    November  2014, 13(6): 2475-2492. doi: 10.3934/cpaa.2014.13.2475

## Global dynamics of a non-local delayed differential equation in the half plane

 1 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Received  January 2014 Revised  May 2014 Published  July 2014

In this paper, we first derive an equation for a single species population with two age stages and a fixed maturation period living in the half plane such as ocean and big lakes. By adopting the compact open topology, we establish some a priori estimate for nontrivial solutions after describing asymptotic properties of the nonlocal delayed effect, which enables us to show the permanence of the equation. Then we can employ standard dynamical system theoretical arguments to establish the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker's birth function and Mackey-Glass's hematopoiesis function, we obtain threshold results for the global dynamics of these two models.
Citation: Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475
##### References:
  K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models,, \emph{J. Math. Biol.}, 39 (1999), 332. doi: 10.1007/s002850050194.  Google Scholar  T. Faria, Asymptotic stability for delayed logistic type equations,, \emph{Math. Comput. Modelling}, 43 (2006), 433. doi: 10.1016/j.mcm.2005.11.006.  Google Scholar  D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded domains and their numerical computations,, \emph{Differ. Equ. Dyn. Syst.}, 11 (2003), 117. Google Scholar  E. Liz, Four theorems and one conjecture on the global asymptotic stability of delay differential equations,, in \emph{The first 60 year of nolinear analysis of Jean Mawhin}, (2004), 117. doi: 10.1142/9789812702906_0010.  Google Scholar  E. Liz and G. Rost, On the global attractor of delay differential equations with unimodal feedback,, \emph{Discrete Contin. Dyn. Syst.}, 24 (2009), 1215. doi: 10.3934/dcds.2009.24.1215.  Google Scholar  E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, \emph{SIAM J. Math. Anal.}, 35 (2003), 596. doi: 10.1137/S0036141001399222.  Google Scholar  J. Metz and O. Diekmann, Dynamics of Physiologically Structured Populations,, Springer-Verlag, (1986). doi: 10.1007/978-3-662-13159-6.  Google Scholar  H. Smith, A structured population model and a related functional-differential equation: global attractors and uniform persistence,, \emph{J. Dyn. Diff. Eqns.}, 6 (1994), 71. doi: 10.1007/BF02219189.  Google Scholar  H. Smith and H. Thieme, Monotone semiflows in scalar non-quasi-monotone functional differential equations,, \emph{J. Math. Anal. Appl.}, 21 (1990), 673. doi: 10.1137/0521036.  Google Scholar  J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains,, \emph{R. Soc. Lond. A}, 457 (2001), 1841. doi: 10.1098/rspa.2001.0789.  Google Scholar  J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay,, \emph{J. Math. Biology}, 43 (2001), 37. doi: 10.1007/s002850100081.  Google Scholar  H. O. Walther, The 2-dimensional attractor of $x'(t)=-\mu x(t)+f(x(t-1))$,, \emph{Mem. Amer. Math. Soc.}, 113 (1995). Google Scholar  J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, \emph{J. Dynam. Diff. Eqns.}, 13 (2001), 651. doi: 10.1023/A:1016690424892.  Google Scholar  D. Xu and X. Zhao, A nonlocal reaction-diffusion population model with stage structure,, \emph{Can. Appl. Math. Q.}, 11 (2003), 303. Google Scholar  T. Yi, Y. Chen and J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains,, \emph{Z. Angew. Math. Phys.}, 63 (2012), 793. doi: 10.1007/s00033-012-0224-x.  Google Scholar  T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case,, \emph{J. Differ. Equ.}, 245 (2008), 3376. doi: 10.1016/j.jde.2008.03.007.  Google Scholar  T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain,, \emph{J. Differ. Equ.}, 251 (2011), 2598. doi: 10.1016/j.jde.2011.04.027.  Google Scholar  T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition,, \emph{Proceedings of the Royal Society A: Mathematical, 466 (2010), 2955. doi: 10.1098/rspa.2009.0650.  Google Scholar

