American Institute of Mathematical Sciences

Advanced
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)

Yoshiaki Muroya 1,

1. 

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

Received  December 2013 Revised  March 2015 Published  July 2015

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, 1967.  Google Scholar

[2]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.  Google Scholar

[3]

F. Chen, The permanence and global attractivity of Lotka-Volterra competition system witheedback controls, Nonlinear Anal. RWA, 7 (2006), 133-143. doi: 10.1016/j.nonrwa.2005.01.006.  Google Scholar

[4]

T. Faria, Asymptotic behabiour for a class of delayed cooperative models with patch structure, Discrete and cont. Dynam. Sys. Series B, 18 (2013), 1567-1579. doi: 10.3934/dcdsb.2013.18.1567.  Google Scholar

[5]

T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure, Appl. Math. Comput., 245 (2014), 575-590. doi: 10.1016/j.amc.2014.08.009.  Google Scholar

[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), 301-330. doi: 10.1017/S0308210513001194.  Google Scholar

[7]

A. Hastings, Spatial heterogeneity and the stability of predator prey systems, Theoret. Popul. Biol., 12 (1977), 37-48. doi: 10.1016/0040-5809(77)90034-X.  Google Scholar

[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), 1105-1118. doi: 10.3934/dcdsb.2014.19.1105.  Google Scholar

[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), 402-413. doi: 10.1016/j.nonrwa.2012.07.004.  Google Scholar

[10]

Y. Muroya, Global stability of a delayed nonlinear Lotka-Volterra system with feedback controls and patch structure, Appl. Math. Comput., 239 (2014), 60-73. doi: 10.1016/j.amc.2014.04.036.  Google Scholar

[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), 341-361. doi: 10.1016/S0252-9602(13)60003-X.  Google Scholar

[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), 415-439. doi: 10.1016/j.jmaa.2014.12.019.  Google Scholar

[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), 295-306. doi: 10.1016/j.mcm.2008.05.004.  Google Scholar

[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), 2214-2226. doi: 10.1016/j.nonrwa.2012.01.016.  Google Scholar

[15]

H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.  Google Scholar

[16]

H. L. Smith and P. Waltman, The Theory of The Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[17]

Y. Takeuchi, W. Wang and Y. Saito, Global stability of population models with patch structure, Nonlinear Analysis RWA, 7 (2006), 235-247. doi: 10.1016/j.nonrwa.2005.02.005.  Google Scholar

show all references

References:
[1]

N. P. Bhatia and G. P. Szegö, Dynamical Systems, Stability Theory and Applications//Lecture Notes in Mathematics. 35. Berlin: Springer, 1967.  Google Scholar

[2]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.  Google Scholar

[3]

F. Chen, The permanence and global attractivity of Lotka-Volterra competition system witheedback controls, Nonlinear Anal. RWA, 7 (2006), 133-143. doi: 10.1016/j.nonrwa.2005.01.006.  Google Scholar

[4]

T. Faria, Asymptotic behabiour for a class of delayed cooperative models with patch structure, Discrete and cont. Dynam. Sys. Series B, 18 (2013), 1567-1579. doi: 10.3934/dcdsb.2013.18.1567.  Google Scholar

[5]

T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure, Appl. Math. Comput., 245 (2014), 575-590. doi: 10.1016/j.amc.2014.08.009.  Google Scholar

[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), 301-330. doi: 10.1017/S0308210513001194.  Google Scholar

[7]

A. Hastings, Spatial heterogeneity and the stability of predator prey systems, Theoret. Popul. Biol., 12 (1977), 37-48. doi: 10.1016/0040-5809(77)90034-X.  Google Scholar

[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), 1105-1118. doi: 10.3934/dcdsb.2014.19.1105.  Google Scholar

[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), 402-413. doi: 10.1016/j.nonrwa.2012.07.004.  Google Scholar

[10]

Y. Muroya, Global stability of a delayed nonlinear Lotka-Volterra system with feedback controls and patch structure, Appl. Math. Comput., 239 (2014), 60-73. doi: 10.1016/j.amc.2014.04.036.  Google Scholar

[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), 341-361. doi: 10.1016/S0252-9602(13)60003-X.  Google Scholar

[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), 415-439. doi: 10.1016/j.jmaa.2014.12.019.  Google Scholar

[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), 295-306. doi: 10.1016/j.mcm.2008.05.004.  Google Scholar

[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), 2214-2226. doi: 10.1016/j.nonrwa.2012.01.016.  Google Scholar

[15]

H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.  Google Scholar

[16]

H. L. Smith and P. Waltman, The Theory of The Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[17]

Y. Takeuchi, W. Wang and Y. Saito, Global stability of population models with patch structure, Nonlinear Analysis RWA, 7 (2006), 235-247. doi: 10.1016/j.nonrwa.2005.02.005.  Google Scholar

[1]

Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177
[2]

Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105
[3]

Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
[4]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057
[5]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[6]

Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151
[7]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[8]

Yasuhisa Saito. A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays. Conference Publications, 2003, 2003 (Special) : 771-777. doi: 10.3934/proc.2003.2003.771
[9]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977
[10]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083
[11]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021071
[12]

Jinliang Wang, Hongying Shu. Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences & Engineering, 2016, 13 (1) : 209-225. doi: 10.3934/mbe.2016.13.209
[13]

Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056
[14]

Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99
[15]

Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109
[16]

Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064
[17]

Hélène Leman, Sylvie Méléard, Sepideh Mirrahimi. Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 469-493. doi: 10.3934/dcdsb.2015.20.469
[18]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059
[19]

Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027
[20]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences