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

Henri Berestycki, Nancy Rodríguez

Research output: Contribution to journalArticle

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.

LanguageEnglish (US)
Pages685-723
Number of pages39
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume37
Issue number2
DOIs
StatePublished - Feb 1 2017

Fingerprint

Reaction-diffusion Equations
Nonlocal Equations
Travelling Fronts
Ecology
Obstruction
Jump Diffusion
Heterogeneous Environment
Physiology
Homogeneous Space
Traveling Wave Solutions
Brownian movement
Brownian motion
Discontinuity
Open Problems
Jump
Scaling
Decay
Propagation
Numerical Simulation
Series

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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

In: Discrete and Continuous Dynamical Systems- Series A, Vol. 37, No. 2, 01.02.2017, p. 685-723.

Research output: Contribution to journalArticle

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