# American Institute of Mathematical Sciences

June  2014, 19(4): 1171-1195. doi: 10.3934/dcdsb.2014.19.1171

## Asymptotic pattern of a migratory and nonmonotone population model

 1 Department of Mathematics, Foshan University, Foshan, 528000, China 2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240 3 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7

Received  April 2012 Revised  July 2013 Published  April 2014

In this paper, we consider a time-delayed and nonlocal population model with migration and relax the monotone assumption for the birth function. We study the global dynamics of the model system when the spatial domain is bounded. If the spatial domain is unbounded, we investigate the spreading speed $c^*$, the non-existence of traveling wave solutions with speed $c\in(0,c^*)$, the existence of traveling wave solutions with $c\geq c^*$, and the uniqueness of traveling wave solutions with $c>c^*$. It is shown that the spreading speed coincides with the minimal wave speed of traveling waves.
Citation: Chufen Wu, Dongmei Xiao, Xiao-Qiang Zhao. Asymptotic pattern of a migratory and nonmonotone population model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1171-1195. doi: 10.3934/dcdsb.2014.19.1171
##### References:
 [1] J. Brown and N. Pavlovic, Evolution in heterogeneous environments: effects of migration on habitat specialization, Evolutionary Ecology, 6 (1992), 360-382. doi: 10.1007/BF02270698. [2] J. Fang, J. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, J. Diff. Equations, 245 (2008), 2749-2770. doi: 10.1016/j.jde.2008.09.001. [3] J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Diff. Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009. [4] S. Gourley and J. Wu, Delayed nonlocal diffusive systems in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200. [5] K. Hadeler and M. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Can. Appl. Math. Quart., 10 (2002), 473-499. [6] J. Hale, Asymptotic Behavior of Dissipative Systems, Math. surveys and monographs, Vol. 25, Amer. Math. Soc., Providence, RI, 1988. [7] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling wave for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. [8] M. Lewis and G. Schemitz, Biological invasion of an organism with separate mobile and stationary states: Modeling and analysis, Forma, 11 (1996), 1-25. [9] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Am. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [10] H. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Math. surveys and monographs, Vol. 41, Amer. Math. Soc., Providence, RI, 1995. [11] H. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [12] H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720. [13] H. Thieme, On a class of Hammerstein integral equations, Manuscripta Math., 29 (1979), 49-84. doi: 10.1007/BF01309313. [14] H. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. (RWA), 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7. [15] H. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Diff. Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [16] Q. Wang and X.-Q. Zhao, Spreading speed and traveling waves for the diffusive logistic equation with a sedentary compartment, Dyn. Cont. Discrete Impulsive Syst. (Ser. A), 13 (2006), 231-246. [17] J. Wu, Theory and applications of partial functional differential equations, Applied Math. Sci., 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [18] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Quart., 11 (2003), 303-319. [19] D. Xu and X.-Q. Zhao, Asymptotic speed of spread and traveling waves for a nonlocal epidemic model, Discrete Cont. Dyn. Syst. (Ser. B), 5 (2005), 1043-1056. doi: 10.3934/dcdsb.2005.5.1043. [20] X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Quart., 17 (2009), 271-281.

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##### References:
 [1] J. Brown and N. Pavlovic, Evolution in heterogeneous environments: effects of migration on habitat specialization, Evolutionary Ecology, 6 (1992), 360-382. doi: 10.1007/BF02270698. [2] J. Fang, J. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, J. Diff. Equations, 245 (2008), 2749-2770. doi: 10.1016/j.jde.2008.09.001. [3] J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Diff. Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009. [4] S. Gourley and J. Wu, Delayed nonlocal diffusive systems in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200. [5] K. Hadeler and M. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Can. Appl. Math. Quart., 10 (2002), 473-499. [6] J. Hale, Asymptotic Behavior of Dissipative Systems, Math. surveys and monographs, Vol. 25, Amer. Math. Soc., Providence, RI, 1988. [7] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling wave for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. [8] M. Lewis and G. Schemitz, Biological invasion of an organism with separate mobile and stationary states: Modeling and analysis, Forma, 11 (1996), 1-25. [9] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Am. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [10] H. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Math. surveys and monographs, Vol. 41, Amer. Math. Soc., Providence, RI, 1995. [11] H. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [12] H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720. [13] H. Thieme, On a class of Hammerstein integral equations, Manuscripta Math., 29 (1979), 49-84. doi: 10.1007/BF01309313. [14] H. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. (RWA), 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7. [15] H. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Diff. Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [16] Q. Wang and X.-Q. Zhao, Spreading speed and traveling waves for the diffusive logistic equation with a sedentary compartment, Dyn. Cont. Discrete Impulsive Syst. (Ser. A), 13 (2006), 231-246. [17] J. Wu, Theory and applications of partial functional differential equations, Applied Math. Sci., 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [18] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Quart., 11 (2003), 303-319. [19] D. Xu and X.-Q. Zhao, Asymptotic speed of spread and traveling waves for a nonlocal epidemic model, Discrete Cont. Dyn. Syst. (Ser. B), 5 (2005), 1043-1056. doi: 10.3934/dcdsb.2005.5.1043. [20] X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Quart., 17 (2009), 271-281.
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