics in Population Biology,, Princeton University Press, (2003). Google Scholar

[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. doi: 10.1142/S021833901250009X. Google Scholar

[26]

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

[27]

D. B. West, Introduction to Graph Theory,, Prentice Hall, (1996). Google Scholar

[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. doi: 10.1016/j.amc.2010.04.004. Google Scholar

[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. doi: 10.1016/j.nonrwa.2006.11.004. Google Scholar

[30]

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

[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. doi: 10.1016/j.apm.2012.10.032. Google Scholar

[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. doi: 10.1016/j.amc.2013.03.048. Google Scholar

[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. doi: 10.1002/mma.2879. Google Scholar

[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. Google Scholar

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. doi: 10.1016/j.amc.2006.06.072. Google Scholar

[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. doi: 10.1016/j.amc.2011.10.015. Google Scholar

[3]

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

[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. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar

[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. doi: 10.3934/dcdsb.2012.17.2413. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[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. doi: 10.1016/j.jmaa.2009.09.017. Google Scholar

[11]

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

[12]

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

[13]

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

[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. doi: 10.1016/j.aml.2012.06.011. Google Scholar

[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. doi: 10.1016/j.cnsns.2011.09.039. Google Scholar

[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. doi: 10.1016/j.sysconle.2013.03.001. Google Scholar

[17]

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

[18]

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

[19]

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

[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. doi: 10.1016/S0252-9602(13)60003-X. Google Scholar

[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. doi: 10.1016/j.nonrwa.2011.11.016. Google Scholar

[22]

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

[23]

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

[24]

H. R. Thieme, Mathematics in Population Biology,, Princeton University Press, (2003). Google Scholar

[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. doi: 10.1142/S021833901250009X. Google Scholar

[26]

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

[27]

D. B. West, Introduction to Graph Theory,, Prentice Hall, (1996). Google Scholar

[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. doi: 10.1016/j.amc.2010.04.004. Google Scholar

[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. doi: 10.1016/j.nonrwa.2006.11.004. Google Scholar

[30]

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

[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. doi: 10.1016/j.apm.2012.10.032. Google Scholar

[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. doi: 10.1016/j.amc.2013.03.048. Google Scholar

[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. doi: 10.1002/mma.2879. Google Scholar

[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. Google Scholar

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