Geometric singular perturbation theory in biological practice

Abstract

Geometric singular perturbation theory is a useful tool in the analysis of problems with a clear separation in time scales. It uses invariant manifolds in phase space in order to understand the global structure of the phase space or to construct orbits with desired properties. This paper explains and explores geometric singular perturbation theory and its use in (biological) practice. The three main theorems due to Fenichel are the fundamental tools in the analysis, so the strategy is to state these theorems and explain their significance and applications. The theory is illustrated by many examples. Mathematics Subject Classification (2000)34e15 · 34c30 · 34c60 · 92b09

A non-local model for cancer stem cells and the tumour growth paradox

“…Hillen et al (2013) used methods from geometric singular perturbation theory (Hek, 2010) and showed that for small δ > 0 there exists a slow manifold inside S which attracts each orbit. The slow manifolds depend monotonically on α in such a way that the population with the larger α value does grow faster hence implying the tumour growth paradox.…”

“…Instead, Hillen et al simplified the iDE model into a (spatially homogeneous) system of ordinary differential equations (ODEs). For this, ODE system they used geometric singular perturbation theory (Hek, 2010) to mathematically prove the existence of a tumour growth paradox. The analysis of the full iPDE model for…”

Delayed phenomenon of loss of stability of solutions in a second-order quasi-linear singularly perturbed boundary value problem with a turning point

“…Both the two theories tell us that the solutions of singularly perturbed problems tend to the stable solutions of the corresponding reduced problems with the small parameter approaching to zero under the normally hyperbolic condition. Since then, under this essential condition of normal hyperbolicity, the theory of singular perturbation finds applications in many problems including boundary value problems [7], existence of solitons [8], and biological models [9], etc. …”

Fenichel's theorems with applications in dynamical systems.

“…For our final example we consider the classic Rosenzweig-MacArthur predatorprey model presented in Rinaldi and Muratori (1992) [19] and given in rescaled form by Hek (2010) [6].…”

“…The resulting equations have a specific structure which, through the application of the following theorems, can be readily understood. When studying such systems, simplifying assumptions may be of great help; if not to understand the full system, then at least to get a first insig;ht into the system's behavior [6].…”