Global-stability problem for coupled systems of differential equations on networks. (English) Zbl 1190.34063

Authors’ abstract: The global stability problem of equilibria is investigated for coupled systems of differential equations on networks. Using results from graph theory, we develop a systematic approach that allows one to construct global Lyapunov functions for building blocks of individual vertex systems. The approach is applied to several classes of coupled systems in engineering, ecology and epidemiology, and is shown to improve existing results.


34D23 Global stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
92D30 Epidemiology
34C40 Ordinary differential equations and systems on manifolds


Full Text: DOI


[1] ()
[2] Awrejcewicz, J., Bifurcation and chaos in coupled oscillators, (1991), World Scientific New Jersey · Zbl 0824.58034
[3] Beretta, E.; Takeuchi, Y., Global stability of single-species diffusion Volterra models with continuous time delays, Bull. math. biol., 49, 431-448, (1987) · Zbl 0627.92021
[4] Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J. math. biol., 33, 250-260, (1995) · Zbl 0811.92019
[5] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1979), Academic Press New York · Zbl 0484.15016
[6] Bhatia, N.P.; Szegö, G.P., Dynamical systems: stability theory and applications, Lecture notes in math., vol. 35, (1967), Springer Berlin · Zbl 0993.37001
[7] Bishop, C., Neural networks for pattern recognition, (1995), Oxford University Press Oxford
[8] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology, Texts appl. math., vol. 40, (2001), Springer Berlin · Zbl 0967.92015
[9] Chow, S.-N.; Conti, R.; Johnson, R.; Mallet-Paret, J.; Nussbaum, R., Dynamical systems, Lecture notes in math., vol. 1822, (2003), Springer Berlin
[10] Chua, L.O.; Roska, T., Cellular neural networks and visual computing: foundations and applications, (2002), Cambridge University Press Cambridge
[11] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J., On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365-382, (1990) · Zbl 0726.92018
[12] Fiedler, B.; Belhaq, M.; Houssni, M., Basins of attraction in strongly damped coupled mechanical oscillators: A global example, Z. angew. math. phys., 50, 282-300, (1999) · Zbl 0919.70012
[13] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023
[14] Guo, H.; Li, M.Y.; Shuai, Z., Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. appl. math. Q., 14, 259-284, (2006) · Zbl 1148.34039
[15] Guo, H.; Li, M.Y.; Shuai, Z., A graph-theoretic approach to the method of global Lyapunov functions, Proc. amer. math. soc., 136, 2793-2802, (2008) · Zbl 1155.34028
[16] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, Appl. math. sci., vol. 99, (1993), Springer New York · Zbl 0787.34002
[17] Harary, F., Graph theory, (1969), Addison-Wesley Reading · Zbl 0797.05064
[18] Hastings, A., Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal rates, J. math. biol., 16, 49-55, (1982) · Zbl 0496.92010
[19] Hoppensteadt, F.C.; Izhikevich, E.M., Weakly connected neural networks, Appl. math. sci., vol. 126, (1997), Springer New York · Zbl 0887.92003
[20] Hsu, S.B., On global stability of a predator – prey systems, Math. biosci., 39, 1-10, (1978) · Zbl 0383.92014
[21] Knuth, D.E., The art of computer programming, vol. 1, (1997), Addison-Wesley Reading · Zbl 0191.17903
[22] Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence, Bull. math. biol., 69, 1871-1886, (2007) · Zbl 1298.92101
[23] Kuang, Y.; Takeuchi, Y., Predator – prey dynamics in models of prey dispersal in two-patch environments, Math. biosci., 120, 77-98, (1994) · Zbl 0793.92014
[24] LaSalle, J.P., The stability of dynamical systems, CBMS-NSF regional conf. ser. in appl. math., (1976), SIAM Philadelphia · Zbl 0364.93002
[25] Li, M.Y.; Muldowney, J.S., Global stability for the SEIR model in epidemiology, Math. biosci., 125, 155-164, (1995) · Zbl 0821.92022
[26] Li, M.Y.; Wang, L., Global stability in some SEIR epidemic models, (), 295-311 · Zbl 1022.92035
[27] Lu, Z.; Takeuchi, Y., Global asymptotic behavior in single-species discrete diffusion systems, J. math. biol., 32, 67-77, (1993) · Zbl 0799.92014
[28] May, R.M., Stability and complexity in model ecosystems, (2001), Princeton University Press Princeton
[29] C.C. McCluskey, Complete global stability for an SIR epidemic model with delay – distributed or discrete, Nonlinear Anal. Real World Appl., in press · Zbl 1185.37209
[30] Moon, J.W., Counting labelled tress, (1970), Canadian Mathematical Congress Montreal · Zbl 0214.23204
[31] Redheffer, R.; Zhou, Z., Global asymptotic stability for a class of many-variable Volterra prey – predator systems, Nonlinear anal., 5, 1309-1329, (1981) · Zbl 0485.92015
[32] Smith, H.L.; Waltman, P., The theory of the chemostat: dynamics of microbial competition, (1995), Cambridge University Press Cambridge · Zbl 0860.92031
[33] Solé, R.V.; Bascompte, J., Self-organization in complex ecosystems, (2006), Princeton University Press Princeton
[34] Thieme, H.R., Mathematics in population biology, (2003), Princeton University Press Princeton · Zbl 1054.92042
[35] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29-48, (2002) · Zbl 1015.92036
[36] West, D.B., Introduction to graph theory, (1996), Prentice Hall Upper Saddle River · Zbl 0845.05001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.