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July  2016, 15(4): 1471-1495. doi: 10.3934/cpaa.2016.15.1471

## Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip

 1 South China Normal University, Guangzhou, China, China 2 School of Mathematics, South China Normal University, Guangzhou 510631

Received  December 2014 Revised  February 2016 Published  April 2016

A time-delayed reaction-diffusion system of mistletoes and birds with nonlocal effect in a two-dimensional strip is considered in this paper. By the background of model deriving, the bird diffuses with a Neumann boundary value condition, and the mistletoes does not diffuse and thus without boundary value condition. Making use of the theory of monotone semiflows and Kuratowski measure of non-compactness, we discuss the existence of spreading speed $c^\ast$. The value of $c^*$ is evaluated by using two auxiliary linear systems accompanied with approximate process.
Citation: Huimin Liang, Peixuan Weng, Yanling Tian. Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1471-1495. doi: 10.3934/cpaa.2016.15.1471
##### References:
 [1] D.G. Aronson and H.F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in: J.A. Goldstein (Ed.), Partial Differential Equations and Related Topics, Lecture in Mathematics, Springer-Verlag, New York, 466 (1975), 5-59. [2] K.C. Chang, Methods in Nonlinear Analysis, Springer, Berlin, 2005. [3] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9. [4] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [5] J. Kuijt, The Biology of Parasitic Flowering Plants, University of California Press, Berkeley, 1969. [6] K. Kishimoto, Instability of non-constant equilibrium solutions of a system of competition-diffusion equations, J. Math. Biol., 13 (1981), 105-114. doi: 10.1007/BF00276869. [7] K. Kishimoto and H.F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8. [8] B.T. Li, H.F. Weinberger and M.A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math Biosci, 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008. [9] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. Erratum: Comm. Pure Appl. Math., 61 (2008), 137-138. doi: 10.1002/cpa.20154. [10] R.H. Martin and H.L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [11] H.L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, in Mathematical Surveys and Monographs, 41 (1995), American Mathematical Society, Providence, RI. [12] H.R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine angew. Math., 306 (1979), 21-30. doi: 10.1515/crll.1979.306.94. [13] H.R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [14] Y.L. Tian and P.X. Weng, Asymptotic patterns of a reaction-diffusion equation with nonlinear-nonlocal functional response, IMA Journal of Applied Mathematics, 78 (2013), 70-101. doi: 10.1093/imamat/hxr038. [15] C.C. Wang, R.S. Liu, J.P. Shi and D.C. Martinez, Spatiotemporal mutualistic model of mistletoes and birds, J. Math. Biol., (2013), 1-42. [16] C.C. Wang, R.S. Liu, J.P. Shi and D.C. Martinez, Traveling waves of a mutualistic model of mistletoes and birds, DCDS-A, 35 (2015), 1734-1765. doi: 10.3934/dcds.2015.35.1743. [17] D.M. Watson, Mistletoe-a keystone resource in forests and woodlands worldwide, Annual Review of Ecology and Systematics, 32 (2001), 219-249. [18] H.F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math Anal, 13 (1982), 353-396. doi: 10.1137/0513028. [19] H.F. Weinberger, M.A. Lewis and B.T. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. [20] H.F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol, 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3. [21] P.X. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type {SIS} epidemic model, J. Differential Equations, 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020. [22] P.X. Weng, H.X. Huang and J.H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409. [23] P.X. Weng and Y.L. Tian, Asymptotic speed of propagation and traveling wave solutions for a lattice integral equation, Nonlinear Analysis: TMA, 70 (2009), 159-175. doi: 10.1016/j.na.2007.11.043. [24] C.F. Wu, D.M. Xiao and X.-Q. Zhao, Spreading speeds of a partially degenerate reaction diffusion system in a periodic habitat, J. Differential Equations, 225 (2013), 3983-4011. doi: 10.1016/j.jde.2013.07.058. [25] J.H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [26] Z.T. Xu and P.X. Weng, Asymptotic speed of propagation and traveling wavefronts for a lattice vector disease model, Nonlinear Analysis: RWA, 12 (2011), 3621-3641. doi: 10.1016/j.nonrwa.2011.06.020. [27] X.-Q. Zhao, Spatial dynamics of some evolution systems in biology, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific Publishing Co. Pte. Ltd., Singapore, (2009), 332-363. doi: 10.1142/9789812834744_0015. [28] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

