Inertial manifolds in biological systems

Abstract

The focus of this thesis is biological systems whose dynamics present an interesting feature: only some dimensions drive the whole system. In our examples, the dynamics is expressed as ODEs, such that the ith equation depends on all the variables xi = f (x1,...,xi,xi+1,...), so that they cannot be solved by classical methods. The authors in the literature found that one could express the variable of order bigger than N as a function of the first N variables, thus closing the differential equations; the approximations obtained were exponentially close to the nonapproximated result. In Nonlinear Dynamics, such functions are called Inertial Manifolds. They are defined as manifolds that are invariant under the flow of the dynamical system, and attract all trajectories exponentially. The first example gives rise to a generalisation of a theorem which, in the literature, is proved for the PDE u= -Au + V(u). We prove existence for the most general case u= -A(u)u + V(u) and consider the validity of the results for the biological parameters. We also present a theoretical discussion, by providing examples. The second example arises from Statistics applied to population biology. The infinite number of differential equations for the moments are approximated using a Moment Closure technique, that is expressing moments of order higher than N as a function of the first moments, generally using the function valid for the normal distribution. The example shows exceptional approximation. Though this technique is often used, there is no complete mathematical justification. We examine the relation between the Moment Closure technique and Inertial Manifolds. We prove that the approximated system can be seen as a perturbation of the original system, that it admits an Inertial Manifold, which is close to the original one for \epsilon \rightarrow 0 and t \rightarrow \infty