Picture of athlete cycling

Open Access research with a real impact on health...

The Strathprints institutional repository is a digital archive of University of Strathclyde's Open Access research outputs. Strathprints provides access to thousands of Open Access research papers by Strathclyde researchers, including by researchers from the Physical Activity for Health Group based within the School of Psychological Sciences & Health. Research here seeks to better understand how and why physical activity improves health, gain a better understanding of the amount, intensity, and type of physical activity needed for health benefits, and evaluate the effect of interventions to promote physical activity.

Explore open research content by Physical Activity for Health...

Non-local dispersal and bistability

Hutson, V. and Grinfeld, M. (2006) Non-local dispersal and bistability. European Journal of Applied Mathematics, 17. pp. 221-232. ISSN 0956-7925

[img]
Preview
Text (strathprints004546)
strathprints004546.pdf - Accepted Author Manuscript

Download (93kB) | Preview

Abstract

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$.