American Institute of Mathematical Sciences

Advanced
August  2013, 18(6): 1567-1579. doi: 10.3934/dcdsb.2013.18.1567

Asymptotic behaviour for a class of delayed cooperative models with patch structure

Teresa Faria 1,

1. 

Departamento de Matemática and CMAF, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

Received  September 2011 Revised  January 2012 Published  March 2013

For a class of cooperative population models with patch structure and multiple discrete delays, we give conditions for the absolute global asymptotic stability of both the trivial solution and -- when it exists -- a positive equilibrium. Under a sublinearity condition, sharper results are obtained. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. As a by-product, we obtain a criterion for the existence of positive traveling wave solutions for an associated reaction-diffusion model with patch structure. Our results improve and generalize criteria in the recent literature.
Keywords: Delay differential equation, patch structure, global asymptotic stability, heteroclinic solution..
Mathematics Subject Classification: Primary: 34K20; Secondary: 34K25, 35C07, 92D2.
Citation: Teresa Faria. Asymptotic behaviour for a class of delayed cooperative models with patch structure. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1567-1579. doi: 10.3934/dcdsb.2013.18.1567
References:
[1]

T. Faria, Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays,, Nonlinear Anal., 74 (2011), 7033.  doi: 10.1016/j.na.2011.07.024.  Google Scholar

[2]

T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks,, J. Differential Equations, 244 (2008), 1049.  doi: 10.1016/j.jde.2007.12.005.  Google Scholar

[3]

T. Faria and S. Trofimchuk, Positive travelling fronts for reaction-diffusion systems with distributed delay,, Nonlinearity, 23 (2010), 2457.  doi: 10.1088/0951-7715/23/10/006.  Google Scholar

[4]

M. Fiedler, "Special Matrices and Their Applications in Numerical Mathematics,", Martinus Nijhoff Publ., (1986).  doi: 10.1007/978-94-009-4335-3.  Google Scholar

[5]

B. Liu, Global stability of a class of delay differential systems,, J. Comput. Appl. Math., 233 (2009), 217.  doi: 10.1016/j.cam.2009.07.024.  Google Scholar

[6]

H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Mathematical Surveys and Monographs, 41 (1995).   Google Scholar

[7]

H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition,", Cambridge Studies in Mathematical Biology, 13 (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[8]

Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of delayed population model with dispersal loss,, Math. Biosci., 201 (2006), 143.  doi: 10.1016/j.mbs.2005.12.012.  Google Scholar

[9]

Y. Takeuchi, W. Wang and Y. Saito, Global stability of population models with patch structure,, Nonlinear Anal. Real World Appl., 7 (2006), 235.  doi: 10.1016/j.nonrwa.2005.02.005.  Google Scholar

[10]

W. Wang, P. Fergola and C. Tenneriello, Global attractivity of periodic solutions of population models,, J. Math. Anal. Appl., 211 (1997), 498.  doi: 10.1006/jmaa.1997.5484.  Google Scholar

[11]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations,, Cann. Appl. Math. Quart., 4 (1996), 421.   Google Scholar

show all references

References:
[1]

T. Faria, Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays,, Nonlinear Anal., 74 (2011), 7033.  doi: 10.1016/j.na.2011.07.024.  Google Scholar

[2]

T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks,, J. Differential Equations, 244 (2008), 1049.  doi: 10.1016/j.jde.2007.12.005.  Google Scholar

[3]

T. Faria and S. Trofimchuk, Positive travelling fronts for reaction-diffusion systems with distributed delay,, Nonlinearity, 23 (2010), 2457.  doi: 10.1088/0951-7715/23/10/006.  Google Scholar

[4]

M. Fiedler, "Special Matrices and Their Applications in Numerical Mathematics,", Martinus Nijhoff Publ., (1986).  doi: 10.1007/978-94-009-4335-3.  Google Scholar

[5]

B. Liu, Global stability of a class of delay differential systems,, J. Comput. Appl. Math., 233 (2009), 217.  doi: 10.1016/j.cam.2009.07.024.  Google Scholar

[6]

H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Mathematical Surveys and Monographs, 41 (1995).   Google Scholar

[7]

H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition,", Cambridge Studies in Mathematical Biology, 13 (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[8]

Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of delayed population model with dispersal loss,, Math. Biosci., 201 (2006), 143.  doi: 10.1016/j.mbs.2005.12.012.  Google Scholar

[9]

Y. Takeuchi, W. Wang and Y. Saito, Global stability of population models with patch structure,, Nonlinear Anal. Real World Appl., 7 (2006), 235.  doi: 10.1016/j.nonrwa.2005.02.005.  Google Scholar

[10]

W. Wang, P. Fergola and C. Tenneriello, Global attractivity of periodic solutions of population models,, J. Math. Anal. Appl., 211 (1997), 498.  doi: 10.1006/jmaa.1997.5484.  Google Scholar

[11]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations,, Cann. Appl. Math. Quart., 4 (1996), 421.   Google Scholar

[1]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[2]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[3]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044
[4]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[5]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025
[6]

Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031
[7]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407
[8]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172
[9]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[10]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159
[11]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[12]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[13]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
[14]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[15]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[16]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[17]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017
[18]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147
[19]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[20]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (2959)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences