# 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
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##### References:
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