# 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 & 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. doi: 10.1007/BF02270698. Google Scholar [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. doi: 10.1016/j.jde.2008.09.001. Google Scholar [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. doi: 10.1016/j.jde.2010.01.009. Google Scholar [4] S. Gourley and J. Wu, Delayed nonlocal diffusive systems in biological invasion and disease spread,, Fields Inst. Commun., 48 (2006), 137. Google Scholar [5] K. Hadeler and M. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Can. Appl. Math. Quart., 10 (2002), 473. Google Scholar [6] J. Hale, Asymptotic Behavior of Dissipative Systems,, Math. surveys and monographs, (1988). Google Scholar [7] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling wave for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776. doi: 10.1137/070703016. Google Scholar [8] M. Lewis and G. Schemitz, Biological invasion of an organism with separate mobile and stationary states: Modeling and analysis,, Forma, 11 (1996), 1. Google Scholar [9] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Am. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590. Google Scholar [10] H. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Math. surveys and monographs, (1995). Google Scholar [11] H. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal., 47 (2001), 6169. doi: 10.1016/S0362-546X(01)00678-2. Google Scholar [12] H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173. doi: 10.1007/BF00279720. Google Scholar [13] H. Thieme, On a class of Hammerstein integral equations,, Manuscripta Math., 29 (1979), 49. doi: 10.1007/BF01309313. Google Scholar [14] H. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model,, Nonlinear Anal. (RWA), 2 (2001), 145. doi: 10.1016/S0362-546X(00)00112-7. Google Scholar [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. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar [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. Google Scholar [17] J. Wu, Theory and applications of partial functional differential equations,, Applied Math. Sci., 119 (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar [18] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure,, Can. Appl. Math. Quart., 11 (2003), 303. Google Scholar [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. doi: 10.3934/dcdsb.2005.5.1043. Google Scholar [20] X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay,, Can. Appl. Math. Quart., 17 (2009), 271. Google Scholar

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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. doi: 10.1007/BF02270698. Google Scholar [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. doi: 10.1016/j.jde.2008.09.001. Google Scholar [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. doi: 10.1016/j.jde.2010.01.009. Google Scholar [4] S. Gourley and J. Wu, Delayed nonlocal diffusive systems in biological invasion and disease spread,, Fields Inst. Commun., 48 (2006), 137. Google Scholar [5] K. Hadeler and M. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Can. Appl. Math. Quart., 10 (2002), 473. Google Scholar [6] J. Hale, Asymptotic Behavior of Dissipative Systems,, Math. surveys and monographs, (1988). Google Scholar [7] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling wave for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776. doi: 10.1137/070703016. Google Scholar [8] M. Lewis and G. Schemitz, Biological invasion of an organism with separate mobile and stationary states: Modeling and analysis,, Forma, 11 (1996), 1. Google Scholar [9] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Am. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590. Google Scholar [10] H. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Math. surveys and monographs, (1995). Google Scholar [11] H. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal., 47 (2001), 6169. doi: 10.1016/S0362-546X(01)00678-2. Google Scholar [12] H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173. doi: 10.1007/BF00279720. Google Scholar [13] H. Thieme, On a class of Hammerstein integral equations,, Manuscripta Math., 29 (1979), 49. doi: 10.1007/BF01309313. Google Scholar [14] H. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model,, Nonlinear Anal. (RWA), 2 (2001), 145. doi: 10.1016/S0362-546X(00)00112-7. Google Scholar [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. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar [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. Google Scholar [17] J. Wu, Theory and applications of partial functional differential equations,, Applied Math. Sci., 119 (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar [18] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure,, Can. Appl. Math. Quart., 11 (2003), 303. Google Scholar [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. doi: 10.3934/dcdsb.2005.5.1043. Google Scholar [20] X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay,, Can. Appl. Math. Quart., 17 (2009), 271. Google Scholar
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