##### References:
 [1] D.G. Aronson and H.F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in: J.A. Goldstein (Ed.), Partial Differential Equations and Related Topics, Lecture in Mathematics, Springer-Verlag, New York, 466 (1975), 5-59. [2] K.C. Chang, Methods in Nonlinear Analysis, Springer, Berlin, 2005. [3] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9. [4] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [5] J. Kuijt, The Biology of Parasitic Flowering Plants, University of California Press, Berkeley, 1969. [6] K. Kishimoto, Instability of non-constant equilibrium solutions of a system of competition-diffusion equations, J. Math. Biol., 13 (1981), 105-114. doi: 10.1007/BF00276869. [7] K. Kishimoto and H.F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8. [8] B.T. Li, H.F. Weinberger and M.A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math Biosci, 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008. [9] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. Erratum: Comm. Pure Appl. Math., 61 (2008), 137-138. doi: 10.1002/cpa.20154. [10] R.H. Martin and H.L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [11] H.L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, in Mathematical Surveys and Monographs, 41 (1995), American Mathematical Society, Providence, RI. [12] H.R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine angew. Math., 306 (1979), 21-30. doi: 10.1515/crll.1979.306.94. [13] H.R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [14] Y.L. Tian and P.X. Weng, Asymptotic patterns of a reaction-diffusion equation with nonlinear-nonlocal functional response, IMA Journal of Applied Mathematics, 78 (2013), 70-101. doi: 10.1093/imamat/hxr038. [15] C.C. Wang, R.S. Liu, J.P. Shi and D.C. Martinez, Spatiotemporal mutualistic model of mistletoes and birds, J. Math. Biol., (2013), 1-42. [16] C.C. Wang, R.S. Liu, J.P. Shi and D.C. Martinez, Traveling waves of a mutualistic model of mistletoes and birds, DCDS-A, 35 (2015), 1734-1765. doi: 10.3934/dcds.2015.35.1743. [17] D.M. Watson, Mistletoe-a keystone resource in forests and woodlands worldwide, Annual Review of Ecology and Systematics, 32 (2001), 219-249. [18] H.F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math Anal, 13 (1982), 353-396. doi: 10.1137/0513028. [19] H.F. Weinberger, M.A. Lewis and B.T. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. [20] H.F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol, 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3. [21] P.X. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type {SIS} epidemic model, J. Differential Equations, 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020. [22] P.X. Weng, H.X. Huang and J.H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409. [23] P.X. Weng and Y.L. Tian, Asymptotic speed of propagation and traveling wave solutions for a lattice integral equation, Nonlinear Analysis: TMA, 70 (2009), 159-175. doi: 10.1016/j.na.2007.11.043. [24] C.F. Wu, D.M. Xiao and X.-Q. Zhao, Spreading speeds of a partially degenerate reaction diffusion system in a periodic habitat, J. Differential Equations, 225 (2013), 3983-4011. doi: 10.1016/j.jde.2013.07.058. [25] J.H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [26] Z.T. Xu and P.X. Weng, Asymptotic speed of propagation and traveling wavefronts for a lattice vector disease model, Nonlinear Analysis: RWA, 12 (2011), 3621-3641. doi: 10.1016/j.nonrwa.2011.06.020. [27] X.-Q. Zhao, Spatial dynamics of some evolution systems in biology, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific Publishing Co. Pte. Ltd., Singapore, (2009), 332-363. doi: 10.1142/9789812834744_0015. [28] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.
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