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Persistence and failure of complete spreading in delayed reaction-diffusion equations

Authors: Guo Lin and Shigui Ruan
Journal: Proc. Amer. Math. Soc. 144 (2016), 1059-1072
MSC (2010): Primary 35C07, 35K57, 37C65
Published electronically: July 24, 2015
MathSciNet review: 3447660
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Abstract: This paper deals with the long time behavior in terms of complete spreading for a population model described by a reaction-diffusion equation with delay, of which the corresponding reaction equation is bistable. When a complete spreading occurs in the corresponding undelayed equation with initial value admitting compact support, it is proved that the invasion can also be successful in the delayed equation if the time delay is small. To spur on a complete spreading, the choice of the initial value would be very technical due to the combination of delay and Allee effects. In addition, we show the possible failure of complete spreading in a quasimonotone delayed equation to illustrate the complexity of the problem.

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Additional Information

Guo Lin
Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China

Shigui Ruan
Affiliation: Department of Mathematics, University of Miami, P.O. Box 249085, Coral Gables, Florida 33124-4250

Received by editor(s): May 27, 2014
Received by editor(s) in revised form: February 5, 2015
Published electronically: July 24, 2015
Additional Notes: The first author was supported by NNSF of China grant 11471149.
The second author was supported by NSF grant DMS-1412454
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society

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