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Spatial dynamics of a lattice population model with two age classes and maturation delay

Published online by Cambridge University Press:  06 November 2014

SHI-LIANG WU ,
PEIXUAN WENG  and
SHIGUI RUAN
SHI-LIANG WU
Affiliation:
School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, People's Republic of China email: slwu@xidian.edu.cn
PEIXUAN WENG
Affiliation:
School of Mathematics, South China Normal University Guangzhou, Guangdong 510631, People's Republic of China email: wengpx@scnu.edu.cn
SHIGUI RUAN
Affiliation:
Department of Mathematics, University of Miami, P.O. Box 249085, Coral Gables, FL 33124–4250, USA email: ruan@math.miami.edu
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Abstract

This paper is concerned with the spatial dynamics of a monostable delayed age-structured population model in a 2D lattice strip. When there exists no positive equilibrium, we prove the global attractivity of the zero equilibrium. Otherwise, we give some sufficient conditions to guarantee the global attractivity of the unique positive equilibrium by establishing a series of comparison arguments. Furthermore, when those conditions do not hold, we show that the system is uniformly persistent. Finally, the spreading speed, including the upward convergence, is established for the model without the monotonicity of the growth function. The linear determinacy of the spreading speed and its coincidence with the minimal wave speed are also proved.

Keywords

population model in 2D lattice stripglobal attractivityspreading speedtravelling waveslinear determinacy
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 1 , February 2015 , pp. 61 - 91
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Al-Omari, J. F. M. & Gourley, S. A. (2005) A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay. Eur. J. Appl. Math. 16, 37–51.Google Scholar
[2]Aronson, D. G. & Weinberger, H. F. (1975) Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J. A. (editor), Lecture Notes in Mathematics, Vol. 446, Springer-Verlag, pp. 5–49.Google Scholar
[3]Cheng, C.-P., Li, W.-T. & Wang, Z.-C. (2008) Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice. IMA J. Appl. Math. 73, 592–618.Google Scholar
[4]Fang, J., Wei, J. & Zhao, X.-Q. (2010) Spreading speeds and travelling waves for non-monotone time-delayed lattice equations. Proc. R. Soc. Lond. Ser. A. 466, 1919–1934.Google Scholar
[5]Fang, J. & Zhao, X.-Q. (2010) Existence and uniqueness of traveling waves for non-monotone integral equations with applications. J. Differ. Eqns 248, 2199–2226.Google Scholar
[6]Freedman, H. I. & Ruan, S. (1995) Uniform persistence in functional differential equations. J. Differ. Eqns 115, 173–192.Google Scholar
[7]Gomez, A. & Trofimchuk, S. (2013) Global continuation of monotone wavefronts. J. Lond. Math. Soc., doi: 10.1112/jlms/jdt050.Google Scholar
[8]Gourley, S. A. & Kuang, Y. (2003) Wavefront and global stability in a time-delayed population model with stage structure. Proc. R. Soc. Lond. Ser. A. 459, 1563–1579.Google Scholar
[9]Gourley, S. A. & Ruan, S. (2000) Dynamics of the diffusive Nicholson's blowfies equation with distributed delay. Proc. R. Soc. Edinburgh Sect. A 130A, 1275–1291.Google Scholar
[10]Hsu, S.-B. & Zhao, X.-Q. (2008) Spreading speeds and traveling waves for nonmonotone integrodifference equations. SIAM J. Math. Anal. 40, 776–789.Google Scholar
[11]Jiang, J. F., Liang, X. & Zhao, X.-Q. (2004) Saddle-point behavior for monotone semiflows and reaction-diffusion models. J. Differ. Eqns 203, 313–330.Google Scholar
[12]Kuang, Y. (1993) Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA.Google Scholar
