### Abstract

Motivated by the propagation of thin bacterial films around planar obstacles, this paper considers the dynamics of travelling wave solutions to the Fisher–KPP equation (Formula presented.) in a planar strip (Formula presented.), (Formula presented.). We examine the propagation of fronts in the presence of a mixed boundary condition (also referred to as a ‘partially absorbing’ or ‘reactive’ boundary) (Formula presented.), with (Formula presented.), at (Formula presented.). The presence of boundary conditions of this kind leads to the development of front solutions that propagate in x but contain transverse structure in y. Motivated by the observation that the speed of propagation in the Fisher–KPP equation is determined (for exponentially decaying initial conditions) by the behaviour at the leading edge, we analyse the linearised Fisher–KPP equation in order to estimate the speed of the stable travelling front, a function of the width L and the imposed boundary conditions. For wide strips the speed estimate based on the linearised equation agrees well with the results of numerical simulations. For narrow channels numerical simulations indicate that the stable front propagates more slowly, and for sufficiently small L or sufficiently large (Formula presented.) the front speed falls to zero and the front collapses. The reason for the collapse is the non-existence, far behind the front, of a stable positive equilibrium solution u(x, y). While existence of these equilibrium states can be demonstrated via phase plane arguments, the investigation of stability is similar to calculations of critical patch sizes carried out in similar ecological models.

Original language | English |
---|---|

Pages (from-to) | 2197–2214 |

Number of pages | 18 |

Journal | Bulletin of Mathematical Biology |

Volume | 79 |

Issue number | 10 |

Early online date | 1 Aug 2017 |

DOIs | |

Publication status | Published - Oct 2017 |

### Fingerprint

### Keywords

- 2D FKPP equation
- Critical patch size
- Mixed boundary conditions
- Population collapse
- Strip invasion

### ASJC Scopus subject areas

- Neuroscience(all)
- Immunology
- Mathematics(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Pharmacology
- Environmental Science(all)
- Agricultural and Biological Sciences(all)
- Computational Theory and Mathematics

### Cite this

*Bulletin of Mathematical Biology*,

*79*(10), 2197–2214. https://doi.org/10.1007/s11538-017-0326-x

**Invasions Slow Down or Collapse in the Presence of Reactive Boundaries.** / Minors, K.; Dawes, J. H.P.

Research output: Contribution to journal › Article

*Bulletin of Mathematical Biology*, vol. 79, no. 10, pp. 2197–2214. https://doi.org/10.1007/s11538-017-0326-x

}

TY - JOUR

T1 - Invasions Slow Down or Collapse in the Presence of Reactive Boundaries

AU - Minors, K.

AU - Dawes, J. H.P.

PY - 2017/10

Y1 - 2017/10

N2 - Motivated by the propagation of thin bacterial films around planar obstacles, this paper considers the dynamics of travelling wave solutions to the Fisher–KPP equation (Formula presented.) in a planar strip (Formula presented.), (Formula presented.). We examine the propagation of fronts in the presence of a mixed boundary condition (also referred to as a ‘partially absorbing’ or ‘reactive’ boundary) (Formula presented.), with (Formula presented.), at (Formula presented.). The presence of boundary conditions of this kind leads to the development of front solutions that propagate in x but contain transverse structure in y. Motivated by the observation that the speed of propagation in the Fisher–KPP equation is determined (for exponentially decaying initial conditions) by the behaviour at the leading edge, we analyse the linearised Fisher–KPP equation in order to estimate the speed of the stable travelling front, a function of the width L and the imposed boundary conditions. For wide strips the speed estimate based on the linearised equation agrees well with the results of numerical simulations. For narrow channels numerical simulations indicate that the stable front propagates more slowly, and for sufficiently small L or sufficiently large (Formula presented.) the front speed falls to zero and the front collapses. The reason for the collapse is the non-existence, far behind the front, of a stable positive equilibrium solution u(x, y). While existence of these equilibrium states can be demonstrated via phase plane arguments, the investigation of stability is similar to calculations of critical patch sizes carried out in similar ecological models.

AB - Motivated by the propagation of thin bacterial films around planar obstacles, this paper considers the dynamics of travelling wave solutions to the Fisher–KPP equation (Formula presented.) in a planar strip (Formula presented.), (Formula presented.). We examine the propagation of fronts in the presence of a mixed boundary condition (also referred to as a ‘partially absorbing’ or ‘reactive’ boundary) (Formula presented.), with (Formula presented.), at (Formula presented.). The presence of boundary conditions of this kind leads to the development of front solutions that propagate in x but contain transverse structure in y. Motivated by the observation that the speed of propagation in the Fisher–KPP equation is determined (for exponentially decaying initial conditions) by the behaviour at the leading edge, we analyse the linearised Fisher–KPP equation in order to estimate the speed of the stable travelling front, a function of the width L and the imposed boundary conditions. For wide strips the speed estimate based on the linearised equation agrees well with the results of numerical simulations. For narrow channels numerical simulations indicate that the stable front propagates more slowly, and for sufficiently small L or sufficiently large (Formula presented.) the front speed falls to zero and the front collapses. The reason for the collapse is the non-existence, far behind the front, of a stable positive equilibrium solution u(x, y). While existence of these equilibrium states can be demonstrated via phase plane arguments, the investigation of stability is similar to calculations of critical patch sizes carried out in similar ecological models.

KW - 2D FKPP equation

KW - Critical patch size

KW - Mixed boundary conditions

KW - Population collapse

KW - Strip invasion

UR - http://www.scopus.com/inward/record.url?scp=85026529044&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s11538-017-0326-x

U2 - 10.1007/s11538-017-0326-x

DO - 10.1007/s11538-017-0326-x

M3 - Article

VL - 79

SP - 2197

EP - 2214

JO - Bulletin of Mathematical Biology

JF - Bulletin of Mathematical Biology

SN - 0092-8240

IS - 10

ER -