# American Institute of Mathematical Sciences

2015, 14(5): 2095-2115. doi: 10.3934/cpaa.2015.14.2095

## A nonlocal diffusion population model with age structure and Dirichlet boundary condition

 1 School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States

Received  February 2014 Revised  August 2014 Published  June 2015

In this paper, we study the global dynamics of a population model with age structure. The model is given by a nonlocal reaction-diffusion equation carrying a maturation time delay, together with the homogeneous Dirichlet boundary condition. The non-locality arises from spatial movements of the immature individuals. We are mainly concerned with the case when the birth rate decays as the mature population size becomes large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of the positive steady states of the model. By establishing an appropriate comparison principle and applying the theory of dissipative systems, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the unique positive steady state.
Citation: Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095
##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space,, \emph{SIAM Review}, 18 (1976), 620. [2] N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model,, \emph{SIAM J. Appl. Math.}, 50 (1990), 1663. doi: 10.1137/0150099. [3] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). [4] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread,, in \emph{Nonlinear Dynamics and Evolution Equations} (H. Brunner, 48 (2006), 137. [5] Z. M. Guo, Z. C. Yang and X. Zou, Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition: a non-monotone case,, \emph{Commun. Pure Appl. Anal.}, 11 (2012), 1825. doi: 10.3934/cpaa.2012.11.1825. [6] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs 25, (1988). [7] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem,, \emph{Commun. Math. Phys.}, 88 (1983), 309. [8] D. Liang, J. W. -H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numerical computations,, \emph{Diff. Eqns. Dynam. Syst.}, 11 (2003), 117. [9] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, \emph{Science}, 197 (1977), 287. [10] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations,, (J. A. J. Metz and O. Diekmann eds.), (1986). doi: 10.1007/978-3-662-13159-6. [11] M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations,, Springer-Verlag, (1984). doi: 10.1007/978-1-4612-5282-5. [12] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, \emph{Indiana Univ. Math. J.}, 21 (1972), 979. [13] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences,, Texts in Applied Mathematics 57, (2011). doi: 10.1007/978-1-4419-7646-8. [14] H. Smith and H. Thieme, Strongly order preserving semi-flows generated by functional differential equations,, \emph{J. Diff. Eqns.}, 93 (1991), 332. doi: 10.1016/0022-0396(91)90016-3. [15] J. W. -H. So, J. Wu and Y. Yang, Numerical steady state and hopf bifurcation analysis on the diffusive Nicholson's blowflies equation,, \emph{Appl. Math. Comput.}, 111 (2000), 33. doi: 10.1016/S0096-3003(99)00047-8. [16] J. W. -H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains,, \emph{Proc. Royal Soc. London. A}, 457 (2001), 1841. doi: 10.1098/rspa.2001.0789. [17] H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model,, \emph{Nonlinear Anal. RWA.}, 2 (2001), 145. doi: 10.1016/S0362-546X(00)00112-7. [18] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models,, \emph{J. Diff. Eqns.}, 195 (2003), 430. doi: 10.1016/S0022-0396(03)00175-X. [19] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Appl. Math. Sci. 119, (1996). doi: 10.1007/978-1-4612-4050-1. [20] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure,, \emph{Canad. Appl. Math. Quart.}, 11 (2003), 303. [21] S. T. Yau and R. Schoen, Lectures on Differential Geometry,, Higher Education Press, (2004). [22] T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain,, \emph{J. Diff. Eqns.}, 251 (2011), 2598. doi: 10.1016/j.jde.2011.04.027. [23] T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality,, \emph{J. Dyn. Diff. Equat.}, 25 (2013), 959. doi: 10.1007/s10884-013-9324-3. [24] X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with time delay,, \emph{Canad. Appl. Math. Quart.}, 17 (2009), 271.

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##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space,, \emph{SIAM Review}, 18 (1976), 620. [2] N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model,, \emph{SIAM J. Appl. Math.}, 50 (1990), 1663. doi: 10.1137/0150099. [3] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). [4] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread,, in \emph{Nonlinear Dynamics and Evolution Equations} (H. Brunner, 48 (2006), 137. [5] Z. M. Guo, Z. C. Yang and X. Zou, Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition: a non-monotone case,, \emph{Commun. Pure Appl. Anal.}, 11 (2012), 1825. doi: 10.3934/cpaa.2012.11.1825. [6] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs 25, (1988). [7] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem,, \emph{Commun. Math. Phys.}, 88 (1983), 309. [8] D. Liang, J. W. -H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numerical computations,, \emph{Diff. Eqns. Dynam. Syst.}, 11 (2003), 117. [9] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, \emph{Science}, 197 (1977), 287. [10] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations,, (J. A. J. Metz and O. Diekmann eds.), (1986). doi: 10.1007/978-3-662-13159-6. [11] M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations,, Springer-Verlag, (1984). doi: 10.1007/978-1-4612-5282-5. [12] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, \emph{Indiana Univ. Math. J.}, 21 (1972), 979. [13] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences,, Texts in Applied Mathematics 57, (2011). doi: 10.1007/978-1-4419-7646-8. [14] H. Smith and H. Thieme, Strongly order preserving semi-flows generated by functional differential equations,, \emph{J. Diff. Eqns.}, 93 (1991), 332. doi: 10.1016/0022-0396(91)90016-3. [15] J. W. -H. So, J. Wu and Y. Yang, Numerical steady state and hopf bifurcation analysis on the diffusive Nicholson's blowflies equation,, \emph{Appl. Math. Comput.}, 111 (2000), 33. doi: 10.1016/S0096-3003(99)00047-8. [16] J. W. -H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains,, \emph{Proc. Royal Soc. London. A}, 457 (2001), 1841. doi: 10.1098/rspa.2001.0789. [17] H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model,, \emph{Nonlinear Anal. RWA.}, 2 (2001), 145. doi: 10.1016/S0362-546X(00)00112-7. [18] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models,, \emph{J. Diff. Eqns.}, 195 (2003), 430. doi: 10.1016/S0022-0396(03)00175-X. [19] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Appl. Math. Sci. 119, (1996). doi: 10.1007/978-1-4612-4050-1. [20] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure,, \emph{Canad. Appl. Math. Quart.}, 11 (2003), 303. [21] S. T. Yau and R. Schoen, Lectures on Differential Geometry,, Higher Education Press, (2004). [22] T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain,, \emph{J. Diff. Eqns.}, 251 (2011), 2598. doi: 10.1016/j.jde.2011.04.027. [23] T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality,, \emph{J. Dyn. Diff. Equat.}, 25 (2013), 959. doi: 10.1007/s10884-013-9324-3. [24] X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with time delay,, \emph{Canad. Appl. Math. Quart.}, 17 (2009), 271.
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