### Abstract

Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a generalized traveling front that approaches a traveling wave solution as t → -∞, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

Language | English (US) |
---|---|

Pages | 685-723 |

Number of pages | 39 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 37 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2017 |

### Fingerprint

### Keywords

- Comparison principle
- Entire solution
- Gap problem
- Non-local diffusion
- Propagation

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*37*(2), 685-723. https://doi.org/10.3934/dcds.2017029

**A non-local bistable reaction-diffusion equation with a gap.** / Berestycki, Henri; Rodríguez, Nancy.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems- Series A*, vol. 37, no. 2, pp. 685-723. https://doi.org/10.3934/dcds.2017029

}

TY - JOUR

T1 - A non-local bistable reaction-diffusion equation with a gap

AU - Berestycki, Henri

AU - Rodríguez, Nancy

PY - 2017/2/1

Y1 - 2017/2/1

N2 - Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a generalized traveling front that approaches a traveling wave solution as t → -∞, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

AB - Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a generalized traveling front that approaches a traveling wave solution as t → -∞, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

KW - Comparison principle

KW - Entire solution

KW - Gap problem

KW - Non-local diffusion

KW - Propagation

UR - http://www.scopus.com/inward/record.url?scp=85006511756&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85006511756&partnerID=8YFLogxK

U2 - 10.3934/dcds.2017029

DO - 10.3934/dcds.2017029

M3 - Article

VL - 37

SP - 685

EP - 723

JO - Discrete and Continuous Dynamical Systems

T2 - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 2

ER -