A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model)

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October 2015, 8(5): 999-1008. doi: 10.3934/dcdss.2015.8.999

A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model)

1.	Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555

Received December 2013 Revised March 2015 Published July 2015

Abstract
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In this paper, for a Lotka-Volterra system with infinite delays and patch structure related to a multi-group SI epidemic model, applying Lyapunov functional techniques without using the form of diagonal dominance of the instantaneous negative terms over the infinite delay terms, we establish the complete global dynamics by a threshold parameter $s(M(0))$, that is, the trivial equilibrium is globally asymptotically stable if $s(M(0)) \leq 0$ and the positive equilibrium is globally asymptotically stable if $s(M(0))>0$, respectively. This offer new type condition of global stability for Lotka-Volterra systems with patch structure.

Keywords: Lyapunov functional, global stability, Lotka-Volterra system, patch structure, multi-group SI epidemic model..

Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 92D3.

Citation: Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999

References:

[1]	N. P. Bhatia and G. P. Szegö, Dynamical Systems,, Stability Theory and Applications//Lecture Notes in Mathematics. 35. Berlin: Springer, 35 (1967).
[2]	A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).
[3]	F. Chen, The permanence and global attractivity of Lotka-Volterra competition system witheedback controls,, Nonlinear Anal. RWA, 7 (2006), 133. doi: 10.1016/j.nonrwa.2005.01.006.
[4]	T. Faria, Asymptotic behabiour for a class of delayed cooperative models with patch structure,, Discrete and cont. Dynam. Sys. Series B, 18 (2013), 1567. doi: 10.3934/dcdsb.2013.18.1567.
[5]	T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure,, Appl. Math. Comput., 245* (2014), 575. doi: 10.1016/j.amc.2014.08.009.*
[6]	T. Faria and Y. Muroya, Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls,, Proceedings of the Royal Society of Edinburgh: Section A, 145* (2015), 301. doi: 10.1017/S0308210513001194.*
[7]	A. Hastings, Spatial heterogeneity and the stability of predator prey systems,, Theoret. Popul. Biol., 12 (1977), 37. doi: 10.1016/0040-5809(77)90034-X.
[8]	T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration,, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 1105. doi: 10.3934/dcdsb.2014.19.1105.
[9]	Z. Li, M. Han and F. Chen, Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays,, Nonlinear Analysis, 14 (2013), 402. doi: 10.1016/j.nonrwa.2012.07.004.
[10]	Y. Muroya, Global stability of a delayed nonlinear Lotka-Volterra system with feedback controls and patch structure,, Appl. Math. Comput., 239 (2014), 60. doi: 10.1016/j.amc.2014.04.036.
[11]	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 Mathematica Scientia, 33 (2013), 341. doi: 10.1016/S0252-9602(13)60003-X.
[12]	Y. Muroya, T. Kuniya and J. Wang, A delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure,, J. Math. Anal. Appl., 425 (2015), 415. doi: 10.1016/j.jmaa.2014.12.019.
[13]	L. Nie, J. Peng and Z. Teng, Permanence and stability in multi-species non-autonomous Lotka-Volterra competitive systems with delays and feedback controls,, Math. Comput. Modelling, 49 (2009), 295. doi: 10.1016/j.mcm.2008.05.004.
[14]	C. Shi, Z. Li and F. Chen, Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls,, Nonlinear Analysis, 13 (2012), 2214. doi: 10.1016/j.nonrwa.2012.01.016.
[15]	H. L. Smith, Monotone Dynamical Systems,, An Introduction to the Theory of Competitive and Cooperative Systems, (1995).
[16]	H. L. Smith and P. Waltman, The Theory of The Chemostat,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043.
[17]	Y. Takeuchi, W. Wang and Y. Saito, Global stability of population models with patch structure,, Nonlinear Analysis RWA, 7 (2006), 235. doi: 10.1016/j.nonrwa.2005.02.005.

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References:

[1]	N. P. Bhatia and G. P. Szegö, Dynamical Systems,, Stability Theory and Applications//Lecture Notes in Mathematics. 35. Berlin: Springer, 35 (1967).
[2]	A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).
[3]	F. Chen, The permanence and global attractivity of Lotka-Volterra competition system witheedback controls,, Nonlinear Anal. RWA, 7 (2006), 133. doi: 10.1016/j.nonrwa.2005.01.006.
[4]	T. Faria, Asymptotic behabiour for a class of delayed cooperative models with patch structure,, Discrete and cont. Dynam. Sys. Series B, 18 (2013), 1567. doi: 10.3934/dcdsb.2013.18.1567.
[5]	T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure,, Appl. Math. Comput., 245* (2014), 575. doi: 10.1016/j.amc.2014.08.009.*
[6]	T. Faria and Y. Muroya, Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls,, Proceedings of the Royal Society of Edinburgh: Section A, 145* (2015), 301. doi: 10.1017/S0308210513001194.*
[7]	A. Hastings, Spatial heterogeneity and the stability of predator prey systems,, Theoret. Popul. Biol., 12 (1977), 37. doi: 10.1016/0040-5809(77)90034-X.
[8]	T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration,, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 1105. doi: 10.3934/dcdsb.2014.19.1105.
[9]	Z. Li, M. Han and F. Chen, Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays,, Nonlinear Analysis, 14 (2013), 402. doi: 10.1016/j.nonrwa.2012.07.004.
[10]	Y. Muroya, Global stability of a delayed nonlinear Lotka-Volterra system with feedback controls and patch structure,, Appl. Math. Comput., 239 (2014), 60. doi: 10.1016/j.amc.2014.04.036.
[11]	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 Mathematica Scientia, 33 (2013), 341. doi: 10.1016/S0252-9602(13)60003-X.
[12]	Y. Muroya, T. Kuniya and J. Wang, A delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure,, J. Math. Anal. Appl., 425 (2015), 415. doi: 10.1016/j.jmaa.2014.12.019.
[13]	L. Nie, J. Peng and Z. Teng, Permanence and stability in multi-species non-autonomous Lotka-Volterra competitive systems with delays and feedback controls,, Math. Comput. Modelling, 49 (2009), 295. doi: 10.1016/j.mcm.2008.05.004.
[14]	C. Shi, Z. Li and F. Chen, Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls,, Nonlinear Analysis, 13 (2012), 2214. doi: 10.1016/j.nonrwa.2012.01.016.
[15]	H. L. Smith, Monotone Dynamical Systems,, An Introduction to the Theory of Competitive and Cooperative Systems, (1995).
[16]	H. L. Smith and P. Waltman, The Theory of The Chemostat,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043.
[17]	Y. Takeuchi, W. Wang and Y. Saito, Global stability of population models with patch structure,, Nonlinear Analysis RWA, 7 (2006), 235. doi: 10.1016/j.nonrwa.2005.02.005.

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