### Abstract

This paper deals with the long time behavior in terms of complete spreading for a population model described by a reaction-diffusion equationwith 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1059-1072 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 144 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Persistence and failure of complete spreading in delayed reaction-diffusion equations.** / Lin, Guo; Ruan, Shigui.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 144, no. 3, pp. 1059-1072. https://doi.org/10.1090/proc/12811

}

TY - JOUR

T1 - Persistence and failure of complete spreading in delayed reaction-diffusion equations

AU - Lin, Guo

AU - Ruan, Shigui

PY - 2016/3/1

Y1 - 2016/3/1

N2 - This paper deals with the long time behavior in terms of complete spreading for a population model described by a reaction-diffusion equationwith 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.

AB - This paper deals with the long time behavior in terms of complete spreading for a population model described by a reaction-diffusion equationwith 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.

UR - http://www.scopus.com/inward/record.url?scp=84954510961&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84954510961&partnerID=8YFLogxK

U2 - 10.1090/proc/12811

DO - 10.1090/proc/12811

M3 - Article

AN - SCOPUS:84954510961

VL - 144

SP - 1059

EP - 1072

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 3

ER -