show all references

##### References:
  K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models,, \emph{J. Math. Biol.}, 39 (1999), 332. doi: 10.1007/s002850050194.  Google Scholar  T. Faria, Asymptotic stability for delayed logistic type equations,, \emph{Math. Comput. Modelling}, 43 (2006), 433. doi: 10.1016/j.mcm.2005.11.006.  Google Scholar  D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded domains and their numerical computations,, \emph{Differ. Equ. Dyn. Syst.}, 11 (2003), 117. Google Scholar  E. Liz, Four theorems and one conjecture on the global asymptotic stability of delay differential equations,, in \emph{The first 60 year of nolinear analysis of Jean Mawhin}, (2004), 117. doi: 10.1142/9789812702906_0010.  Google Scholar  E. Liz and G. Rost, On the global attractor of delay differential equations with unimodal feedback,, \emph{Discrete Contin. Dyn. Syst.}, 24 (2009), 1215. doi: 10.3934/dcds.2009.24.1215.  Google Scholar  E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, \emph{SIAM J. Math. Anal.}, 35 (2003), 596. doi: 10.1137/S0036141001399222.  Google Scholar  J. Metz and O. Diekmann, Dynamics of Physiologically Structured Populations,, Springer-Verlag, (1986). doi: 10.1007/978-3-662-13159-6.  Google Scholar  H. Smith, A structured population model and a related functional-differential equation: global attractors and uniform persistence,, \emph{J. Dyn. Diff. Eqns.}, 6 (1994), 71. doi: 10.1007/BF02219189.  Google Scholar  H. Smith and H. Thieme, Monotone semiflows in scalar non-quasi-monotone functional differential equations,, \emph{J. Math. Anal. Appl.}, 21 (1990), 673. doi: 10.1137/0521036.  Google Scholar  J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains,, \emph{R. Soc. Lond. A}, 457 (2001), 1841. doi: 10.1098/rspa.2001.0789.  Google Scholar  J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay,, \emph{J. Math. Biology}, 43 (2001), 37. doi: 10.1007/s002850100081.  Google Scholar  H. O. Walther, The 2-dimensional attractor of $x'(t)=-\mu x(t)+f(x(t-1))$,, \emph{Mem. Amer. Math. Soc.}, 113 (1995). Google Scholar  J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, \emph{J. Dynam. Diff. Eqns.}, 13 (2001), 651. doi: 10.1023/A:1016690424892.  Google Scholar  D. Xu and X. Zhao, A nonlocal reaction-diffusion population model with stage structure,, \emph{Can. Appl. Math. Q.}, 11 (2003), 303. Google Scholar  T. Yi, Y. Chen and J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains,, \emph{Z. Angew. Math. Phys.}, 63 (2012), 793. doi: 10.1007/s00033-012-0224-x.  Google Scholar  T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case,, \emph{J. Differ. Equ.}, 245 (2008), 3376. doi: 10.1016/j.jde.2008.03.007.  Google Scholar  T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain,, \emph{J. Differ. Equ.}, 251 (2011), 2598. doi: 10.1016/j.jde.2011.04.027.  Google Scholar  T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition,, \emph{Proceedings of the Royal Society A: Mathematical, 466 (2010), 2955. doi: 10.1098/rspa.2009.0650.  Google Scholar
  Jin-Liang Wang, Zhi-Chun Yang, Tingwen Huang, Mingqing Xiao. Local and global exponential synchronization of complex delayed dynamical networks with general topology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 393-408. doi: 10.3934/dcdsb.2011.16.393  Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037  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  Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203  Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825  Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029  Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701  A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35  Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107  Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126  Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741  Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225  Tahir Bachar Issa, Rachidi Bolaji Salako. Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3839-3874. doi: 10.3934/dcdsb.2017193  Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115  Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935  Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079  Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036  Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511  Nikos I. Karachalios, Athanasios N Lyberopoulos. On the dynamics of a degenerate damped semilinear wave equation in \mathbb{R}^N : the non-compact case. Conference Publications, 2007, 2007 (Special) : 531-540. doi: 10.3934/proc.2007.2007.531  Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088

2018 Impact Factor: 0.925