[13]Kyrychko, Y., Gourley, S. A. & Bartuccelli, M. V. (2006) Dynamics of a stage-structured population model on an isolated finite lattice. SIAM J. Math. Anal. 37, 1688–1708.Google Scholar
[14]Li, W. T., Ruan, S. & Wang, Z. C. (2007) On the diffusive Nicholson's Blowflies equation with nonlocal delays. J. Nonlinear Sci. 17, 505–525.Google Scholar
[15]Li, B., Lewis, M. A. & Weinberger, H. F. (2009) Existence of traveling waves for integral recursions with nonmonotone growth functions. J. Math. Biol. 58, 323–338.Google Scholar
[16]Liang, X. & Zhao, X.-Q. (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40. (Erratum: Commun. Pure Appl. Math. 61 (2008), 137–138.)Google Scholar
[17]Ma, S. (2007) Traveling waves for nonlocal delayed diffusion equations via auxiliary equations. J. Differ. Eqns 237, 259–277.Google Scholar
[18]Martin, R. H. & Smith, H. L. (1990) Abstract functional differential equations and reaction-diffusion system. Trans. Am. Math. Soc. 321, 1–44.Google Scholar
[19]Ou, C. & Wu, J. (2007) Persistence of wavefronts in delayed nonlocal reaction-diffusion equations. J. Differ. Eqns 235, 219–261.Google Scholar
[20]Smith, H. L. (1995) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., Vol. 41, American Mathematical Society, Providence, RI.Google Scholar
[21]Smith, H. & Thieme, H. (1991) Strongly order preserving semiflows generated by functional differential equations. J. Differ. Eqns 93, 332–363.Google Scholar
[22]So, J. W.-H., Wu, J. & Zou, X. (2001) A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on the unbounded domains. Proc. R. Soc. Lond. Ser. A. 457, 1841–1853.Google Scholar
[23]Thieme, H. R. & Zhao, X.-Q. (2001) A non-local delayed and diffusive predator-prey model. Nonlinear Anal. RWA 2, 145–160.Google Scholar
[24]Thieme, H. R. & Zhao, X.-Q. (2003) Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. J. Differ. Eqns 195, 430–470.Google Scholar
[25]Wang, H.-Y. (2011) Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems. J. Nonlinear Sci. 21, 747–783.Google Scholar
[26]Wang, Z.-C. & Li, W.-T. (2010) Dynamics of a nonlocal delayed reaction-diffusion equation without quasi-monotonicity. Proc. R. Soc. Edinburgh Sect. A 140, 1081–1109.Google Scholar
[27]Weng, P.-X., Huang, H. & Wu, J. (2003) Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J. Appl. Math. 68, 409–439.Google Scholar
[28]Weng, P.-X. (2009) Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete Contin. Dyn. Syst. Ser. B 12, 883–904.Google Scholar
[29]Weng, P.-X. & Zhao, X.-Q. (2011) Spatial dynamics of a nonlocal and delayed population model in a periodic habitat. Discrete Contin. Dyn. Syst. Ser. A 29 (1), 343–366.Google Scholar
[30]Wu, S.-L. & Liu, S.-Y. (2010) Existence and uniqueness of traveling waves for non-monotone integral equations with application. J. Math. Anal. Appl. 365, 729–741.Google Scholar
[31]Wu, J. & Zhao, X.-Q. (2002) Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations. J. Differ. Eqns 186, 470–484.Google Scholar
[32]Wu, J. & Zou, X. (2001) Traveling wave fronts of reaction-diffusion systems with delay. J. Dyn. Differ. Eqns. 13, 651–687.Google Scholar
[33]Yang, Y. & So, J.W.-H. (1996) Dynamics for the diffusive Nicholson's blowflies equation. In: Procedings of the International Conference on Dynamical Systems and Differential Equations, 29 May–1 June, 1996, Springfield, Missouri.Google Scholar
[34]Yi, T. & Zou, X. (2008) Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case. J. Differ. Eqns 245, 3376–3388.Google Scholar
[35]Zhao, H.-Q., Wu, S.-L. & Liu, S.-Y. (2013) Entire solutions of a monostable age-tructured population model in a 2D lattice strip. J. Math. Anal. Appl. 401, 85–97.Google Scholar
[36]Zou, X. (2002) Delay induced traveling wave fronts in reaction diffusion equations of KPP–Fisher type. J. Comput. Appl. Math. 146, 309–321.Google Scholar