Issue Math. Model. Nat. Phenom. Volume 3, Number 7, 2008 Special issue dedicated to Glenn Webb Page(s) 115 - 125 DOI 10.1051/mmnp:2008044 Published online 23 October 2008

Math. Model. Nat. Phenom. Vol. 3, No. 7, 2008, pp. 115-125
DOI: 10.1051/mmnp:2008044

Homogeneous Systems with a Quiescent Phase

K.P. Hadeler

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA

hadeler@uni-tuebingen.de

Published online: 23 October 2008

Abstract
Recently the effect of a quiescent phase (or dormant/resting phase in applications) on the dynamics of a system of differential equations has been investigated, in particular with respect to stability properties of stationary points. It has been shown that there is a general phenomenon of stabilization against oscillations which can be cast in rigorous form. Here we investigate, for homogeneous systems, the effect of a quiescent phase, and more generally, a phase with slower dynamics. We show that each exponential solution of the original system produces two exponential solutions of the extended system whereby the stability properties can be controlled.

Mathematics Subject Classification. 34C14, 34D08, 92D25, 92D40

Key words: quiescence -- homogeneous system -- exponential solution -- stability -- non-linear eigenvalue problem -- predator-prey system

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