### Abstract

Language | English |
---|---|

Pages | 221-232 |

Number of pages | 11 |

Journal | European Journal of Applied Mathematics |

Volume | 17 |

Early online date | 22 Feb 2006 |

DOIs | |

Publication status | Published - 30 Apr 2006 |

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### Keywords

- bistability
- dispersal
- mathematics
- applied mathematics

### Cite this

*European Journal of Applied Mathematics*,

*17*, 221-232. https://doi.org/10.1017/S0956792506006462

}

*European Journal of Applied Mathematics*, vol. 17, pp. 221-232. https://doi.org/10.1017/S0956792506006462

**Non-local dispersal and bistability.** / Hutson, V.; Grinfeld, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Non-local dispersal and bistability

AU - Hutson, V.

AU - Grinfeld, M.

PY - 2006/4/30

Y1 - 2006/4/30

N2 - The scalar initial value problem [ u_t = ho Du + f(u), ] is a model for dispersal. Here $u$ represents the density at point $x$ of a compact spatial region $Omega in mathbb{R}^n$ and time $t$, and $u(cdot)$ is a function of $t$ with values in some function space $B$. $D$ is a bounded linear operator and $f(u)$ is a bistable nonlinearity for the associated ODE $u_t = f(u)$. Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with $D=Delta$ is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case $D=Delta$). We develop a technique for proving that indeed convergence does hold for small $ ho$ and show by constructing a counter-example that this result does not hold in general for all $ ho$.

AB - The scalar initial value problem [ u_t = ho Du + f(u), ] is a model for dispersal. Here $u$ represents the density at point $x$ of a compact spatial region $Omega in mathbb{R}^n$ and time $t$, and $u(cdot)$ is a function of $t$ with values in some function space $B$. $D$ is a bounded linear operator and $f(u)$ is a bistable nonlinearity for the associated ODE $u_t = f(u)$. Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with $D=Delta$ is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case $D=Delta$). We develop a technique for proving that indeed convergence does hold for small $ ho$ and show by constructing a counter-example that this result does not hold in general for all $ ho$.

KW - bistability

KW - dispersal

KW - mathematics

KW - applied mathematics

U2 - 10.1017/S0956792506006462

DO - 10.1017/S0956792506006462

M3 - Article

VL - 17

SP - 221

EP - 232

JO - European Journal of Applied Mathematics

T2 - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

ER -