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:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Metz and O. Diekmann, Dynamics of Physiologically Structured Populations,, Springer-Verlag, (1986).  doi: 10.1007/978-3-662-13159-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[14]

D. Xu and X. Zhao, A nonlocal reaction-diffusion population model with stage structure,, \emph{Can. Appl. Math. Q.}, 11 (2003), 303.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Metz and O. Diekmann, Dynamics of Physiologically Structured Populations,, Springer-Verlag, (1986).  doi: 10.1007/978-3-662-13159-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[14]

D. Xu and X. Zhao, A nonlocal reaction-diffusion population model with stage structure,, \emph{Can. Appl. Math. Q.}, 11 (2003), 303.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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