# American Institute of Mathematical Sciences

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
 [1] Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106 [2] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [3] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [4] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448 [5] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [6] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 [7] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [8] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [9] Telmo Peixe. Permanence in polymatrix replicators. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020032 [10] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [11] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376 [12] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [13] Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269 [14] Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 [15] Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 [16] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [17] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [18] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [19] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [20] Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

2019 Impact Factor: 1.105