The paper concerns a class of n-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Bélair, J., Mackey, M., Mahaffy, J.: Age-structured and two-delays models for erythropoiesis. Math. Biosci. 241, 109–119 (2006)
Berezansky, L., Braverman, E.: Boundedness and persistence of delay differential equations with mixed nonlinearity. Appl. Math. Comput. 279, 154–169 (2016)
Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model. 34, 1405–1417 (2010)
Berezansky, L., Idels, L., Troib, L.: Global dynamics of Nicholson-type delay systems with applications. Nonlinear Anal. RWA 12, 436–445 (2011)
Coppel, W.A.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978)
Cushing, J.M.: An Introduction to Structured Population Dynamics, Conference Series in Applied Mathematics, vol. 71. SIAM, Philadelphia (1998)
Faria, T.: Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays. Nonlinear Anal. 74, 7033–7046 (2011)
Faria, T.: Global dynamics for Lotka-Volterra systems with infinite delay and patch structure. Appl. Math. Comput. 245, 575–590 (2014)
Faria, T.: Persistence and permanence for a class of functional differential equations with infinite delay. J. Dyn. Differ. Equ. 28, 1163–1186 (2016)
Faria, T., Röst, G.: Persistence, permanence and global stability of an \(n\)-dimensional Nicholson system. J. Dyn. Differ. Equ. 26, 723–744 (2014)
Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff Publ. Kluwer, Dordrechit (1986)
Fink, A.M.: Almost Periodic Differential Equations. Lecture Notes in Math, vol. 377. Springer, Berlin (1974)
Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)
Hale, J.K.: Ordinary Differential Equations. Wiley, New York (1969)
Liu, B.: Global stability of a class of delay differential equations. J. Comput. Appl. Math. 233, 217–223 (2009)
Liu, B.: Global stability of a class of Nicholson’s blowflies model with patch structure and multiple time-varying delays. Nonlinear Anal. RWA 11, 2557–2562 (2010)
Liu, B.: The existence and uniqueness of positive periodic solutions of Nicholson-type delay systems. Nonlinear Anal. RWA 12, 3145–3451 (2011)
Liu, B.: Global dynamic behaviors for a delayed Nicholson’s blowflies model with a linear harvesting term. Electron. J. Qual. Theory Differ. Equ. 2013(45), 1–13 (2013)
Mackey, M.C., Glass, L.: Oscillations and chaos in physiological control systems. Science 197(4300), 287–289 (1997)
Metz, J.A.J., Diekmann, O.: The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomath, vol. 68. Springer, Berlin (1986)
Novo, S., Obaya, R., Sanz, A.M.: Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows. Nonlinearity 26, 1–32 (2013)
Obaya, R., Sanz, A.M.: Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems. J. Differ. Equ. 261, 4135–4163 (2016)
Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Differ. Equ. 27, 320–358 (1978)
Smith, H.L.: Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs. Providence, American Mathematical Society (1995)
Smith, H.L.: An Introduction to Delay Differential Equations with Applications to Life Sciences, Texts in Applied Mathematics, vol. 57. Springer, Berlin (2011)
Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. American Mathematical Society, Providence (2011)
Takeuchi, Y., Cui, J., Miyazaki, R., Saito, Y.: Permanence of delayed population model with dispersal loss. Math. Biosci. 201, 143–156 (2006)
Wang, L.: Almost periodic solution for Nicholson’s blowflies model with patch structure and linear harvesting terms. Appl. Math. Model. 37, 2153–2165 (2013)
Wang, W., Wang, L., Chen, W.: Existence and exponential stability of positive almost periodic solution for Nicholson-type delay systems. Nonlinear Anal. RWA 12, 1938–1949 (2011)
Zhang, A.: New results on almost periodic solutions for a Nicholson’s blowflies model with a linear harvesting term. Electron. J. Qual. Theory Differ. Equ. 2014(37), 1–14 (2014)
Zhou, Q.: The positive periodic solution for Nicholson-type delay system with linear harvesting terms. Appl. Math. Model. 37, 5581–5590 (2013)
This work was partially supported by Fundação para a Ciência e a Tecnologia under project UID/MAT/04561/2013 (T. Faria) and by Ministerio de Economía y Competitividad under project MTM2015-66330, and the European Commission under project H2020-MSCA-ITN-2014 (R. Obaya and A. M. Sanz). The authors are very grateful to the referee, whose careful reading and valuable comments led to significant improvements of the manuscript.
About this article
Cite this article
Faria, T., Obaya, R. & Sanz, A.M. Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications. J Dyn Diff Equat 30, 911–935 (2018). https://doi.org/10.1007/s10884-017-9572-8
- Delay differential equation
- Non-autonomous Nicholson system
- Quasi-monotone condition
- Global asymptotic stability
Mathematics Subject Classification