# American Institute of Mathematical Sciences

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:
 [1] 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. [2] T. Faria, Asymptotic stability for delayed logistic type equations,, \emph{Math. Comput. Modelling}, 43 (2006), 433. doi: 10.1016/j.mcm.2005.11.006. [3] 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. [4] 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. [5] 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. [6] 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. [7] J. Metz and O. Diekmann, Dynamics of Physiologically Structured Populations,, Springer-Verlag, (1986). doi: 10.1007/978-3-662-13159-6. [8] 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. [9] 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. [10] 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. [11] 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. [12] H. O. Walther, The 2-dimensional attractor of $x'(t)=-\mu x(t)+f(x(t-1))$,, \emph{Mem. Amer. Math. Soc.}, 113 (1995). [13] 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. [14] D. Xu and X. Zhao, A nonlocal reaction-diffusion population model with stage structure,, \emph{Can. Appl. Math. Q.}, 11 (2003), 303. [15] 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. [16] 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. [17] 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. [18] 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.

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##### References:
 [1] 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. [2] T. Faria, Asymptotic stability for delayed logistic type equations,, \emph{Math. Comput. Modelling}, 43 (2006), 433. doi: 10.1016/j.mcm.2005.11.006. [3] 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. [4] 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. [5] 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. [6] 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. [7] J. Metz and O. Diekmann, Dynamics of Physiologically Structured Populations,, Springer-Verlag, (1986). doi: 10.1007/978-3-662-13159-6. [8] 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. [9] 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. [10] 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. [11] 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. [12] H. O. Walther, The 2-dimensional attractor of $x'(t)=-\mu x(t)+f(x(t-1))$,, \emph{Mem. Amer. Math. Soc.}, 113 (1995). [13] 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. [14] D. Xu and X. Zhao, A nonlocal reaction-diffusion population model with stage structure,, \emph{Can. Appl. Math. Q.}, 11 (2003), 303. [15] 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. [16] 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. [17] 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. [18] 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.
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