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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), 7033-7046. 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), 1049-1079. 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), 2457-2481. doi: 10.1088/0951-7715/23/10/006.  Google Scholar

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B. Liu, Global stability of a class of delay differential systems, J. Comput. Appl. Math., 233 (2009), 217-223. doi: 10.1016/j.cam.2009.07.024.  Google Scholar

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H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, Amer. Math. Soc., Providence, RI, 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, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

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Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of delayed population model with dispersal loss, Math. Biosci., 201 (2006), 143-156. 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), 235-247. 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), 498-511. doi: 10.1006/jmaa.1997.5484.  Google Scholar

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X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Cann. Appl. Math. Quart., 4 (1996), 421-444.  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), 7033-7046. 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), 1049-1079. 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), 2457-2481. doi: 10.1088/0951-7715/23/10/006.  Google Scholar

[4]

M. Fiedler, "Special Matrices and Their Applications in Numerical Mathematics," Martinus Nijhoff Publ., Dordrecht, 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), 217-223. 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, Amer. Math. Soc., Providence, RI, 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, Cambridge University Press, Cambridge, 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), 143-156. 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), 235-247. 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), 498-511. 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), 421-444.  Google Scholar

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