A singular limit for an age structured mutation problem

Abstract

The spread of a particular trait in a cell population often is modelled by an appropriate system of ordinary differential equations describing how the sizes of subpopulations of the cells with the same genome change in time. On the other hand, it is recognized that cells have their own vital dynamics and mutations, leading to changes in their genome, mostly occurring during the cell division at the end of its life cycle. In this context, the process is described by a system of McKendrick type equations which resembles a network transport problem. In this paper we show that, under an appropriate scaling of the latter, these two descriptions are asymptotically equivalent.

Generalized network transport and Euler-Hille formula

“…The main aim of this paper is to generalize the above results, both on the asymptotic behaviour of solutions to (6) and on the Euler-Hille type formula (7), to operators of the form B = K + C, where K is a contraction having 1 as an isolated eigenvalue, as well as to allow for the perturbation M in the transport terms. In particular, this generalization enabled an asymptotic analysis of a boundary perturbation of any network flow (1) with the node exchange rule given by the Kirchhoff rule.…”

“…(12)), it follows that the regular convergence of the semigroups solving (6) is equivalent to (7). We emphasize the role played by the regular convergence of the semigroup solving (6) in the proof of (7). As illustrated in Example 4, a direct proof would require a detailed knowledge of the fine structure of eigenvalues of the perturbed operator that seems to be difficult to obtain in a general case.…”

“…Since the semigroup solving (6) is given as the composition of the iterates of B and the translation semigroup (see e.g. (12)), it follows that the regular convergence of the semigroups solving (6) is equivalent to (7). We emphasize the role played by the regular convergence of the semigroup solving (6) in the proof of (7).…”

“…As mentioned above our interest is to determine, when the process described by (6) can be approximately aggregated to a macromodel; that is, to a system of ordinary differential equations modelling the transfer of mass from one edge to another. In recent papers, [7,8] (see also [8,14], where transport was replaced by diffusion), the authors were able to show the convergence of solutions to (6) with fast transport (that is, with ∂ x u(x, t) replaced by −1 ∂ x u(x, t)) balanced by slow exchange between the states (modelled by B = I + C, where C is an arbitrary matrix and I is the identity) to solutions of an appropriate system of differential equations.…”

“…The analysis of (6) with B = I + C and of (7) has been carried out in [8]. The ideas of that paper were extended in [7,12] to prove a similar result with I replaced by a cyclic operator T but the price to pay was that only the projections of the semigroups onto the eigenspace of T belonging to the eigenvalue 1 converge.…”