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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.

34D23Global stability of ODE
92D25Population dynamics (general)
34C40ODE on manifolds
Full Text: DOI
[1] , The handbook of brain theory and neural network (1989)
[2] Awrejcewicz, J.: Bifurcation and chaos in coupled oscillators, (1991) · 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 · doi:10.1007/BF00169563
[5] Berman, A.; Plemmons, R. J.: Nonnegative matrices in the mathematical sciences, (1979) · Zbl 0484.15016
[6] Bhatia, N. P.; Szegö, G. P.: Dynamical systems: stability theory and applications, Lecture notes in math. 35 (1967) · Zbl 0155.42201
[7] Bishop, C.: Neural networks for pattern recognition, (1995) · Zbl 0868.68096
[8] Brauer, F.; Castillo-Chavez, C.: Mathematical models in population biology and epidemiology, Texts appl. Math. 40 (2001) · Zbl 1302.92001
[9] Chow, S. -N.; Conti, R.; Johnson, R.; Mallet-Paret, J.; Nussbaum, R.: Dynamical systems, Lecture notes in math. 1822 (2003)
[10] Chua, L. O.; Roska, T.: Cellular neural networks and visual computing: foundations and applications, (2002)
[11] Diekmann, O.; Heesterbeek, J. A. P.; Metz, J. A. J.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. math. Biol. 28, 365-382 (1990) · Zbl 0726.92018 · doi:10.1007/BF00178324
[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 · doi:10.1007/s000330050151
[13] Freedman, H. I.: Deterministic mathematical models in population ecology, (1980) · 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 · doi:10.1090/S0002-9939-08-09341-6
[16] Hale, J. K.; Lunel, S. M. Verduyn: Introduction to functional differential equations, Appl. math. Sci. 99 (1993) · Zbl 0787.34002
[17] Harary, F.: Graph theory, (1969) · Zbl 0182.57702
[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 · doi:10.1007/BF00275160
[19] Hoppensteadt, F. C.; Izhikevich, E. M.: Weakly connected neural networks, Appl. math. Sci. 126 (1997) · 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) · Zbl 0895.68055
[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 · doi:10.1016/0025-5564(94)90038-8
[24] Lasalle, J. P.: The stability of dynamical systems, CBMS-NSF regional conf. Ser. in appl. Math. (1976)
[25] Li, M. Y.; Muldowney, J. S.: Global stability for the SEIR model in epidemiology, Math. biosci. 125, 155-164 (1995) · Zbl 0821.92022 · doi:10.1016/0025-5564(95)92756-5
[26] Li, M. Y.; Wang, L.: Global stability in some SEIR epidemic models, IMA vol. Math. appl. 126, 295-311 (2002) · 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 · doi:10.1007/BF00160375
[28] May, R. M.: Stability and complexity in model ecosystems, (2001) · Zbl 1044.92047
[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 · doi:10.1016/j.nonrwa.2008.10.014
[30] Moon, J. W.: Counting labelled tress, (1970) · 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 · doi:10.1016/0362-546X(81)90108-5
[32] Smith, H. L.; Waltman, P.: The theory of the chemostat: dynamics of microbial competition, (1995) · Zbl 0860.92031
[33] Solé, R. V.; Bascompte, J.: Self-organization in complex ecosystems, (2006)
[34] Thieme, H. R.: Mathematics in population biology, (2003) · Zbl 1054.92042
[35] Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29-48 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[36] West, D. B.: Introduction to graph theory, (1996) · Zbl 0845.05001