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), 7033-7046. 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

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

[1]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[2]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[3]

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
[4]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991
[5]

Sze-Bi Hsu, Ming-Chia Li, Weishi Liu, Mikhail Malkin. Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1465-1492. doi: 10.3934/dcds.2003.9.1465
[6]

Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2417-2442. doi: 10.3934/dcdsb.2018259
[7]

A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373-380. doi: 10.3934/proc.2011.2011.373
[8]

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
[9]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[10]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[11]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3659-3683. doi: 10.3934/dcdss.2021023
[12]

Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053
[13]

Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139
[14]

Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks & Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002
[15]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040
[16]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111
[17]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[18]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215
[19]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401
[20]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (3023)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences